Dissipative dynamics at first-order quantum transitions
DDissipative dynamics at first-order quantum transitions
Giovanni Di Meglio, Davide Rossini, and Ettore Vicari Dipartimento di Fisica dell’Universit`a di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy (Dated: September 24, 2020)We investigate the effects of dissipation on the quantum dynamics of many-body systems atquantum transitions, especially considering those of the first order. This issue is studied within theparadigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamicsarising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by aLindblad master equation, with either local or global spin operators acting as dissipative operators.Analogously to what happens at continuous quantum transitions, we observe a regime where thesystem develops a nontrivial dynamic scaling behavior, which is realized when the dissipation pa-rameter u (globally controlling the decay rate of the dissipation within the Lindblad framework)scales as the energy difference ∆ of the lowest levels of the Hamiltonian, i.e., u ∼ ∆. However,unlike continuous quantum transitions where ∆ is power-law suppressed, at first-order quantumtransitions ∆ is exponentially suppressed with increasing the system size (provided the boundaryconditions do not favor any particular phase). I. INTRODUCTION
The recent progress in atomic physics and quantumoptical technologies has enabled great opportunities fora thorough investigation of the interplay between the co-herent quantum dynamics and the (practically unavoid-able) dissipative effects, due to the interaction with anexternal environment [1–4]. The competition betweencoherent and dissipative dynamic mechanisms may orig-inate a nontrivial interplay, which can be responsible forthe emergence of further interesting phenomena in many-body systems, in particular close to a quantum phasetransition where the many-body systems develop pecu-liar quantum correlations [5].Certain issues related to the competition between co-herent and dissipative dynamics have been addressedat continuous quantum transitions (CQTs) [6–8], wherequantum correlations develop diverging length scales ξ ,and the gap ∆ closes as a power law of ξ , i.e. ∆ ∼ ξ − z , z being the universal dynamic exponent. These studiesconsidered a class of dissipative mechanisms which canbe reliably described by a Lindblad master equation gov-erning the time evolution of the system’s density matrix.It was argued, and numerically checked, that a dynamicscaling limit exists at a CQT even in the presence ofdissipation, whose main features are controlled by theuniversality class of the quantum transition. However,such a dynamic scaling limit requires a particular tuningof the dissipative interactions, whose decay rate u shouldscale as u ∼ ∆ ∼ ξ − z .In this paper we extend the above studies to first-orderquantum transitions (FOQTs), which have their own pe-culiarities, in particular related to the emergence of anexponentially suppressed gap and to their sensitivity tothe boundary conditions in finite systems [9–11]. Be-sides that, FOQTs are of great phenomenological inter-est, since they occur in a large variety of many-body sys-tems, including quantum Hall samples [12], itinerant fer-romagnets [13], heavy-fermion metals [14–16], disordered systems [17, 18], and infinite-range models [19, 20].We address the interplay between the critical coherentdynamics and dissipative mechanisms, when the Hamil-tonian parameters are close to a FOQT. To this pur-pose, we consider dynamic protocols that start fromground states close to FOQTs, and then analyze theout-equilibrium dynamics arising from a instantaneousquench of the Hamiltonian parameters and the dissi-pative interaction with the environment. We take, asa paradigmatic example, the one-dimensional spin-1 / FIG. 1: The quantum spin-chain model discussed in this work.Neighboring spins are coupled through a coherent Hamilto-nian ˆ H (bidirectional blue arrows). Each spin is also homo-geneously and weakly coupled to some external bath B via aset of dissipators D (vertical red arrows), whose effect is toinduce incoherent dissipation. The environment is modeledeither as a sequence of local independent baths, each for anyspin of the chain (top drawing), or as a single common bath towhich each spin is supposed to be uniformly coupled (bottomdrawing). a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p quantum Ising model, exhibiting a FOQT line in its zero-temperature phase diagram, in the presence of either lo-cal or global homogeneous dissipative mechanisms—seeFig. 1. Their effects are assumed to be be well capturedby a Lindblad master equation of the density matrix ofthe open system, which we have numerically integratedfor systems with up to O (10) spins.Analogously to what happens at CQTs [6, 7], the quan-tum Ising chain along the FOQT line unveils a regimewhere a nontrivial dynamic scaling behavior is developed.This is observed when the dissipation parameter u (glob-ally controlling the decay rate of the dissipation withinthe master Lindblad equation) scales as the energy dif-ference ∆ of the lowest levels of the Hamiltonian of themany-body system, i.e., u ∼ ∆. However, unlike CQTswhere ∆ is power-law suppressed, at FOQTs ∆ is ex-ponentially suppressed with increasing the system size(when the boundary conditions do not favor any par-ticular phase). The dynamic scaling behavior turns outto become apparent for relatively small systems already,such as chains with L (cid:46)
10. This makes such dynamicscaling phenomena particularly interesting even from anexperimental point of view, where the technical difficul-ties in manipulating and controlling such systems can beprobably faced with up-to-date methods.Here the out-of-equilibrium quantum dynamics asso-ciated with the above-mentioned protocol is numericallymonitored by considering standard observables, such asthe longitudinal magnetization, as well as the averagework and heat characterizing the quantum thermody-namic properties of the out-of-equilibrium phenomenon.The paper is organized as follows. In Sec. II we intro-duce the one-dimensional quantum Ising chain, the mod-elization of the dissipative interactions through a mas-ter Lindblad equation, and the out-of-equilibrium pro-tocol discussed hereafter. Sec. III presents a summaryof the main features of the dynamic scaling theory forclosed systems, and their extension to allow for dissipa-tive interactions; in particular we address the behaviorexpected at FOQTs. In Sec. IV we numerically studythe dynamics of quantum Ising chains arising from theabove-mentioned protocol, up to size of order L = 10,along the FOQTs and at the CQT, showing that theresults support the general dynamic scaling theory. Fi-nally in Sec. V we summarize and draw our conclusions.In App. A we solve the analogous problem for a singlequantum spin. In App. B we address some related ques-tions for the out-of-equilibrium behavior of the Kitaevfermionic wire (related to the quantum Ising chain by aJordan-Wigner mapping) in the presence of local dissipa-tion due to particle pumping or decay, at its CQT; theseinclude exact analytic results for the quantum work andthe heat interchange during its time evolution. II. THE OUT-OF-EQUILIBRIUM PROTOCOL
The Hamiltonian of the one-dimensional ferromagneticquantum Ising model, reads:ˆ H ( g, h ) = − J L (cid:88) x =1 (cid:104) ˆ σ (1) x ˆ σ (1) x +1 + g ˆ σ (3) x + h ˆ σ (1) x (cid:105) , (1)where ˆ σ ( k ) x are spin-1 / x th site of the chain, J > g ≥
0. We alsoconsider spin systems of size L with periodic boundaryconditions.It is well known that, at g = g c = 1 and h = 0, themodel undergoes a CQT separating a disordered phase( g >
1) from an ordered ( g <
1) one [5]. The corre-sponding quantum critical behavior belongs to the two-dimensional Ising universality class, characterized by adiverging length scale ( ξ ∼ | g − g c | − ν , with ν = 1) and thepower-law suppression of the gap ∆ between the two low-est energy levels (as ∆ ∼ ξ − z with z = 1, or as ∆ ∼ L − z in finite-size systems at the critical point).The FOQT line, located at g < g c = 1, is related tothe level crossing of the two lowest states | ↑(cid:105) and | ↓(cid:105) for h = 0, such that (cid:104)↑ | ˆ σ (1) x | ↑(cid:105) = m and (cid:104)↓ | ˆ σ (1) x | ↓(cid:105) = − m (independently of x ), with m >
