Dynamic Kibble-Zurek scaling framework for open dissipative many-body systems crossing quantum transitions

We study the quantum dynamics of many-body systems, in the presence of dissipation due to the interaction with the environment, under Kibble-Zurek (KZ) protocols in which one Hamiltonian parameter is slowly, and linearly in time, driven across the critical value of a zero-temperature quantum transition. In particular we address whether, and under which conditions, open quantum systems can develop a universal dynamic scaling regime similar to that emerging in closed systems. We focus on a class of dissipative mechanisms whose dynamics can be reliably described through a Lindblad master equation governing the time evolution of the system's density matrix. We argue that a dynamic scaling limit exists even in the presence of dissipation, whose main features are controlled by the universality class of the quantum transition. This requires a particular tuning of the dissipative interactions, whose decay rate $u$ should scale as $u\sim t_s^{-\kappa}$ with increasing the time scale $t_s$ of the KZ protocol, where the exponent $\kappa = z/(y_\mu+z)$ depends on the dynamic exponent $z$ and the renormalization-group dimension $y_\mu$ of the driving Hamiltonian parameter. Our dynamic scaling arguments are supported by numerical results for KZ protocols applied to a one-dimensional fermionic wire undergoing a quantum transition in the same universality class of the quantum Ising chain, in the presence of dissipative mechanisms which include local pumping, decay, and dephasing.


I. INTRODUCTION
The recent experimental progress in the control and manipulation of quantum many-body systems has led to great achievements, opening the door for the realization of quantum simulators [1][2][3][4][5][6]. However, the effective isolation of a quantum system remains a challenge, since interactions with the environment can have a significant impact in the dynamics, even when they interact weakly. It is thus important to understand the effects of dissipative interactions, irrespectively of their strength. This issue is of particular relevance for systems at quantum transitions [7], where the above mentioned effects are generally relevant, thus tending to suppress the critical quantum correlations [8,9].
Slow passages through quantum transitions allow us to probe some universal features of quantum fluctuations in such circumstances. In this respect, we mention the Kibble-Zurek (KZ) problem [10][11][12][13][14], related to the amount of final defects, after slow (quasi adiabatic) passages through continuous quantum transitions, from the disorder phase to the order phase. Its scaling predictions have been confirmed by experiments for various physically interesting systems, see e.g. Refs. [15][16][17][18][19][20]. KZ-like protocols have been largely employed to investigate the critical dynamics of closed systems, subject to unitary time evolutions only [21,22]. The open nature quantum systems, however, may lead to a departure from the dynamic scaling behavior predicted for the isolated case [23][24][25][26][27][28][29][30][31][32][33][34]. In particular, it has been observed that slower quenches in open systems, or subject to noisy controls, may generate an overabundance of defects when approaching the adiabatic limit in KZ protocols, sometimes named anti-KZ behavior [35].
Since dissipative mechanisms are expected to give rise to relevant perturbations at the quantum criticality of closed systems [8,9,36] (such as the temperature), they do not generally preserve the universal dynamic properties of quantum transitions. From this point of view, the above-mentioned anti-KZ behavior should not be considered as unexpected. Indeed, due to the general relevance of the perturbations associated with dissipative mechanisms, slower protocols favor the dissipation effects, in that they give them more time to act. Therefore, unlike closed systems, the dynamic behaviors arising from slow changes of the Hamiltonian parameters, across their critical values, do not anymore develop universal critical features controlled by the quantum transition of the closed system. Only an appropriate tuning of the dissipation strength may give rise to a nontrivial interplay with the critical unitary dynamics, developing a dynamic scaling behavior in KZ protocols controlled by the universality class of the quantum transition.
The issue we address in this paper is whether, and under which conditions, open dissipative systems may still present a universal regime controlled by the universality class of the quantum transition of the closed system. We focus on a class of dissipative mechanisms whose dynamics can be reliably described through a Lindblad master equation [37,38] governing the time evolution of the density matrix of the system [39][40][41]. We argue that, in the presence of weak dissipation, the dynamics of many-body systems may still develop a scaling behavior under KZ protocols (i.e., slow changes of one Hamiltonian parameter across its critical value), thus extending the dynamic KZ scaling of closed systems [14]. Its main features, in the presence of weak dissipation, are still controlled by the universality class of the quantum transition, provided the system-environment interaction strength is suitably tuned. This allows us to define a dynamic KZ scaling limit in the presence of dissipation. In particular, we argue that the decay rate u of the dissipative interactions must scale as a power law u ∼ t −κ s with increasing the time scale t s of the slow variations of the KZ protocol, where κ < 1 is an appropriate positive exponent, depending on the universal critical exponents of the quantum transition. The suppression of the dissipation rate is necessary to observe universal dynamic scaling and it is analogous to that found in other dynamic problems with dissipation at quantum transitions [8,9].
To check our general framework, we present a numerical analysis of KZ protocols applied to the fermionic Kitaev wire [42] across its quantum transition (belonging to the same universality class of the one appearing in quantum Ising chains) in the presence of dissipative mechanisms including local pumping, decay, and dephasing. This model can be exactly and fully solved (i.e. with respect to its full excitation spectrum) even with a large number of sites, up to a few thousands, thus enabling up to perform an accurate numerical investigation of the dynamic KZ scaling behavior put forward. Our results nicely confirm the emerging of a dynamic scaling in the limit of slow passages across the quantum transition and in the presence of weak dissipation.
The paper is organized as follows. In Sec. II we describe our dynamic KZ protocol and discuss the Lindblad modelization of a dissipative system-environment interaction. In Sec. III we summarize the main features of the dynamic KZ scaling limit, and the emerging scaling laws in the limit of large time scale of the KZ protocol. In Sec. IV we extend the dynamic KZ scaling laws to allow for the presence of dissipation, thus achieving a unique framework to discuss the interplay between (critical) coherent and dissipative drivings. Subsequently we introduce the open-system Kitaev quantum wire, which represents our theoretical laboratory to check the phenomenological dynamic KZ scaling we put forward (Sec. V), and present extensive numerical analyses of KZ protocols for that model at its zero-temperature quantum transition in the presence of dissipation (Sec. VI). Finally, Sec. VII contains a brief summary and some concluding remarks.

