Fisher information approach to non-equilibrium phase transitions in quantum XXZ spin chain with boundary noise
FFisher information approach to non-equilibrium phase transitions in quantum XXZspin chain with boundary noise
Ugo Marzolino a,b , Tomaž Prosen b a Ruđer Bošković Institute, HR-10000 Zagreb, Croatia b Department of Physics, Faculty of Mathematics and Physics,University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia (Dated: August 28, 2017)We investigated quantum critical behaviours in the non-equilibrium steady state of a
XXZ spinchain with boundary Markovian noise using the Fisher information. The latter represents the dis-tance between two infinitesimally close states, and its superextensive size scaling witnesses a criticalbehaviour due to a phase transition, since all the interaction terms are extensive. Perturbatively inthe noise strength, we found superextensive Fisher information at anisotropy | ∆ | (cid:54) and irrational arccos ∆ π irrespective of the order of two non-commuting limits, i.e. the thermodynamic limit and thelimit of sending arccos ∆ π to an irrational number via a sequence of rational approximants. From thisresult we argue the existence of a non-equilibrium quantum phase transition with a critical phase | ∆ | (cid:54) . From the non-superextensivity of the Fisher information of reduced states, we infer thatthis non-equilibrium quantum phase transition does not have local order parameters but has non-local ones, at least at | ∆ | = 1 . In the non-perturbative regime for the noise strength, we numericallycomputed the reduced Fisher information which lower bounds the full state Fisher information, andis superextensive only at | ∆ | = 1 . Form the latter result, we derived local order parameters at | ∆ | = 1 in the non-perturbative case. The existence of critical behaviour witnessed by the Fisherinformation in the phase | ∆ | < is still an open problem. The Fisher information also representsthe best sensitivity for any estimation of the control parameter, in our case the anisotropy ∆ , and itssuperextensivity implies enhanced estimation precision which is also highly robust in the presenceof a critical phase. PACS numbers: 05.30.Rt, 03.65.Yz,75.10.Pq,06.20.Dk
I. INTRODUCTION
One of the paradigms for non-equilibrium statisti-cal physics consists in the study of non-thermalisingnoisy dynamics : non-equilibrium phase transitions arenon-analytic changes of non-equilibrium steady states(NESS). This kind of transitions has a much richerphenomenology than equilibrium phase transitions be-cause NESSes lack a universal description in termsof thermodynamic potentials. From a methodologi-cal point of view, this situation results in a large va-riety of universality classes without general tools fortheir characterisation . For instance, algebraically de-caying correlation functions are not peculiar of criti-cal phenomena . Also the specral gap of the Liouvil-lian, an open system generalisation of the Hamiltoniangap, may vanish in the thermodynamic limit for all pa-rameters, with critical points resulting only in a fasterconvergence .The broad interest on non-equilibrium phase transi-tions and on the search, pursued in our approach, foruniversal tools to characterise them is also manifest fromtheir emergence in a large variety of settings, from com-plex systems, both physical and biological , to so-cial sciences and economics . Furthermore, quantum-like models have been developed to fit phenomena in so-cial sciences and economics .For quantum systems, dynamics with Markovian noiseare represented by Lindblad master equations . Re- cently, many investigations enlightened complex and crit-ical behaviours of quantum NESSes . An inter-esting paradigmatic master equation consisting of ananisotropic Heisenberg (XXZ) spin chain driven withan unequal noise at its boundaries has a non-trivialsteady state with transitions manifested in transportproperties .Exactly solvable models are forming one of the mainpillars of classical statistical mechanics, both in and outof equilibrium. Among important general concepts whichare amenable to exact solutions, are the non-equilibriumsteady states (NESS), important nontrivial examples ofwhich are the simple exclusion processes with boundarydriving . Similar does not yet hold for quantum statis-tical mechanics, as the number of exact solutions for in-teracting models, in particular out of equilibrium, is verylimited. The example of boundary driven XXZ model isone of the very few. Nevertheless, the behaviour of non-equilibrum partition function for a few other models thatcan be exactly solved using a similar boundary noise pro-tocol (e.g., boundary driven Fermi-Hubbard model andan integrable SU (3) chain ) is qualitatively identical tothe one for the isotropic Heisenberg model. This leads usto believe that the boundary driven XXZ model discussedhere may represent an important out-of-equilibrium uni-versality class and the same