0. The degeneracyof these states is lifted by the longitudinal field, withstrength h . Therefore, h = 0 is a FOQT point, wherethe ground-state longitudinal magnetization M = (cid:10) ˆΣ (1) (cid:11) , ˆΣ (1) ≡ L (cid:88) x ˆ σ (1) x , (2)becomes discontinuous in the infinite-volume limit. Thetransition separates two different phases characterized byopposite values of the magnetization m , i.e. [21]lim h → ± lim L →∞ M = ± m , m = (cid:0) − g (cid:1) / . (3)In a finite system of size L with either periodic or openboundary conditions (which do not favor any of the twomagnetized phases separated by the FOQT), the loweststates are superpositions of the two magnetized states | ↑(cid:105) and | ↓(cid:105) . Due to tunneling effects, the energy gapat h = 0 vanishes exponentially as L increases [9, 22],as ∆ L ∼ e − cL d . In particular, for the one-dimensionalcase of the (1), the gap at h = 0 behaves as [21, 23]∆ L ≈ − g ) g L for open boundary conditions, and∆ L ≈ (cid:114) − g πL g L (4)for periodic boundary conditions. The differences E i − E for the higher excited states ( i >
1) are finite for L → ∞ .We model the dissipative interaction with the envi-ronment by Lindblad master equations for the densitymatrix of the system [24, 25], ∂ρ∂t = − i (cid:125) [ ˆ H, ρ ] + u D [ ρ ] , (5)where the first term in the r.h.s. provides the coherentdriving, while the second term accounts for the couplingto the environment. Its form depends on the natureof the dissipation arising from the interaction with thebath, which is effectively described by a set of dissipa-tors D , and a global coupling u >
0. In quantum op-tical implementations, the conditions leading to such aframework to study dissipative phenomena are typicallysatisfied [26], therefore this formalism constitutes a stan-dard choice for theoretical investigations of such kind ofsystems.In the following, we restrict to homogeneous dissipa-tion mechanisms, preserving translational invariance. Wemostly consider local dissipative mechanisms such as theone sketched in the top drawing of Fig. 1, whose trace-preserving superoperator D [ ρ ] can be written as [27, 28] D [ ρ ] = L (cid:88) x =1 ˆ L x ρ ˆ L † x − (cid:0) ρ ˆ L † x ˆ L x + ˆ L † x ˆ L x ρ (cid:1) . (6)The Lindblad jump operator ˆ L x associated to the localsystem-bath coupling scheme is chosen to beˆ L ± x ≡ ˆ σ ± x = (cid:2) ˆ σ (1) x ± i ˆ σ (2) x (cid:3) , (7)corresponding to mechanisms of incoherent raising (+)or lowering ( − ) for each spin of the chain. We shall alsoconsider an alternative global dissipative interaction withthe environment (bottom drawing of Fig. 1), described by D [ ρ ] = ˆ Lρ ˆ L † − (cid:0) ρ ˆ L † ˆ L + ˆ L † ˆ Lρ (cid:1) , (8)with a single raising, or lowering, Lindblad operatorˆ L ± ≡ ˆΣ ± , ˆΣ ± ≡ L (cid:88) x ˆ σ ± x . (9)In the following, we address the interplay between thecoherent dynamics and dissipative mechanisms, focussingin particular on the cases g ≤ | h | (cid:28)
1, correspond-ing to situations close to the transition line. For thispurpose, we consider dynamic protocols that start fromground states of the quantum Ising Hamiltonian close tothe transition line, and then analyze the out-equilibriumdynamics driven by the master Lindblad equation (5).More precisely, we adopt the following protocol: (i) thesystem starts, at t = 0, from the ground state of theHamiltonian (1) with transverse field parameter g ≤ h i ; (ii) then the systemevolves according to Eq. (5), where the coherent driv-ing is provided by the Hamiltonian for the same value g and a longitudinal parameter h (which may differ from h i , giving rise to a sudden quench), while the dissipativedriving is controlled by the parameter u (for u = 0, onerecovers the unitary dynamics of closed systems).The out-of-equilibrium evolution, for t >
0, is moni-tored by measuring certain fixed-time observables, suchas the longitudinal magnetization M ( t, h i , h, L ) = Tr (cid:2) ρ ( t ) ˆΣ (1) (cid:3) , (10) where the spin operator ˆΣ (1) is defined in Eq. (2) and ρ ( t ) is the density matrix of the evolving system at time t . Analogously, one may also consider fixed-time spincorrelations.We are also interested in the quantum thermodynamicproperties associated with this dissipative dynamics. Thefirst law of thermodynamics describing the energy flowsof the global system, including the environment, can bewritten as [29–31] dE s dt = w ( t ) + q ( t ) , (11)where E s is the average energy of the open system E s = Tr (cid:2) ρ ( t ) ˆ H ( t ) (cid:3) , (12)and w ( t ) ≡ dWdt = Tr (cid:20) ρ ( t ) d ˆ H ( t ) dt (cid:21) , (13) q ( t ) ≡ dQdt = Tr (cid:20) dρ ( t ) dt ˆ H ( t ) (cid:21) , (14)with W and Q respectively denoting the average workdone on the system and the heat interchanged with theenvironment.In our quench protocol, a nonvanishing work is onlydone at t = 0, when the longitudinal-field parameter sud-denly changes from h i to h (cid:54) = h i . Since, after quenchingthe field, the Hamiltonian is kept fixed [thus w ( t ) = 0,for t > W = (cid:104) h i | ˆ H ( h ) − ˆ H ( h i ) | h i (cid:105) = ( h i − h ) L (cid:104) h i | ˆΣ (1) | h i (cid:105) , (15)where | h i (cid:105) is the starting ground state associated withthe longitudinal parameter h i . Note that the averagework of this protocol is the same of that arising at suddenquenches of closed Ising chains, whose scaling behaviorat the CQT and FOQTs has been analyzed in Ref. [32].On the other hand, the heat interchange with the envi-ronment is strictly related to the dissipative mechanism,indeed one can easily derive the relation q ( t ) = u Tr (cid:2) D [ ρ ] ˆ H ( t ) (cid:3) , (16)by replacing the r.h.s. of the Lindblad equation (5) intothe expression (14). III. DYNAMIC SCALING AT QUANTUMTRANSITIONS
We now summarize the main features of the dynamicscaling framework that we will exploit to analyze theout-of-equilibrium quantum dynamics of closed and openmany-body systems. The scaling hypothesis is based onthe existence of a nontrivial large-size limit, keeping ap-propriate scaling variables fixed.We focus on the quantum Ising chain (1) along its tran-sition line, thus for g ≤ | h | (cid:28)
1, corresponding toFOQTs for g < g = 1. Then we discussthe scaling behaviors arising from a longitudinal externalfield h , i.e. ˆ H h = − h (cid:88) x ˆ σ (1) x . (17)The corresponding scaling variable controlling the equi-librium properties of isolated many-body systems at bothCQTs and FOQTs can be generally written as the ra-tio [9, 34] κ ( h ) = E h / ∆ L , (18)between the L -dependent energy variation E h associatedwith the ˆ H h term and the energy difference ∆ L ≡ E − E of the lowest-energy states at the transition point h = 0.Nonzero temperatures could be taken into account aswell, by adding a further scaling variable τ = T / ∆ L .Dynamic behaviors, exhibiting nontrivial time dependen-cies, also require a scaling variable associated with thetime variable, which is generally given by θ = ∆ L t . (19)The equilibrium and dynamic scaling limits are definedas the large-size limit, keeping the above scaling variablesfixed. Within this framework, the differences betweenCQTs and FOQTs are basically related to the functionaldependence of the above scaling variables on the size:typically, power laws arise at CQTs, while exponentiallaws emerge at FOQTs.Specializing Eq. (18) to the FOQTs of the quantumIsing chain, we obtain the scaling variable [9] κ ( h ) = 2 m hL/ ∆ L , (20)since 2 m hL quantifies the energy associated with thecorresponding longitudinal-field perturbation ˆ H h , and∆ L ∼ g L . For example, in the equilibrium finite-sizescaling limit, the magnetization is expected to behaveas [9] M ( h, L ) = m M ( κ ), where M is a suitable scalingfunction. We point out that the FOQT scenario basedon the avoided crossing of two levels is not realized forany boundary condition [9, 11]: in fact, in some cases theenergy difference ∆ L of the lowest levels may even dis-play a power-law dependence on L . However, the scal-ing variable κ ( h ) obtained using the corresponding ∆ L turns out to be appropriate, as well [9]. In the rest ofthe paper we shall restrict our study to quantum Isingmodels with boundary conditions that do