II. DYNAMIC KZ PROTOCOL IN THE PRESENCE OF DISSIPATION
We consider a many-body system presenting a quantum transition driven by the Hamiltonian parameters. For simplicity, we assume that the HamiltonianĤ depends on a single relevant parameter µ, whose variation drives a quantum transition separating two different quantum phases. The deviation quantifies the distance from the critical point, located at µ = µ c . We also suppose that negative valuesμ < 0 correspond to the gapped quantum disordered phase. Quasi-adiabatic passages through the quantum transition, slowly varying µ across µ = µ c , give rise to peculiar out-of-equilibrium phenomena, such as the one related to the so-called KZ problem [11-13, 21, 22] addressing the formation of defects when passing through quantum critical points, from the gapped disordered phase to the ordered phase. A standard KZ protocol would proceed as follows: i) One starts from the ground state of the many-body system atμ i < 0, or alternatively from a statistical state described by the Gibbs distribution ∝ e −Ĥ(μi)/k B T at small temperature T ; (ii) Then the quantum dynamics is driven by slow variations of the relevant parameterμ associated with the quantum transition, for example linearly asμ up to a valueμ f > 0. The parameter t s > 0 denotes the time scale of the slow variations of the Hamiltonian parameterμ. The time evolution is unitary, i.e.
where ρ(t) is the density matrix of the many-body system. Even in the limit of very slow changes, corresponding to t s → ∞, infinite-volume systems cannot satisfy the adiabatic dynamic condition when passing through the transition point, thus developing out-of-equilibrium behaviors. The resulting evolution of the system is usually investigated by monitoring observables obtained by taking expectation values at fixed time. For example, in the case of lattice spin models, one may consider the magnetization, the two-point function of local operators related to the order parameter, etc . . . Here we want to study the effects of weak dissipative mechanisms on the slow dynamics across the quantum transition. Therefore, beside the changes of the Hamiltonian parameters, we suppose that the many-body system is also subject to some interaction with the environment. The time dependence of its density matrix ρ can be reasonably described by the Lindblad master equation [39] where the first term in the right-hand side provides the coherent driving, while the second term accounts for the coupling to the environment, characterized by a global coupling constant u > 0. We restrict to homogeneous dissipation mechanisms, preserving translational invariance. In the case of systems weakly coupled to Markovian baths, the tracepreserving superoperator can be written as a sum of local terms, such as [37,38] whereL o is the Lindblad jump operator associated with the local system-bath coupling scheme, and o denotes an appropriate spatial coordinate. In quantum optical implementations, the conditions leading to Eqs. (4)-(6) are typically satisfied [41], therefore this formalism constitutes the standard choice for theoretical investigations of this kind of systems. In the following we analyze the dynamic scaling behavior arising from dynamic protocols of quantum manybody systems in the presence of weak dissipation, thus evolving according to Eq. (4), when the parameter µ is slowly varied across its critical value µ c associated with the quantum transition driven by the Hamiltonian, starting from the gapped disordered phase, analogously to the standard KZ protocol for closed systems.