type of phase transition maylater be seen in other models. This may not be related tointegrability, but in non-integrable systems the numeri-cal analysis required to apply our approach in the NESSwill be much harder. a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Many equilibrium phase transitions are detected bythe Bures metric, also known as fidelity susceptibility .The latter is proportional to the Fisher information except in the presence of pathological behaviours how-ever producing only removable singularities . Whilethis quantity reduces to standard susceptibilities for ther-mal phase transitions , it represents a more sophisti-cated tool for quantum phase transitions (QPT), bothsymmetry-breaking and topological ones. It isworth to mention another non-standard approach tophase transitions in equilibrium statistical mechanicsand in chaotic dynamics, which is based on topologicalchanges of isoenergetic manifolds in the phase space .For non-equilibrium steady-state quantum phase transi-tions (NESS-QPT), as will be discussed here, the studyof Fisher information is in the very early stage .The rationale of our approach relies upon the geometricinterpretation of the Fisher information as the distancebetween two infinitesimally close states with respect toa varying control parameter. Indeed, when all interac-tion terms are extensive, superextensive metric impliesinstability with respect to small changes, e.g. due tocritical points separating different phases. In this pa-per, we exemplify this approach with deep characterisa-tions of the NESS-QPT in the XXZ chain with bound-ary noise. The above geometric interpretation providesa universal and unifying approach for both equilibriumand non-equilibrium, and possibly unknown, phase tran-sitions, with a clear advantage over the aforementionednon-universal tools.We also investigate, to best of our knowledge, origi-nal relations between non-local or local order parametersand the Fisher information of the full state or of reducedstates, respectively. This relation is general and doesnot depend on the model and, as such, can be appliedto any phase transition detected by the Fisher informa-tion. Moreover, it replies upon the Cramér-Rao bound,i.e. a result from estimation theory , while previousstudies on the fidelity susceptibility only focuses on theintuition behind the geometric interpretation, thus miss-ing the connection with order parameters. The latter isintuitive for symmetry breaking phase transitions wherelocal order parameters are known, and so signatures ofthe phase transition can be found in reduced states. Themost interesting application is in phase transitions with-out known order parameters, like in our case. This re-verses the usefulness argument for the fidelity suscepti-bility in topological phase transitions: while size scalingof the Fisher information were used to detect transitionswithout local order parameters, in our case we infer thelocal/non-local nature of order parameters from the sizescaling of the Fisher information.Basing on the above arguments, we endorse our ap-proach as a powerful tool to characterise general non-equilibrium quantum phase transitions in other systemsfar beyond previously considered cases, according to thefollowing recipe: superextensivity of the Fisher informa-tion, in systems with extensive interactions which scale lineraly with the volume, detects general critical be-haviours with at least non-local order parameters, su-perextensivity of the reduced state Fisher informationfurther proves local order parameters. Our study opensa new avenue of research on NESS-QPT, illustrating thatcomplex structures and relevant features can be extractedby the Fisher information in highly non-trivial systems.Fisher information is also intimately connected tometrology, being the inverse of the smallest variancein the estimation of the varying parameters . Su-perextensivity implies extraordinary enhanced metrolog-ical performances. Thus, beyond the aim of NESS-QPT,our study deepens the connection between quantum noisydynamics and metrology , as well as general relationsbetween NESS and quantum information .The paper is organised as follows. We define the spinchain model with boundary Markovian dissipation in sec-tion II, and the Fisher information with properties rel-evant for our analysis in section III. In section IV, wediscuss the size scaling of Fisher information for pertur-batively small dissipation strength, and implications onnon-equilibrium phase transition, including the existenceof a critical phase and of (non-)local order parameters.In section V, we report on the Fisher information andproperties of the non-equilibrium phase transition non-perturbatively in the dissipation strength, and in sectionVI we conclude. II. SPIN MODEL
