Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions
EE ff ective Theory for the Measurement-Induced Phase Transition of Dirac Fermions M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, D-50937 Cologne, Germany Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria (Dated: February 18, 2021)A wave function exposed to measurements undergoes pure state dynamics, with deterministic unitary andprobabilistic measurement induced state updates, defining a quantum trajectory. For many-particle systems,the competition of these di ff erent elements of dynamics can give rise to a scenario similar to quantum phasetransitions. To access it despite the randomness of single quantum trajectories, we construct an n -replica Keldyshfield theory for the ensemble average of the n -th moment of the trajectory projector. A key finding is that thisfield theory decouples into one set of degrees of freedom that heats up indefinitely, while n − ff ective non-Hermitian Hamiltonian. This decoupling isexact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermionsin (1 +
1) dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths.For this model, the non-Hermitian Hamiltonian corresponds to a quantum Sine-Gordon model with complexcoe ffi cients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglemententropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. Thephysical picture emerging here is a pinning of the trajectory wave function into eigenstates of the measurementoperators upon increasing the monitoring rate. I. INTRODUCTION
In quantum mechanics, there are two fundamentally distinctdynamical evolutions: First, a pure state can evolve determin-istically according to the Schr¨odinger equation, with dynam-ics generated by a Hamiltonian operator ˆ H . Second however,it can be updated in a stochastic fashion, when the quantumsystem is subject to observation. In the case of a strong, pro-jective measurement of an observable ˆ M , the wave functionabruptly collapses into one of the eigenstates | m (cid:105) of ˆ M , witha probability determined by the overlap of the state before themeasurement with | m (cid:105) . If the Hamiltonian commutes with themeasurement operator, after a single collapse into a certainstate | m (cid:105) the system will be confined to it indefinitely. In con-trast, if [ ˆ H , ˆ M ] (cid:44)
0, generically the competition of ˆ H and ˆ M will not allow the evolution to come to rest.In a many-body context, such a competition of two oper-ators can give rise to fundamental macroscopic phenomenasuch as quantum phase transitions. In this case, two non-commuting terms, e.g. kinetic and potential energy, each sep-arately stabilize ground states with macroscopically distinctproperties. While finite systems can only undergo a grad-ual change of properties upon tuning the ratio of competingenergy scales, in the thermodynamic limit a phase transitionseparating qualitatively distinct phases of matter will occur. Inthis light, it is a natural question whether a many-body systemundergoing competing Hamiltonian and measurement dynam-ics likewise may undergo an abrupt change in behavior, andwhich quantities may host this information. This question wasanswered in the a ffi rmative in [1–4] for projectively measuredrandom unitary circuits. In these setups, the dynamics of aone-dimensional spin chain is generated either by randomlyselected quasilocal entangling unitary gates, or by quasilocalmeasurements. Since the sets of operators of the entanglinggates and the measurements do not commute, a competition isrealized, with a strength tuneable via the ratio of applied uni-taries per time unit vs. applied measurements per time unit. For Haar random unitaries, moderate size numerics [5–7]and analytical studies [3, 4] have been conducted. Choosingtime evolution randomly from the Cli ff ord- and the measure-ment operators from the Pauli-group, e ffi cient numerical anal-ysis of even large systems is possible [8–10]. The two extremecases are clearly distinct in their dynamics: The unitarilyevolving circuit is characterized by unbounded growth of theentanglement entropy with system size (volume law) [1, 11–17]. On the other hand, for any random initial state local mea-surements of the spins e.g. in the z -basis will collapse the stateinto a pure product state of some configuration of the z pro-jection of the spins, thus characterized by a saturation of theentanglement entropy to an area law behavior. It was shownin Refs. [1, 2] that the respective entanglement growth aver-aged over the ensemble of trajectories is a good witness forthe phase transition between volume and area law growth ata finite competition ratio between unitary and measurementdynamics.This discovery has sparked significant research on the na-ture of this transition, its proper description, and its general-ity in terms of models hosting such behavior. The entangle-ment growth quantifier suggests a physical picture in termsof information scrambling vs. information localization. Thishas been made more precise in [7, 18–20], which explain therobustness of the volume law phase within a quantum errorcorrection picture, where the fast information spreading isprotected from errors realized as the readout of the measure-ments. Giving up the focus on pure state evolution Ref. [21]characterizes combined unitary and measurement dynamicsby its potential to purify a maximally mixed initial state. Al-ternatively to random circuits, models of fermions hopping ona one-dimensional lattice and exposed to local density mea-surements have been suggested [22]. These do not show avolume to area law transition [22], but rather a transition froma critical phase with a logarithmic scaling of the entanglemententropy to an area law [23] (the transition is absent in non-unitary circuit models not corresponding to physical measure- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ments [24–26]). For free fermion models, a general corre-spondence between non-unitary circuit dynamics and Hamil-tonian dynamics subject to measurements in ( d +
1) dimen-sions has been shown to enable a classification of the mea-surement dynamics in terms of symmetries [26]. These resultssuggest a finer structure in the phenomenology of measure-ment induced phase transitions.In this work, we approach the measurement-induced phasetransition from the perspective of non-equilibrium quantumstatistical mechanics, asking in particular what the properdegrees of freedom are to capture it. To this end, we de-velop a replica field theory approach for a minimal model ofDirac fermions in one spatial dimension, undergoing contin-uous measurements. This model enables an alternative repre-sentation via bosonization at long wavelength, amounting toa measured non-linear Luttinger liquid. In our analysis, weare guided by the physical picture of a pinning or localizationtransition upon increasing the measurement strength: On thelevel of a single, pure state quantum trajectory, the delocaliza-tion due to kinetic energy competes with pinning of fermionsdue to the measurement-induced collapse of the wave func-tion.Care has to be exercised when taking the ensemble average.The binary measurement outcomes on each point in space leadto an extensive configurational entropy. The averages thatare usually considered, which are linear in the state, corre-spond at long times to infinite temperature states and maskthe transition in the trajectory ensemble. In contrast, averagesof non-linear functions of the state do witness a transition, aswe demonstrate on our concrete model: The trajectory aver-aged n -replica state hosts one structureless ’hot’ mode, butalso n − +
1) dimensions. We expect the structures revealedhere to be useful for even broader classes of measurement-induced phase transitions.Applied to the problem at hand, it allows us to makeprogress in terms of the bosonized theory. In particular, wedistill an e ff ective non-Hermitian sine-Gordon model for den-sity fluctuations, and show that it undergoes a Berezinskii-Kosterlitz-Thouless (BKT) pinning transition. Its gaplessphase is protected by current conservation. The pinning in-duced by measurements, into the eigenstates of the mea-surement operators, drives a gap opening. As witnesses ofthe transition, we compute the 2-replica correlation functionsshowing algebraic and exponential behavior in the gapless andin the gapped phases, respectively. Furthermore, we com-pute the entanglement entropy in the weakly and stronglymonitored regimes, showing logarithmic growth and satura-tion with system size L , respectively. A measurement ratedependent e ff ective central charge, appearing as the coe ffi -cient of the log( L ) growth term, is found in the gapless phase.This scenario is reminiscent to previous numerical resultsfor a related model of monitored lattice fermions [22–24].Here, however, we find that the e ff ective central charge growsmonotonously with the measurement rate, in the weak mea-surement regime. We believe that such a pinning mechanism,and the BKT universality class itself, could underlie broaderclasses of measurement-induced phase transitions in 1 + Structure of the paper:
In the following section, we intro-duce our setup and provide a summary of our main results. InSection III, we discuss general properties of the measurementmodel and draw the temporal and spatial continuum limit,introducing a field theory for continuously monitored Diracfermions. In Section IV, a two-replica field theory for a gen-eral continuous monitoring setup is derived and then appliedto our concrete setup. It provides the basis for the identifica-tion of the relevant degrees of freedom at the measurement-induced phase transition. Then, in Section V, we perform arenormalization group analysis of the phase transition. In Sec-tion VI, the replica theory is extended to an arbitrary numberof replicas n in a Keldysh real-time path integral framework.This serves as a basis for the computation of higher order cor-relation functions, and we include a discussion of the R´enyientropy and the von Neumann entanglement entropy. We con-clude in Section VII. Several technical details of our analysisare presented in the appendix. II. EXTENDED SYNOPSIS AND KEY RESULTS
Setup – Our model for the measurement dynamics in onespatial dimension is based on the stochastic Schr¨odinger equa-tion for the trajectory wave function | ψ t (cid:105) [28–30], or alter-natively the conditioned trajectory projector ˆ ρ ( c ) t = | ψ t (cid:105) (cid:104) ψ t | [31–33], which amounts to a continuous time description ofweak measurements. The competition between unitary dy-namics and measurements is described by the dimensionlessparameter g = γ/ J , where J describes the kinetic energy, and γ the rate of measurement. This formulation is able to cap-ture the collapse of the wave function characteristic of physi-cal measurements. Previous work on measurement-inducedphase transitions has mainly focused on the case of strong(projective) measurements [33–35]; however, the weak mea-surement scenario is smoothly connected to the latter. In fact,Ref. [34] has established that the transition persists upon in-terpolating between the limiting cases of strong projective andweak continuous measurements in random circuits. For ourpurpose of constructing an analytically tractable model for thecompeting dynamics, the continuous time formulation provesadvantageous. Quantifying the phase transition – Continuously measur-ing a set of local, mutually commuting operators ˆ O j , e.g., thelocal particle density on some lattice site j , pins the systemonto their eigenstates. Competing with this is the unitary dy-namics, which leads to the spreading of the local operatorsvia particle propagation, and therefore to a depinning fromeigenstates of the measurement operators. The competition oftwo non-commuting operators driving a phase transition mustthus be encoded in the evolution of the pure state wave func-tion, similar to a quantum phase transition in a ground stateproblem. Importantly however, the pure state considered herecorresponds to a random variable, due to the stochastic na-ture of measurements. To acquire information independent ofthis randomness, suitable ensemble averages need to be con-sidered. Taking the statistical average over the trajectory en-semble (which we denote with . . . ) in the way familiar fromquantum statistical physics for instance yields (cid:104) ψ t | ˆ O i ˆ O j | ψ t (cid:105) = tr[ ˆ O i ˆ O j ˆ ρ ( c ) t ] = tr[ ˆ O i ˆ O j ˆ ρ ( c ) t ] , (1)where ˆ ρ ( c ) t is the trajectory averaged density matrix. For ameasurement dynamics, however, ˆ ρ ( c ) t = N , which corre-sponds to evaluating the product of the measured operatorsˆ O i ˆ O j in an infinite temperature state – thus the correlationswill be trivial except for g =
0, where there is no averaging.The masking e ff ect of such trajectory ensemble average canbe mitigated by considering correlation functions which arenon-linear in the state – such as, for example, the equal-timeproduct (cid:104) ψ t | ˆ O i | ψ t (cid:105) × (cid:104) ψ t | ˆ O j | ψ t (cid:105) for the same stochastic wavefunction | ψ t (cid:105) in both quantum mechanical expectation values.We note that the entanglement entropy falls into this class ofquantifiers which are non-linear in the state, involving eveninfinite powers in the case of the von Neumann entropy.In this work, we focus on the non-linear connected covari-ance matrix for the measured operators C i j ( t ) = (cid:104) ψ t | ˆ O i ˆ O j | ψ t (cid:105) − (cid:104) ψ t | ˆ O i | ψ t (cid:105) (cid:104) ψ t | ˆ O j | ψ t (cid:105) . (2)in the limit t → ∞ . Here, the second part ∼(cid:104) ψ t | ˆ O i | ψ t (cid:105) (cid:104) ψ t | ˆ O j | ψ t (cid:105) is nonlinear in ˆ ρ ( c ) and therefore doesn’tcorrespond to an infinite temperature average. It is partic-ularly well suited to distinguish between strong monitoring,where the system remains close to an eigenstate of the op-erators ˆ O j (such that C i j is short-ranged or even zero for i (cid:44) j ), and weak monitoring, where longer-ranged correla-tions between the measured operators can be established bythe Hamiltonian. Indeed, such a scenario is realized in ourmodel. For g (cid:28)
1, the problem becomes linear in the bosonicformulation. The correlator can be computed analytically,demonstrating an algebraic decay C i j ( t ) ∼ | i − j | − . In theopposite regime g − (cid:28)
1, the pinning of particle density dueto the measurements cuts o ff this correlation function moreseverely. In the measurement-only limit g − =
0, the wavefunction evolves into a product state of onsite occupations0 , ff ective massive bosonic theory with exponentially decay-ing correlations. This behavior is backed up by a complemen-tary perturbative calculation for g − (cid:28) ff erent from each other, pointing at the exis-tence of a phase transition at a finite competition g c . Replica approach – The ability to capture the qualitative be-havior for finite values of g motivates us to study the bosonicnon-linear Luttinger model in more detail. To this end, we findit convenient to work in a replica formalism. We introduce aproduct of replicated, conditioned density matrices ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t ,which allows us to rewrite C i j ( t ) = Tr( ˆ O i ˆ O j ˆ ρ ( c ) t )Tr ˆ ρ ( c ) t − Tr( ˆ O i ˆ ρ ( c ) t )Tr( ˆ O j ˆ ρ ( c ) t ) =
12 Tr (cid:20) ( ˆ O (1) i − ˆ O (2) i )( ˆ O (1) j − ˆ O (2) j ) ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:21) (3)with ˆ O (1) j = ˆ O j ⊗ and ˆ O (2) j = ⊗ ˆ O j . This formulation o ff erstwo key advantages: First, it encodes the correlation informa-tion in an expectation value linear in the trajectory averaged,replicated state. We then construct a quantum master equationdirectly for ˆ ρ ( R ) t ≡ ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t . It features a non-linear state de-pendence, which however becomes unimportant at late times.As a net result, we obtain an evolution linear in the replicated,averaged state ˆ ρ ( R ) t . We emphasize that we do not considerquenched disorder in our system, and statistical averages areexclusively reserved for taking the average over random mea-surement trajectories.Second, it allows us to introduce new degrees of freedomwhich capture the physics of the transition more transparently.In the bosonic model, the replicas are acted on by the mea-sured operators ˆ O ( l ) = f ( ˆ φ ( l ) x ) with l = , f either a lin-ear or a nonlinear function of the bosonic operator ˆ φ ( l ) x , whichis associated to density fluctuations. For a linear problem(the problem can be linearized close to g , g − = φ ( a , r ) x = √ ( ˆ φ (1) x ± ˆ φ (2) x ).The center-of-mass coordinates follow an evolution whichheats them up indefinitely, while the relative ones obey aSchr¨odinger-type evolution under an e ff ective non-HermitianHamiltonian, | ψ ( r ) t (cid:105) = e − i ˆ H e ff t | ψ ( r )0 (cid:105) .Including the non-linear terms, the decoupling is no longerexact. However, assuming that the center-of-mass degrees offreedom still heat up to infinity irrespectively of the behav-ior of the relative ones, the center-of-mass degrees of freedomcan be traced out. The remaining relative coordinate evolutionstill takes the above form, and can be written as a path inte-gral governed by the action of a non-Hermitian sine-Gordonmodel, S = (cid:90) X π [( ∂ t φ ( r ) X ) − η ( ∂ x φ ( r ) X ) ] − i λ cos φ ( r ) X ] , (4)where we combined X = ( x , t ), and with η being complexand λ real. We derive the flow equations for this model, andfind that it exhibits a BKT type phase transition as a functionof varying the competition ratio g , which enters the couplingconstants η ( g ) , λ ( g ). This places the measurement-inducedphase transition into the BKT universality class.This result may be rationalized by the fact that a gaplessphase in (1 +
1) dimensions quite generically is left via theBKT mechanism by the generation of a mass [36]. Here thegapless phase is protected by current conservation. Indeed,ˆ φ in the original model describes density fluctuations, result-ing in gapless relative modes ˆ φ ( r ) at weak monitoring. Uponincreasing the measurement strength, these modes generate amass, and thus get pinned.We also study the generalization from two to n replicaswithin a multicontour Keldysh approach, representing Z ( n ) = tr ˆ ρ ( c ) n in terms of a functional integral with 2 n contours – twofor each copy of the state, as in the usual Keldysh path integral( n = n powers of the quantum trajectory pro-jector. The noise average then generates correlations betweenall the copies of the state. In this framework, and for generalGaussian problems in the long time limit, we find that thereis one center-of-mass mode (the symmetric superposition, orFourier mode k =
0) which heats up indefinitely, while thereare n − n = n within an Abelian bosonization framework. Entanglement entropy in n-replica Keldysh approach – Fi-nally, we embark on a calculation of the entanglement entropy.To this end, we consider a bipartition of the system into a sub-system A of length L and R \ A which is traced out, and decom-pose the partition function of the subsystem Z A ( n ) = tr ˆ ρ ( c ) nA into a noiseless and a noisy contribution. For free theories,including the Gaussian fixed points of our measurement the-ory, we show that the noiseless contribution to entanglemententropy exclusively determines the scaling with the bipartitionsize, while the noisy contribution amounts to a size indepen-dent constant. In the gapless regime we find S vN ( L ) = c ( γ ) log( L ) , (5)with a γ -dependent e ff ective central charge c ( γ ). It dependson the renormalization group flow of the non-Hermitian sine-Gordon theory. In the non-monitored limit, c ( γ → → c ( γ ) growsmonotonously, while saturating for intermediate measurementstrength. In contrast, in the massive regime g − (cid:28)
1, theentanglement entropy saturates to an area law. Only for γ =
0a thermal initial state produces a volume law entanglemententropy.
III. MEASUREMENT MODEL
In this section, we construct a model for measurements inmany-body systems in the temporal and spatial continuumlimit. This model captures the essence of measurement in-duced phase transitions, realizing non-commuting operatorsby means of a kinetic energy Hamiltonian competing with ex-actly local measurements, and lends itself to a field theoryanalysis.