not favor anyof the two magnetized phases, such as periodic or openboundary conditions, which generally lead to exponentialfinite-size scaling laws. Thus also the scaling variable θ related to the time dependence, cf. Eq. (19), is subjectto an exponential rescaling.The CQT at g = 1 and h = 0 is characterized by powerlaws, irrespective of the boundary conditions (see, e.g., Ref. [33]). The corresponding scaling variable turns outto be κ ( h ) ∝ L y h h , (21)where y h = 15 / h [see, e.g., Ref. [34] for itsderivation from the general expression given in Eq. (18)].Moreover, the energy difference between the two-loweststates behaves as ∆ L ∼ L − z where z = 1.For example, let us consider a quench of the longitudi-nal field of a closed quantum Ising chain at t = 0, from h i (starting from the corresponding ground state) to h (cid:54) = h i .At both the FOQTs and the CQT, we expect that thequantum coherent evolution of the longitudinal magneti-zation (10) develops the dynamic scaling behavior M ( t, h i , h, L ) ≈ L − ζ F m ( θ, κ i , κ ) , (22)where κ i ≡ κ ( h i ) , κ ≡ κ ( h ) , (23)the exponent ζ = 1 / ζ = 0 at the FOQTs, and F M is an appro-priate scaling function. The approach to such asymptoticbehavior is generally characterized by power-law correc-tions [9, 33, 34].As discussed in Refs. [6, 7], at CQTs the dissipator D [ ρ ] typically drives the system to a noncritical steadystate, even when the Hamiltonian parameters are closeto those leading to a quantum transitions. However, onemay identify a regime where the dissipation is sufficientlysmall to compete with the coherent evolution driven bythe critical Hamiltonian, leading to potentially novel dy-namic behaviors. At such low-dissipation regime, a dy-namic scaling framework can be observed after appro-priately rescaling the global dissipation parameter u , cf.Eq. (5). Indeed, the master Lindblad equation (5) at theCQTs of the coherent Hamiltonian driving develops ascaling behavior as well [6–8], with a further dependenceon the dissipation scaling variable γ ≡ u/ ∆ L , (24)thus γ ∼ uL z . Therefore, in the presence of dissipationthe dynamic scaling behavior (22) after quenching h isexpected to change into M ( t, h i , h, u, L ) ≈ L − ζ F m ( θ, κ i , κ, γ ) . (25)Our working hypothesis for the study of analogous is-sues at FOQTs is that the dynamic scaling behavior (25)applies as well, with the same definition (24) of the dis-sipation scaling variable γ . In the following we challengethis scenario by means of numerical computations. Forconvenience, we calculate the rescaled longitudinal mag-netization, defined as (cid:102) M ( t, h i , h, u, L ) = M ( t, h i , h, u, L ) M (0 , h i , h, u, L ) , (26)which is expected to behave as (cid:102) M ( t, h i , h, u, L ) ≈ (cid:101) F m ( θ, κ i , κ, γ ) , (27)at both the CQT and FOQTs. We recall that, accordingto our protocol, the initial longitudinal magnetization at t = 0 corresponds to that of the equilibrium ground-state expectation value for h i . Therefore it satisfies theasymptotic scaling behavior [9, 33] M (0 , h i , h, u, L ) ≡ M ( h i , L ) ≈ L − ζ f m ( κ i ) , (28)with ζ = 1 / ζ = 0 at the FOQTs, and f M being a universal scaling function (apart from a multi-plicative normalization and a normalization of the argu-ment) which depends on the type of transition, being aCQT or a FOQT. IV. NUMERICAL RESULTS
In this section we present numerical results for thequantum Ising chain subject to the protocol describedin Sec. II, when the system is close to the FOQT line,i.e. for g < | h | (cid:28)
1. We also report some results atthe CQT, for g = 1 and | h | (cid:28)
1, extending the study al-ready reported in Ref. [6], which focussed on the Kitaevfermionic wire (see also App. B). The latter is somehowrelated to quantum Ising chain, although they are notequivalent, in particular in the presence of local dissipa-tion. In all our simulations we set (cid:125) = 1, and J = 1 asthe energy scale.Numerically solving the Lindblad master equation (5)for a system as the one in Eq. (1) generally requires ahuge computational effort, due to the large number ofstates in the many-body Hilbert space H , which increasesexponentially with the system size (dim H = 2 L ). Moreprecisely, the time evolution of the density matrix ρ ( t ),which belongs to the space of the linear operators on H ,can be addressed by manipulating a Liouvillian superop-erator of size 2 L × L . This severely limits the accessiblesystem size to L (cid:46)
10 sites, unless the model is amenableto a direct solvability. A notable example in this respectis the Kitaev chain with one-body Lindblad operators,whose corresponding Liouvillian operator is quadratic inthe fermionic creation and annihilation operators (see,e.g., Ref. [6]). Unfortunately this is not the case for adissipative Ising spin chain, in which the Jordan-Wignermapping of Lindblad operators as those in (7) and (9)produces a nonlocal string operator forbidding an an-alytic treatment. Therefore, in this work we resort toa brute-force numerical integration of Eq. (5) througha fourth-order Runge-Kutta method, with a time step dt = 10 − , sufficiently small to ensure convergence forall our purposes. M L = 4 L = 5 L = 6 L = 7 L = 8 L = 9 θ -0.6-0.4-0.200.20.40.60.81 M κ : 1 →
1 [no quench] κ : 1 →
0 [quench] ∼∼ FIG. 2: Time behavior of the longitudinal magnetization fora quantum Ising ring close to the FOQT, in the presence ofhomogenous dissipation described by local Lindblad operatorsˆ L − x . We show the rescaled quantity (cid:102) M in Eq. (26) versus therescaled time θ , for different values of the system size L (seelegend). With increasing L , we keep the scaling variables κ i = 1 and γ = 0 . κ i = κ (i.e.,without the quench of the Hamiltonian parameter h ), whilethe lower panel is for κ = 0. A. Dynamics along the FOQT line
In the following, we numerically challenge the dynamicscaling behavior put forward in Sec. III at the FOQTs.We provide results for quantum Ising chains with g = 0 . L = O (10). Computations for othervalues of g < h i ; then thesystem evolves according to the Lindblad master equa-tion (5), with Hamiltonian parameter h (in principle dif-ferent from h i ) and dissipative strength u .Figure 2 displays the time evolution of the longitudinalmagnetization (10), in particular the rescaled one definedin Eq. (26), for some selected values of dynamic scalingvariables. The system-bath coupling has been modeledthrough local dissipative mechanisms as in the top draw-ing of Fig. 1, where the Lindblad operator associated toeach site induces incoherent lowering of the correspond-ing spin ( ˆ L − x = ˆ σ − x ). The scaling behavior (27) is checkedby varying the Hamiltonian and the dissipation param-eters of the protocol with increasing size L , so that thescaling variables κ , κ i , and γ [as defined in Eqs. (20),(23), and (24)] are kept fixed. For the gap ∆ L , enter-ing the definition of the scaling variables θ = ∆ L t and γ = u/ ∆ L , we do not use its asymptotic behavior (4),but the actual energy difference of the lowest levels of thequantum Ising ring at h = 0, with L spins. In both casesconsidered in Fig. 2, namely for κ i = κ (top panel, with-out quenching the Hamiltonian) and for κ i (cid:54) = κ (bottompanel, in the presence of a Hamiltonian quench), the lon-gitudinal magnetization appears to asymptotically van-ish in the large-time limit, although with different qual-itative trends. Actually this turns out to be a generalfeature for any nonzero value of the dissipation variable γ . Analogous results are obtained using Lindblad opera-tors inducing incoherent raising of the corresponding spin( ˆ L + x = ˆ σ + x ).Although limited to small system sizes, L ≤