We discuss a n -spin chain with XXZ Hamiltonian H XXZ = n − (cid:88) j =1 (cid:0) σ xj σ xj +1 + σ yj σ yj +1 + ∆ σ zj σ zj +1 (cid:1) , (1)which is an archetypical nearest neighbour interaction incondensed matter , with σ αj being Pauli matrices ofthe j -th spin. In addition, we consider a uniform mag-netic field along the z direction and Markovian dissipa-tion at the boundary of the chain, arriving thus at the fol-lowing dynamical equation for the density matrix, calledmaster equation, ddt ρ ( t ) = − i (cid:20) Ω2 M z + JH XXZ , ρ ( t ) (cid:21) + λ (cid:88) k =1 (cid:18) L k ρ ( t ) L † k − (cid:110) L † k L k , ρ ( t ) (cid:111)(cid:19) , (2)where L , = (cid:114) ± µ σ ± , L , = (cid:114) ∓ µ σ ± n (3)are the so-called Lindblad operators, and M z = (cid:80) nj =1 σ zj is the total magnetization along the z direction .While the first line of (2) reproduces the standardSchrödinger equation, the second line is the prototypi-cal form of quantum Markovian dissipation, under theminimal physical assumption that the resulting time-evolution γ t be a semigroup, i.e. γ t γ s = γ s + t ∀ t, s (cid:62) ,trace preserving, and completely positive, i.e. that pre-serves positivity of any initial density matrix even whenarbitrarily correlated with ancillary systems.Markovian master equations can be derived from mi-croscopic models with system-environment interactionthat is linear in the Lindblad operators . In partic-ular, Markovian master equations with local Lindbladoperators, i.e. each environment interacting with a sin-gle particle as in (2), derive from the so-called singularcoupling approximation or from the weak system-environment coupling if the system Hamiltonian is dom-inated by the interaction-free part , in our case Ω (cid:29) J .Our model has an exactly solvable steady state densityoperator, i.e., a fixed point ρ ∞ = lim t →∞ ρ t of (2), whichcan be represented in terms of a matrix product ansatz(see Ref. for a review). This structure will be essentialto make our computations efficient. III. FISHER INFORMATION
Given the aforementioned analytic solution, we com-pute the Fisher information for variations of theanisotropy ∆ F ∆ = 8 lim δ → − (cid:113) F (cid:0) ρ ∞ (∆) , ρ ∞ (∆ + δ ) (cid:1) δ == 2 (cid:90) ∞ ds Tr (cid:34)(cid:18) ∂ρ ∞ ∂ ∆ e − sρ ∞ (cid:19) (cid:35) , (4)with the Uhlmann fidelity F ( ρ, σ ) = (cid:0) Tr (cid:112) √ σρ √ σ (cid:1) .Defining the eigenvalues { p j } j and the correspondingeigenvectors {| j (cid:105)} j of the state ρ ∞ , the definition (4) ofthe Fisher information reads F ∆ = 2 (cid:88) j,l |(cid:104) j | ∂ ∆ ρ ∞ | l (cid:105)| p j + p l . (5)The connection between Fisher information F ∆ and es-timation theory is summarised in the Cramér-Rao boundwhich bounds any estimation variance of ∆ . If ∆ isestimated by the measurement of the observable O , theCramér-Rao bound readsVar (∆) = ∆ O (cid:0) ∂∂ ∆ (cid:104) O (cid:105) (cid:1) (cid:62) F ∆ , (6)where ∆ O is the variance of the observable O , andVar (∆) follows from error propagation. A property of the Uhlmann fidelity, useful in the fol-lowing, is that it is non-decreasing under the action oftrace preserving and completely positive maps on boththe arguments . The partial trace, namely the aver-age over the degrees of freedom of subsystems, is a tracepreserving and completely positive map. Therefore, theFisher information computed from (4) but using reducedstates, i.e. resulting from partial traces of the full state ρ ∞ , is a lower bound to the Fisher information of ρ ∞ .In next sections, we shall use the relation between localorder parameters and the Fisher information computedwith reduced states instead of full states, that we aregoing to explain here.Good order parameters for phase transitions are non-analytic quantities at critical points. Consider local ex-pectations (cid:104) O (cid:105) with O = (cid:88) R O R , (cid:104) O R (cid:105) = Tr (cid:0) ρ R∞ O R (cid:1) , (7)where R are subsystems with finite, n -independent, size, ρ R∞ = Tr ¯ R ρ ∞ (8)the reduced state resulting from the partial trace over thecomplement ¯ R of the subsystem R , and O R an observ-able of the subsystem R . Divergences of the derivatives of (cid:104) O (cid:105) are related to the Fisher information F R ∆ computedfrom equation (4) using the state ρ R∞ . Suppose that theanisotropy ∆ has to be estimated via measuments of localexpectations (cid:104) O R (cid:105) . The Cramér-Rao bound is a boundfor any estimation variance :Var (∆) = ∆ O R (cid:0) ∂∂ ∆ (cid:104) O R (cid:105) (cid:1) (cid:62) F R ∆ , (9)where ∆ O R is the variance of the observable O R , andVar (∆) follows from error propagation. Suppose, instead,to estimate ∆ via experimental measurements of the k -thderivative ∂ k ∂ ∆ k (cid:104) O R (cid:105) . The Cramér-Rao bound readsVar (∆) = Var (cid:16) ∂ k − ∂ ∆ k − (cid:104) O R (cid:105) (cid:17)(cid:16) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:17) (cid:62) F R ∆ , (10)where Var (cid:16) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:17) is the variance of the experimen-tal measurements of ∂ k ∂ ∆ k (cid:104) O R (cid:105) . Such a quantity dependson