A. Temporal continuum limit: Continuous measurements inquantum state di ff usion approach Projective vs. continuous measurements – Here we describethe framework of continuous measurements that we utilize tomodel the monitored fermion and boson dynamics. Comparedto discrete projective measurements, the continuous measure-ments are often referred to as weak measurements. Broadlyspeaking, continuous measurements correspond to the limitwhere the amount of information extracted per measurement(in the sense of an incomplete readout), and the duration ofeach measurement, are reduced to zero simultaneously, in away that their ratio γ is kept fixed [33, 37, 38]. This comeswith the advantage of enabling us to straightforwardly takethe continuum limit in time, which in turn is useful for map-ping the problem into a field theory in Sec. IV. In the follow-ing we refer to this type of measurement exclusively as ’con-tinuous measurement’ or ’monitoring’ and reserve the terms’weak’ and ’strong’ measurements (or monitoring) to situa-tions where γ is small or large compared to the scale of theHamiltonian. We emphasize that continuous measurementsdo not exclude the existence of a phase transition [34, 35]. Infact, we can replace in the competition ratio from the frame-work of discrete unitary circuits and projective measurements g = / time / time → γ J , (6)where γ is the time scale of the monitoring and J the scaleof the Hamiltonian, which can take arbitrary values. Accord-ingly, we will distinguish weak monitoring γ (cid:28) J and strongmonitoring γ (cid:29) J . Indeed, Ref. [34] has demonstrated thatthe existence of a phase transition in random circuits uponincreasing g is preserved all along the way from strong pro-jective to weak continuous measurements. Stochastic Schr¨odinger equation – The starting point of ourapproach is a stochastic Schr¨odinger equation in the quantumstate di ff usion framework [32, 37–39]. It corresponds to amonitoring protocol in which the expectation value of an oper-ator (cid:104) ˆ O l (cid:105) t is continuously measured, for instance a homodynedetection scheme [37, 40, 41]. In the quantum state di ff usion,the evolution equation d | ψ t (cid:105) = | ψ t + dt (cid:105) − | ψ t (cid:105) for the pure state,normalized quantum trajectory wave function | ψ t (cid:105) is, d | ψ t (cid:105) = − idt [ ˆ H − i γ (cid:88) i ˆ M i , t ] | ψ t (cid:105) + (cid:88) i dW i ˆ M i , t | ψ t (cid:105) , (7)for which we define the measurement operators ˆ M i , t = ˆ O i − (cid:104) ˆ O i (cid:105) t (8)for a general set of local measured operators ˆ O i , labeled by anindex i (e.g. a lattice site index). The dynamical update of thestate | ψ t (cid:105) for an infinitesimal time interval dt is generated by(i) a deterministic contribution due to a non-Hermitian Hamil-tonian ˆ H nH = ˆ H − i γ (cid:80) i ˆ M i , t involving both the Hamiltonian ˆ H and the measurement operators. These contributions are asso-ciated with the scale J for the Hamiltonian as anticipated, and γ for the measurements, which is implemented by the mea-surement operators ˆ M i , t . In addition, (ii), there is a stochasticcontribution: The Wiener process dW i is a local, real valuedGaussian noise increment, with zero mean dW i = dW i dW j = γ dt δ i j [33, 37, 39].The measurement operators ˆ M i , t involve the quantum me-chanical expectation value (cid:104) ˆ O i (cid:105) t ≡ (cid:104) ψ t | ˆ O i | ψ t (cid:105) evaluated at thetime before the update, describing the measurement feedback.This term makes the quantum trajectory evolution non-linearin the state of the system. It ensures preservation of the normof the trajectory wave function: It is easily verified that all mo-ments of the norm N ( n ) ≡ (cid:104) ψ | ψ (cid:105) n are conserved, ∂ t N ( n ) = n . This is an important condition for describing aphysical measurement. For example, the ’raw’ quantum statedi ff usion protocol, obtained by the replacement ˆ M i , t → ˆ O i discarding the feedback term, preserves the norm only on av-erage, ∂ t N ( n ) ∼ γ n ( n − N ( n ), i.e. for n =
1, and does notqualify for the description of a physical quantum trajectoryundergoing measurements.A useful alternative representation of the stochasticSchr¨odinger equation utilizes the conditioned projectorˆ ρ ( c ) t = | ψ t (cid:105) (cid:104) ψ t | , which evolves according to d ˆ ρ ( c ) t = dt ( − i [ ˆ H , ˆ ρ ( c ) t ] − γ (cid:88) i [ ˆ O i , [ ˆ O i , ˆ ρ ( c ) t ]]) (9) + (cid:88) i dW i { ˆ M i , t , ˆ ρ ( c ) t } , and where we used the anti-commutator { ˆ A , ˆ B } = ˆ A ˆ B + ˆ B ˆ A . Connection to projective measurements – This becomesparticularly transparent when focusing on the stochastic weakmeasurement evolution alone ( ˆ H = t → ∞ where dW i acts as a multi-plicative noise (multiplied with the state dependent term ˆ M i , t ),and becomes inactive only once the conditionˆ O i | ψ t (cid:105) = (cid:104) ˆ O i (cid:105) t | ψ t (cid:105) = O i | ψ t (cid:105) . (10)is fulfilled – in other words, once the system has reached aneigenstate of the measurement operator with eigenvalue O i .In this case, also the double commutator in Eq. (9) vanishes.Thus, this evolution describes a ‘continuous collapse’ of aninitial wavefunction into an eigenstate of the measurement op-erator, with the stochastic element provided by the di ff erentnoise realizations. We will refer to these eigenstates of themeasurement operators in a quantum optics language as darkstates , since the measurement evolution stops once the systemhas reached such state. Dark state configurational entropy – As already mentionedin the introduction, measurements come with a large config-urational entropy (as opposed to the entanglement entropy,building up due to the entangling operations along each purestate trajectory) related to the spectrum of eigenstates of themeasurement operators and the their random realization in themeasurement process. This is best illustrated by means ofa concrete lattice model of N continuously monitored spin-less fermions. Consider an entangling tight-binding hoppingHamiltonian ˆ H = − J (cid:88) l c † i c i + + H.c. , (11) where c † i , c i are fermionic creation and annihilation operatorson a lattice with sites i = , ..., L , and a continuous measure-ment implemented by monitoring the local fermion densitiesˆ O i = ˆ n i = c † i c i . (12)For each of the local fermionic density measurements, thereare two dark eigenstates | σ i (cid:105) with σ i = ,
1. A random ini-tial state will ultimately collapse into a pure product state |{ σ i }(cid:105) = (cid:81) i | σ i (cid:105) i , where the configuration { σ i } is random, sub-ject only to the constraint (cid:80) i σ i = N , the total number offermions in the system. However, for large systems there areexponentially many such pure states, e.g. (cid:32) NN (cid:33) at half fill-ing. Of course, this reasoning generalizes to the quantum non-demolition case, where [ ˆ H , ˆ M i ] = i ; however, the non-commuting hopping Hamiltonian will generate competitionbetween measurement and Hamiltonian evolution, and gener-ically prevent the system from reaching an eigenstate of themeasurement operators. Indeed, this model exhibits a mea-surement induced phase transition, established in Ref. [23]based on numerical simulations.The large configurational entropy is seen even more directlyupon taking the statistical average over the noise realizations.This yields the Lindblad quantum master equation, i.e. theevolution equation for the unconditioned density matrix ˆ ρ t = ˆ ρ ( c ) t , d ˆ ρ t = dt ( − i [ ˆ H , ˆ ρ t ] − γ (cid:88) i [ˆ n i , [ˆ n i , ˆ ρ t ]]) . (13)The right hand side indeed describes a completely posi-tive Lindblad operator, for the special case of HermitianLindblad jump operators ˆ n l (generalized measurements usingnon-Hermitian operators, and subsequently general Lindbladequations, can be described in the POVM (positive operatorvalued measurement) framework, cf. [42]). Clearly, while ˆ ρ ( c ) t describes a pure state, ˆ ρ t represents a mixed state ensemble ofpure states. Equation (13) is a deterministic equation which islinear in the state ˆ ρ t , as required by the foundations of prob-ability theory for the generator of motion for any completestatistical representation of a system [43].In terms of a physical interpretation, the Lindblad equationcorresponds to unread (or averaged over) measurements. In-deed, this averaging over measurement outcomes introducesan extensive configurational entropy, e.g. S = log ( N ! N ! ) → N for a large, half filled system. For H (cid:44)
0, the Lindbladequation (13) is solved by a totally mixed, infinite tempera-ture state ˆ ρ ∼ (here is the identity in the Hilbert spaceof fixed total particle number N ). In fact, this indefinite heat-ing directed towards a stable, totally mixed dynamical fixedpoint is generic for Hermitian measurement operators com-peting with a Hamiltonian. This can be illustrated by consid-ering the e ff ect of the Hamiltonian and the Lindblad operatorsseparately. The Hermitian Lindblad operators evolve the sys-tem towards a density matrix, which is diagonal in Fock space(or diagonal in the basis of eigenstates of general Hermitianmeasurement operators ˆ O l ). The Hamiltonian, due to the non-commutativity [ H , ˆ n l ] =
0, induces transitions between dif- h ˆ n ih ˆ n i h ˆ n ˆ n ih ˆ n ˆ n i h ˆ n i . . g h ˆ n i t
00 7 . . a ) t h ˆ n i t
00 7 . . b )( c ) t Figure 1. Illustration of competing Hamilton and measurement dy-namics in the two-site toy model. (a),(b) Single trajectory dynamics:(a) For strong monitoring g =
10 the system is pinned to dark statesfor most of the time, with weak, short-time fluctuations. (b) A dom-inant Hamilton dynamics ( g = .
5) leads to depinning, with almostno time spent in the dark states. (c) Trajectory averaged correlationfunctions: Quantities linear in the state are insensitive to the com-petition ratio, while the correlation function nonlinear in the statewitnesses a monitoring dependence, which develops a non-analyticbehavior in the thermodynamic limit. ferent eigenstates and mixes di ff erent matrix elements until atotally mixed state ∼ is reached. Two-site toy model – Before delving into the analysis ofthe many-body problem, we illustrate the competition be-tween Hamiltonian and measurement operators in a minimaltoy model, a single fermion on two sites, L = , N =
1. Theproblem is equivalent to a monitored two-level system, withits Hilbert space spanned by the two states {|↑(cid:105) = | , (cid:105) , |↓(cid:105) = | , (cid:105)} , and ˆ H = σ x , and the operator valued part of ˆ M i , t givenby ( × ∓ σ z ) / i = ,
2, respectively. For g − = |↑(cid:105) , |↓(cid:105) , while for g =
0, it describes Rabi oscillations. In Fig. 1(a,b) we plot thetrajectory expectation value of the fermion occupation on site i =
1. For strong monitoring g − (cid:28)
1, the system is pinnedfor long times in the dark states, with rare jumps into the otherdark state induced by the Hamiltonian, and small fluctuationsin the rest periods [44, 45]. Conversely, for weak monitoring g (cid:28)
1, the evolution is still closely reminiscent of Rabi oscil-lations, and there is a vanishing amount of time spent in thedark states. We will be guided by this picture of depinningfrom the dark states induced by the Hamiltonian dynamics inour analysis of the many-body problem.The toy model also illustrates the importance of a suit-able choice of trajectory averaged quantities to detect thechanges in the trajectories upon varying the competition ra-tio g . Fig. 1(c) plots averages evaluated after long evolutiontime. The observables (cid:104) ˆ n (cid:105) and (cid:104) ˆ n ˆ n (cid:105) are linear in the con-ditional state ˆ ρ ( c ) t : Due to the indefinite heating, they are notsensitive to the value of g . In contrast, the trajectory averaged covariance matrix for the occupations C = (cid:104) ˆ M , t ˆ M , t (cid:105) t = (cid:104) ˆ n ˆ n (cid:105) t − (cid:104) ˆ n (cid:105) t (cid:104) ˆ n (cid:105) t (14)is non-linear in the state and tracks a quantitative change asfunction of g , rooted in the non-commuting nature of theHamilton operator and the measurement operators. In thesmall Hilbert space considered here, such changes are onlyof a quantitative nature. In Sec. III C, we will show that corre-lators of this type may behave qualitatively di ff erently in theweak and strong monitoring regimes in a many-body system,which provides a clear indication of a phase transition at afinite competition ratio g c . B. Spatial continuum limit: Measured Dirac fermions andbosonization
Measured Dirac fermions – The above microscopic modelof measured fermions o ff ers a measurement induced phasetransition, however in light of the heating for the average mea-surement a direct low energy reduction to a continuum limitmay not be fully justified. Here we take a more conserva-tive approach, and formulate our starting point directly in thespatial continuum in terms of (1 +
1) dimensional monitoredmassless Dirac model. The key ingredient shared with thelattice model is a competition between a kinetic Hamiltoniangiving rise to delocalization of fermions and creating spatialentanglement, and local, mutually commuting measurements,which localize or pin the fermions, and by themselves drivethe system into dark eigenstates of product form.The massless Dirac Hamiltonian reads in terms of thespinor ˆ Ψ x = ( ˆ ψ R , x , ˆ ψ L , x ) T ˆ H = iv (cid:90) x ˆ Ψ † x σ z ∂ x ˆ Ψ x , (15)where σ z is the Pauli matrix. We choose two distinct localmeasurement operators ˆ O , x , ˆ O , x ,ˆ O , x = Ψ † x Ψ x = ˆ J (0) x , ˆ O , x = Ψ † x σ x Ψ x , (16)which are both measured independently but with the samerate γ . The measurement operators commute with each other,but they do not commute with the Hamiltonian, thus realizingthe desired competition. They are local, and stabilize productdark states | σ a , x (cid:105) = (cid:81) a , x | σ a , x (cid:105) x with a = ± , and σ a , x = , σ x is diagonal.The associated stochastic Schr¨odinger equation then fea-tures a non-Hermitian Hamiltonian,ˆ H nH = ˆ H − i γ (cid:88) s = , (cid:90) x ˆ M s , x , t , (17)with ˆ M s , x , t = ˆ O s , x − (cid:104) ˆ O s , x (cid:105) t and uncorrelated noise increments dW s , x dW s (cid:48) , x (cid:48) = δ s , s (cid:48) δ ( x − x (cid:48) ) for both measurements. In or-der to characterize the transition we consider the correlationfunctions C y = (cid:104) ˆ M x , t ˆ M x + y , t (cid:105) , ˆ M x , t = ˆ M , x , t + ˆ M , x , t (18)with the symmetric sum of the measurement operators. Thiscombination of the measurement operators corresponds to thelocal fermion density.We note that this Hamiltonian and the measurement oper-ators would obtain in a naive low energy approach (at halffilling), where the continuum limit is performed for the latticemodel defined with Eqs. (11,12), and the continuum fermionfield is decomposed as ˆ ψ x = ˆ ψ R , x e i πρ x + ˆ ψ L , x e − i πρ x , where ρ here indicates the average fermion density. For the mainresults in this paper, we will not rely on this connection, andconsider the continuum model in autonomy. Bosonization – The monitored Dirac model lends itself tobosonization, where the fermion bilinears are replaced by bo-son fields. The resulting bosonic model is well suited for afurther practical evaluation. The Hamiltonian maps to the Lut-tinger Liquid Hamiltonianˆ H = v π (cid:90) x [( ∂ x ˆ θ x ) + ( ∂ x ˆ φ x ) ] . (19)It is quadratic in the Hermitian operator ˆ φ x associated to den-sity fluctuations, and its conjugate ˆ θ x connected to phase fluc-tuations. These operators fulfill the canonical commutationrelations [ ∂ x ˆ θ x , ˆ φ x (cid:48) ] = − i δ ( x − x (cid:48) ). The measurement opera-tors transform intoˆ O , x = − π ∂ x ˆ φ x , ˆ O , x = m cos(2 ˆ φ x ) (20)with a ’mass’ m = O (1) that depends on the normal order-ing prescription. Clearly, also in bosonized language ˆ O s , x donot commute with ˆ H . The measurement operators both arefunctions of the operator ˆ φ x and therefore pairwise commut-ing [ ˆ O α, x , ˆ O β, x (cid:48) ] = α, β ∈ { , } , and thus stabilize theeigenstates of ˆ φ x . In the measurement-only limit (i.e., forˆ H =
0) this gives rise to a set of dark states ˆ M s , x , t | Ψ D (cid:105) = φ x | Ψ D (cid:105) = φ x | Ψ D (cid:105) for all positions x . The only restriction to the set of real valuedeigenvalues { φ x } is to match the condition for a fixed total par-ticle number, which for periodic boundary conditions reads as (cid:82) x cos(2 πρ x + φ x ) =