10, ournumerical results substantially support the dynamic scal-ing behavior conjectured in Eq. (27), especially for suffi-ciently small values of θ . Most likely, the large- L con-vergence is not uniform and tends to be slower withincreasing θ . In particular, the results reported in thelower panel of Fig. 2 (when the dynamics arises bothfrom a quench of the Hamiltonian parameter and fromthe presence of dissipation) display oscillations whose ze-roes nicely scale with the time scaling variable θ , even forquite large θ . On the other hand, the values at the max-ima and minima undergo larger corrections, likely requir-ing larger lattice sizes to clearly observe the asymptoticscaling behavior. The apparent asymptotic convergenceto the conjectured dynamic scaling is also suggested bythe various plots in Fig. 3, showing data at fixed θ = 1,for various values of κ and γ (panels on the left corre-spond to the two situations in Fig. 2, while panels on theright are for the analogous cases with a larger dissipationstrength γ = 0 . L limit isgenerally compatible with 1 /L corrections, as hinted bythe dashed red lines, denoting 1 /L fits of numerical dataextrapolated at the largest available sizes. B. Comparison with the single-spin problem
In the absence of dissipation, the dynamic scaling be-havior of a quantum Ising chain along the FOQT turnsout to be well described by an effective two-level prob-lem [8, 9, 32, 35], defined by the single-spin Hamiltonianˆ H s = a ˆ σ (1) + a ˆ σ (3) . (29)Indeed the dynamic scaling functions of the quantumIsing chain, when the boundary conditions do not favorany phase separated by the FOQT, match the dynamicsof the single-spin problem (see App. A for an analyticdiscussion of the single-spin problem). In a sense, closedsystems behave rigidly at FOQTs. We now want to un-derstand whether that kind of property persists in thepresence of dissipative interaction with an environment,such as in the two schemes sketched in Fig. 1.In the presence of local dissipation (top drawing inFig. 1), we observe that the asymptotic scaling behavior L M L M L L κ :1 → γ =0.1 κ :1 → γ =0.1 ~~ κ :1 → γ =0.5 κ :1 → γ =0.5 FIG. 3: Approach to the asymptotic dynamic scaling behav-ior of the longitudinal magnetization at the FOQT, in thepresence of dissipation ˆ L − x . Here we set g = 0 . κ i = 1, and θ = 1, and show results for κ = 0 , γ = 0 .
1, 0 . /L approach to the asymptotic dynamic scal-ing (the dashed red lines are 1 /L fits of the numerical data,black circles, for L ≥ does not apparently display the rigidity property men-tioned above, in that the scaling functions are not repro-duced by the single-spin model. The corresponding datafor the magnetization as a function of time are shown inthe top panel of Fig. 4. The initial oscillating behaviorat finite lengths is reasonably captured by the single-spinprediction with dissipation coupling γ s = γ [thick-dashedblack line — see Eqs. (A17), (A21)]. However, alreadyfor θ (cid:38)
2, the curves for (cid:102) M exhibit large finite-size correc-tions and seem to approach an asymptotic overdampedbehavior for L → ∞ , quickly reaching the zero value. Inany case, the frequencies of oscillations for finite L valuesmatch those of the single-spin model, even for large θ .The rigidity of the dynamic scaling behavior can berecovered if a global dissipative mechanism is considered(bottom drawing in Fig. 1). This is the case of the datareported in the bottom panel of Fig. 4. The curves fordifferent L appear to approach an asymptotic dynamicscaling behavior, as well. Such convergence is much fasterthan that observed for the local dissipation scheme. In-terestingly, the L → ∞ behavior turns out to be well ap-proximated by the solution of the single-spin problem (atleast for θ (cid:46) γ s = γ/ -1-0.500.51 M L =2 L =4 L =6 L =8 θ -1-0.500.51 M L =2 L =3 L =4 L =5 L =6 ∼∼ Local dissipationGlobal dissipation γ s = γ γ s = γ /2 FIG. 4: Time behavior of the longitudinal magnetization at g = 0 .
5, in quench protocols with κ i = 100, κ = 0, and γ = 0 .
1, for various values of L (see legends). The systembath coupling is implemented in the form of either local dis-sipation operators (6), with ˆ L x = ˆ σ − x (top panel), or a singleglobal dissipation operator (8), with ˆ L = Σ − (bottom panel).The numerical results are compared with the single-spin prob-lem (thick-dashed curves): in the upper case, with increasing L , the curves tend to depart from the single-spin behavior; inthe lower case, the curves appear to approach an asymptoticdynamic scaling behavior, which is well approximated by thesolution of the single-spin problem with renormalized dissipa-tion coupling γ s = γ/ .
05. In both cases, the frequenciesof oscillations reasonably match.
C. Quantum work and heat
We now discuss the quantum thermodynamics arisingfrom the out-of-equilibrium protocol. As already men-tioned in Sec. II, a nonzero average work is only requiredat t = 0, when the Hamiltonian parameter suddenlychanges from h i to h . Therefore, it is the same as thatof systems subject to quench protocols without dissipa-tion (i.e., for u = 0). The dynamic scaling behavior ofthe work fluctuations for analogous quenches at the FO-QTs of the quantum Ising chain was studied in Ref. [32],showing that they reproduce the analogous quantities ofthe single-spin problem. In particular, the average workis given by W ≈ ∆ L W ( κ i , κ ) , W = ( κ i − κ ) κ i (cid:112) κ i . (30)Ref. [32] also reports results for the higher moments ofwork fluctuations and a discussion of the correction tothe asymptotic behavior. u t q ( t ) L =4, u =0.1 L =6, u =0.1 L =8, u =0.1 L q ( ) / u L =8, u =0.2 L =8, u =0.4 L u =0.1 u =0.2 u =0.4 g q ( ) / ( u L ) u =0.1 u =0.2 u =0.4 g = 0.5 FIG. 5: Behavior of the heat interchanged between the quan-tum Ising chain and the environment, per unit of time, as afunction of the variables of the system. Top panel: the func-tion q ( t ) with respect to the rescaled time u t , for different val-ues of the system size L and of the dissipation strength u (seelegend). In the inset we show the behavior of q ( t = 0) /u with L , for various values of u . The transverse field strength is keptfixed at g = 5. Bottom panel: the function q ( t = 0) / ( u L )with respect to g , for L = 6 and various values of u . Allthe data reported in this figure as for κ i = κ = 0, while thesystem-bath coupling is taken in the form of local and uniformLindblad operators ˆ L − x = ˆ σ − x . Coming to the heat interchanged between the systemand the environment, for the sake of convenience herewe focus on its time derivative q ( t ) = dQ/dt , defined inEq. (14). Unlike for the average work, this quantity doesnot present any relevant scaling property. In particular,our numerical results for the dynamics of the quantumIsing chain homogeneously coupled to local incoherentlowering operators, cf. Eq. (6) with ˆ L − x = ˆ σ − x , clearly in-dicate an exponential decay of q ( t ) with time t , as shownin the top panel of Fig. 5. More specifically, we found thefollowing functional dependence on the various parame-ters of the system: q ( t ) = C ( g, κ i , κ ) u L e − u t . (31)The coefficient C ( g, κ i , κ ) depends nearly exponentiallyon the transverse field g , although minor quantitative de-viations are present (bottom panel). On the other hand,the dependence of C on both κ i and κ is very weak andtends to vanish for increasing L (on the scale of the figure,it becomes unappreciable already for L (cid:38) q ( t ) with the systemvolume ( q ∝ L , as indicated in the top inset of Fig. 5) θ -0.8-0.6-0.4-0.200.20.40.60.81 M L = 4 L = 6 L = 8 L = 10 L -0.8-0.7-0.6-0.5-0.4-0.3 M θ = 3 ∼ ∼ FIG. 6: Time behavior of the longitudinal magnetization fora quantum Ising chain at the CQT, in the presence of ho-mogenous dissipation described by local Lindblad operatorsˆ L − x . We show the rescaled magnetization (cid:102) M , cf. Eq. (26),versus the rescaled time θ , for different values of the systemsize L (see legend). With increasing L we keep the scalingvariables κ i = 1, κ = 0 and γ = 0 . θ = 3, while the dashed redcurve denotes the corresponding 1 /L fit for L ≥ hints at the fact that such quantity is essentially relatedto a one-body mechanism, and not to a collective behav-ior of the system. In fact, the exponential time depen-dence q ( t ) ∼ e − ut is identical to the single-spin modelbehavior, as emerging from the analysis in App. A [see,e.g., Eq. (A22)]. D. Dynamics at the CQT
We now discuss the dynamic scaling behavior at theCQT (i.e., at g = 1 and | h | (cid:28) (cid:102) M versus the rescaled time θ = tL − z , for various systems sizes, up to L = 10. Thecurves with increasing L are obtained keeping the scal-ing variables κ i ∼ h i L y h , κ ∼ hL y h , and γ ∼ uL z (with y h = 15 / z = 1) fixed. The displayed data are for κ i = 1, κ = 0, γ = 0 .