III. DYNAMIC KZ SCALING FOR CLOSED QUANTUM SYSTEMS
Before discussing the effects of dissipation, we recall the main features of the dynamic scaling behavior developed by many-body systems unitarily evolving at quantum transitions [14,[43][44][45], and in particular when they are slowly driven across its quantum transition, according to the KZ protocol described in Sec. II, cf. Eqs. (2)-(3).

A. Homogeneous scaling laws
At the critical point, the low-energy unitary Hamiltonian dynamics develops long-distance correlations, characterized by a diverging length scale ξ ∼ |μ| −ν [where ν = 1/y µ and y µ is the renormalization-group (RG) dimension of the relevant parameter] and the suppression of the gap (energy difference between the lowest states) ∆ ∼ ξ −z . The correlation-length exponent ν and the dynamic exponent z are the critical exponents associated with the universality class of the quantum transition. The dynamics at continuous quantum transitions develop homogeneous scaling laws [12,14,[43][44][45][46][47][48][49][50][51], even in the presence of interactions with an environment [8,9,36,52].
For example, in the case of instantaneous quenches of closed systems, arising from the instantaneous variation of the Hamiltonian parameter fromμ i toμ, starting from the ground state atμ i , the evolution of a generic observable B, such as the expectation value of a local operator B (assuming translation invariance), satisfies the homogeneous scaling relation [44] Here |Ψ(t) indicates the quantum many-body state after the quench, b is an arbitrary positive parameter, y B is the RG dimension of the operatorB, L is the size of the system, and B is a universal scaling function apart from normalizations. Eq. (7) is expected to provide the asymptotic power-law behavior in the large-b limit.
The KZ protocol focuses on the opposite quasiadiabatic regime, where the driving parameterμ is slowly varied across the quantum transition, starting from the ground state at a givenμ i < 0 and then changingμ linearly in time, as in Eq. (2) (thus the initial conditionμ i corresponds to the initial time t i = t sμi ). A phenomenological scaling theory is obtained by assuming the homogeneous scaling law where, again, b is an arbitrary positive parameter. Analogous scaling equations can be written down for the fixedtime correlations G AB of two local operatorsÂ andB at a distance x. Assuming translation invariance, where ϕ = y A + y B and y A , y B are the RG dimensions of the operatorsÂ andB, respectively.
The dynamic KZ scaling framework can be extended to situations where the initial condition is given by a Gibbs ensemble at temperature T , by adding a further dependence on the product T b z in the KZ scaling functions of Eqs. (8a) and (8b).