0. This gives rise to an exponentiallygrowing number of dark states with pinned field operator ex-pectation values (cid:104) ˆ φ x (cid:105) t = φ x , analogous to the lattice fermiondark states with a pinned fermion particle number.As soon as the Hamiltonian is switched on ( ˆ H (cid:44) M , x , t ,which tends to push the system towards one of the measure-ment dark states, and the conjugate field ˆ θ x in the Hamiltonian,which favors a depinning of ˆ φ x . We refer to the linear mea-surement operators ˆ M , x . t ∼ ∂ x ( ˆ φ x − (cid:104) ˆ φ x (cid:105) t ) as gapless, due totheir vanishing in the long wavelength limit (taking q → (cid:104) ˆ M x , t ˆ M x + y , t (cid:105) ∼ | y | − .What drives the competition between the pinning and de-pinning of the field operator ˆ φ x is the interplay between thelocal, nonlinear appearance of ˆ φ x in the measurement opera-tors ˆ M , x and the conjugate fields ∂ x ˆ θ x in the Hamiltonian. As we detail below in Eq. (21), for strong monitoring, the mea-surement operators can be linearized around the dark state,yielding a measurement operator ˆ M , x , t ∼ m ( ˆ φ x −(cid:104) ˆ φ x (cid:105) t ), whichdoes not vanish in the long wavelength limit. It causes expo-nentially decaying correlations C y , with a correlation lengththat is comparable to the strong-monitoring correlation lengthin the fermion lattice model Eq. (A4). We therefore denotethe linear approximation of M , x , t as a gapped measurementoperator. C. Limits of weak and strong monitoring: Riccati approach
We proceed by analyzing the connected correlation func-tions in two limiting cases, where either M , x , t is irrelevantand can be neglected, or M , x , t is dominating and obtains thegapped form ˆ M , x , t ∼ m ( ˆ φ x − (cid:104) ˆ φ x (cid:105) t ). In both cases the mea-surement operators are linear in the fields ˆ φ x . For linear mea-surement operators, the correlation function (18) a ff ords ananalytical solution. This allows us to demonstrate that indeedfor a gapless measurement operator, the correlations functionsobtain a scale-invariant form, while, in contrast, correlationswill not be scale invariant if the measurement operators aregapped.We identify the limit in which M , x , t is irrelevant with thelimit of weak monitoring, i.e., when the Hamiltonian domi-nates the dynamics γ (cid:28) v . Then, the pinning of ˆ φ x inducedby the nonlinearity in ˆ M , x , t cannot overcome the depinninge ff ect from the Hamiltonian in Eq. (17). The field operatorsˆ φ x will explore an extended region of phase space, and thebounded nonlinear terms are irrelevant and average to zero.We therefore drop M , x , t from the evolution.In the opposite limit of strong measurements, i.e., v (cid:28) γ ,the state of the system will remain pinned in a measurementdark state | Ψ D (cid:105) similar to Fig. 1, where this is shown for thefermion density. At random times, it may jump from thevicinity of one dark state to another dark state with a dif-ferent eigenvalue. These jumps are, however, uncorrelatedand exponentially rare in time in the limit of dominant mea-surements. We take the pinning into account by assumingthat the state | ψ t (cid:105) remains close to a measurement dark state | Ψ D (cid:105) with ˆ φ x | Ψ D (cid:105) = φ x | Ψ D (cid:105) . For instance one may assume | ψ t (cid:105) = | Ψ D (cid:105) + | δψ t (cid:105) with (cid:104) δψ t | ( ˆ φ x − φ x ) | δψ t (cid:105) / (cid:104) δψ t | δψ t (cid:105) (cid:28) φ x → φ x + ˆ φ x with respect to theeigenvalue φ x in the dark state and expand the measurementoperators ˆ M , x , t linearly in the small deviation ˆ φ x , yieldingˆ M , x , t ∼ ( ˆ φ x − (cid:104) ˆ φ x (cid:105) t ) . (21)The more accurate proportionality constant in Eq. (21) is2 m sin(2 φ x ) and depends on the eigenvalues φ x . We neglectit here (see Appendix B for more details, including the case ofinhomogeneously distributed φ x ).We consider now the dynamics with the quadratic Hamil-tonian (19) and one dominant measurement operator ˆ M x , t which is linear in ˆ φ x , i.e., either M x , t = ∂ x ( ˆ φ x − (cid:104) ˆ φ x (cid:105) t ) orˆ M x , t = ˆ φ x − (cid:104) ˆ φ x (cid:105) t , reflecting the two di ff erent regimes. Thisyields the stochastic Schr¨odinger equation d | ψ t (cid:105) = − dt (cid:104) i ˆ H + γ (cid:90) x (cid:16) ˆ M x , t (cid:17) (cid:105) | ψ t (cid:105) + (cid:90) x dW x ˆ M x , t | ψ t (cid:105) . (22)The connected two-point correlation function for linear mea-surement problems can be solved exactly by methods devel-oped in the context of Kalman filtering [46] in classical con-trol theory [47, 48], also used in quantum optics [38, 49, 50].The problem maps to a Riccati equation, which we solve in thestationary state in both limits (see Appendix B for details).In the weak monitoring limit, this yields correlation func-tions in real and momentum space C y = C | y | , C k = C | k | . (23)Here C = v γ (cid:104)(cid:16) γ π v + (cid:17) − (cid:3) . In the limit γ → C = /π . In the opposite limit γ (cid:29) v it assumes theasymptotic form C = (cid:112) v / γπ .This result shows that linear measurement operators ˆ M x , t ∼ ∂ x ˆ φ x − (cid:104) ∂ x ˆ φ x (cid:105) t stabilize a scale invariant covariance matrixwith a 1 / | y | scaling behavior. Precisely this scaling wasalso detected numerically in the case of measured latticefermions [23]. The only assumption leading to this result inthe present framework is that the nonlinear terms in the mea-surement operators are negligible (or technically, m = M x , t = ˆ φ x − (cid:104) ˆ φ x (cid:105) t yields C y ∼ (cid:32) v πγ (cid:33) | y | = e −| y | /ξ . (24)This correlation function decays exponentially with the dis-tance and with a correlation length ξ − = log(2 πγ/ v ). Thisindicates that the extreme limit v =
0, which is uncorrelatedexcept for y =
0, extends smoothly to a short range correlatedstate at non-zero v . To further back up these result beyondthe linear approximation performed here, we have computedthe correlation length in the related fermionic lattice modeldefined by Eqs. (11,12) perturbatively in Appendix A. Thisyields a similar correlation length of ξ − ∼ log( γ/ν ).To summarize, we obtain a scale invariant correlation func-tion with characteristic 1 / | y | decay at weak monitoring, ver-sus exponentially decaying correlations with a logarithmi-cally growing correlation length at strong monitoring. Strictlyspeaking, to obtain the first result, we needed to set ˆ M , x , t = M , x , t in the opposite limit. Theviability of this procedure reduces to the question whether ornot the nonlinearity in the measurement operator is relevant orirrelevant. This question will be addressed in the next section. IV. REPLICA FIELD THEORY FOR MONITOREDLUTTINGER LIQUIDS
In this section, we will provide a theoretical description,which is not restricted to the two limiting cases and also incor-porates intermediate monitoring strength. To this end, we de-velop a field theory description of a two-replica setup, which provides a tool to analyze whether an initial nonlinearity inthe measurement operators becomes relevant or irrelevant atlarge distances. The replica approach also grants valuable in-sights and gives access to the relevant, nontrivial degrees offreedom in the measurement evolution, which turn out to bethe relative replica fluctuations.
A. Structure of measurement operator correlation functions
In order to motivate the replica approach, we revisit themeasurement operator correlation function on the lattice C i j ( t ) = (cid:104) ˆ M i , t ˆ M j , t (cid:105) t , where again ˆ M i , t = ˆ O i − (cid:104) ˆ O i (cid:105) t for themeasured operator ˆ O i . In Eq. (3), we expressed the correla-tion function C i j in terms of the conditioned projector ˆ ρ ( c ) t andthe trajectory average of its product ˆ ρ ( R ) t = ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t as C i j ( t ) =
12 Tr (cid:104) ( ˆ O (1) i − ˆ O (2) i )( ˆ O (1) j − ˆ O (2) j ) ˆ ρ ( R ) (cid:105) . (25)The density matrix ˆ ρ ( R ) t can no longer be written as a prod-uct of pure states and encodes the correlations, and therefore C i j ( t ) can behave nontrivially.Equation (25) shows that (i) the measurement correlationfunction is the product of two relative observables ˆ O (1) − ˆ O (2) (or their fluctuation) in the replicated space and (ii) that thiscorrelation function must be encoded in the linear statisticalaverage ˆ ρ ( R ) t . For this and the following section, we introducea shorthand notation for expectation values as in Eq. (25). Weset (cid:104)(cid:104) ... (cid:105)(cid:105) = Tr[ ... ˆ ρ ( R ) ] , (26)which is the expectation value of an operator (or operatorproduct) ... with respect to ˆ ρ ( R ) . It combines the quantummechanical average as well as the trajectory average from theprevious sections. B. Two-replica master equation
The two-replica density matrix ˆ ρ ( R ) t is no longer a stochas-tic object but evolves according to a deterministic evolutionequation. The infinitesimal increment d ˆ ρ ( R ) t is obtained fromtaking the statistical average of the product increment d ˆ ρ ( R ) t = ˆ ρ ( R ) t + dt − ˆ ρ ( R ) t = ˆ ρ ( c ) t + dt ⊗ ˆ ρ ( c ) t + dt − ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t = d ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t + ˆ ρ ( c ) t ⊗ d ˆ ρ ( c ) t + d ˆ ρ ( c ) t ⊗ d ˆ ρ ( c ) t . (27)The stochastic increments { dW i } at time-step t → t + dt arenot correlated with the increments at earlier times. There-fore the trajectory average in Eq. (27) sets any incrementswhich are only linear in dW to zero and replaces averages overterms quadratic in dW by ( ... ) dW i dW j = ( ... ) δ i j γ dt . Here ( ... )represents any other stochastic variable, such as (cid:104) ˆ O i (cid:105) t whichdoes not depend on dW but on stochastic increments at earliertimes.The first and the second term in Eq. (27) only include theincrement of one single density matrix and are thus identicalto taking the statistical average over a single-replica masterequation for d ˆ ρ ( c ) , i.e., d ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t = dt L (1) (cid:16) ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:17) + (cid:88) i dW i (cid:110) ˆ M (1) i , t , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111) = dt L (1) (cid:18) ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:19) = dt L (1) ˆ ρ ( R ) t . (28)Here, we have defined the intra-replica Liouvillian L ( l ) = i [ ˆ H ( l ) , · ] − γ (cid:80) i [ ˆ O i , [ ˆ O i , · ]] and measurement operator ˆ M ( l ) i , t = ˆ O ( l ) − (cid:104) ˆ O i (cid:105) t . Here, (cid:104) ˆ O i (cid:105) t = Tr (cid:104) ˆ O i ˆ ρ ( c ) t (cid:105) is still a stochastic ob-ject, which does, however, not depend on dW as mentionedabove. This yields the familiar Lindblad form as in Eq. (13),acting on each single replica separately.The third term, however, builds up correlations betweenboth replicas and yields an unconventional contribution to theevolution of ˆ ρ ( R ) t . The replica-coupling evolution term is d ˆ ρ ( c ) t ⊗ d ˆ ρ ( c ) t = (cid:88) i , j (cid:110) ˆ M (1) i , t , (cid:110) ˆ M (2) j , t , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111)(cid:111) dW i dW j = γ dt (cid:88) i (cid:110) ˆ M (1) i , t , (cid:110) ˆ M (2) i , t , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111)(cid:111) . (29)It contains three di ff erent classes of couplings, two of whichwe have to consider with care. The first class are terms com-posed only of operators, i.e., the ones summarized by (cid:110) ˆ O (1) i , (cid:110) ˆ O (2) i , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111)(cid:111) = (cid:110) ˆ O (1) i , (cid:110) ˆ O (2) i , ˆ ρ ( R ) t (cid:111)(cid:111) . (30)Here the stochastic average acts only on the density matrixand does not a ff ect the operators at all. The second class ofincrements is (cid:110) (cid:104) ˆ O i (cid:105) t , (cid:110) (cid:104) ˆ O i (cid:105) t , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111)(cid:111) = (cid:104) ˆ O i (cid:105) t ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t . (31)Here we face a problem with computing the stochastic aver-age, since the two replica density matrix ˆ ρ ( R ) t does not giveaccess to the individual trajectories of (cid:104) ˆ O i (cid:105) t . It only allowsus to compute the stochastic averages (cid:104)(cid:104) ˆ O ( l ) (cid:105)(cid:105) = (cid:104) ˆ O i (cid:105) t or (cid:104)(cid:104) ˆ O (1) j ˆ O (2) i (cid:105)(cid:105) = (cid:104) ˆ O i (cid:105) t (cid:104) ˆ O j (cid:105) t , but this does not include informa-tion on the statistical correlations between (cid:104) ˆ O i (cid:105) t and ˆ ρ ct ⊗ ˆ ρ ct ,which are required in the average in Eq. (31). This also holdsfor the third type of increment, which is (cid:110) ˆ O ( l ) i , (cid:110) (cid:104) ˆ O i (cid:105) t , ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:111)(cid:111) = (cid:26) ˆ O ( l ) i , (cid:104) ˆ O i (cid:105) t ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t (cid:27) . (32)This problem is indicating the initial point of an infinitecoupled hierarchy of replica correlation functions: Indeed,the statistical averages in Eqs. (32) and (31) can again be for-mulated as linear problems in a three- or four-replica frame-work, respectively (e.g., (cid:104) ˆ O i (cid:105) t ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t = Tr (3) [ ˆ O (3) i ˆ ρ ( R ) t ] canbe written as a partial trace over the third replica in a three-replica density matrix ˆ ρ ( R ) t ). This illustrates that once therealm of a single replica description of the problem is left, an infinite hierarchy of replica equations emerges. In order tofind a solution for ˆ ρ ( R ) , this hierarchy has to be truncated atfinite order.Here we perform the truncation on the level of the two-replica density matrix, which amounts to treating the stochas-tic correlations between (cid:104) ˆ O i (cid:105) t and ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t in a mean-fieldtype decoupling by setting (cid:104) ˆ O i (cid:105) t ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t ≡ (cid:104)(cid:104) ˆ O (1 , i (cid:105)(cid:105) t ˆ ρ ( R ) t , (33) (cid:104) ˆ O i (cid:105) t ˆ ρ ( c ) t ⊗ ˆ ρ ( c ) t ≡ (cid:104)(cid:104) ˆ O (1) i ˆ O (2) i (cid:105)(cid:105) t ˆ ρ ( R ) t . (34)We will see later that if the operators ˆ O l are linear func-tions of boson or fermion fields (as it is the case for the op-erators ˆ M , x , t in the bosonized framework), then the mean-field decoupling in Eqs. (33), (34) has no e ff ect on the rela-tive correlation function C i j , i.e., the mean field approxima-tion is exact for these correlators. The measurement expecta-tion values then only show up in a center-of-mass coordinateˆ O (1) i + ˆ O (2) i , which completely decouples from relative coor-dinates ˆ O (1) i − ˆ O (2) i and e ff ectively heats up to infinite temper-ature. This heating of the center-of-mass coordinate remainsoperational when the linear operator ˆ M , x , t and the non-linearoperator ˆ M , x , t are measured simultaneously; in turn this jus-tifies the mean-field approximation also in this case. The Her-mitian nature of the measurement operators, and the subse-quent heating, simplifies the problem significantly here. Fora general set of only nonlinear and non-Hermitian measure-ment operators, including nonlinear feedback operations, themean-field decoupling is, however, not immediately justified.After performing the statistical average including the mean-field decoupling, one obtains a deterministic evolution equa-tion for ˆ ρ ( R ) , the two-replica master equation ∂ t ˆ ρ ( R ) t = L (1) ˆ ρ ( R ) t + L (2) ˆ ρ ( R ) t − γ ˆ ρ ( R ) ˜ C ( t ) (35) + γ (cid:88) l (cid:110) ˆ O (2) i − (cid:104)(cid:104) O (2) i (cid:105)(cid:105) , (cid:110) ˆ O (1) i − (cid:104)(cid:104) ˆ O (1) i (cid:105)(cid:105) , ˆ ρ ( R ) t (cid:111)(cid:111) . Here, the normalization function is ˜ C t = (cid:80) i (cid:16) (cid:104)(cid:104) ˆ O (1) i ˆ O (2) i (cid:105)(cid:105) t − (cid:104)(cid:104) ˆ O (1) i (cid:105)(cid:105) t (cid:104)(cid:104) ˆ O (2) i (cid:105)(cid:105) t (cid:17) . The first line is theregular master equation for two uncoupled replicas evolvingunder the Liouvillians L ( l ) . If no additional terms werepresent, the Liouvillians would heat up each replica towardsan infinite temperature state.The second line in Eq. (35) is what distinguishes the two-replica setup from a single replica, which would only heat upto infinite temperature. It leads to the build-up of inter-replicacorrelations, and transforms the replica master equation intoa non-Lindblad form. This equation is still non-linear in thestate due to the expectation values (cid:104) ˆ O i (cid:105) t , contained in the op-erators ˆ M i , t and in ˜ C t . However, the equation is determin-istic, and the expectation values are no longer stochastic vari-ables. In the stationary state, they can be replaced by a number (cid:104) ˆ O i (cid:105) t = o i . We will analyze the consequences of this evolutionequation in the following section.0 C. Two-replica master equation in bosonized framework