1, but analogous qualitative conclu-sions can be drawn by changing the specific values of suchrescaled quantities. Our results substantially support thedynamic scaling behavior conjectured in Eq. (27), in par-ticular for sufficiently small values of θ . The approach tothe asymptotic behavior is again compatible with 1 /L corrections, as expected (see the inset, where numericaldata for the rescaled magnetization at fixed θ = 3 areplotted against the inverse system size, and the dashedred line denotes a 1 /L fit of such data at large L ). Wepoint out that analogous dynamic scaling behaviors have been reported for the Kitaev fermionic wire (see App. B),where much larger sizes can be reached, allowing us toachieve a definitely more robust check of the dynamicscaling behavior at a CQT [6, 7].Again the quantum thermodynamics arising from thedynamic protocol is characterized by an initial work at t = 0, due to the quench of the Hamiltonian parameter.Its scaling behavior was already discussed in Ref. [32] forclosed systems. As demonstrated there, the average workshows the scaling behavior W ≈ L − z W ( κ i , κ ) . (32)In App. B we discuss the average work within the Ki-taev fermionic wire at its CQT, where the quench is per-formed over the chemical potential, corresponding to thetransverse field g of the Ising chain. As shown there, theleading contribution to the average work is provided byanalytical terms, while the scaling part, such as that inEq. (32), turns out to be subleading.On the other hand, even at the CQT, the heat inter-changed with the environment does not exhibit scalingproperties. We have observed an exponential decay intime and a trivial linear dependence with L , for the func-tion q ( t ), analogously as in Eq. (31). The only differencewith respect to the g < C ( g = 1 , κ i , κ ) on the rescaled longitu-dinal field parameters, κ i and κ , which however rapidlyreduces with increasing L . V. CONCLUSIONS
We have investigated the effects of dissipation on thequantum dynamics of many-body systems close to aFOQT (that is, whose Hamiltonian parameters are thoseleading to a FOQT for the closed system), arising fromthe interaction with the environment, as for examplesketched in Fig. 1. The latter is modeled through aclass of dissipative mechanisms that can be effectively de-scribed by Lindblad equations (5) for the density matrixof the system [24, 25], with local or global homogenousLindblad operators, such as those reported in Eqs. (6)-(7)or (8)-(9), respectively. This framework is of experimen-tal interest, indeed the conditions for its validity are typ-ically realized in quantum optical implementations [26].We have analyzed how homogenous dissipative mech-anisms change the dynamic scaling laws developed byclosed systems at FOQTs (see, e.g., Refs. [11, 35]). Wealso mention that analogous issues have been addressedat CQTs (see, e.g., Refs. [6–8]).We study the above issues within the paradigmaticone-dimensional quantum Ising model, cf. Eq. (1), whichprovides an optimal theoretical laboratory for the inves-tigation of phenomena emerging at quantum transitions.Indeed its zero-temperature phase diagram presents aline of FOQTs driven by a longitudinal external field,ending at a continuous quantum transition. To investi-gate the interplay between coherent and dissipative driv-ings at the quantum-transition line (FOQTs for g < | h | (cid:28) g ≈ | h | (cid:28) t = 0, into the ground state of the Hamiltonian (1),for a given longitudinal parameter h i ; then, for t > H ( g, h ), with h gen-erally different from h i , and the system-bath interactioneffectively described by the dissipator D [ ρ ] with a fixedcoupling strength u .Analogously to what happens at CQTs, we observe aregime where the system develops a nontrivial dynamicscaling behavior, which is realized when the dissipationparameter u scales as the energy difference ∆ of the low-est levels of the Hamiltonian of the many-body system.However, unlike CQTs where ∆ is power-law suppressed,at FOQTs ∆ is exponentially suppressed when boundaryconditions do not favor any particular phase [9, 11]. Nu-merical solutions of the Lindblad equations up to latticesizes with L ≈
10 substantially confirm the existence ofsuch dynamic scaling behavior and pave the way towardexperimental testing in the near future through quantumsimulation platforms for spin systems of small size, wherethe required resources are less demanding.We also compare the emerging asymptotic scaling be-havior with the single-spin problem interacting with anenvironment modeled by a corresponding Lindblad equa-tion. Unlike closed systems where the unitary dynamicsis well described by the two-level single-spin problem, inthe presence of dissipation the system loses such a rigid-ity, and the scaling behavior turns out to significantlydiffer. Only in the case of global dissipators the dynamicscaling resembles that of the single-spin problem.The arguments leading to this scaling scenario at FO-QTs are quite general. Analogous phenomena are ex-pected to develop in any homogeneous d -dimensionalmany-body system at a continuous quantum transition,whose Markovian interactions with the bath can be de-scribed by local or extended dissipators within a Lindbladequation (5). Appendix A: The single-spin model
We consider the following single-spin Hamiltonian:ˆ H s = ∆ ˆ H r , ˆ H r = 12 ˆ σ (3) − κ σ (1) , (A1)where ∆ is the energy scale (the gap for κ = 0), κ is therescaled parameter related to the intensity of an appliedexternal magnetic field, and ˆ σ ( k ) are the usual spin-1/2Pauli matrices. We discuss the dynamics of that single-spin system subject to dissipation, as described by theLindblad master equation ∂ρ∂t = − i (cid:125) (cid:2) ˆ H s , ρ (cid:3) + u D ( ρ ) , (A2) D ( ρ ) = ˆ Lρ ˆ L † − (cid:0) ρ ˆ L † ˆ L + ˆ L † ˆ Lρ (cid:1) . (A3) The protocol starts again from the ground state asso-ciated with an initial value κ i , that is, the pure state | Ψ( t = 0) (cid:105) = cos( α i / | + (cid:105) + sin( α i / |−(cid:105) , (A4)where |±(cid:105) are the eigenstates of ˆ σ (1) and tan( α i ) = κ − i .The corresponding density matrix reads ρ ( t = 0) = 12 (cid:104) ˆ I + c k ˆ σ ( k ) (cid:105) , (A5) c = cos α i , c = 0 , c = sin α i , where ˆ I is the 2 × κ (thus for κ (cid:54) = κ i we have also a quench).We may generally define the energy of the system as E = Tr (cid:2) ρ ˆ H s (cid:3) , (A6)whose time derivative allows us to