the measured observables and on instrumental param-eters, e.g., if derivatives are estimated via difference quo-tients, the increment of ∆ . Therefore, the size scaling ofthe reduced Fisher information bounds from above thedegree of divergence of the derivatives of local expecta-tions (7): (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ ∆ (cid:104) O (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:88) R (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ ∆ (cid:104) O R (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:88) R (cid:113) F R ∆ ∆ O R , (11) (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂ ∆ k (cid:104) O (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:88) R (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:88) R (cid:115) F R ∆ Var (cid:18) ∂ k − ∂ ∆ k − (cid:104) O R (cid:105) (cid:19) . (12)We shall use these bounds to infer the existence of localorder parameters. IV. PERTRUBATIVE ANALYSIS IN THEDISSIPATION STRENGTH
We start our investigation with the perturbative anal-ysis for small noise strength λJ . It is worth to stressthat this analysis does not correspond to a perturbationaround equilibrium. The zeroth order of the NESS is thecompletely mixed state which does not depend on anyparameter. Therefore, there is neither a notion of tem-perature nor of other equilibrium properties, nor traces ofphase transitions in the zeroth order. As a consequence,our perturbative analysis already captures genuine non-equilibrium phase transitions. In this case, the NESS is: ρ ∞ = 12 n (cid:18) + i λµ J (cid:0) Z − Z † (cid:1) + λ µ J (cid:16)(cid:2) Z, Z † (cid:3) − µ (cid:0) Z − Z † (cid:1) − Tr (cid:0) ZZ † (cid:1) (cid:17) (cid:19) + O (cid:18) λJ (cid:19) , (13)where Z is a matrix product operator Z = (cid:88) { s ,...,s N }∈{ , + , −} N (cid:104) L | n (cid:89) j =1 A s j | R (cid:105) n (cid:79) j =1 σ s j j . (14) A s j are tridiagonal matrices on the auxiliaryHilbert space spanned by the orthonormal basis {| L (cid:105) , | R (cid:105) , | (cid:105) , | (cid:105) , . . . , |(cid:98) n (cid:99)(cid:105)} : A = | L (cid:105)(cid:104) L | + | R (cid:105)(cid:104) R | + (cid:98) n (cid:99) (cid:88) k =1 cos( ηk ) | k (cid:105)(cid:104) k | ,A + = | (cid:105)(cid:104) R | − (cid:98) n (cid:99) (cid:88) k =1 sin( ηk ) | k + 1 (cid:105)(cid:104) k | ,A − = | L (cid:105)(cid:104) | + (cid:98) n (cid:99) (cid:88) k =1 sin( η ( k + 1)) | k (cid:105)(cid:104) k + 1 | , (15) and η = arccos ∆ ∈ R ∪ i R . The expansion (13) holdsas soon as the zeroth order is larger than the first orderin λJ . Estimating the magnitude of each order with itsHilbert-Schmidt norm ( || O || HS = (cid:112) Tr ( OO † ) ), the valid-ity condition for (13) reads λJ < √ n +1 µ || Z || − HS = (cid:40) O (cid:16) √ n (cid:17) if | ∆ | < , O (cid:0) n (cid:1) if | ∆ | = 1 . . (16)For ∆ > the upper bound in (16) is the inverse of a su-perexponential function, thus the perturbative expansion(13) is not useful. A. Non-commuting limits for the Fisherinformation
At the lowest order in λJ , the Fisher information is F ∆ = λ µ n +1 J Tr (cid:0) ∂ ∆ Z ∂ ∆ Z † (cid:1) + O (cid:18) λJ (cid:19) = λ µ J (cid:0) (cid:101) F ∆ + (cid:98) F ∆ (cid:1) + O (cid:18) λJ (cid:19) (17)with the two non-negative contributions (cid:101) F ∆ = 12(1 − ∆ ) n (cid:88) j =1 (cid:104) L | A j − DA n − j | R (cid:105) , (18) (cid:98) F ∆ = 18(1 − ∆ ) · d dη (cid:104) L | A n | R (cid:105) , (19)and the matrices A and D on the auxiliary space of thematrix product structure A = (cid:88) k,k (cid:48) = L,R, ,..., (cid:98) n (cid:99) (cid:18) ( A ) k,k (cid:48) + 12 ( A + ) k,k (cid:48) + 12 ( A − ) k,k (cid:48) (cid:19) | k (cid:105)(cid:104) k (cid:48) | , (20) D = (cid:98) n (cid:99) (cid:88) k =1 (cid:16) sign (1 − ∆ ) k | k (cid:105)(cid:104) k | + k | k + 1 (cid:105)(cid:104) k | + ( k + 1) | k (cid:105)(cid:104) k + 1 | (cid:17) . (21)The trace in (17) equals a transition amplitudein the doubled auxiliary space, spanned by {| k (cid:105) ⊗| k (cid:48) (cid:105)} k,k (cid:48) = L,R, , ,..., (cid:98) n (cid:99) , e.g. the left(right)-most state inright-hand-sides of equations (18) and (19) is actually (cid:104) L | ⊗ (cid:104) L | ( | R (cid:105) ⊗ | R (cid:105) ). Nevertheless, only the subspacespanned by {| k (cid:105) ⊗ | k (cid:105)} k = L,R, , ,..., (cid:98) n (cid:99) contributes, andthen we have applied the mapping | k (cid:105) ⊗ | k (cid:105) → | k (cid:105) toreduce the dimension of the auxiliary space.Now, we briefly reread results of Refs. , originallyfocused on metrology but not on NESS-QPT, and thenwe shall report original results in order to end up with afull description of the NESS-QPT. The system undergoesa NESS-QPT at | ∆ | = 1 , detected by superextensiveFisher information in the leading order F ∆ (cid:39) λ µ J n , for λµJ < n and large n. (22)When the rescaled anisotropy parameter ηπ = arccos ∆ π is rational and | ∆ | < , the Fisher information in theleading order is F ∆ (cid:39) λ µ J (cid:0)(cid:101) ξ n + ξ n (cid:1) , for λµJ < √ n and large n, (23)with size-independent coefficients (cid:101) ξ and ξ . Thus, F ∆ can-not be superextensive. Keeping only the leading contri-bution of the Fisher information in the thermodynamiclimit, and only afterwards setting ηπ to an irrational num-ber results in F ∆ = λ µ J O ( n ) , with some oscillations in n damped for more irrational ηπ . The latter approachcatches the superextensive size scaling of the Fisher in-formation, i.e. the divergent degree of the Fisher informa-tion density, when the limit of ηπ approaching irrationalsis taken after the thermodynamic limit.Keeping in mind the just mentioned results of ,we now present original results aiming to complete thecharacterisation of the NESS-QPT. We shall show thatthe limit of ηπ approaching an irrational number doesnot commute with the thermodynamic limit n → ∞ for | ∆ | < . Consider first the thermodynamic limit andthen the limit of ηπ approaching an irrational numbervia a sequence of rational approximants, say η m π = f m +1 f m with { f m } m the Fibonacci sequence for m (cid:62) which ap-proaches the golden ratio ϕ = √ as m → ∞ . Thecoefficient ξ , plotted in figure 1, shows the divergence for m → ∞ fitted by ξ = (0 . ± . f . ± . m (24)When | f m | (cid:62) (cid:4) n (cid:5) + 1 , ξ ( | f m | ) = ξ ( (cid:4) n (cid:5) + 1) , thus ξ = O ( n ) in the limit m → ∞ in agreement with the aboveresults.We now show the numerical computation of the Fisherinformation with the opposite order of limits, namelyat irrational ηπ without any assumption on the particlenumber n . The log-log plot of the rescaled contribution J λ µ (cid:101) F ∆ to the Fisher information, with (cid:101) F ∆ < F ∆ , isshown in figure 2. We are particularly interested in su-perextensivity of F ∆ , as a signature of a critical phase,and the remaining contribution to F ∆ , i.e. (cid:98) F ∆ , can onlyscale linearly with n . This plot shows a slower overallgrowth, as compared to the Fisher information with thelimit order exchanged, with fits given in table I. (cid:230) (cid:230)(cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:43) ΗΠ Ξ (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
10 10 f m Ξ FIG. 1: Semi-log plot of the coefficient ξ for ηπ = f m +1 f m with { f m } m the Fibonacci sequence. The inset shows the log-logplot of ξ as a function of f m , which is perfectly fitted by (0 . ± . f . ± . m .
10 10 n J F (cid:142) (cid:68) Λ Μ FIG. 2: Log-log plots of the contribution J λ µ (cid:101) F ∆ to the Fisherinformation as function of n for irrational ηπ : η = πϕ with ϕ = √ the golden ratio (dark, continuous), η = π √ (dotted), η = π (dashed), η = πe (dotdashed). For a comparison wealso plotted the slopes of power laws − n and n (greycontinuous lines). ηπ = fit: J λ µ (cid:101) F ∆ (cid:39) ϕ = √ (2 . ± . − n . ± . √ (3 . ± . − n . ± . π (1 . ± . − n . ± . e (3 . ± . − n . ± . TABLE I: Fits of the size scalings plotted in figure 2. The fitsare more precise when the oscillations are smaller.
We have thus shown the non-commutativity of the twolimits, but also the superextensivity of the Fisher infor-mation for both limit orders, for irrational ηπ which are allcritical points. This indicates that the model has a crit-ical phase for | ∆ | (cid:54) with a highly singular behaviour. B. Reduced Fisher information and the absence oflocal order parameters
A critical phase detected by the Fisher informationwas also observed in the XY model with boundary noisewhich is mapped to a free Fermion model . Our modelhas Fermion interactions, i.e. the anisotropy term, andthe above novel singular behaviour. A critical phase withseveral peaks of the Fisher information was also found inthe topological phase transion of the Kitaev honeycombmodel without local order parameter. This analogydemands a deeper understanding of the NESS-QPT interms of order parameters. We undertake this investiga-tion basing on the Fisher information of reduced states.Defining the set R = {R j } j =1 ,... |R| made of |R| spinsat increasing positions R j , the reduced NESS of thischain portion is ρ R∞ = 12 |R| (cid:18) |R| + i λµ J (cid:16) Z R − Z †R (cid:17)(cid:19) + O (cid:18) λJ (cid:19) , (25)where R is the R × R identity matrix, and Z R = (cid:88) { s R j } j =1 ,... |R| ∈{ , + , −} |R| (cid:104) L | A s R |R| (cid:89) j =2 A R j −R j − − A s R j | R (cid:105) |R| (cid:79) j =1 σ s R j R j . (26)In equation (26), we have used the fact that A | R (cid:105) = | R (cid:105) and (cid:104) L | A = (cid:104) L | .We shall show upper bounds for the reduced Fisher in-formation and non-increasing n dependence of local ex-pections, thus of ∆ O R and Var (cid:16) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:17) . These re-sults, together with equations (11) and (12), imply thatthere are no local order parameters, i.e. non-analytic ex-pectations (7). For instance, extensive expectations (cid:104) O (cid:105) ,e.g. with a