We now formulate the bosonized measurement setup intro-duced in Sec. III B in terms of the replica master equation (35).This will allow us to establish the connection between the gen-erator of dynamics, and the correlation functions. To this end,we extend the bosonized Hamiltonian and the measurementoperators to replica space by introducingˆ H ( l ) = v π (cid:90) x ( ∂ x ˆ φ ( l ) x ) + ( ∂ x ˆ θ ( l ) x ) , (36)ˆ M ( l )1 , x , t = − π ∂ x ( ˆ φ ( l ) x − (cid:104) ˆ φ ( l ) x (cid:105) t ) , (37)ˆ M ( l )2 , x , t = m (cos(2 ˆ φ ( l ) x ) − (cid:104) cos(2 ˆ φ ( l ) x ) (cid:105) t ) , (38)where again the label l indicates how the operator acts inreplica space (see the definition above Eq. (25)). The two-replica Hamiltonian is ˆ H = ˆ H (1) + ˆ H (2) . Due to indepen-dent noise increments for ˆ M , x , t and ˆ M , x , t , the two di ff erentmeasurements are uncorrelated and contribute independentlyto the replica master equation. This can be implemented in thetwo-replica master equation by replacing the index i → ( s , x )with s = , x the continuous position.To gain a better understanding of the two-replica dynamics,we again start by considering only measurements of the lin-ear operator ˆ M , x , t without measuring ˆ M , x , t . Apart from theconstant terms, this yields the two-replica master equation ∂ t ˆ ρ ( R ) = i (cid:104) ˆ ρ ( R ) , ˆ H (cid:105) − γ π (cid:88) l = , (cid:90) x (cid:104) ∂ x ˆ φ ( l ) x , (cid:104) ∂ x ˆ φ ( l ) x , ˆ ρ ( R ) (cid:105)(cid:105) (39) + γπ (cid:90) x (cid:110) ∂ x ˆ φ (2) x − (cid:104)(cid:104) ∂ x ˆ φ (2) x (cid:105)(cid:105) , (cid:110) ∂ x ˆ φ (1) x − (cid:104)(cid:104) ∂ x ˆ φ (1) x (cid:105)(cid:105) , ˆ ρ ( R ) (cid:111)(cid:111) . The master equation is quadratic in the field operators, butcouples operators with di ff erent replica index l . The termscan be decoupled by performing a coordinate transforma-tion into a replica center-of-mass field ˆ φ ( a ) x , ˆ θ ( a ) x and a rel-ative field ˆ φ ( r ) x , ˆ θ ( r ) x (describing inter-replica fluctuations) ac-cording to the unitary transformation ˆ φ ( a , r ) x = ( ˆ φ (1) x ± ˆ φ (2) x ) / √ θ ( a , r ) x = (ˆ θ (1) x ± ˆ θ (2) x ) / √
2. It preserves the bosonic commutationrelations between the operators [ ∂ x ˆ θ ( l ) x , ˆ φ ( l (cid:48) ) y ] = − i δ l , l (cid:48) δ ( x − y )for l , l (cid:48) = a , r .The quadratic Hamiltonian ˆ H remains a sum ˆ H = ˆ H ( r ) + ˆ H ( a ) with ˆ H ( l ) as defined in Eq. (36), but with center-of-massand relative operators. The measurement part of the evolution,however, is transformed such that the relative and center-of-mass decouple in the new basis. The master equation for ˆ ρ ( R ) is therefore separable, which suggests a product ansatz ˆ ρ ( R ) = ˆ ρ ( r ) ⊗ ˆ ρ ( a ) . Inserting this ansatz into Eq. (39) yields ∂ t ˆ ρ ( r ) = i [ ˆ ρ ( r ) , ˆ H ( r ) ] − γπ (cid:90) x (cid:110) ( ∂ x ˆ φ ( r ) x ) , ˆ ρ ( r ) (cid:111) , (40) ∂ t ˆ ρ ( a ) = i [ ˆ ρ ( a ) , ˆ H ( a ) ] (41) + γπ (cid:90) x (cid:16) ∂ x ˆ φ ( a ) − (cid:104)(cid:104) ∂ x ˆ φ ( a ) (cid:105)(cid:105) (cid:17) ˆ ρ ( a ) (cid:16) ∂ x ˆ φ ( a ) − (cid:104)(cid:104) ∂ x ˆ φ ( a ) (cid:105)(cid:105) (cid:17) . This decomposition is a remarkable result, which gives thetwo replica master equation a new interpretation. The den- sity matrix ˆ ρ ( r ) of the fluctuations evolves according to a non-Hermitian Schr¨odinger equation with an e ff ective Hamilto-nian ˆ H ( r )e ff = v π (cid:90) x ( ∂ x ˆ θ ( r ) x ) + (1 − i γ v π )( ∂ x ˆ φ ( r ) x ) , (42)but without feedback from a contour-coupling term (i.e., nocoupling between the left and the right side of ˆ ρ ( R ) ). Thedensity matrix of the relative degrees is thus evolving to-wards a stationary state ˆ ρ ( r ) t = e − i ˆ H ( r )e ff t ˆ ρ ( r )0 e i ˆ H ( r )e ff t → | ψ ( r ) D (cid:105)(cid:104) ψ ( r ) D | which approaches a dark state of the e ff ective Hamiltonianˆ H ( r )e ff | ψ ( r ) D (cid:105) =
0. In Appendix E, we demonstrate that H ( r )e ff in-deed has one unique dark state, which is reached from anyinitial state during the time evolution. We show that the corre-lation functions of the replica field operators in the dark state, (cid:104) ψ ( r ) D | ∂ x ˆ φ ( r ) x ∂ y ˆ φ ( r ) y | ψ ( r ) D (cid:105) are identical to the correlation functions(23) obtained from the Riccati approach, which provides theexact solution for the covariance matrix.This observation is independent of the mean-field decou-pling of the statistical fluctuations in Eqs. (33), (34). Thereason is that due to the replica symmetry (i.e. invarianceof the theory under exchange ˆ φ (1) ↔ ˆ φ (2) ), independently ofthe noise realization we have (cid:104) ˆ φ (1) (cid:105) = (cid:104) ˆ φ (2) (cid:105) . This implies √ (cid:104) ˆ φ ( r ) (cid:105) = (cid:104) ˆ φ (1) − ˆ φ (2) (cid:105) =
0. Therefore, the neglected noisecorrelations are only a ff ecting ˆ ρ ( a ) , but not the density matrixof the fluctuations ˆ ρ ( r ) . The dark state therefore describes theexact steady state of the replica fluctuations.We now turn to the measurement correlation functionEq. (25), here under the assumption that the nonlinear termsare irrelevant, ˆ M , x , t =
0. The relative measurement operatoris ˆ M (1) x , t − ˆ M (2) x , t = √ π ∂ x ˆ φ ( r ) x , yielding the measurement corre-lation function C y = π (cid:104) ψ ( r ) D | ∂ x ˆ φ ( r ) ∂ x ˆ φ ( r ) x + y | ψ ( r ) D (cid:105) , the expectationvalue of ∂ x ˆ φ ( r ) ∂ x ˆ φ ( r ) x + y in the dark state. The scale invariance ofthe e ff ective Hamiltonian ˆ H ( r )e ff implies that also the measure-ment correlation functions in this limit become scale invariantand algebraically decaying, confirming the observation fromSec. III C.We can also infer what happens in the limit when ˆ M , x , t ismeasured with a large mass γ m (cid:29) v , where the linearizationˆ M , x , t → ˆ φ ( l ) x − (cid:104) ˆ φ ( l ) x (cid:105) t is justified. Using the same arguments asfor the case m =
0, this yields a separable evolution equationfor l = r , a with a modified e ff ective Hamiltonian in Eq. (40)ˆ H ( r )e ff → ˆ H ( r )mass and ˆ H ( r )mass = ˆ H ( r )e ff − i γ (cid:90) x ( ˆ φ ( r ) x ) . (43)In ˆ H ( r )mass the additional dissipative mass term ∼ γ introduces alength scale. The measurement correlations are therefore stilldescribed by a pure quantum state but now correlations func-tions in the state | ˜ ψ ( r ) D (cid:105) are exponentially decaying in space.The e ff ective cooling and the evolution towards a singledark state in the relative degrees of freedom ˆ θ ( r ) x , ˆ φ ( r ) x have tobe contrasted with the behavior of the center-of-mass fieldsˆ θ ( a ) x , ˆ φ ( a ) x described by Eq. (41). Their evolution does not fol-low an e ff ective Schr¨odinger equation but instead displays en-hanced statistical fluctuations. In order to determine the fate1of the local correlation functions for the center-of-mass de-grees of freedom, we consider their evolution equation ∂ t (cid:104)(cid:104) ˆ φ ( a ) x ˆ φ ( a ) x (cid:105)(cid:105) = Tr (cid:104) ( ˆ φ ( a ) x ) ∂ t ˆ ρ ( a ) (cid:105) = − i (cid:104)(cid:104) [ ˆ H ( a ) , ˆ φ ( a ) x ˆ φ ( a ) x ] (cid:105)(cid:105) + γπ (cid:90) x (cid:48) (cid:104)(cid:104) ( ∂ x ˆ φ ( a ) x (cid:48) − (cid:104)(cid:104) ∂ x ˆ φ ( a ) x (cid:48) (cid:105)(cid:105) ) ( ˆ φ ( a ) x ) (cid:105)(cid:105) . (44)Due to the quadratic evolution equation for ˆ ρ ( a ) t , higher or-der correlation functions such as in second line of Eq. (44)can be decoupled via Wick’s theorem. This yields a nonlin-ear, strictly positive growth of (cid:104)(cid:104) ˆ φ ( a ) x ˆ φ ( a ) x (cid:105)(cid:105) in time. The com-mutator with the Hamiltonian yields a term proportional to (cid:104)(cid:104) ˆ φ ( a ) x ˆ θ ( a ) x (cid:105)(cid:105) , which, however, has a similar non-linear evolutionequation (just like (cid:104)(cid:104) ˆ θ ( a ) x ˆ θ ( a ) x (cid:105)(cid:105) ). In consequence this yields anasymptotic value (cid:104)(cid:104) ˆ φ ( a ) x ˆ φ ( a ) x (cid:105)(cid:105) → ∞ , which is consistent withan infinite temperature state in the center-of-mass degrees of freedom. D. Nonlinear master equation for the replica fluctuations
Now, we will focus on the general case in which the mea-surement operator ˆ M , x , t cannot be neglected and is a nonlin-ear function of ˆ φ ( l ) x . We will use the insights from the previousdiscussion to obtain the master equation for the replica fluc-tuations. We demonstrate that also in this case the time evo-lution of the relative degrees of freedom is described by ane ff ective Hamiltonian. It is a non-Hermitian analogue of thesine-Gordon Hamiltonian.We start with the full two-replica master equation in thebasis of relative and center-of-mass degrees of freedom ∂ t ˆ ρ ( R ) = − i [ ˆ H ( r )e ff ˆ ρ ( R ) − ˆ ρ ( R ) ( ˆ H ( r )e ff ) † ] − i [ ˆ H ( a ) , ˆ ρ ( R ) ] + γπ (cid:90) x (cid:16) ∂ x ˆ φ ( a ) − (cid:104)(cid:104) ∂ x ˆ φ ( a ) x (cid:105)(cid:105) (cid:17) ˆ ρ ( R ) (cid:16) ∂ x ˆ φ ( a ) x − (cid:104)(cid:104) ∂ x ˆ φ ( a ) x (cid:105)(cid:105) (cid:17) (45) − γ m (cid:90) x (cid:88) σ = ± (cid:104) cos (cid:16) √
2( ˆ φ ( a ) x + σ ˆ φ ( r ) x ) (cid:17) , (cid:104) cos (cid:16) √
2( ˆ φ ( a ) x + σ ˆ φ ( r ) x ) (cid:17) , ˆ ρ ( R ) (cid:105)(cid:105) + γ m (cid:110) cos (cid:16) √
2( ˆ φ ( a ) x + ˆ φ ( r ) x ) (cid:17) − (cid:104)(cid:104) cos (cid:16) √
2( ˆ φ ( a ) x + ˆ φ ( r ) x ) (cid:17) (cid:105)(cid:105) , (cid:110) cos (cid:16) √
2( ˆ φ ( a ) x − ˆ φ ( r ) x ) (cid:17) − (cid:104)(cid:104) cos (cid:16) √
2( ˆ φ ( a ) x − ˆ φ ( r ) x ) (cid:17) (cid:105)(cid:105) , ˆ ρ ( R ) (cid:111)(cid:111) . This equation may look overwhelming at first sight, but wewill now show how its complexity can be drastically reducedby making use of the insights from the previous section. Wewill eliminate the coupling of the relative degrees of free-dom to the center-of-mass modes, to obtain an e ff ective masterequation for ˆ ρ ( r ) .In order to do this, we trace out the center-of-mass degreesof freedom in Eq. (45), i.e., we set ∂ t ˆ ρ ( r ) ≡ tr ( a ) ∂ t ˆ ρ ( R ) , wheretr ( a ) denotes the trace over the center-of-mass modes. Wealso set (cid:104)(cid:104) ... (cid:105)(cid:105) a = tr ( a ) ( ...ρ ( R ) ). In order to integrate out thecenter-of-mass modes, we have to consider that they coupleto the relative modes nonlinearly. However, we have seenabove that the quadratic part of Eq. (45) is separable, i.e., ρ ( R ) = ρ ( r ) ⊗ ρ ( a ) , and pushes the correlation functions to-wards (cid:104)(cid:104) ˆ φ ax ˆ φ ax (cid:105)(cid:105) a → ∞ , corresponding to a density matrix ˆ ρ ( a ) in an infinite temperature state.The bounded vertex operators exp( i √ φ ( a ) x ) alone do notpush the center-of-mass mode away from an infinite tempera-ture state. They rather become irrelevant in this limit, which follows from (cid:104)(cid:104) exp( i √ φ ( a ) x ) (cid:105)(cid:105) a = exp( −(cid:104)(cid:104) ˆ φ ( a ) x ˆ φ ( a ) x (cid:105)(cid:105) a ) → ρ ( R ) may be unknown, this suggests very weak correlations be-tween the two sets of modes. We therefore approximate thedensity matrix by a product ˆ ρ ( R ) ∼ ˆ ρ ( r ) ⊗ ˆ ρ ( a ) . The the onlyassumption we need regarding ˆ ρ ( a ) is that it is in a Gaussianstate with (cid:104)(cid:104) ˆ φ ( a ) x ˆ φ ( a ) x (cid:105)(cid:105) a → ∞ [51].In this case, tracing out the center-of-mass modes fromEq. (45) can be performed analytically. First, the dependenceon the expectation values (cid:104)(cid:104) cos (cid:16) √
2( ˆ φ ( a ) x ± ˆ φ ( r ) x ) (cid:17) (cid:105)(cid:105) = ∼ exp( i ... ),which are independent of ˆ φ ( a ) x , persist upon taking the tracewith respect to the average degrees of freedom. Thus only theterms in the third line of Eq. (45) yield a non-trivial contri-bution to the evolution equation of ρ ( r ) . We illustrate this bytracing out the center-of-mass mode in the cross term (cid:104)(cid:104) cos (cid:16) √
2( ˆ φ ( a ) x + ˆ φ ( r ) x ) (cid:17) cos (cid:16) √
2( ˆ φ ( a ) x − ˆ φ ( r ) x ) (cid:17) (cid:105)(cid:105) a = (cid:104)(cid:104) cos (cid:16) √ φ ( a ) x (cid:17) + cos (cid:16) √ φ ( r ) x (cid:17) (cid:105)(cid:105) a =
12 cos (cid:16) √ φ ( r ) x (cid:17) . (46)From this procedure, we obtain the evolution equation for the relative replica fluctuations. In the following we will only2focus on this degree of freedom and drop the label ( r ) [52].After keeping the most relevant nonlinear terms only, the mas-ter equation for the fluctuations is ∂ t ˆ ρ = i ( ˆ ρ ˆ H † sg − ˆ H sg ˆ ρ ) , (47)where λ = γ m , and the non-Hermitian sine-Gordon Hamilto-nian reads ˆ H sg = v π (cid:90) x (cid:104) ( ∂ x ˆ θ x ) + (cid:16) − i γ v π (cid:17) ( ∂ x ˆ φ x ) (cid:105) (48) − i λ (cid:90) x (cos( √ φ x ) − . Again we encounter a non-Hermitian Hamiltonian evolutionfor the relative degrees of freedom.On large length- and long time-scales, the irrelevance ofhigher order perturbative corrections (scaling ∼ cos( ζ √ φ ) at ζ -th order) again leads to an e ff ective cooling of the replicafluctuations towards the dark state of the non-HermitianHamiltonian ˆ H sg . Depending on the strength of the nonlin-earity and the imaginary part in the kinetic energy term, thisdark state is scale invariant at large distances, or has built ina length-scale yielding either algebraically or exponentiallydecaying correlations. For a Hermitian sine-Gordon Hamil-tonian, a renormalization group treatment predicts a phasetransition between a scale invariant ground state and a gappedground state according to the Berezinskii-Kosterlitz-Thoulessparadigm [53–55], and we may conjecture that a similar sce-nario applies to the dark state of the non-Hermitian Hamil-tonian. We will shed light on this scenario in the followingsection and perform a renormalization group analysis of ˆ H sg .We conclude this section with an additional remark. Thee ff ective Hamiltonian ˆ H sg is not PT symmetric [56] and there-fore its eigenvalues are generally complex. However, theimaginary part of the eigenvalues is always negative, whichguarantees dynamical stability of the time-evolution. An evo-lution with a non-Hermitian Hamiltonian with complex eigen-values, however, does not preserve the norm of the state. Thisis possible because there is no constraint that ensures the normof ˆ ρ ( r ) and ˆ ρ ( a ) to be conserved individually. The norm of ˆ ρ ( R ) is preserved as argued in Sec. III. V. RENORMALIZATION GROUP APPROACH TO THENON-HERMITIAN SINE-GORDON THEORY
The sine-Gordon Hamiltonian in Eq. (48) describes thelong-time and large-distance behavior of the replica fluc-tuations. In order to determine whether or not the cos-nonlinearity is relevant or irrelevant at large distances, i.e.,whether it generates a mass term ∼ m ˆ φ x or vanishes andrestores a scale-invariant action, we perform a renormaliza-tion group analysis. While the renormalization group (RG)flow of the sine-Gordon model is well established in ther-mal equilibrium (i.e., for a Hermitian Hamiltonian), only fewnon-Hermitian situations have been considered previously[57, 58]. | K | A Gaussianfixed-point fixed-pointstrong coupling t h e r m a l e q u ili b r i u m - - s s | ( s ) | A | ( s ) | A Phase Diagram( a ) ( b )( c ) ( i ) ( ii ) ( iii )( iv )( v )( vi ) ( i )( iii )( iv )( v ) non-Hermitian Hermitian Figure 2. (a) Phase diagram of the non-Hermitian sine-Gordonfield theory, obtained from numerical integration of the flow equa-tions (51), (52). In the grey region the theory flows towards theGaussian fixed point located at the origin. The phase boundary issignalled by the black dotted line. For comparison we also show thetransition line for the Hermitian (thermal equilibrium) theory. Fig-ures (b) and (c) display the renormalization group flow towards thestrong coupling fixed point (b) and the Gaussian fixed point (c) forthe non-Hermitian sine-Gordon theory and the corresponding Her-mitian sine-Gordon theory (with comparably larger starting valueof K ). The initial parameters of the curves (i) and (iii)-(v) are dis-played in (a), the remaining curves correspond to | K | = . | K | = . λ/ A = A convenient starting point for a renormalizationgroup treatment is the non-Hermitian sine-Gordonaction S sg . It is derived by writing the propagator U ( t , = (cid:104){ φ x , f }| e − i ˆ H sg ( t f − t i ) |{ φ x , i }(cid:105) in terms of a path inte-gral U ( t , = (cid:90) φ x , t = φ x , f φ x , = φ x , i D [ { φ x , t } ] e iS ( { φ x , t } ) (49)over the real fields { φ x , t } by the conventional Trotterizationprocedure, e.g., see Appendix C.The real-time action for the non-Hermitian sine-GordonHamiltonian (48) can be brought into the Lagrangian form(with X = ( x , t )) S = (cid:90) X (cid:40) K π (cid:34) η ( ∂ t φ X ) − η ( ∂ x φ X ) (cid:35) − i λ cos( φ X ) (cid:41) . (50)Here, K , λ are the flow parameters for the renormalizationgroup treatment with the microscopic value of K = η and η = − i γ v π . The parameter η can been eliminated from theaction (50) by a complex Wick rotation ( x , t ) → ( η x , i η − t ).This yields an imaginary time path integral, which is well-defined for the here realized case Im( η ) <
0, i.e., when thereis no dynamical instability.We already note that the cos-nonlinearity here becomes ir-relevant in the limit γ → K , η → ∼ π , which yields strong fluc-tuations of the field φ X . This prefactor arises from rescaling3the fields with the large factor φ X → φ X / √ φ X in the cos( √ φ X )-term in the sine-Gordon Hamiltonian Eq. (48). It guarantees that the Gaussianfixed point is robust against an infinitesimal measurement rate.The renormalization group flow equations for the couplings K , λ are derived in the conventional way, but here we considerthe parameters K , λ, η to be arbitrary complex numbers. Thefields φ X = φ ( < ) X + φ ( > ) X are decomposed into long-distance ( φ ( > ) X )and short-distance ( φ ( < ) X ) fluctuations, where the long-distancefluctuations correspond to momenta | k | < Λ /ξ and the shortdistance fluctuations correspond to momenta Λ /ξ < | k | < Λ .Here Λ is a short-distance cuto ff ( Λ = π in units of the lat-tice spacing) and ξ = e s , where s is the rescaling parametercontrolling the renormalization group flow. The renormaliza-tion group flow is then obtained by integrating out the short-distance modes perturbatively in λ and rescaling x → x /ξ .This yields the flow equations ∂ s λ = (cid:16) − π K (cid:17) λ, (51) ∂ s K = − λ A , (52)where A is a positive number of order O (1) determined by thepropagator of the Gaussian theory [54, 59] (see App. D).The perturbative flow equations for the non-Hermitian sine-Gordon model in Eq. (48) take the same form as those in theBKT scenario, but are more general due to the generally com-plex nature of the couplings K , λ . They include thermal equi-librium and some established non-Hermitian theories in par-ticular limits, which can be immediately reproduced: If K isreal and λ is purely imaginary, i.e., λ = i | λ | , then the non-linearity λ is generally relevant for an initial K > π , and K is monotonously growing under coarse graining, reproduc-ing the conventional KT-flow diagram known from thermalequilibrium [54, 59]. On the other hand, if K and λ are bothreal (corresponding to a purely imaginary nonlinearity but realGaussian part), then K is continuously decreasing, such thatthe only asymptotic fixed point of the renormalization groupflow is the Gaussian one [57].In our situation, where the parameter of the Gaussian the-ory K is generally complex, both K , λ will not only expe-rience a flow of their magnitude, but also of their complexphase. In this case, the magnitude of the parameter K inEq. (52) initially experiences both periods of growth and pe-riods of shrinking during the RG flow, depending on the com-plex phase of λ . Since λ is real initially, one starts with anRG flow that reduces K , shifting the critical point at which λ becomes relevant to larger values of K compared to the corre-sponding equilibrium RG flow. Figure 2(a) displays the phasediagram obtained from solving Eqs. (51), (52) numerically forthe initial condition K = η .The analysis shows that, while the initial, short distance be-havior of the RG is a ff ected by the complex valued nature ofthe coe ffi cients, the asymptotic flow channels in on the onefamiliar from the Hermitian case. In consequence, the phaseboundary is deformed compared to the latter, but the two fa-miliar phases and their large distance behavior is reproduced,cf. Fig. 2. In more detail, after an initial period of rapid phase oscillations, λ either flows to zero, or the complex phases ϕ K , ϕ λ of K = | K | e i ϕ K , λ = | λ | i ϕ λ get pinned with respect toeach other, fulfilling the relation 2 ϕ λ − ϕ K = ± π . This hap-pens typically at s = O (1) and from then on, the flow equa-tions (51), (52) describe the familiar thermal KT flow. SeeFig. 2(b) for examples of RG flow curves towards the strongcoupling fixed point and Fig. 2(c) for examples flowing to-wards the Gaussian fixed point. VI. N-REPLICA KELDYSH CONSTRUCTION
Above we have studied two replicas, identifying center-of-mass and relative coordinates as useful degrees of freedom tocapture the transition: In particular, for Gaussian problems,these coordinates decouple exactly. The center-of-mass co-ordinate undergoes indefinite heating, the relative coordinateis governed by a non-Hermitian Hamiltonian. In this section,we address the question how this decomposition generalizesto n replicas. Extending the theory to n replicas enables usto compute higher order correlation functions, which are of n -th order in the conditioned density matrix ρ ( c ) . In particular, itgives access to the computation of R´enyi entropies and the vonNeumann entanglement entropy, which we will discuss in thissection. The construction is done within a replicated Keldyshfunctional integral formalism [60–62], and yields two key re-sults: First, we show that the transition established previouslyin the 2-replica setup extends to the n replica scenario. Sec-ond, we find that for Gaussian problems there continues to bean exact decoupling into one mode analogous to the center-of-mass coordinate, and n − A. General construction
Starting point is again Eq. (9) for the conditioned trajectoryprojector. We are then interested in the update (see Eq. (27)) d ˆ ρ ( R n ) = d (cid:104) ⊗ nl = ˆ ρ ( c ) (cid:105) (53)of the trajectory average over n identical copies of the system.We can iterate this update in time, and represent the processin terms of a path integral by inserting coherent state resolu-tions of identity after each time step in the usual way. Theresulting path integral is still conditioned on the noise realiza-tion, we finally take the noise average, yielding the n -replicapartition function Z ( n ) = Z ( n , { dW } ). This bears similaritiesto the Keldysh double- or multi-contour constructions, whichhave been introduced to study out-of-time ordered correlationfunctions (OTOCs), with the major di ff erence that the noiseaverage introduces additional contour couplings compared toa Hermitian or purely Lindbladian evolution [60–63]. Theprocedure is illustrated in Fig. 3.We consider again the general scenario of a continuouslymonitored quantum system, whose wave function evolutionis described by the stochastic Schr¨odinger equation (7). The4 t f t f Z (1 , { ⇠ } ) = tr( ⇢ t f !1 ) = tt x contour+ contourtr ... Z ( n, { dW } ) = A l = 1 l + 1 = 2 n = 2 Z A ( n, { dW } ) = tr ⇢ nA = ( a )( b )( c ) Z (1 , { dW } ) = tr tr tr Figure 3. (a) Keldysh representation of a single replica. Time evolu-tion of the noise averaged density matrix is captured by two Keldyshcontours running horizontally; the spatial dimension is representedby the vertical lines. The trace operation is represented by a blackline connecting final times t f , for a selected position x . The noise av-erage (lower blue line) yields the usual path integral representationof a Lindblad equation, with characteristic contour coupling terms.(b) Keldysh representation of n replicas. In addition to the couplingof contours within each replica, the average over the stochastic termcouples all replicas (upper red lines). The trace operation acts oneach replica individually, and is indicated by black lines. present construction is general, and encompasses fermionsor bosons alike; for the sake of concreteness and nota-tion, we first focus on a (1 + H = ˆ H [ ˆ ψ † x , ˆ ψ x ] and local, Hermitian measurement operatorsˆ M s , x , t = ˆ O s , x − (cid:104) ˆ O s , x (cid:105) t , where ˆ O s , x = ˆ O s , x [ ˆ ψ † x , ˆ ψ x ]. Here x isthe position and the index s distinguishes potentially di ff erenttypes of local measurements. We leave implicit internal de-grees of freedom of the fermion field, like L , R for the Diracfermions considered above.We illustrate the construction of the n -replica path integralstep by step. For the sake of clarity, we will drop the spatialand measurement indices x , s for now, set γ =
1, and restorethese quantities at the end. The evolution operator ˆ V dt evolv-ing the state | ψ t (cid:105) → | ψ t + dt (cid:105) = ˆ V dt | ψ t (cid:105) then isˆ V dt = exp (cid:104) − ( i ˆ H + ˆ M t ) dt + dW ˆ M t (cid:105) , (54)which when expanded up to second order (since dW = dt )yields the stochastic Schr¨odinger equation for the increment d | ψ t (cid:105) = | ψ t + dt (cid:105) − | ψ t (cid:105) .The single replica partition function for the conditionedprojector Z (1 , { dW } ) t →∞ = Tr( ˆ ρ ( c ) t ) t →∞ = Tr( | ψ t (cid:105)(cid:104) ψ t | ) then re-quires two time strings or contours, corresponding to actingˆ V dt from the left ( + contour) and ˆ V † dt from the right ( − con-tour) onto ˆ ρ ( c ) t at each time step. This can be expressed viaa two-contour path integral over the (Grassmann valued, forfermions) fields ψ σ , ¯ ψ σ , which carry a contour index σ = ± .The n -replica partition function Z ( n , { dW } ) = Tr[ ⊗ nn = ˆ ρ ( c ) t ]is obtained from the product over n independent trajectoryprojectors ˆ ρ ( c ) , yielding the partition function Z ( n , { dW } ) = [ Z (1 , { dW } )] n . In order to express this via a single path inte-gral, we add a replica index l to the fields ψ ( l ) σ , ¯ ψ ( l ) σ , accountingfor each individual copy ˆ ρ ( c ) . The path integral expression for the stochastic n -replica partition function then is Z ( n , { dW } ) = (cid:90) D [ Ψ ] exp (cid:2) i ( S n , H [ Ψ ] + S n , dW [ Ψ ]) (cid:3) . (55)The action is composed of a Hamiltonian part S n , H [ Ψ ] and ameasurement part S n , dW [ Ψ ], S n , H [ Ψ ] = (cid:88) σ = ± n (cid:88) l = σ (cid:90) t (cid:16) ¯ ψ ( l ) σ, t i ∂ t ψ ( l ) σ, t − H [ ¯ ψ ( l ) σ, t , ψ ( l ) σ, t ] (cid:17) , (56) S n , dW [ Ψ ] = i (cid:88) σ = ± n (cid:88) l = (cid:90) t (cid:16) [ M ( l ) σ, t ] − dW t M ( l ) σ, t (cid:17) . (57)Here we have bestowed dW t a temporal index, since in-crements at di ff erent times are uncorrelated and the aver-aging prescription in the operator formalism is replaced by dW t dW t (cid:48) = δ ( t − t (cid:48) ). For fermions, the functional integral hasanti-periodic boundary conditions in time, which we do notmake explicit in the notation here.The partition function Z ( n , { dW } ) is the product of 2 n inde-pendent path integrals ( n forward and n backward contours).They share in common that all contours couple to the samenoise increment dW t , which acts as a contour-independentsource term. Performing the noise average via integration overthe Gaussian distributed increments dW t , the partition func-tion Z ( n ) = Z ( n , { dW } ) results, which is no longer of productstructure. The corresponding path integral is Z ( n ) = (cid:90) D Ψ exp (cid:2) i ( S n , H [ Ψ ] + S n , M [ Ψ ]) (cid:3) , (58)where the unitary part S n , H [ Ψ ] (56) remains unchanged by thenoise average. The measurement action S n , M [ Ψ ] will now bediscussed for the cases n = n > ± )-contours, as illustrated in Fig. 3(a),and yields S , M [ Ψ ] = i (cid:90) t M + , t + M + , t −
12 ( M + , t + M − , t ) = i (cid:90) t ( O + , t − O − , t ) . (59)It therefore removes any state-dependent term ( ∼ (cid:104) ˆ O s , x (cid:105) t ), andmakes the noise averaged single replica evolution linear in thestate. This is then equivalent to the path integral representa-tion of a Lindblad equation. In particular, for the HermitianLindblad operators considered here, the stationary state de-scribed by it is at infinite temperature.Next we consider the case n >
1, which is illustrated inFig. 3(b). In addition to intra-replica contour couplings, thenoise average produces additional inter-replica couplings (redlines in Fig. 3(b)). Here, the measurement expectation valuesgenerally do not drop out, and we have to perform the aver-age in the mean-field decoupling approximation outlined inSec. IV B. In the path integral description this amounts to theapproximation that M ( l ) σ, t is independent of the history of dW t .Importantly, for the same reason as in the two-replica oper-ator formulation, the thus obtained action functional features5linear, noise averaged trajectory expectation values, such as (cid:104) ˆ O s , x (cid:105) t . These are now averages obtained from the path in-tegral and, focusing on the long time limit, we replace themdirectly by their stationary values, e.g. (cid:104) ˆ O s , x (cid:105) t → o s . The action obtained from this average is S n , M [ Ψ ] = i (cid:90) t n (cid:88) l = (cid:16) [ M ( l ) + , t ] + [ M ( l ) + , t ] (cid:17) − n (cid:88) l = M ( l ) + , t + M ( l ) − , t . (60)Restoring the spatial and measurement indices inˆ O s , x , ˆ ψ † x , ˆ ψ x and considering arbitrary γ , the measurement-induced action is S n , M [ Ψ ] = − i γ (cid:90) X n (cid:88) l = (cid:88) s (cid:104) ( O ( l ) s , + , X ) + ( O ( l ) s , − , X ) (cid:105) − Ω s , + , X + Ω s , − , X ) o s − (cid:2) Ω s , + , X + Ω s , − , X − n · o s (cid:3) , (61)with Ω s , ± , X = (cid:80) nl = O ( l ) s , ± , X . Eq. (61) yields an interesting struc-ture. While the first term is of the form of an e ff ective non-Hermitian Hamiltonian acting on each replica individually,the second term features inter-contour couplings only betweentwo collective degrees of freedom Ω s , ± , X . For n > Ω s , ± , X (seeSec. VI B below), which are free of contour coupling termsand only evolve according to an e ff ective Hamiltonian. B. Bosonization of the n -replica theory and decoupling of theGaussian theory We apply this setup to the continuum measurement modelpresented in Sec. III B. In the fermionic formulation definedby Eqs. (15,16), we first have the massless Dirac action asso-ciated to unitary evolution S n , H [ Ψ ] = n (cid:88) l = (cid:88) σ = ± σ (cid:90) X ¯ Ψ ( l ) σ, X i ( ∂ t − v σ z ∂ x ) Ψ ( l ) σ, X . (62)We consider two independent sets of measurement operators O ( l )1 ,σ, X = ¯ Ψ ( l ) σ, X Ψ ( l ) σ, X , O ( l )2 ,σ, X = ¯ Ψ ( l ) σ, X σ x Ψ ( l ) σ, X , (63)each measured with rate γ . In the bosonized formulation,Eqs. (19,20), the action is defined with the expressions S n , H [ φ ] = − π n (cid:88) l = (cid:88) σ = ± σ (cid:90) X φ ( l ) σ, X ∂ φ ( l ) σ, X , ˆ O ,σ, X [ φ ] = − π ∂ x φ ( l ) σ, X , ˆ O ,σ, X = m cos(2 φ ( l ) σ, X ) . (64)Before we discuss the general, nonlinear theory and in or-der to reveal further structures, we now focus on the caseof a Gaussian bosonic setting, relevant to the cases of weakand strong monitoring in the above model. It is defined witha quadratic Hamiltonian and measurement operators that arelinear functions of Hermitian (bosonic) field operators, i.e., O ( l ) σ, X = D φ ( l ) σ, X , where D = m π, ∂ x can be either a mass or aderivative. We then find a decoupling analogous to the one in the 2-replica case discussed in Sec. IV C. To demonstrate this,we first introduce a Fourier expansion in replica space, φ ( k ) σ, X = √ n n (cid:88) l = e i π kln φ ( l ) σ, X , k = , ..., n − . (65)The action is then expressed in terms of bosonic fields φ = ( φ (0) , ..., φ ( n − ) T , φ ( k ) X = ( φ ( k ) + , X , φ ( k ) − , X ), as a sum S n [ φ ] = n − (cid:88) k = S ( k ) [ φ ( k ) ] . (66)The above action has one center-of-mass or Fourier k = φ ( k = = √ n (cid:80) l φ ( l ) , which is described by S (0) [ φ (0) ] = − π (cid:90) X (cid:110) (cid:88) σ = ± (cid:20) σφ (0) σ, X ∂ φ (0) σ, X − γ i π v ( D φ (0) σ, X ) (cid:21) (67) + n γ i π v (cid:104) D φ (0) − , X + D φ (0) + , X (cid:105) (cid:111) , where ∂ ≡ ∂ t − ∂ x . Due to the intra-replica contour couplingterm (last contribution in Eq. (61)) there is heating to an infi-nite temperature state. This can be inferred from the Green’sfunction of the average mode in Fourier space ( D = m π, q )( G (0) q ,ω ) − = × ( ω − q + i γ ( n − v π D ) + i γ n v π D σ x . (68)It has four poles at frequencies ω ± , ± = ± (cid:114) q ± D γπ v √ n − . (69)For n >
1, this implies that at least two of the poles lie onthe real axis [64]. They cause all the matrix elements of theequal-time correlation function for a given momentum mode q to diverge G (0) ( q , t = = (cid:90) ω G (0) ( q , ω ) → ∞ . (70)6This indicates unbounded fluctuations of φ (0) X , characteristicof the infinite temperature state, and in line with the findingsdiscussed around Eq. (44).In addition, the action S n [ φ ] features n − φ ( k ) , k = , ..., n −
1. These do not involve intra-replica contour coupling in their action ( k > S ( k ) [ φ ( k ) ] = − π (cid:90) X (cid:88) σ = ± (cid:20) σφ ( k ) σ, X ∂ ˜ φ ( k ) σ, X − γ i π v ( D φ ( k ) σ, X D ˜ φ ( k ) σ, X ) (cid:21) , (71)where we have introduced the shortcut ˜ φ ( k ) σ, X = φ ( n − k ) σ, X .Each set of relative fields φ ( k > X evolves according to a non-Hermitian action, yielding a Keldysh Green’s function iG K ( k > ( Q ) = (cid:104) φ ( k > c , Q ˜ φ ( k > c , − Q (cid:105) = − i ˜ γ D ( q − ω ) + D ˜ γ . (72)The corresponding equal-time correlation functions G K ( k > ( q , t = = (cid:104)(cid:0) i ˜ γ D − q (cid:1) − (cid:105) (73)show the same scaling behavior as for a Luttinger Liquid inthe ground state, i.e., G K ( k > ( q , t = ∼ | q | for D ∼ q , butwith a di ff erent, γ -dependent amplitude.As a consequence of the decoupling, the n -replica partitionfunction for linear measurement operators O ( l ) X ∼ D φ ( l ) X anda quadratic Hamiltonian factorizes into n independent prod-ucts. This remains true when coupling the replica fields φ X linearly to local sources h X . For later purposes we intro-duce such coupling to source terms via (cid:82) X h TX φ X , with h = ( h (0) , ..., h ( n − ) T , h ( k ) X = ( h ( k ) c , X , h ( k ) q , X ) T . The sourced n -replicapartition function then obtains by Gaussian integration andreads Z ( n )[ h ] = n − (cid:89) k = Z ( k ) [ h ] , (74) Z ( k ) [ h ] = (cid:104) e i (cid:82) X h TX φ X (cid:105) = Z ( k ) [0] exp (cid:34) − (cid:90) X , X (cid:48) h ( k ) σ, X (cid:104) φ ( k ) σ, X ˜ φ ( k ) σ (cid:48) , X (cid:48) (cid:105) ˜ h ( k ) σ (cid:48) X (cid:48) (cid:35) . This allows us to e ffi ciently compute n -replica correlationfunctions of arbitrary power in the fields φ X , including the n -thorder R´enyi entropies, which we will discuss in the followingsection.Based on the bosonic formulation, and our understandingof the Gaussian theory, now we are in the position to dis-cuss the measurement induced phase transition imposed bya nonlinear measurement operator O ,σ, X . For n = n >