define quantities anal-ogous to the heat q and work w in the time unit, i.e. dEdt = Tr (cid:20) dρdt ˆ H s (cid:21) + Tr (cid:20) ρ d ˆ H s dt (cid:21) ≡ q + w . (A7)Using the Lindblad master equation we can easily derivethe relation q = dQdt = Tr (cid:20) dρdt ˆ H s (cid:21) = u Tr (cid:2) D ( ρ ) ˆ H s (cid:3) . (A8)One can easily prove that q = 0, if [ ˆ H, ˆ L ] = 0.Moreover we define the purity P = Tr (cid:2) ρ (cid:3) , (A9)which equals one for pure systems. Its time derivativecan be written as dPdt = 2 Tr (cid:20) dρdt ρ (cid:21) = 2 u Tr (cid:2) D ( ρ ) ρ (cid:3) . (A10)We may rescale the parameters and the time variableso that ρ (cid:48) ( θ ) ≡ ∂ρ∂θ = − i (cid:125) (cid:2) ˆ H r , ρ (cid:3) + γ D ( ρ ) , (A11) θ = t ∆ , γ = u/ ∆ . (A12)The time dependent density matrix can be generallyparametrized as ρ ( θ ) = 12 (cid:104) ˆ I + A k ( θ )ˆ σ ( k ) (cid:105) , (cid:88) k A k ( θ ) ≤ , (A13)where A i are real functions of the rescaled time θ . Notethat Tr[ ρ ] = 1 , Tr (cid:2) ρ (cid:3) = 12 (cid:16) (cid:88) k A k (cid:17) . (A14)0The Lindblad equation (A2) can be turned into cou-pled differential equations of the functions A k . Let usconsider the Lindblad operatorsˆ L ± ≡ ˆ σ ± ≡ ˆ σ (1) ± i ˆ σ (2) , (A15)corresponding to the sign + and − respectively. Straight-forward computations lead to the coupled differentialequations A (cid:48) = − A − γ A ,A (cid:48) = A + κA − γ A ,A (cid:48) = − κA − γ ( A ∓ . (A16)The upper/lower signs correspond to the cases ˆ L ± . Thevarious observables can be written in terms of the func-tion A i . For example, the longitudinal magnetization M reads M ( θ ) = Tr (cid:2) ˆ σ (1) ρ ( θ ) (cid:3) = A ( θ ) . (A17)The heat per unit of rescaled time is given by q r ≡ Q (cid:48) = Tr (cid:2) ρ (cid:48) ˆ H r (cid:3) = 12 A (cid:48) − κ A (cid:48) = − γ A ∓
1) + 14 γκA . (A18)The time dependence of the purity (A9) can be easily de-rived using Eq. (A14) and the above solutions, obtaining P ( θ ) = 12 (cid:16) (cid:88) k A k ( θ ) (cid:17) ≤ , (A19) P (cid:48) ( θ ) = (cid:88) k A k ( θ ) A (cid:48) k ( θ ) == − γ A + A ) − γA ( A ∓ . (A20)In the case κ = 0, one can easily find the solution A ( θ ) = e − γθ/ [ A (0) cos( θ ) − A (0) sin( θ )] , (A21) A ( θ ) = e − γθ/ [ A (0) sin( θ ) + A (0) cos( θ )] ,A ( θ ) ∓ e − γθ [ A (0) ∓ , in terms of the initial density matrix ρ (0), and in partic-ular of its coefficients A i (0). Therefore we have that q r = − γ e − γθ [ A (0) ∓ , (A22) Q = ∆ (cid:90) ∞ dθ q r = − ∆2 [ A (0) ∓ . (A23)Note that Q is positive/negative for pumping/decay (pos-itive Q means that the system is getting energy from thebath). The purity (A9) turns out to exponentially ap-proach one, reflecting the fact the the system relaxes toa pure state. In the case of dephasing Lindblad operatorˆ L d ≡ ˆ σ (3) , (A24)we obtain the coupled differential equations A (cid:48) = − A − γA ,A (cid:48) = A + κA − γA ,A (cid:48) = − κA . (A25)Using the above equations, we derive the heat per unitof rescaled time is given by q r = Tr[ ρ (cid:48) ˆ H r ] = γκA . (A26)Again, for κ = 0 the solution is quite simple, obtaining A ( θ ) = e − γθ [ A (0) cos( θ ) − A (0) sin( θ )] ,A ( θ ) = e − γθ [ A (0) sin( θ ) + A (0) cos( θ )] ,A ( θ ) = A (0) . (A27)No heat transmission occurs, because [ ˆ H, ˆ L ] = 0 when κ = 0. On the other hand the purity changes P ( θ ) = 12 (cid:110) A (0) + e − γθ (cid:2) A (0) + A (0) (cid:3)(cid:111) . (A28)approaching exponentially the asymptotic value P ( θ → ∞ ) = 1 + A (0) . (A29)Therefore, the spin system relaxes to a mixed state underdephasing. Appendix B: Quantum thermodynamics of afermionic wire coupled to local baths
In this appendix we focus on the quantum thermo-dynamics of fermionic quantum wires coupled to localMarkovian baths. In particular, we consider a Kitaevquantum wire defined by the Hamiltonian [36]ˆ H K = − J L (cid:88) x =1 (cid:0) ˆ c † x ˆ c x +1 + δ ˆ c † x ˆ c † x +1 +h . c . (cid:1) − µ L (cid:88) x =1 ˆ n x , (B1)where ˆ c x is the fermionic annihilation operator associatedwith the sites of the chain of size L , ˆ n x ≡ ˆ c † x ˆ c x is thedensity operator, and δ >
0. We set (cid:125) = 1, and J = 1as the energy scale. We consider antiperiodic boundaryconditions, ˆ c L +1 = − ˆ c , and even L for computationalconvenience.The Hamiltonian (B1) can be mapped into a spin-1/2XY chain, through a Jordan-Wigner transformation [5].In the following we fix δ = 1 (without loss of general-ity), so that the corresponding spin model is the quantumIsing chain (1). Note however that the non-local Jordan-Wigner transformation of the Ising chain with periodicor antiperiodic boundary conditions does not map into1the fermionic model (B1) with periodic or antiperiodicboundary conditions. Indeed further considerations ap-ply [21, 37], leading to a less straightforward correspon-dence, depending on the parity of the particle numbereigenvalue. Therefore, the Kitaev quantum wire cannotbe considered completely equivalent to the quantum Isingchain (see, e.g., the discussion in the appendix of Ref. [8]).However, analogously to the quantum Ising chain, theKitaev quantum wire undergoes a continuous quantumtransition at µ = µ c = −
2, between a disordered ( µ < µ c )and an ordered quantum phase ( | µ | < | µ c | ). This tran-sition belongs to the two-dimensional Ising universalityclass [5], characterized by the length-scale critical expo-nent ν = 1, related to the renormalization-group dimen-sion y µ = 1 /ν = 1 of the Hamiltonian parameter µ (moreprecisely of the difference ¯ µ ≡ µ − µ c ). The dynamic ex-ponent associated with the unitary quantum dynamics is z = 1.We focus on the out-of-equilibrium thermodynamic be-havior of the Fermi lattice gas close to its continuousquantum transition in the presence of homogeneous dis-sipation mechanisms described by the Lindblad equa-tion (5). We consider local dissipative mechanisms, sothat D [ ρ ] = (cid:80) x D x [ ρ ] is given by a sum of local (single-site) terms (top drawing of Fig. 1). The onsite Lindbladoperators ˆ L x describe the coupling of each site with anindependent bath B , associated with particle loss (l) orpumping (p), thusˆ L l ,x = ˆ c x , ˆ L p ,x = ˆ c † x , (B2)respectively. With this choice of dissipators, the fullopen-system many-body fermionic master equation en-joys a particularly simple treatment, see e.g. Ref. [6].Our protocol starts from the ground state | µ i (cid:105) of ˆ H K for a generic µ i . We then