number O ( n ) of subsystems such as R la-beling single spins or neighbouring couples, cannot havesuperextensive derivatives.We start this analysis by bounding the reducedFisher information of arbitrary subsystems R with a n -independent size |R| at order λ J : F R ∆ (cid:54) O (cid:18) n (cid:19) (27)which follows from the matrix operator structure of ρ ∞ and ρ R∞ through the following logical steps. • Coefficients of ρ ∞ expanded in the tensor basismade of Pauli matrices are generated by productsof sequences of n tridiagonal matrices on an auxil-iary space , as shown in equation (14). • The dependence of these coefficients on n entersthrough the number of matrices in the sequencegenerating a (cid:98) n (cid:99) -dimensional auxiliary subspace. • The coefficients of ρ R∞ in the Pauli tensor basis haveanalogous structure, as shown in equation (26), butwith the diagonal matrix A in the matrix productat positions corresponding to traced-out spins. • This diagonal matrix A does not have raising andlowering operators, and thus the dimension of thegenerated auxiliary subspace equals |R| . As a con-sequence, the dependence of ρ R∞ on n is manifestonly from the exponents R j − R j − + 1 < n . • The modulus of A is strictly upper bounded bythe identity matrix at | ∆ | < . Therefore, the co-efficients of ρ R∞ in the Pauli tensor basis are upperbounded in modulus by an exponentially decayingfunction due to A R j −R j − − if some R j − R j − − grow with n , or by an n -independent contributionif R j − R j − − O ( n ) for j ∈ (cid:2) , |R| (cid:3) . This al-ready proves that local expections, and thus ∆ O R and Var (cid:16) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:17) , do not increase with n . Theabove n -dependence, together with the definition(4) and the range λJ < √ n of the perturbation ex-pansion, implies the bound (27) for | ∆ | < . • At | ∆ | = 1 , the eigenvalues of A are ± , and localexpections remain non-increasing with n . Further-more, the derivative in the definition (4) gives anadditional multiplicative factor upper bounded by n , when deriving the term A R j −R j − − , but withthe exponential damping suppressed in the limit | ∆ | → . This multiplicative factor is however in-sufficient to compensate the smallness of λJ < n which is the validity range of the perturbation ex-pansion at | ∆ | = 1 . This again implies the bound(27).Remarkable examples are reduced states ρ R∞ of con-tiguous blocks of spins, i.e. R = [ R min , R max ] , so withall traced-out spins near the boundaries. These reducedstates, i.e. equation (25) with R j − R j − − ,have exactly the same analytic form of the full steadystate ρ ∞ with n replaced by the number of spins |R| = R max − R min in the subsystem. The system is thus self-similar.Summing up, k -th derivatives of any observable with k = O ( n ) lack divergent behaviours. If both k and R j − R j − − grow with n , k -th derivatives can divergebecause of repeated derivations of terms A R j −R j − − .Since the lower bound in (10) does not depend on k , thisdivergence must be compensated by the numerator of theleft-hand-side. Intuitively, to measure derivatives at in-creasing orders, e.g. via different quotients, we need todistinguish many measurements all at values of ∆ that liewithin a very, ideally vanishingly, small interval. Thus,the measurements of such derivatives become very hard,witnessed by large Var (cid:16) ∂ k ∂ ∆ k (cid:104) O R (cid:105) (cid:17) , and so is the conse-quent determination of ∆ as imposed by the Cramér-Raobound (10).The above discussion is insufficient for the reducedstates of a single spin, i.e. R = { j } , because it is com-pletely mixed at order λJ , i.e. ρ R = { j }∞ = + O (cid:0) λJ (cid:1) , andthe next order is ρ R = { j }∞ = 12 (cid:18) + λ µ J γ j σ zj (cid:19) + O (cid:18) λJ (cid:19) , (28)with γ j = (cid:104) L | A j − A z A n − j | R (cid:105) , (29) A z = 12 (cid:88) k,k (cid:48) = L,R, ,..., (cid:98) n (cid:99) (cid:0) ( A + ) k,k (cid:48) − ( A − ) k,k (cid:48) (cid:1) | k (cid:105)(cid:104) k (cid:48) | , (30)and A defined in equation (20). As before, we have ap-plied the mapping | k (cid:105)⊗| k (cid:105) → | k (cid:105) to reduce the dimensionof the auxiliary space because only the subspace spannedby {| k (cid:105) ⊗ | k (cid:105)} k = L,R, , ,..., (cid:98) n (cid:99) contributes.The corresponding single spin reduced Fisher informa-tion is F R = { j } ∆ = (cid:0) ∂∂ ∆ (cid:104) σ zj (cid:105) (cid:1) ∆ σ zj = λ µ J (cid:18) ∂γ j ∂ ∆ (cid:19) + O (cid:18) λJ (cid:19) . (31)From numerical computations, we found intensive, andeven very small F R = { j } ∆ at | ∆ | < . This, together withthe validity condition λJ < √ n of the perturbative ex-pansion at | ∆ | < , implies a bound tighter than (27),namely F R = { j } ∆ (cid:54) O (cid:18) λ J n (cid:19) < O (cid:18) n (cid:19) . (32)The zeroth and the first orders of γ j around | ∆ | = 1 can be analytically computed truncating the auxiliaryspace to the