2, we can proceed in thesame manner and bring the Gaussian part of the action, i.e.,the sum S n , H + S (0) n , M , into a replica diagonal form by apply-ing the replica-Fourier transform described in Eq. (65). The’hot’ or infinite-temperature mode φ ( k = σ, X can then again be in-tegrated out, yielding the remaining action for the n − φ ( k > σ, X . These modes are coupledby the cos-nonlinearities, which contain their sum as argu-ments, e.g., the term cos( √ φ (1) X ) → cos( (cid:80) k > a (1) k ˆ φ ( k ) X ) after integrating out the k = a ( l ) k = (cid:113) n e i π lk / n ∈ C . The Gaussian part of theaction for k >
0, however, does not couple di ff erent replicasectors k , k (cid:48) and furthermore is identically the same for each k >
0. Therefore the common BKT perturbative renormaliza-tion group scheme boils down to the individual but identical,multiplicative renormalization of the factors e ± i φ ( k ) σ, X for each k >
0. While deriving the set of flow equations for n > ff erent replicaindices k , k (cid:48) can be generated. This in turn ensures that theGaussian fixed point for n > n =
2, describing n − C. Entanglement transition
In this section, we discuss how the measurement inducedphase transition leaves its fingerprints in the entanglement en-tropy. On both sides of the transition, the e ff ective theory isa theory of free bosons, which we examine to quantify theirentanglement. The structural di ff erence between the n − k >
0. With this result, we are able to numerically com-pute the von Neumann entanglement entropy for our systemfrom the boson covariance matrix. In the weakly and stronglymonitored limits, it exhibits subextensive logarithmic growthor area law saturation, respectively, confirming a phase tran-sition in the entanglement entropy. We use RG arguments tointerpolate between these regimes. We also determine the ef-fective central charge in the weak monitoring regime and findthat it behaves non-universally. Surprisingly, our approachpredicts an increase of the e ff ective central charge as a func-tion of the measurement strength γ in the weak monitoringregime. This is in contrast to previous works on monitoredfree fermions on the lattice, which reported a decreasing pref-actor [23, 24]. General approach. – To determine the properties of the en-tanglement entropy in the measurement problem, we leveragethe techniques put forward in Ref. [65–67], which computethe entanglement entropy of a free problem – completely de-scribed by a Gaussian density matrix – from its covariancematrix.For a bipartition of the one-dimensional system into twodisjoint subsystems A = [ x , x + L ] and B = R \ A , the R´enyientropy for the measured n − replica system is given by the ex-pression S n ( L ) = − n log Z A ( n , { dW } ) , Z A ( n , { dW } ) ≡ tr[( ˆ ρ ( c ) A ) n ] . (75)Here, Z A ( n , { dW } ) is the reduced partition function for the sub-set A , which is obtained by first performing a partial trace over7 B to obtain ˆ ρ ( c ) A = tr B ˆ ρ ( c ) , and then taking the trace of the prod-uct ( ˆ ρ ( c ) A ) n . In the path integral, the matrix multiplication is im-plemented by the boundary conditions φ ( l ) + , x , t f = φ ( l + − , x , t f if x ∈ A and t f is the time at which ˆ ρ ( c ) A is evaluated [68, 69].The boundary conditions and the coupling to the noisein Z A ( n , { dW } ) simplify in the Fourier representation of thereplica modes φ ( k ) x in Eq. (65). This splits the action into a sumas in Eq. (66). We also show in Appendix F that the boundaryconditions do not modify the action of the k = Z A ( n , { dW } ), which yields Z A ( n , { dW } ) = Z ( r ) A ( n ) × z √ ndW . (76)Here Z ( r ) A ( n ) = Z A ( n ,
0) is the reduced partition function for n replicas with vanishing noise dW = z √ ndW is a noise de-pendent quantity. It is, however, independent of A and there-fore does not contribute to the scaling of the entropy withthe bipartition size. Up to a constant, the R´enyi entropy inEq. (76) is then determined exclusively by the k > S n ( L ) = − n log Z ( r ) A ( n ) . (77)We will now focus on the von Neumann entanglement en-tropy, obtained by taking the limit n →
1. In analogy to theground state entanglement entropy of Hermitian free bosons,we expect the entanglement entropy in Eq. (77) to either growlogarithmically with the system size, or to saturate at a finitelength scale set by the mass gap. However, since the theoryis non-Hermitian, the usual tools of conformal field theory arenot immediately applicable. Instead, a feasible approach is tocompute the entanglement entropy directly from the covari-ance matrix of the subsystem A [65–67], evaluated at t = t f ˜ C x , y = (cid:42) φ ( r ) σ, x φ ( r ) σ, y π (cid:110) φ ( r ) σ, x , ∂ t φ ( r ) σ, y (cid:111) π (cid:110) φ ( r ) σ, x , ∂ t φ ( r ) σ, y (cid:111) π ∂ t φ ( r ) σ, x ∂ t φ ( r ) σ, y (cid:43) S ( r ) , (78)with x , y ∈ A and σ either + or − . Here the index S ( r ) indicatesthat the expectation value is taken with respect to the Keldyshaction in Eq. (74), with r being any k >
0. Equivalently, thecovariance matrix can be obtained from an average with re-spect to the dark state | ψ D (cid:105) (see Appendix E) after replacing( φ ( r ) σ, x , t , π ∂ t φ ( r ) σ, x , t ) → ( ˆ φ ( r ) x , ∂ x ˆ θ ( r ) x ) [70].For free bosons in the ground state, the scaling behaviorof the entanglement entropy with the size L of the subsystem A can already be inferred from the scaling of the upper leftblock of ˜ C x , y , ∼ (cid:104) ˆ φ ( r ) x ˆ φ ( r ) x + L (cid:105) . We verify that this is also true hereby evaluating the covariance matrix ˜ C x , y on a grid, and deter-mining its eigenvalues { λ i } . The von Neumann entanglemententropy S vN can then readily be computed from the eigenval-ues (see e.g. [67, 71, 72] for a detailed discussion of bosonicsystems) according to S vN ( L ) = lim n → S n ( L ) = (cid:88) i (cid:34) f (cid:32) λ i + (cid:33) − f (cid:32) λ i − (cid:33)(cid:35) . (79)Here, f ( x ) = x log x and we sum over all λ i >
0. We discussthe entropies separately for the gapless ( γ (cid:28) v ) and gapped ( γ (cid:29) v ) cases, as well as for the limit of zero measurements γ = Weak monitoring entropy, γ (cid:28) v – If the nonlinearity isirrelevant, the renormalization group flow approaches a Gaus-sian fixed point at which λ =
0, and the theory is fully parame-terized by the fixed point value of K . The correlation functionat the Gaussian fixed point then is (cid:104) φ ( r ) c ; q , t φ ( r ) c ; − q , t (cid:105) = Re (cid:16) π K | q | (cid:17) ⇒ (cid:104) φ ( r ) c , x , t φ ( r ) c , y , t (cid:105) ∼ log( | x − y | ) . (80)In this case Eq. (79) predicts the entanglement entropy and amonotonously growing e ff ective central charge S vN ( L ) = c ( γ )3 log( L ) , c ( γ ) = + ∆ c ( γ ) , (81)with ∆ c ( γ ) ≥
0. In the limit γ, λ →
0, i.e., in the absence ofmeasurements, we have K → S = log( L ) with c ( γ → → λ = γ >
0, the sys-tem is already initialized at a Gaussian fixed point with K = (cid:113) − i γπ v . In this case ∆ c ( γ ) behaves ∼ γ for small γ (cid:28) v and then slowly approaches an upper bound ∆ c ( γ → ∞ ) ≈ .For the general case γ > , λ >
0, the nonlinear terms lead toa renormalization of the Gaussian part of the theory, i.e., of ane ff ective measurement strength entering K at the fixed point.Generally, we observe a growth of the parameter K during therenormalization group flow. Strong measurement entropy, γ (cid:29) v – For strong measure-ments, the problem is gapped. Repeating the calculation fromabove but with non-zero mass m yields, for a bipartition of thesystem with subsystem sizes L (cid:29) m − , S vN ( L ) = c ( γ )3 log (cid:32) L √ + m L (cid:33) ≈ c ( γ )3 log m − . (82)The entanglement entropy saturates and does no longer de-pend on L . The overall constant prefactor of the entropy hasthe same monotonous behavior as reported above. Here, how-ever, the inverse mass m − is a measurement induced corre-lation length. It is also γ -dependent, approaching the short-distance cuto ff ( = γ → ∞ . Zero monitoring, γ = γ → c (0) =
1. We confirm thatthis remains true for an interacting system with a Luttingerparameter di ff erent from K =
1. However, this limit has tobe taken with care, since without measurements the stationarystate depends on the initial state of the system.For instance, for a thermal state at temperature T , the initialcondition in the Keldysh framework is usually implementedvia an infinitesimal contribution to the quantum-quantum sec-tor [73] S qq = (cid:90) P i (cid:15)ω coth( ω T ) φ q , − P φ q , P , (83)8with (cid:15) = + infinitesimal, and φ q , P = φ + , P − φ − , P √ . This ini-tial contribution appears in the quadratic sector and is iden-tical for each replica, such that it is directly carried over tothe relative and center-of-mass coordinates. For any non-vanishing measurement rate γ > ∼ (cid:15)ω coth( ω T ), however, is overwritten by a finiteimaginary contribution ∼ γ in Eq. (72). Only at the isolatedpoint γ =
0, the initial state matters, and yields the corre-lation functions of a Luttinger Liquid at finite temperature (cid:104) φ ( r ) σ, x , t φ ( r ) σ, y , t (cid:105) ∼ | x − y | on distances | x − y | > v / T . For a non-zerotemperature (or energy density), we thus recover the expectedvolume law of the entanglement entropy. This is in line withnumerical results, which show that the volume law is onlyasymptotically stable in the absence of monitoring [22], andis replaced by a logarithmic scaling at any non-zero monitor-ing rate [23]. Summary of the entanglement entropy – In total, the en-tanglement entropy in our setup reflects the scaling behav-ior, which was numerically observed for monitored latticefermions [23, 24], including the isolated point of zero mon-itoring. In contrast to the lattice simulations, however, wedetermine a moderate growth of the e ff ective central chargewith the measurement strength, while Refs. [23, 24] found asuppression of c ( γ ) with γ . This is an intriguing result, forwhich we currently have no clear explanation. It may, e.g.,be traced back to the microscopic origin of the correspondinglattice model, which also predicts a diverging e ff ective centralcharge in the limit γ →
0, or to the fact that the fermion entan-glement is encoded in transcendental functions of the bosonfields, and therefore di ffi cult to determine away from Hermi-tian free theory. We have verified that in general for free non-Hermitian bosons, the e ff ective central charge obtained fromthe covariance matrix is bounded from below by c = VII. CONCLUSIONS AND OUTLOOK
In this work, we have constructed a general field theory ap-proach to measurement-induced phase transitions, and appliedit to demonstrate a BKT type phase transition in a concretemodel of measured massless Dirac fermions in (1 +
1) dimen-sions.In general measurement problems, the measured wavefunction evolves in a pure state, however in a combined de-terministic and random way. The pure state character en-ables transitions similar to quantum phase transitions, result-ing from the competition of non-commuting operators. Therandomness of the state in each realization of the quantumtrajectory however requires statistical analysis to assess suchtransitions. Averages linear in the quantum trajectory projec-tor are equivalent to expectation values in an infinite tempera-ture state density matrix, at least for Hermitian measurementoperators. We thus introduce the generating functionals forthe n th moment of the state, Z ( n ) = tr[ ˆ ρ ( c ) ] n , worked out forboth the operatorial (for n =
2) and a Keldysh functional in-tegral formulation. This allows us to identify the relevant de-grees of freedom for the description of measurement induced phase transitions. A particularly strong structure emerges forfree theories, which often form the basis for more sophisti-cated analysis: There, we find an exact decoupling of the gen-erating functional into one ’hot’ center-of-mass mode, whichindeed heats up indefinitely, and n − ff ective Hamiltonian, describing a kindof cooling. Beyond free theories, this framework allows us todefine and compute correlation functions which are non-linearin the state. It also provides a formula for the computation ofentanglement entropies.A key advantage of this approach is that it is formulatedin terms of (replicated) microscopic degrees of freedom. Weexemplify the strengths of this formalism in the bosonizedversion of the fermion model, leveraging powerful bosoniza-tion and RG techniques to the many-body measurement prob-lem: The e ff ective Hamiltonian for relative modes is givenby a quantum sine-Gordon model with complex coe ffi cients,and the non-linearity is found irrelevant or relevant for weakor strong monitoring, respectively. Conceptually, the flowtowards a Gaussian theory – either a gapless, or a gappedscalar boson – established in this way leads to an e ff ectivedecoupling into ’hot’ and ’cold’ modes at long wavelength.Practically, this allows us to connect the microscopic physicsto macroscopic phases, establishing a gapless and a gappedphase, with a logarithmic scaling and an area law saturationof the R´enyi entropy. The gapless phase asymptotically ex-hibits an emergent conformal invariance. The critical point isin the BKT universality class.We expect the toolbox developed here to provide furtherinsight into the nature of measurement-induced phase tran-sitions. For example, building on the striking parallels ofthe phenomenology found here with spinless fermionic lat-tice models suggests that the transition from a gapless CFTphase to a gapped one in (1 +
1) dimensions will persist inthe presence of interactions between lattice fermions, at leastassuming the validity of a naive connection between latticefermions and the interacting Luttinger liquid: In that case, aninteracting fermion Hamiltonian would simply manifest in arenormalized Luttinger parameter, and none of the qualitativeconclusions of the present work would be changed. More gen-erally, our approach should help to construct further modelsshowing such transitions, including in higher dimensions. Inparticular, an important goal is to identify a volume-to-arealaw transition: A possible mechanism for such a transition isan incomplete decoupling into ’hot’ and ’cold’ modes at longwavelength, leaving the cooling of the ’cold’ modes imper-fect, in turn enhancing entanglement growth. In turn, con-necting such a possible mechanism to the presence or absenceof integrability of the dynamics would be intriguing [74].Conceptually, it will be interesting to explore the re-lation of the damping dynamics under the non-HermitianHamiltonian to pure states to the purification scenario devel-oped in Refs. [21, 74]. Finally, our analysis clarifies that,while entanglement entropies provide a hallmark signature ofmeasurement-induced transitions, they are not at the root of it,and correlation functions in the n th moment do provide simi-lar information on the existence of the transition. The picture9of a depinning transition from the eigenstates of the measure-ment operators developed here sparks the hope that the tran-sition could be witnessed in experiment by suitable statisticalanalysis of quantum trajectories in highly controlled quantumsystems undergoing monitoring. ACKNOWLEDGMENTS
We thank O. Alberton, T. Bintener, T. Botzung, P. Cal-abrese A. Daley, M. Gullans, D. Huse, K. von Keyserlingk,M. Knap, J. Knolle, B. Ladewig, M. McGinley, Y. Malo,M. M¨uller, T. M¨uller, A. Nahum, F. Pollmann, A. Roschand S. Roy for fruitful discussions. We acknowledge supportfrom the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) under Germany’s Excellence StrategyCluster of Excellence Matter and Light for Quantum Comput-ing (ML4Q) EXC 2004 / Appendix A: Correlation functions in the strong measurementlimit