study the time evolution after aquench of the Hamiltonian parameter to µ at t = 0, anda simultaneous turning on of the interaction with the en-vironment controlled by the dissipation coupling u . Theout-of-equilibrium behavior of the fermionic correlationfunctions resulting from this protocol has been alreadystudied in Refs. [6, 7].Before presenting the analysis of the quantum thermo-dynamic properties during the time evolution associatedwith the above protocol, let us discuss some results forthe following correlation function at distance r : P ( r, t ) = Tr (cid:2) ρ ( t ) (ˆ c † x ˆ c † x + r + ˆ c x + r ˆ c x ) (cid:3) . (B3)This is expected to converge, in the large- L limit, to theasymptotic dynamic scaling behavior [6] P ( r, t, ¯ µ i , ¯ µ f , u, L ) ≈ L − P ( R, θ, κ i , κ, γ ) , (B4)where R = r/L and the other scaling variables are de-fined as usual. The data reported in Fig. 7 display theapproach to such scaling behavior, after keeping the scal-ing variables fixed, where θ = tL − z , γ = uL z , κ ( µ ) = ¯ µL y µ , (B5)¯ µ = µ − µ c , κ i ≡ κ ( µ i ) , κ ≡ κ ( µ ) . L P ( r , L ) L L P ( r , L ) θ L P ( r , L ) L = 4 L = 6 L = 8 L = 10 L = 20 L = 80 L L P ( r , L ) κ = 0 κ = 2 θ = 1 θ = 1 FIG. 7: Time behavior of the correlation function P ( r, t ) forthe Kitaev quantum wire close to the CQT point ( δ = 1, µ = − γ = 0 .
1. The top figure shows data for κ i = κ = 0 (i.e., the Hamiltonian is exactly at the CQT),while the bottom figure is for κ i = κ = 2. The various curvesin the two main panels show L P ( r, t ) for r = L/ L (see legend), as a function of the rescaled time θ . In the insets we report the same quantity as a function of L − (top) or of L − (bottom), for a fixed value of θ = 1. Toperform this analysis, we have first washed out the wiggles in θ (see main plot), by fitting the original curves in the interval θ ∈ [0 ,
5] with a fifth-order polynomial.
The large- L asymptotic behavior turns out to be ap-proached with power-law suppressed corrections, anal-ogously to what has been hinted in the main text for thedissipative quantum Ising chain. The approach to theasymptotic behavior is generally characterized by O (1 /L )corrections, except at the critical point where they mayget suppressed by a larger O (1 /L ) power [7, 8, 33], asalso shown by the insets of Fig. 7. Further results for theasymptotic large- L behavior of the fermionic correlationsfunctions can be found in Refs. [6, 7].We now extend the analysis of the out-of-equilibriumdynamics arising from the protocol described above tothe quantum thermodynamics, focussing on the quantumwork and heat interchange during the out-of-equilibriumevolution. We discuss the case of particle decay as adissipative mechanism; the case of pumping can be easilyobtained by analogous computations.The first law of thermodynamics (11) describes the en-ergy flows of the global system, including the environ-2ment. In our protocol, a nonvanishing work is only doneat t = 0, if the Hamiltonian parameter µ is suddenlychanged from µ i to µ (cid:54) = µ i . Since after quenching theHamiltonian is kept fixed, thus w ( t ) = 0 for t >
0, andthe average work is just given by W = (cid:104) µ i | ˆ H K ( µ ) − ˆ H K ( µ i ) | µ i (cid:105) = ( µ i − µ ) L (cid:104) µ i | ˆ n x | µ i (cid:105) , (B6)where last expression is obtained because the antiperi-odic boundary conditions respect translational invari-ance. The matrix element of the particle density ˆ n x canbe computed analytically, in particular when µ i = µ c , wehave (cid:104) µ c | ˆ n x | µ c (cid:105) = D ( L ) , (B7) D ( L ) = 12 − sin (cid:0) π L (cid:1) L (cid:2) − cos (cid:0) πL (cid:1)(cid:3) = π − π + π L + O ( L − ) . (B8)More generally, for µ in the critical region, so that ¯ µ ≡ µ − µ c (cid:28)
1, we obtain the asymptotic expansion [33] (cid:104) µ | ˆ n x | µ (cid:105) = f a (¯ µ ) + L − ζ f s ( κ ) + O ( L − ) , (B9)where ζ = 1 + z − y µ = 1, f a and f s are appropriatefunctions.Concerning the heat interchanged with the environ-ment, we derive the nontrivial relation q ( t ) = − u Tr (cid:2) ρ ( t ) ˆ H K ( µ ) (cid:3) , (B10)obtained by replacing ∂ t ρ ( t ) using the Lindblad equa-tion for the density matrix, and further manipulationsrelated to the particular structure of the Lindblad dissi-pator D j [ ρ ]. Moreover, since the Hamiltonian is indepen-dent of the time for t >
0, we also have q ( t ) = Tr (cid:20) dρ ( t ) dt ˆ H K ( µ ) (cid:21) = ddt Tr (cid:2) ρ ( t ) ˆ H K ( µ ) (cid:3) = dE s dt . (B11)Then using Eqs. (B10) and (B11), we obtain q ( t ) = − u Tr (cid:2) ρ (0) ˆ H K ( µ ) (cid:3) e − ut = − u (cid:104) µ i | ˆ H K ( µ ) | µ i (cid:105) e − ut . (B12)Note that (cid:104) µ i | ˆ H K ( µ ) | µ i (cid:105) = E µi + W , (B13)where E µi = (cid:104) µ i | ˆ H K ( µ i ) | µ i (cid:105) is the ground-state en-ergy at µ i , and W is the average work done at t = 0, cf.Eq. (B6). In particular for ¯ µ i = 0 (cid:104) µ c | ˆ H K ( µ ) | µ c (cid:105) = L [4 D ( L ) − µD ( L )] (B14)= L (cid:20) π − π − ¯ µ π − π + O ( L − ) (cid:21) . Equilibrium computations around the critical point givethe general structure [33] (cid:104) µ i | ˆ H K ( µ ) | µ i (cid:105) = Lg a (¯ µ i , ¯ µ ) + L − g s ( κ i , κ ) + O ( L − ) . (B15)Finally, we obtain Q ( t ) = (cid:90) t dt q ( t ) = (cid:104) µ i | ˆ H K ( µ ) | µ i (cid:105) (cid:0) e − ut − (cid:1) . (B16)The above results have been also carefully checked nu-merically, since very accurate results for large lattice sizes[ L = O (10 )] can be easily obtained exploiting the par-ticular structure of the Lindblad equations for the systemconsidered [6, 7].Let us finally address the scaling behavior at the criti-cal point of the above results. In particular, we note thatthe average work asymptotically behaves as W = L ¯ µ i f a (¯ µ i )+ L − ( κ i − κ ) f s ( κ i )+ O ( L − ) . (B17)Its structure does not apparently agree with the generalscaling behaviors put forward in Ref. [32]. Indeed, takinginto account that y n = 1 is the renormalization-groupdimension of the perturbation involved by the quench ofthe parameter µ , which is the density operator ˆ n x , onewould expect W ≈ L − z F s ( κ i , κ ) , (B18)with z = 1, which matches the subleading term in theexpansion (B17). Equation (B18) is supposed to be theasymptotic behavior keeping κ i and κ fixed. This ap-parent contradiction is explained by contributions aris-ing from short-ranged fluctuations like those giving riseto the