B. Dynamic scaling in the infinite-volume limit
We now concentrate on KZ protocols. To derive a dynamic scaling theory for infinite-volume systems, it is possible to exploit the arbitrariness of the scale parameter b in the general homogeneous power laws (8a) and (8b). To this purpose we set where λ is the length scale associated with the KZ protocol, and take the limit L/λ → ∞ (corresponding to taking the so-called thermodynamic limit). This leads to the dynamic KZ scaling ansatz where τ is the rescaled time: The dynamic KZ scaling limit, where the above asymptotic behaviors apply, is obtained by taking t s → ∞ keeping the arguments of the dynamic scaling functions B i and G i fixed. Actually, introducing a time scaling variable related to initial time of the KZ protocol, we may rewrite the scaling Eqs. (10a) and (10b) as Note that the scaling functions B i , G i and B i , G i in Eqs. (10a)-(10b) and (13a)-(13b) do not coincide, but are trivially related by the change of scaling variables.
Since the KZ protocol starts fromμ i < 0 corresponding to the gapped phase, whose gap decreases as ∆ ∼ ξ −z and the ground-state length scale ξ diverges only at the critical pointμ = 0, the emerging dynamic KZ scaling should be independent of the actual finite value ofμ i < 0, if this is kept fixed in the dynamic KZ scaling limit. This is essentially due to the fact that, in a gapped phase, the evolution arising from slow changes of the parameters is essentially adiabatic, fromμ i to the relevant scaling interval δ µ aroundμ = 0, which effectively decreases as in the dynamic KZ scaling limit. Therefore, when increasing t s , keeping µ i < 0 constant and finite, the dynamic KZ scaling must be independent ofμ i , corresponding to the τ i → −∞ limit of the relations (13a) and (13b). Therefore this leads to the dynamic scaling ansatz The dynamic scaling functions introduced above are expected to be universal with respect to changes of the microscopic details of the Hamiltonian within the given universality class. Of course, like any scaling function at quantum transitions, such a universality holds apart from a multiplicative overall constant and normalizations of the scaling variables. The approach to the asymptotic dynamic scaling behavior is expected to be generally characterized by power-law suppressed corrections.
We finally mention that the so-called KZ problem genuinely addresses the formation of defects when slowly crossing the quantum transition, from the disordered to the ordered phase. The above scaling arguments in the dynamic KZ limit (see Refs. [11,14,22]) lead to the expectation that the density of defects arising after crossing the transition scales as the inverse scaling volume λ −d , cf. Eq. (9), that is This scaling behavior has been verified in experiments, see, e.g., Refs. [15-18, 20, 35, 53, 54].

C. Dynamic finite-size scaling
The scaling Eqs. (8a) and (8b) also allow us to derive dynamic finite-size scaling (FSS) relations, which are valid far from the thermodynamic limit, and which extend those predicted by the FSS theory for systems at equilibrium [55][56][57]. For example, by setting b = L in Eq. (8b), we obtain (17) This dynamic FSS behavior is expected to be obtained by taking L → ∞, while keeping the arguments of the scaling function G L fixed. One may introduce more convenient scaling variables, which are combinations of those entering Eq. (17). For example, one can write it as where and τ, τ i are defined in Eqs. (11) and (12), respectively. Assuming again that the KZ protocol starts from the gapped disordered phase and the initialμ i < 0 is kept fixed in the dynamic scaling limit, the same dynamic FSS is expected to hold, irrespective of the value ofμ i . Thus, the dynamic FSS in Eq. (18) simplifies into Indeed, with increasing L, the dynamic FSS occurs within a smaller and smaller interval of values of |µ| aroundμ = 0: since the time interval of the dynamic process scales as t sca ∼ t κ s , the relevant interval of values of |μ| shrinks as t sca /t s ∼ L −yµ . Note that, in the limit υ → ∞, the evolution as a function ofμ(t) = t/t s corresponds to an adiabatic dynamics. Indeed, since the finite size L guarantees the presence of a gap between the lowest states, one may adiabatically cross the critical point in the limit υ → ∞, passing through the ground states of the finite-size system forμ(t). The adiabatic evolution across the transition point is prevented only when L → ∞ (before the limit t s → ∞), i.e., when the time scale of the critical correlations diverges, since τ cr ∼ ∆ −1 ∼ L z .