Here we demonstrate the emergence of a gap in the strongmeasurement limit via the exponential decay of the connectedmeasurement correlation function C i j = (cid:104) ˆ M i , t ˆ M j ( t ) (cid:105) . To thisend, we work on a lattice, utilizing the microscopic fermionmodel defined in Eqs. (11,12), a modification which shouldnot a ff ect the long range decay properties that we are inter-ested in. For strong measurements 0 < J (cid:28) γ , we can de-termine the solution for C i j perturbatively in J /γ . Its explicittime evolution before performing the trajectory average is ob-tained by applying Eq. (7) to the definition of C i j . It yields dC i j = − dt γ (cid:88) k C ik C k j + (cid:104) [ i ˆ H , ˆ M i , t ˆ M j , t ] (cid:105) (A1) + (cid:88) k dW k (cid:16) (cid:104) c † i c k (cid:105)(cid:104) c † k c j (cid:105)(cid:104) c † j c i (cid:105) + (cid:104) c † i c j (cid:105)(cid:104) c † j c k (cid:105)(cid:104) c † k c i (cid:105) (cid:17) . The second row of this equation is obtained by applyingWick’s theorem to the higher order correlation functions oc-curring in the equation of motion. It contains the randomWiener increments and is proportional to a product of threenon-local fermion bilinears. Each individual expectationvalue in the second row vanishes in the steady state if J = O ( J /γ ). This yields a subleadingscaling of the entire product ∼ dW J /γ .In order to proceed with the perturbative expansion, itis convenient to introduce the correlation function B i j ≡ (cid:104) ˆ M i , t [ i ˆ H , ˆ M j ( t )] (cid:105) . It evolves according to dB i j = dtJ (cid:104) ˆ M i , t (cid:16) ˆ M j + , t − M j , t + ˆ M j − , t (cid:17) (cid:105) (A2) − dt γ B i j + (cid:88) l C il B l j + nonlocal terms . The combination of Eqs. (A1) and (A2) provides importantinformation. First, any nonlocal function, which cannot beexpressed exclusively via fermion densities, will be expo-nentially suppressed in its time evolution, and its stationarystate expectation value is at most of order O ( J /γ ). This iscontrasted by the evolution of the correlation function C i j inEq. (A1), which is dominated by the product C ik C k j and yieldsa slow ∼ t − decay over time. Second, in both equations,the remaining stochastic terms are products of three nonlo-cal operator expectation values, and therefore at least of order O ( J /γ ). They therefore will be neglected in a perturbativetreatment.Without the remaining stochastic increments, Eqs. (A1) and(A2) yield the stationary state solution C i j = γ (cid:104)(cid:16) γ + J T (cid:17) (cid:105) i j − δ i j . (A3)Here T i j = − δ i , j + − δ i , j − + δ i j is the discrete lattice Laplacian.The square root is a non-local function, but for small J /γ it canbe expanded in a Taylor series. The leading order contributionconnecting two sites of separation | i − j | is the | i − j | -th termof the expansion, yielding C i j = ( − | i − j + | (2 | i − j | − | i − j + | | i − j | ! (cid:32) J γ (cid:33) | i − j | ∼ | i − j | / e −| i − j | /ξ . (A4)This describes exponentially decaying correlations with a per-turbative correlation length ξ given by 1 /ξ = γ J √ . Appendix B: Riccati equation for linearly monitored bosons
For a quadratic Hamiltonian and linear measurement oper-ators, the steady state covariance functions C ABi j of the form C ABi j = (cid:104){ ˆ A i − (cid:104) ˆ A i (cid:105) , ˆ B j − (cid:104) ˆ B j (cid:105)}(cid:105) for a set of bosonic operatorsˆ A i , ˆ B j can be determined analytically. This is a consequenceof three key properties of their evolution equation: (i) theHamiltonian part of the evolution equation is a linear functionof the C ABi j , (ii) the Lindblad-type contribution to the evolutionequation yields double commutators with the measurementoperators ∼ [ ˆ M Qk , [ ˆ M Qk , ˆ A i ˆ B j ]], which due to bosonic commu-tation relations are either proportional to the identity ∼ orvanish, (iii) importantly, for all stochastic terms ∼ dW i (cid:104) ... (cid:105) that occur in the evolution, the quantum mechanical expec-tation values (cid:104) ... (cid:105) = dW i . This is true for any Gaussian state (includinginitial product states) | ψ t (cid:105) . Consequently, the stochastic terms ∼ dW i drop out, and the equation for the covariance matrixbecomes deterministic.0Here, we demonstrate how one can solve for the station-ary state covariance functions in such a setting explicitly. Asa first step, we intermittently re-discretize the linear bosonicproblem, taking the continuum limit at the end – this willnot a ff ect the long-distance properties we are interested in.We thus start from a general set of Hermitian, bosonic op-erators Q i , P i on a lattice with the sites i . Their spec-trum is real and continuous, and they form a pair of conju-gate variables, [ Q k , P j ] = i δ k , j . We will then later identifythese operators with the discretized counterparts ˆ φ i , ˆ θ i fromthe bosonized measurement setup introduced in Sec. III B.The correspondence will be either ( Q i , P i ) = ( ˆ φ i , ˆ θ i + − ˆ θ i − ) or( Q i , P i ) = ( ˆ φ i + − ˆ φ i − , ˆ θ i ) depending on whether we considerweak or strong monitoring.We assume a quadratic Hamiltonian, which is expressed bythe Hermitian matrices V , W viaˆ H = (cid:88) i , j (cid:16) ˆ Q i V i j ˆ Q j + ˆ P i W i j ˆ P j (cid:17) . (B1)The measurement operator shall be linear in ˆ Q i , i.e., ˆ M Qi = ˆ Q i − (cid:104) ˆ Q i (cid:105) t yielding the stochastic Schr¨odinger equation d | ψ t (cid:105) = − dt (cid:104) i ˆ H + γ (cid:88) i (cid:16) ˆ M Qi (cid:17) (cid:105) | ψ t (cid:105) + (cid:88) i dW i ˆ M Qi | ψ t (cid:105) . (B2)We are interested in the covariance functions C ABi j ≡ (cid:104){ ˆ A i − (cid:104) ˆ A i (cid:105) t , ˆ B j − (cid:104) ˆ B j (cid:105) t }(cid:105) , where the operators ˆ A i , ˆ B i ∈{ ˆ Q i , ˆ P i } . Starting from Eq. (B2), the evolution equation forthe operator expectation value (cid:104) ˆ O (cid:105) for a general operator ˆ O according to the ˆIto product rule [38] is d (cid:104) ψ t | ˆ O | ψ t (cid:105) = ( d (cid:104) ψ t | ) ˆ O | ψ t (cid:105) + (cid:104) ψ t | ˆ O | d | ψ t (cid:105) + ( d (cid:104) ψ t | ) ˆ O ( d | ψ t (cid:105) ) . (B3)Combining Eq. (B2) and Eq. (B3) we obtain d (cid:104) ˆ O (cid:105) = i (cid:104) [ ˆ O , ˆ H ] (cid:105) dt − γ dt (cid:88) k (cid:104) [ ˆ Q k , [ ˆ Q k , ˆ O ]] (cid:105) (B4) + (cid:88) j dW k (cid:104) (cid:104) (cid:110) ˆ Q k , ˆ O (cid:111) (cid:105) − (cid:104) ˆ Q k (cid:105)(cid:104) ˆ O (cid:105) (cid:105) . In the limit where (i) the measurement ˆ Q k is linear and (ii)the Hamiltonian ˆ H and the operator ˆ O is at most a quadraticfunction of ˆ P k , ˆ Q k , we stay withing the manifold of states fullycharacterized by the covariance matrix [75].For the covariance functions we’re interested in, the secondline in Eq. (B4) vanishes, which we argue below. Insertingthen the operator ˆ O = { ˆ A i − (cid:104) ˆ A i (cid:105) , ˆ B j − (cid:104) ˆ B j (cid:105)} into the first linein Eq. (B4) yields the Riccati equation ddt C ABi j = i (cid:104) [ ˆ H , { A i − (cid:104) A i (cid:105) , B j − (cid:104) B j (cid:105)} ] (cid:105) (B5) + γ δ i j δ B , P δ A , P − (cid:88) k C AQik C QBk j , (B6)where δ i , j is a Kronecker- δ in the operator index and the prod-uct δ B , P δ A , P equals 1 if the operators A = B = P and zero otherwise. Matrix Riccati equations [46, 49, 76] of this typehave a unique and well defined steady state. The quadraticform of the Hamiltonian ensures that the unitary part Eq. (B5)remains linear in C AB , while the nonlinear part ensures con-vergence of the evolution towards a well-defined stationarystate. This gives rise to three types of correlation functions, C QQ , C PP and C QP ( C PQ = ( C QP ) † ). Their stationary valuescan be solved from ddt C AB = C AB become deter-ministic. Dropping this line is, however, not an approxima-tion. It turns out that for the choice of ˆ O = { ˆ A i − (cid:104) ˆ A i (cid:105) , ˆ B j −(cid:104) ˆ B j (cid:105)} the second line is identical to zero, independently of thevalue of the increments dW k . This can be seen by focusing ona single term k of the sum, and ˆ o = ( ˆ A i − (cid:104) ˆ A i (cid:105) )( ˆ B j − (cid:104) ˆ B j ). Thisyields (cid:104) (cid:110) ˆ Q k , ˆ o (cid:111) (cid:105) − (cid:104) ˆ Q k (cid:105)(cid:104) ˆ o (cid:105) = (cid:104){ ˆ Q k , ˆ A i ˆ B j }(cid:105) − (cid:104){ ˆ Q k , ˆ A i }(cid:105)(cid:104) ˆ B j (cid:105) (B7) − (cid:104){ ˆ Q k , ˆ B j }(cid:105)(cid:104) ˆ A i (cid:105) − (cid:104) ˆ Q k (cid:105)(cid:104) ˆ A i ˆ B j (cid:105) . The RHS of this equation is identical to zero if the cubicexpectation value is evaluated in a Gaussian state, i.e., in astate where Wick’s theorem for the decoupling of higher or-der expectation values can be applied. Since the stochasticSchr¨odinger equation (B2) is quadratic in the bosonic opera-tors, this applies to any initial Gaussian state, which we as-sume here. Equations (B5), (B6) are therefore the exact evo-lution equations for the covariance matrix for any initial stateof this type.Inserting the Hamiltonian (B1) into the equation of motionand solving for ddt C ABi j =
0, we find for the cases A = B = P and A = B = Q the following equations0 = − VC QP − C PQ V − γ C PQ C QP + γ , (B8)0 = WC PQ + C QP W − γ C QQ C QQ , . (B9)The two equations can be solved straightforwardly, yielding C QP = (cid:20) + γ V (cid:21) − γ V , (B10) C QQ = (cid:40) W γ (cid:20)(cid:16) γ + V (cid:17) − V (cid:21)(cid:41) , (B11)which depend only on the matrices V , W defining the Hamil-tonian. Next we discuss the implications for the weak andstrong measurement limits.At weak monitoring, we identify ˆ Q l = ˆ φ l + − ˆ φ l − , which is thelattice analogue of ∂ x ˆ φ x . A direct comparison shows that thelattice equivalent of the Hamiltonian (19) is described by (B1)with matrices V i j = δ i j π v and W i j = v π (cid:16) δ i j − δ i , j + − δ i , j − (cid:17) ,i.e. a diagonal matrix and a lattice Laplacian. Inserting thisexpression into the correlation function (B11) and taking thecontinuum limit yields Eqs. (23) from the main text.For strong monitoring we only consider the gapped mea-surement operator ˆ M , x , t = − ˜ m x ( ˆ φ x − (cid:104) ˆ φ x (cid:105) t ) , (B12)1where we inserted the mass ˜ m x = m sin(2 φ x ), which de-pends on the eigenvalues φ x of ˆ φ x in the corresponding darkstate. The corresponding lattice measurement operator is ˆ Q l = − ˜ m l ˆ φ l with ˜ m l = m sin(2 φ l ) and its conjugate ˆ P l = ˆ θ l + − ˆ θ l .Compared to the weak monitoring limit, this requires the ex-change of V ↔ W , yielding V = v π T and W i j = δ i j π v with thefamiliar lattice Laplacian T i j = δ i j − δ i , j + − δ i , j − .To treat the general case of spatially inhomogeneousmasses ˜ m l , we have to modify the measurement part of theRiccati equation (B6) according to ddt C ABi j = i (cid:104) [ ˆ H , { A i − (cid:104) A i (cid:105) , B j − (cid:104) B j (cid:105)} ] (cid:105) (B13) + γ ˜ m i δ i j δ B , P δ A , P − (cid:88) k ˜ m k C AQik C QBk j . (B14)Solving this equation along the lines of the homogeneous Ric-cati equation yields the solution C = v γ (cid:104)(cid:16) T + π γ v D (cid:17) − T (cid:105) . (B15)Here D is a diagonal matrix D l , s = δ l , s ˜ m l containing themasses ˜ m l . For the case of homogeneously distributed masses˜ m l = ˜ m , the square root can be expanded in a Taylor seriesin T in the limit γ (cid:29) v . This yields the covariance matrix inEq. (24) in the main text, which decays exponentially with thedistance with a correlation length ξ − = log(2 πγ ˜ m / v ).We can also convince ourselves that the more realistic sce-nario including inhomogeneously distributed masses ˜ m l doesnot alter this picture qualitatively. In Eq. (B15), a spatiallydependent mass enters multiplicatively. The algebraic equa-tions can then be solved numerically and one finds that thismodifies the correlation length approximately such that that˜ m is replaced by the lattice average ˜ m = N (cid:80) Ni = m i . Thisstill leads to an exponential suppression of the covariance ma-trix with the distance. In the numerical confirmation we useda random, uniform distribution of the masses ˜ m l . Appendix C: Review of constructing the sine-Gordon pathintegral
Here we briefly review the construction of the bosonizedpath integral for the Hamiltonian (48). For clarity, we use a haton quantum operators and no hat for real fields. For the fieldoperator ˆ φ x and its conjugate ˆ Θ x = ∂ x ˆ θ x /π , the kinematicsfollow from the commutation relation[ ˆ Θ y , ˆ φ x ] = − i δ ( x − y ) ⇒ (cid:104) φ x | Θ x (cid:105) = e i Θ x φ x , (C1)where ˆ φ x | φ x (cid:105) = φ x | φ x (cid:105) and ˆ Θ x | Θ x (cid:105) = Θ x | Θ x (cid:105) , and the Hamil-tonianˆ H = v π (cid:90) x (cid:104) ( π ˆ Θ x ) + η ( ∂ x ˆ φ ) + i λ (cos( √ φ ) − (cid:105) . (C2)We perform the usual Trotterization and the l -th trotterizedtime step is computed by using the completeness relations = (cid:82) d φ x | φ x (cid:105)(cid:104) φ x | = (cid:82) d Θ x | Θ x (cid:105)(cid:104) Θ x | . This yields for an in-finitesimal step e iS l , x = (cid:104) φ l + , x | e − i ˆ H sg δ t | φ l , x (cid:105) (C3) = (cid:90) d Θ x (cid:104) φ l + , x | Θ x (cid:105)(cid:104) Θ x | ( − i δ t ˆ H ) | φ l , x (cid:105) = (cid:90) d Θ x e i Θ x ( φ l + , x − φ l , x ) − i δ t v π ( π Θ x + η ( ∂ x φ x ) + i λ (cos( √ φ x ) − = e i δ t π (cid:32) ( φ l + , x − φ l , x )2 δ t − η ( ∂ x φ x ) − i πλ cos( φ x ) . (cid:33) · const. (C4)In the last step, we performed a rescaling of the field φ x → φ x / √ v δ t → δ t , and factored out an overall normalizationconstant.Multiplying the Trotter increments and taking the limit ofinfinitesimal time steps δ t → dt yields the non-Hermitian ac-tion ( X = ( x , t )) S = (cid:90) X π (cid:104) ( ∂ t φ X ) − η ( ∂ x φ X ) (cid:105) − i λ cos φ X . (C5) Appendix D: Renormalization group equations for thesine-Gordon model
In order to derive the renormalization group equations forthe couplings K , λ we decompose the fields into short and longdistance modes, φ X = φ ( < ) X + φ ( > ) X . The long distance modescorrespond to momenta | k | ≤ Λ /ξ for a short-distance cuto ffΛ = π (in units of the lattice spacing) and a dimensionlessparameter ξ = e s . The short-distance modes correspond tothe small interval Λ /ξ < | k | ≤ Λ . In each RG step, the short-distance modes are integrated out perturbatively, and then mo-mentum and position are rescaled as k → k ξ, x → x /ξ .The RG equations for the sine-Gordon model are most eas-ily derived in (2 + x , t ) → ( η x , i η − t )towards imaginary time action. This yields the (2 + S = (cid:90) K π ( ∇ φ X ) + i λ cos( φ X ) . (D1)The free Green’s function of the theory in momentum spacethen is G Ω , k = (cid:104) φ Ω , k φ − Ω , − k (cid:105) = π K ( Ω + k ) , (D2)where Ω denotes the imaginary time frequency.Integrating out the short-distance modes perturbativelyyield the renormalized action S ( > ) = − log( (cid:104) e − S (cid:105) < ) = S ( > )0 − log (cid:104) e − δ S (cid:105) < , (D3)where we have denoted δ S = (cid:82) X i λ π cos φ X and S = S − δ S .Up to second order in λ one finds S ( > ) = S ( > )0 + (cid:104) δ S (cid:105) < − (cid:16) (cid:104) δ S (cid:105) < − (cid:104) δ S (cid:105) < (cid:17) . (D4)2The linear term yields the correction (cid:104) δ S (cid:105) < = i λ (cid:90) X (cid:104) cos( φ ( < ) X + φ ( > ) X ) (cid:105) < = i λ (cid:90) X cos( φ ( > ) X ) e − (cid:104) ( φ ( < ) X ) (cid:105) s . After rescaling space and time ( x , τ ) → ( x , τ ) /ξ this leads to λ → λξ − π K . (D5)Equation (D5) shows that up to first order in λ , the renormal-ization group flow predicts that, due to K = − i γπ v , in the limitof weak monitoring γ (cid:28) v the coupling always evolves to-wards zero and recovers the free theory. A dissipation strengthof at least γ > π v √ π − λ rele-vant.The second order correction in λ is a bit more subtle butstill manageable. It yields (cid:104) δ S (cid:105) s − (cid:104) δ S (cid:105) s = − λ ξ − π K (cid:90) X , Y (cid:104) cos( φ ( > ) X − φ ( > ) Y ) (cid:16) e (cid:104) φ ( < ) X φ ( < ) Y (cid:105) < − (cid:17) + cos( φ ( > ) X + φ ( > ) Y ) (cid:18) e −(cid:104) φ ( < ) X φ ( < ) Y (cid:105) < − (cid:19) (cid:105) . (D6)The average (cid:104) φ ( < ) X φ ( < ) Y (cid:105) < being performed over the short-distance modes only ensures that it remains local and that thedi ff erences in Eq. (D6) are only non-zero for X ≈ Y . For thesecond term, this yields cos( φ ( > ) X + φ ( > ) Y ) → cos(2 φ ( > ) X ), which isless relevant than the cos( φ ( > ) X ) term (due to the additional ex-ponential suppression in each RG step) and is commonly ne-glected. The derivative term is expanded around δ X = Y − X ,which yields cos( φ ( > ) X − φ ( > ) Y ) ≈ − ( δ X ∇ φ ( > ) X ) . The Green’sfunction in Eq. (D2) is even and symmetric in Ω , k , and thuscross averages x · τ vanish. We thus finally obtain (cid:104) δ S (cid:105) s − (cid:104) δ S (cid:105) s = − ( λξ − π K ) (cid:90) X A ( ξ )( ∇ φ X ) (D7)with A ( ξ ) = (cid:90) x ,τ x (cid:18) e G ( < ) x ,τ − (cid:19) . (D8)To leading order, the new function A ( ξ ) can be seen as aconstant, typically of order O (1). Performing the substitution ξ = e s with 0 < s (cid:28)
1, we find the flow equations from themain text.