analytic part of the free energy at the criticalpoint [33]. In other words, this is related to the mixingof the perturbation considered, i.e. the particle densityoperator (cid:80) x ˆ n x , with the identity operator, which leadsto the leading term in Eq. (B9). On the other hand,Ref. [32] considered perturbations not showing such prob-lems, such as the longitudinal spin operator in the quan-tum Ising chain. Generally, quenches associated withperturbations vanishing at the critical point, by symme-try, give rise to an average works satisfying the scalinglaws reported in Ref. [32].Analogous considerations apply to the behavior of theheat interchange around the critical point. Indeed, us-ing Eq. (B15), we may rewrite the expression of q , cf.Eq. (B12), in terms of dynamic scaling variables (B5),obtaining q ( t ) ≈ L G a ( t, µ i , µ, u ) + L − G s ( θ, κ i , κ, γ ) , (B19)where the scaling part is again the subleading term asso-ciated with the scaling function G s ( θ, κ i , κ, γ ) = − γe − γθ g s ( κ i , κ ) . (B20)However, we should observe that disentangling the O ( L − ) scaling part from the leading O ( L ) term turns3out to be a very hard task in practice, due also to thefact that the different variables of the two terms, withtheir different L dependence, makes such distinction of doubtful value. Therefore, we do not pursue this issuefurther. [1] A. A. Houck, H. E. T¨ureci, and J. Koch, On-chip quan-tum simulation with superconducting circuits, Nat. Phys. , 292 (2012).[2] M. M¨uller, S. Diehl, G. Pupillo, and P. Zoller, Engineeredopen systems and quantum simulations with atoms andions, Adv. At. Mol. Opt. Phys. , 1 (2012).[3] I. Carusotto and C. Ciuti, Quantum fluids of light, Rev.Mod. Phys. , 299 (2013).[4] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,Cavity optomechanics, Rev. Mod. Phys. , 1391 (2014).[5] S. Sachdev, Quantum Phase Transitions , (CambridgeUniversity, Cambridge, England, 1999).[6] D. Nigro, D. Rossini, and E. Vicari, Competing coher-ent and dissipative dynamics close to quantum criticality,Phys. Rev. A , 052108 (2019).[7] D. Rossini and E. Vicari, Scaling behavior of the station-ary states arising from dissipation at continuous quantumtransitions, Phys. Rev. B , 174303 (2019).[8] D. Rossini and E. Vicari, Dynamic Kibble-Zurek scalingframework for open dissipative many-body systems cross-ing quantum transitions, Phys. Rev. Research , 023211(2020).[9] M. Campostrini, J. Nespolo, A. Pelissetto, and E. Vi-cari, Finite-size scaling at first-order quantum transi-tions, Phys. Rev. Lett. , 070402 (2014).[10] A. Pelissetto, D. Rossini, and E. Vicari, Finite-size scal-ing at first-order quantum transitions when boundaryconditions favor one of the two phases, Phys. Rev. E ,032124 (2018).[11] A. Pelissetto, D. Rossini, and E. Vicari, Scaling proper-ties of the dynamics at first-order quantum transitionswhen boundary conditions favor one of the two phases,Phys. Rev. E , 012143 (2020).[12] V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider,T. Jungwirth, and A. H. MacDonald, First-order phasetransitions in a quantum Hall ferromagnet, Nature (Lon-don) , 638 (1999).[13] T. Vojta, D. Belitz, T. R. Kirkpatrick, and R. Narayanan,Quantum critical behavior of itinerant ferromagnets,Ann. Phys. (Leipzig) , 593 (1999).[14] M. Uhlarz, C. Pfleiderer, and S. M. Hayden, Quantumphase transitions in the itinerant ferromagnet ZrZn ,Phys. Rev. Lett. , 256404 (2004).[15] C. Pfleiderer, Why first order quantum phase transi-tions are interesting, J. Phys. Condens. Matter , S987(2005).[16] W. Knafo, S. Raymond, P. Lejay, and J. Flouquet, An-tiferromagnetic criticality at a heavy-fermion quantumphase transition, Nat. Phys. , 753 (2009).[17] T. J¨org, F. Krzakala, J. Kurchan, and A. C. Maggs, Sim-ple glass models and their quantum annealing, Phys. Rev.Lett. , 147204 (2008).[18] T. J¨org, F. Krzakala, G. Semerjian, and F. Zamponi,First-order transitions and the performance of quantumalgorithms in random optimization problems, Phys. Rev.Lett. , 207206 (2010). [19] A. P. Young, S. Knysh, and V. N. Smelyanskiy, First or-der phase transition in the quantum adiabatic algorithm,Phys. Rev. Lett. , 020502 (2010).[20] Y. Seki and H. Nishimori, Quantum annealing with an-tiferromagnetic fluctuations, Phys. Rev. E , 051112(2012).[21] P. Pfeuty, The one-dimensional Ising model with a trans-verse field, Ann. Phys. , 79 (1970).[22] V. Privman and M. E. Fisher, Finite-size effects at first-order transitions, J. Stat. Phys. , 385 (1983).[23] G. G. Cabrera and R. Jullien, Role of boundary condi-tions in the finite-size Ising model, Phys. Rev. B , 7062(1987).[24] H.-P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (Oxford University Press, New York,2002).[25] A. Rivas and S. F. Huelga,
Open Quantum System: AnIntroduction (Springer, New York, 2012).[26] L. M. Sieberer, M. Buchhold, and S. Diehl, Keldyshfield theory for driven open quantum systems, Rep. Prog.Phys. , 096001 (2016).[27] G. Lindblad, On the generators of quantum dynamicalsemigroups, Commun. Math. Phys. , 119 (1976).[28] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Completely positive dynamical semigroups of N-level sys-tems, J. Math. Phys. , 821 (1976).[29] S. Deffner and S. Campbell, Quantum Thermodynamics:An introduction to the thermodynamics of quantum in-formation , (Morgan and Claypool, San Rafael, 2019).[30] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G.Adesso,
Thermodynamics in the Quantum Regime. Fun-damental Theories of Physics , (Springer InternationalPublishing, Cham, 2018).[31] J. Gemmer, M. Michel, and G. Mahler,
Quantum Ther-modynamics: Emergence of Thermodynamic BehaviorWithin Composite Quantum Systems , (Springer Verlag,Berlin, 2004).[32] D. Nigro, D. Rossini, and E. Vicari, Scaling propertiesof work fluctuations after quenches near quantum tran-sitions, J. Stat. Mech. (2019) 023104.[33] M. Campostrini, A. Pelissetto, and E. Vicari, Finite-sizescaling at quantum transitions, Phys. Rev. B , 094516(2014).[34] D. Rossini and E. Vicari, Scaling of decoherence andenergy flow in interacting quantum spin systems, Phys.Rev. A , 052113 (2019).[35] A. Pelissetto, D. Rossini, and E. Vicari, Dynamic finite-size scaling after a quench at quantum transitions, Phys.Rev. E , 052148 (2018).[36] A. Yu. Kitaev, Unpaired Majorana fermions in quantumwires, Phys. Usp. , 131 (2001).[37] S. Katsura, Statistical mechanics of the anisotropic linearHeisenberg model, Phys. Rev.127