IV. DYNAMIC KZ SCALING FOR OPEN QUANTUM SYSTEMS
A. Dynamic scaling allowing for dissipation In this section, we extend the dynamic scaling theory outlined in Sec. III to systems subject to dissipative interactions with the environment, so that the time dependence of the density matrix ρ is described by the Lindblad master equation (4). Namely, we assume that the quantum evolution arising from the KZ protocol occurs in the presence of dissipation with the effective coupling u > 0, thus being ruled by Eq. (4). The resulting dynamic KZ scaling framework will provide a unique framework to discuss the interplay between (critical) coherent and dissipative drivings.
The dynamic behavior in the presence of weak dissipation has been addressed within a phenomenological dynamic scaling theory in Refs. [8,9], extending the dynamic scaling scenario holding for closed systems. This has been obtained by adding a further dependence associated with the dissipation parameter u in the dynamic scaling relations (8a) and (8b), through a power law ub z , where the dynamic exponent z ensures the substantial balance (i.e., competition) with the critical coherent driving. We recall that this hypothesis has been put forward after noting that the parameter u of the dissipator in Eq. (4) plays the role of a decay rate, i.e., of an inverse relaxation time, of the associated dissipative process [39]. Thus, to observe a nontrivial competition between critical coherent dynamics and dissipation, the dissipative coupling must be comparable to the gap of the critical Hamiltonian, therefore its scaling variable must be controlled by the dynamic exponent z.
Following the above reasoning, we conjecture that KZ protocols in the presence of dissipation develop homogeneous laws, such as similar to those in Eqs. (8a) and (8b), but with one additional scaling variable associated with u.

B. Dynamic scaling in the infinite-volume limit
Analogously to the dynamics of closed systems, it is possible to derive scaling laws in the thermodynamic limit, by fixing b as in Eq. (9) and taking L/λ → ∞. One can easily show that Eqs. (21a) and (21b) imply the dynamic KZ scaling ansatz where we introduced the scaling variable γ associated with the dissipation parameter, The above scaling laws are expected to provide the asymptotic behavior in the t s → ∞ limit while keeping the scaling variables fixed, including γ. Note that, like for closed systems, the large-t s limit of KZ protocols starting from finite and fixedμ i < 0 should correspond to the limit τ i → −∞ in the right-hand side of Eqs. (22a) and (22b). Indeed, the dissipation with coupling strength u ∼ λ −z is not expected to play any relevant role at finiteμ i < 0, where the gap is ∆ = O(1), while it should compete with the unitary evolution only very close toμ = 0 where u ∼ ∆ ∼ λ −z . Therefore, under such conditions we expect the scaling behavior We mention that, in the above KZ scaling limit allowing for dissipation, the scaling law associated with the number of defects, cf. Eq. (16), should be replaced with where the dependence on the dissipative coupling u enters the scaling function D through the scaling variable γ. Of course, one must recover the scaling law (16) for γ = 0.

C. Dynamic finite-size scaling
The dynamic FSS behavior can be obtained by setting b = L in Eqs. (21a) and (21b), thus extending the results contained in Sec. III C to allow for the dissipation term of the Lindblad equation. Namely, The above scaling law can be obtained in the L → ∞ limit while keeping the scaling variables fixed, including γ L . Moreover, assuming again that the quantum phase for µ < 0 is gapped, with ∆ ∼ ξ −z and the ground-state length scale ξ diverges only at the critical pointμ = 0, KZ protocols associated with any finite initialμ i < 0 develop the same dynamic FSS independently of their actual values. Thus, the dynamic FSS can be written as Indeed, similarly to the infinite-volume case, the dissipation with coupling strength u ∼ L −z is not expected to play any relevant role at finiteμ i < 0, where the gap is ∆ = O(1), while it should compete with the unitary evolution only very close to the critical point where u ∼ ∆ ∼ L −z . Like for closed systems, the dynamic scaling limit thus involves smaller and smaller intervals of values of |μ| aroundμ = 0 with increasing L: since the time interval of the dynamic process scales as t sca ∼ t κ s , the relevant interval of values of |µ| shrinks as t sca /t s ∼ L −yµ .