subsystem spanned by {| L (cid:105) , | R (cid:105) , | (cid:105) , | (cid:105)} : γ j | ∆= ± = 14 ( n − j +1) (cid:0) ± ( n − ∓ (cid:1) + O (∆ ∓ . (33)Therefore, the Fisher information of the reduced state(28) at | ∆ | = 1 is F R = { j } ∆= ± = λ µ J ( n − ( n − j + 1) + O (cid:18) λJ (cid:19) . (34)The Fisher information of the single spin reduced state(34) exhibits an apparent superextensive behaviour, i.e.a power law size scaling with exponent between and , depending on the spin position j . Nevertheless, thispower law is reduced by the validity range of the pertur-bative expansion, namely λJ < n at | ∆ | = 1 . Thereforethe bound to the reduced Fisher information is F R = { j } ∆ (cid:54) O (cid:18) λ J n (cid:19) < O ( n ) . (35)Although the above bound does not allow for local orderparameters, its increase with respect to (27) suggests thatsuperextensivity of F R = { j } ∆= ± might gradually emerge wheninceasing the order of λJ and at non-perturbative regime,as we will discuss in section V.These results imply that there are no local order pa-rameters detecting the NESS-QPT, as for tolopogicalphase transitions. C. Non-local order parameters
Although there are no local order parameters at thelowest order in λJ , there are non-local order parameterswhich detect at least the onset of the critical phase | ∆ | =1 , for instance the expectation of O ∆ = 2 n +1 Jµ · ∂∂λ ρ ∞ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 (36)or its limit O ∆ →± if one prefers a ∆ -independent oper-ator. The expectation of (36) satisfies (cid:104) O ∆ (cid:105) = λµJ (cid:104) L | A n | R (cid:105) + O (cid:18) λJ (cid:19) −−−−→ ∆ →± λµ J n ( n −
1) + O (cid:18) λJ (cid:19) (37)and ∂∂ ∆ (cid:104) O ∆ (cid:105) −−−−→ ∆ →± ∓ λµ J n ( n − n −
2) + O (cid:18) λJ (cid:19) . (38)The variance of O ∆ is ∆ O ∆ = 2 (cid:104) L | A n | R (cid:105) + O (cid:18) λJ (cid:19) −−−−→ ∆ →± n ( n −
1) + O (cid:18) λJ (cid:19) . (39)The non-local order parameter has then superexten-sive derivative, i.e. divergent density of the deriva-tive in the thermodynamic limit. The density deriva-tive is n ∂∂ ∆ O ∆ = O ( n − α ) for λµJ = O (cid:0) n α (cid:1) with α ∈ (1 , , compatibly with the range of validity λµJ < n of the perturbative expansion for | ∆ | = 1 . The ra-tio (cid:0) ∂∂ ∆ (cid:104) O ∆ (cid:105) (cid:1) / ∆ O ∆ has also the same scaling of theFisher information F ∆ (cid:12)(cid:12) | ∆ | =1 (22) almost saturating theCramér-Rao bound (6) with O = O ∆ . V. NON-PERTRUBATIVE ANALYSIS IN THEDISSIPATION STRENGTH
In order to investigate the non-perturbative behaviourof the Fisher information, we consider the steady state ofthe master equation (2) with µ = 1 which is known forany λ : ρ ∞ = SS † Tr ( SS † ) , S = (cid:88) { s ,...,s n }∈{ , + , −} n (cid:104) | n (cid:89) j =1 B s j | (cid:105) n (cid:79) j =1 σ s j j , (40)with the matrix product operator S and tridiagonal ma-trices B s j on the auxiliary Hilbert space spanned by theorthonormal basis {| (cid:105) , | (cid:105) , | (cid:105) , . . . , |(cid:98) n (cid:99)(cid:105)} B = (cid:98) n (cid:99) (cid:88) k =0 sin( η ( s − k ))sin( ηs ) | k (cid:105)(cid:104) k | ,B + = − (cid:98) n (cid:99) (cid:88) k =0 sin( η ( k + 1))sin( ηs ) | k (cid:105)(cid:104) k + 1 | ,B − = (cid:98) n (cid:99) (cid:88) k =0 sin( η (2 s − k ))sin( ηs ) | k + 1 (cid:105)(cid:104) k | , (41)and with s given by i sin η cot( sη ) = λ .While the numerical or analytical computation of thefull state Fisher information (4) is very hard, the compu-tation of reduced states of small subsytems and their re-duced Fisher information is feasible. The reduced Fisherinformation is a lower bound of the full state Fisher infor-mation because the Uhlmann fidelity is a non-decreasingfunction under the action of trace preserving and com-pletely positive maps, like the partial trace, on both thearguments . Therefore, superextensivity of the reducedFisher information immediately implies superextensivityof the full state’s Fisher information, which is the moregeneral signature of the phase transition. As explained insection III, superextensivity of the reduced Fisher infor-mation also provides additional knowledge, e.g. provingthe existence and deriving local order parameters.The j -th spin reduced state is diagonal in the σ zj basis: ρ ( j ) ∞ = 12 (cid:0) + γ j σ zj (cid:1) , (42) γ j = (cid:104) σ zj (cid:105) = (cid:104) | B j − B z B n − j | (cid:105)(cid:104) | B n | (cid:105) , (43)and B = (cid:98) n (cid:99) (cid:88) k,k (cid:48) =0 (cid:18) | ( B ) k,k (cid:48) | + 12 | ( B + ) k,k (cid:48) | + 12 | ( B − ) k,k (cid:48) | (cid:19) × | k (cid:105)(cid:104) k (cid:48) | , (44) B z = 12 (cid:98) n (cid:99) (cid:88) k,k (cid:48) =0 (cid:16) | ( B + ) k,k (cid:48) | − | ( B − ) k,k (cid:48) | (cid:17) | k (cid:105)(cid:104) k (cid:48) | . (45)We have again applied the mapping | k (cid:105) ⊗ | k (cid:105) → | k (cid:105) toreduce the dimension of the auxiliary space because onlythe subspace spanned