Appendix E: Dark state wave function and correlations
Here we analytically compute the dark state | ψ D (cid:105) of the non-Hermitian Hamiltonian (42) in the main text and show that | ψ D (cid:105) is unique and can be reached from any initial state. Thenon-Hermitian Hamiltonian in momentum space is H = π (cid:90) q q (cid:16) ˆ θ q ˆ θ − q + (1 − i γ ) ˆ φ q ˆ φ − q (cid:17) , (E1) where we have set v = γ → γπ/ | ψ D (cid:105) of this Hamil-tonian, i.e. in the state H | ψ D (cid:105) =
0. Therefore, we introduceboson operators b † q , b q with [ b q , b † k ] = δ q , k viaˆ φ q = − i (cid:114) π | q | sgn( q )( b † q + b − q ) , (E2)ˆ θ q = i (cid:114) π | q | ( b † q − b − q ) . (E3)Inserting them into the Hamiltonian yields H = (cid:90) q | q | (cid:16) b † q b q + b − q b †− q − i γ ( b † q + b − q )( b †− q − b q ) (cid:17) . (E4)If the Hamiltonian were Hermitian, one would now diago-nalize the problem by a canonical transformation, preserv-ing bosonic commutation relations. For the present non-Hermitian Hamiltonian, a canonical transformation can befound that brings it into a tri-diagonal form, which still al-lows us to extract the dark state properties. Here, this requiresa particularly simple Bogoliubov transformation, which is in-dependent of the momentum mode q , i.e. we can reduce theproblem to tri-diagonalizing the Hamiltonian h = ( b † q , b − q ) (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) = B † q (cid:32) − i γ − i γ − i γ − i γ (cid:33)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = M (cid:32) b q b †− q (cid:33)(cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) = B q = B † q MB q . (E5)Any transformation of the form B q = VC q through some ma-trix V and onto a new set of bosonic operators C q = ( c q , c †− q ) T has to respect bosonic commutation relations1 = B † q σ z B q = C † q V † σ z VC q ! = C † q σ z C q ⇒ V − = σ z V † σ z . The RHS of this equation yields a representation of the matrix V and its inverse (cid:32) b q b †− q (cid:33) = (cid:32) α ∗ − β − β ∗ α (cid:33) (cid:32) c q c †− q (cid:33) ⇔ (cid:32) c q c †− q (cid:33) = (cid:32) α ββ ∗ α ∗ (cid:33) (cid:32) b q b †− q (cid:33) , (E6)with the constraint | α | − | β | =
1. We identify the matrixelements α = + (cid:112) − i γ ) − i γ (cid:113) | + (cid:112) − i γ ) − i γ | − γ ,β = − i γ (cid:113) | + (cid:112) − i γ ) − i γ | − γ , (E7)which yield the Hamiltonian in the basis of the new bosonoperators c q , c † q h = (cid:15) ( c † q c q + c †− q c − q ) + η c q c − q . (E8)3Here, (cid:15) = (cid:112) − i γ and η = − i γ + √ − i γ − √ + i γ are complexnumbers. Although the canonical transformation does not di-agonalize the non-Hermitian Hamiltonian, Eq. (E8) has oneunique dark state | ψ D (cid:105) , which is the vacuum in the basis of the c q , c − q states, c q | ψ D (cid:105) = , ∀ q . The diagonal terms ∼ (cid:15) c † q c q ensure that any other state is exponentially suppressed overtime due to the imaginary part of (cid:15) , Im( (cid:15) ) < ∼ η ensures that the vacuum is reached from any other state,even if the initial state has no overlap with the vacuum. Thecondition c q | ψ D (cid:105) = | ψ D (cid:105) = | (cid:105)(cid:104) | b † q b q | (cid:105) = | β | = (cid:104) | b q b † q | (cid:105) − , (E9) (cid:104) | b − q b q | (cid:105) = − α ∗ β = (cid:104) | b † q b †− q | (cid:105) ∗ . (E10)With this result, we can determine the correlation functions ofthe Luttinger Liquid field operators (remember that [ ˆ φ q , ˆ θ − q ] = π q ), (cid:104) | ˆ φ q ˆ φ − q | (cid:105) = π | q | (cid:16) + | β | − α ∗ β − β ∗ α (cid:17) = π | q | γ (cid:114) (cid:113) γ + − , (E11)12 (cid:104) |{ ˆ φ q , ˆ θ − q }| (cid:105) = π q ( α ∗ β − β ∗ α ) = − i π q γ + (cid:112) + γ , (cid:104) | ˆ θ q ˆ θ − q | (cid:105) = π | q | (cid:112) + γ Re( (cid:112) − i γ )1 + (cid:112) + γ . (E12)The result for the correlation function in Eq. (E11) is identi-cal to the correlation function of the ˆ φ field operators in theRiccati approach (23) (after rescaling of γ and multiplicationˆ φ q → iq /π ˆ φ q , which was measured in the Ricatti equations).One can easily show, that the remaining correlation functionsare also identical in both approaches. Since the Riccati resultsare obtained without any approximation, this demonstratesthat the dark state of the non-Hermitian Hamiltonian indeeddescribes the steady state of the replica fluctuations. Appendix F: Entanglement Entropy for a noisy Gaussian system
Here we provide a derivation for Eqs. (76) and (77) in themain text, which is based on the path integral for the partitionreduced function Z A ( n , { dW } ). In the bosonized framework,the latter is obtained in two steps. First, we bosonize the pathintegral for the general n -replica partition function in Eq. (55)at the Gaussian fixed point. This yields Z ( n , { dW } ) = (cid:90) D [ { φ ( l ) X } ] exp( iS n [ { φ ( l ) X , dW } ]) , (F1)with the action S n [ { φ ( l ) X , dW } ] = − π (cid:90) X n (cid:88) l = (cid:88) σ = π (cid:16) σφ ( l ) σ, X ∂ φ ( l ) σ, X − i ( D φ ( l ) σ, X ) (cid:17) − i π (cid:90) X dW t (cid:88) σ = ± n (cid:88) l = D φ ( l ) σ, X (F2) and D = ∂ x , π m depending on whether we consider the scaleinvariant or the gapped Gaussian fixed point.In a second step, we implement the boundary condition forthe matrix product in tr[( ρ ( c ) A ) n ] in Eq. (75). Consider the traceand the partial trace to be evaluated after some evolution time t f at which the density matrix has reached a steady state. Thisyields the boundary conditions for the two-contour path inte-gral [68, 77] φ ( l ) + , t f , x = φ ( l ) − , t f , x if x (cid:60) A φ ( l + − , t f , x if x ∈ A , (F3)where we set φ ( n + − , t f , x = φ (1) − , t f , x . The first condition is the com-mon boundary condition for the partition function of a singlebosonic system evaluated at time t = t f . It is realized for fieldsoutside the subset A . The second condition, which applies forfields with x ∈ A implements the multiplication of the reduceddensity matrices. In the Lagrangian formulation of the parti-tion function, we need to apply the same boundary conditionsas in Eq. (F3) also for the temporal derivatives ∂ t φ ( l ) X .The additional replica index translations in (F3) are com-monly implemented via so-called branch point twist fieldsT [68, 77], which translate the fields by one replica index T A φ ( l ) + , X = φ ( l + + , X if x ∈ A and t = t f and else act as the identity T A φ ( l ) + , X = φ ( l ) + , X . We will not elaborate further on the branchpoint twist fields. but we will use two important properties of T . First, since T is a translation operator on the replicas, it isdiagonal in the replica Fourier basis T A φ ( k ) + , X = exp( i ϕ ( k ) X ) φ ( k ) + , X , (F4)where ϕ ( k ) X = π k / n if x ∈ A and t = t f and ϕ ( k ) X = ϕ ( k ) X that T A acts as the identity on the absolute replica mode with k = dW ,but decouples from the boundary conditions of the reduceddensity matrix. Conversely, the k > Z A ( n , { dW } ) in Eq. (75) via Z A ( n , { dW } ) = n − (cid:89) k = (cid:104) T A (cid:105) S ( k ) × Z A (1 , { √ ndW } ) , (F5)where (cid:104) T A (cid:105) S ( k ) = (cid:90) D [ φ ( k ) X ] T A exp( iS ( k ) [ φ ( k ) ]) (F6)with S ( k ) [ φ ( k ) ] as defined in Eq. (66), and Z A (1 , { √ ndW } ) isthe reduced partition function for the absolute mode. It istherefore identical to the reduced partition function of a sin-gle replica, but with an increased measurement noise dW →√ ndW .The expression in Eq. (F5) has been derived in several pre-vious works for Hermitian systems [68, 77, 78]. In our case, itgrants an additional important insight, namely the decoupling4of the branch point twist fields from the measurement noise.This makes the entanglement entropy a noise insensitive mea-sure for the Gaussian fixed points. The direct computationof the expectation values of T A is, however, di ffi cult in mostcases. We therefore modify Eq. (F5) by multiplying it with1 = Z A (1 , Z A (1 , , where Z A (1 ,
0) is a shorthand for the reduced par-tition function of an absolute mode with zero measurementnoise. We can then reverse the steps leading to Eq. (F5) and arrive at Z A ( n , { dW } ) = Z A ( n , × Z (1 , { √ ndW } ) Z (1 , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ≡ z √ ndW . (F7)In the last equation, we removed the index A from the reducedpartition function, since for a single replica Z A is just a num-ber, i.e., the normalization, which is independent of A . Thisproves that the reduced partition function for a Gaussian the-ory of n -replica with measurement noise is identical to thereduced partition function of the equivalent system with zeromeasurement noise, multiplied by a bipartition independentnumber. This yields Eq. (76) in the main text. [1] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X ,031016 (2017).[2] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B , 205136(2018).[3] C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Phys.Rev. B , 104302 (2020).[4] Y. Bao, S. Choi, and E. Altman, Physical Review B (2020),10.1103 / physrevb.101.104301.[5] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys.Rev. B , 224307 (2019).[6] B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X , 031009(2019).[7] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Phys. Rev. Lett. ,030505 (2020).[8] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B , 205136(2018).[9] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B , 134306(2019), arXiv:1901.08092.[10] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, andV. Khemani, (2020), arXiv:2004.09560.[11] P. Calabrese and J. Cardy, Journal of Statistical Mechanics:Theory and Experiment , P04010 (2005).[12] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal ofHigh Energy Physics (2016).[13] T. Zhou and A. Nahum, Phys. Rev. B , 174205 (2019).[14] H. Kim and D. A. Huse, Phys. Rev. Lett. , 127205 (2013).[15] C. Jonay, D. A. Huse, and A. Nahum, (2018),arXiv:1803.00089.[16] A. A. Akhtar and Y.-Z. You, Phys. Rev. B , 134203 (2020).[17] B. Bertini and L. Piroli, Phys. Rev. B , 064305 (2020).[18] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, (2020),arXiv:2002.12385.[19] L. Fidkowski, J. Haah, and M. B. Hastings, Quantum , 382(2021).[20] Y. Li and M. P. A. Fisher, (2020), arXiv:2007.03822.[21] M. J. Gullans and D. A. Huse, Phys. Rev. X , 041020 (2020).[22] X. Cao, A. Tilloy, and A. De Luca, SciPost Physics (2019),10.21468 / scipostphys.7.2.024.[23] O. Alberton, M. Buchhold, and S. Diehl, (2020),https: // arxiv.org / abs / , 033017 (2020).[25] Y. Li, X. Chen, A. W. W. Ludwig, and M. P. A. Fisher, (2020),arXiv:2003.12721. [26] C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Ludwig,(2020), arXiv:2012.04666.[27] A. Nahum, S. Roy, B. Skinner, and J. Ruhman, (2020),arXiv:2009.11311.[28] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. , 580(1992).[29] R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A , 4879 (1992).[30] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. , 580(1992).[31] V. P. Belavkin, Journal of Physics A: Mathematical and General , L1109 (1989).[32] H. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).[33] K. Jacobs and D. A. Steck, Contemporary Physics , 279–303(2006).[34] M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. B , 064204 (2019).[35] M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev.Lett. , 210602 (2020).[36] D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Phys.Rev. D , 125005 (2009).[37] H. M. Wiseman and G. J. Milburn, Phys. Rev. A , 1652(1993).[38] H. M. Wiseman and G. J. Milburn, Quantum Measurement andControl (Cambridge University Press, 2009).[39] N. Gisin and I. C. Percival, Journal of Physics A: Mathematicaland General , 5677 (1992).[40] D. Yang, C. Laflamme, D. V. Vasilyev, M. A. Baranov, andP. Zoller, Phys. Rev. Lett. , 133601 (2018).[41] D. Yang, D. V. Vasilyev, C. Laflamme, M. A. Baranov, andP. Zoller, Phys. Rev. A , 023852 (2018).[42] M. A. Nielsen and I. Chuang, Quantum Computation andQuantum Information (Cambridge University Press, Cam-bridge, 2000).[43] H. Primas,
Sixty Two Years of Uncertainty, edited by A. I. Miller (Plenum, New York, 1990).[44] P. Zoller, M. Marte, and D. F. Walls, Phys. Rev. A , 198(1987).[45] T. Sauter, R. Blatt, W. Neuhauser, and P. E. Toschek, in Inter-national Quantum Electronics Conference (Optical Society ofAmerica, 1987) p. THDD2.[46] R. E. Kalman and R. S. Bucy, Journal of Basic Engineering ,95 (1961).[47] P. S. Maybeck, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, 2017).[48] S. L. Brunton and J. N. Kutz,
Data-Driven Science and Engi-neering: Machine Learning, Dynamical Systems, and Control (Cambridge University Press, 2019).[49] A. C. Doherty and K. Jacobs, Phys. Rev. A , 2700 (1999).[50] K. Jacobs, Quantum Measurement Theory and its Applications (Cambridge University Press, 2014).[51] It is also possible to perturbatively incorporate correlations be-tween the center-of-mass and the relative degrees of freedombeyond a product state assumption by applying time-dependentperturbation theory in the Mori-Zwanzig formalism [79]. How-ever, these corrections are always less relevant than the first or-der correction used here.[52] The trace has to be evaluated with care. Due to the noise averagethere is no straightforward cyclic permutation rule for operatorsin the absolute and relative basis.[53] J. Zinn-Justin,
Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989).[54] D. J. Amit, Y. Y. Goldschmidt, and S. Grinstein, Journal ofPhysics A: Mathematical and General , 585 (1980).[55] J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: SolidState Physics , 1181 (1973).[56] Both ( ∂ x ˆ φ x ) and cos( √ φ x ) are PT symmetric and thereforemultiplying them with an imaginary unit generates PT non-symmetric terms.[57] P. Fendley, H. Saleur, and A. B. Zamolodchikov, InternationalJournal of Modern Physics A , 5717–5750 (1993).[58] Y. Ashida, S. Furukawa, and M. Ueda, Nature Communications (2017), 10.1038 / ncomms15791.[59] J. V. Jos´e, L. P. Kadano ff , S. Kirkpatrick, and D. R. Nelson,Phys. Rev. B , 1217 (1977).[60] I. L. Aleiner, L. Faoro, and L. B. Io ff e, Annals of Physics ,378 (2016).[61] N. Tsuji, P. Werner, and M. Ueda, Phys. Rev. A , 011601(2017).[62] S. H. Shenker and D. Stanford, Journal of High Energy Physics , 132 (2015). [63] M. Ansari and Y. V. Nazarov, JETP , 3 (2016).[64] This behavior persists under introducing a regularization: Thelatter would enter Eq. (68) as an additional matrix i (cid:15)ωσ z , thee ff ect of which is overwritten by any non-zero measurementrate γ , and does not shift the poles away from the real axis.[65] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. , 227902 (2003).[66] I. Peschel, , L205 (2003).[67] M. Cramer, J. Eisert, M. B. Plenio, and J. Dreißig, Phys. Rev.A , 012309 (2006).[68] P. Calabrese and J. Cardy, Journal of Physics A: Mathematicaland Theoretical , 504005 (2009).[69] P. Calabrese and J. Cardy, Journal of Statistical Mechanics:Theory and Experiment , P06002 (2004).[70] The covariance matrix has to be evaluated always at the coordi-nates corresponding to the operator ˆ φ x and its conjugate ∂ x ˆ θ x . Inthe Lagrangian formulation of the path integral ∂ x ˆ θ x → K θ π ∂ t φ X ,where K θ is determined in the Hamiltonian formulation.[71] I. Peschel and V. Eisler, Journal of Physics A MathematicalGeneral , 504003 (2009), arXiv:0906.1663 [cond-mat.stat-mech].[72] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A ,022318 (2004).[73] A. Kamenev, Field Theory of Non-Equilibrium Systems (Cam-bridge University Press, 2011).[74] O. Lunt and A. Pal, Phys. Rev. Research , 043072 (2020).[75] S. Habib, (2004), https: // arxiv.org / abs / quant-ph / Quantum Continuous Variables: A Primer of The-oretical Methods (CRC Press, 2017).[77] J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon, Journal ofStatistical Physics , 129–168 (2007).[78] H. Casini, C. D. Fosco, and M. Huerta, Journal of StatisticalMechanics: Theory and Experiment , P07007 (2005).[79] R. Zwanzig,