by {| k (cid:105) ⊗ | k (cid:105)} k =0 , , ,..., (cid:98) n (cid:99) con-tributes.Therefore, the j -th spin reduced Fisher information is F R = { j } ∆ = (cid:0) ∂∂ ∆ (cid:104) σ zj (cid:105) (cid:1) ∆ σ zj , (46)saturating the Cramér-Rao bound (9) with O R = { j } = σ zj .The derivative ∂∂ ∆ (cid:104) σ zj (cid:105) and F { j } ∆ are both superextensiveat | ∆ | = 1 , as shown in figure 3 for ∆ = 1 . The superex-tensive size scalings of F { j } ∆ are fitted with power lawslisted in table II. The case ∆ = − gives similar results.The reduced Fisher information F { j } ∆ is also symmetricwith respect to reflection of the spin chain around itscenter.As a consequence of the above superextensivity, themagnetisation profile (cid:104) σ zj (cid:105) is an intensive, local order pa-rameter for the critical points | ∆ | = 1 , with divergingderivative ∂∂ ∆ (cid:104) σ zj (cid:105) , plotted in figure 4. The finite sizescaling of ∂∂ ∆ (cid:104) σ zj (cid:105) equals the square root of that of the re-duced Fisher information F R = { j } ∆ , from (46) because thevariance in the denominator is ∆ σ zj = 1 −(cid:104) σ zj (cid:105) = O ( n ) in agreement with the plot in figure 4. Extensive local or-der parameters are the magnetisations (cid:80) j ∈R (cid:104) σ zj (cid:105) for anymacroscopic but not centrosymmetric portion R of thechain. For centrosymmetric portions, the divergences atspin positions j and n − j cancel with each other. Otherextensive local order parameters are (cid:80) j ∈R f (cid:0) (cid:104) σ zj (cid:105) (cid:1) witheven functions f ( · ) and for any set R , even centrosym-metric ones. spin position: j = fit: F { j } ∆=1 (cid:39) . ± . − n . . ± . (cid:4) n (cid:5) (8 . ± . − n . . ± . (cid:4) n (cid:5) (6 . ± . − n . . ± . TABLE II: Fits of the size scalings plotted in figure 3. (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:131) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156) (cid:156)
10 100 1000 10 n F (cid:68)(cid:61) (cid:56) j (cid:60) FIG. 3: Upper panel: Plot of the Fisher information F { j } ∆=1 ofthe j -th spin reduced state as a function of n and j . The su-perextensivity is manifest from the comparison with the plane n . Lower panel: Log-log plots of the F { j } ∆=1 as function of n , set to powers of , for λ = 1 and j = 1 (circles), j = (cid:4) n (cid:5) (squares), and j = (cid:4) n (cid:5) (diamonds). The continuous lines arethe corresponding fits, see figure II, excluding the first threepoints of each line, which clearly deviates from the large n behaviour. The reduced Fisher information does not show su-perextensive size scaling at | ∆ | < . Therefore, thesuperextensivity of the full state Fisher information at | ∆ | < , and thus the presence of a critical phase in thenon-perturbative regime, is still an open question. VI. CONCLUSIONS
We derived characterizations of the NESS-QPT ofthe XXZ model with boundary noise, starting from theFisher information. We identified a critical phase definedby the anisotropy range | ∆ | (cid:54) , with irrational ηπ beingcritical points, for small dissipation. For instance, we ob-served a clear divergence for ηπ approaching the goldenratio through the Fibonacci sequence, and superextensive Fisher information at different irrational ηπ . This critical FIG. 4: Plot of the magnetisation profile (cid:104) σ zj (cid:105) of the j -th spinas a function of j and ∆ for n = 1000 and λ = 1 . behaviour lacks local order parameters but exhibits non-local ones. Moreover, it is observed for a small dissipa-tion strength which vanishes for infinite particle number.This limit might be considered similar to reducing theXYZ model to the XY model which yet exhibits a phasetransition. Moreover, other topological characterisationsof phase transitions already revealed critical points withnon-analytic microcanonical entropy at finite size whichbecomes smoother as the particle number grows and an-alytic in the thermodynamic limit .At non-perturbative dissipation, the reduced Fisherinformation provides a superextensive lower bound tothe full state Fisher information at | ∆ | = 1 togetherwith local order parameters, e.g. the magnetisation pro-file. Since the reduced Fisher information of the non-perturbative NESS is not superextensive for | ∆ | < , itis still an open question whether the Fisher informationis superextensive.We have proved the power of the Fisher informa-tion approach to characterise NESS-QPT. We suggestthat this approach will be useful for many other criti-cal phenomena, such as classical non-equilibrium phasetransitions , in quenched and dynamical systems ,or in chaotic systems . Superextensive Fisher in-formation also identifies probes for the estimation ofthe control parameter with enhanced performances .Our system, having a NESS with a very low or vanishingentanglement, is also relevant for enhanced metrologicalschemes without entanglement . Acknowledgments.
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