V. KZ PROTOCOLS FOR THE KITAEV QUANTUM WIRE SUBJECT TO DISSIPATION
To verify the dynamic KZ scaling laws put forward in the previous sections, and in particular in Sec. IV, we consider a Kitaev quantum wire defined by the Hamiltonian [42] whereĉ j is the fermionic annihilation operator on the jth site of the chain,n j ≡ĉ † jĉ j is the density operator, and δ > 0. We set = 1, and t = 1 as the energy scale. Moreover we fix δ = 1. We consider antiperiodic boundary conditions,ĉ L+1 = −ĉ 1 , and even L for computational convenience.
The fermionic system described by the Hamiltonian (29) undergoes a continuous quantum transition at µ = µ c = −2, independently of δ, belonging to the same universality class of that of the quantum Ising chain (when δ > 0), i.e. the two-dimensional Ising universality class [7,42], characterized by the length-scale critical exponent ν = 1, related to the RG dimension y µ = 1/ν = 1 of the Hamiltonian parameter µ (more precisely of the differenceμ ≡ µ−µ c ). The dynamic exponent associated with the unitary quantum dynamics is z = 1. Moreover, the RG dimension of the fermionic operatorsĉ j andĉ † j is yĉ = yĉ † = 1/2, and that of the density operatorn j is yn = 1 [7]. Details on the correspondence with the quantum Ising chain are provided in App. A.
We focus on the dynamic behavior of the Fermi lattice gas (29) close to its quantum transition, in the presence of homogeneous dissipation mechanisms following the Lindblad equation (4). The dissipator D[ρ] is defined as a sum of local (single-site) terms of the form whereL j denotes the Lindblad jump operator associated with the system-bath coupling scheme, and the index j corresponds to a lattice site [thus replacing the index o in Eqs. (5), (6)]. The onsite Lindblad operatorsL j describe the coupling of each site with an independent bath. We consider dissipation mechanisms associated with either particle losses (l), pumping (p), or dephasing (d), respectively [8,29,[58][59][60][61][62]: The choice of such dissipators turns out to be particularly convenient for the numerical analysis, allowing us to scale the difficulty of the problem linearly with L and thus to obtain results for the Kitaev wire with thousands of sites (see Sec. VI). This is important, in view of the necessity to perform adequate numerical checks of a new scaling theory lying on phenomenological grounds.
The KZ protocol that we consider starts from the ground state ofĤ K for a genericμ i < 0, where the system is gapped, ∆ = |μ| + O(L −2 ), while ∆ ∼ L −1 at µ = 0 (see App. A). Then the system evolves according to Eq. (4) with a time dependent parameterμ(t) = t/t s , starting from t i < 0 such thatμ i = t i /t s . To characterize the dynamic properties of the evolution described by the Lindblad equation, and in particular the corresponding asymptotic large-time behavior, we consider the fixedtime correlations The dynamic KZ scaling of the above correlation functions is expected to be given by the general scaling laws reported for the generic two-point function G AB in Sec. IV, taking into account that ϕ = 1 for the correlations P and C (since yĉ = yĉ † = 1/2), while ϕ = 2 for G (since yn = 1). This scaling scenario should hold for all the considered dissipation mechanisms, cf. Eq. (31). Of course, the corresponding scaling functions are expected to differ.

VI. NUMERICAL RESULTS
We now present the results of a series of numerical computations we have performed on the Kitaev quantum wire. As stated above, this model is amenable to a direct solvability for systems with O(10 3 ) sites, thus representing the ideal playground for open quantum lattice problems, given the remarkable difficulty to simulate the dynamics of interacting many-body quantum systems coupled to an external bath.
For the specific choice of dissipators in Eq. (31), the exponential complexity of the Kitaev chain can be semianalytically reduced to a polynomial one [8,29,58,63]. In particular, in the presence of particle losses (l) or pumping (p) and for translationally invariant systems, the driven-dissipative quantum dynamics ruled by the master equation (4) can be exactly solved by decoupling in Fourier space the various sectors with different momenta, analogously to fermionic Gaussian Hamiltonian models. Similar strategies can be adopted for more general inhomogeneous (disordered) lossy dynamics, provided the Liouvillian operator remains quadratic in the creation and annihilation operators for fermions. On the other hand, although the quantum dynamics with a dephasing (d) mechanism cannot be simply obtained, two-point observables are still fully captured by a set of coupled linear differential equations, whose number increases linearly with the number of sites L (see, e.g., the appendix in Ref. [8] for details). The latter can be integrated, e.g., with a standard fourth-order Runge-Kutta method.

A. Dynamic KZ scaling in the infinite-volume limit
We first discuss systems in the thermodynamic limit. To ensure that finite-size corrections are negligible on the scale of all the numerics presented below for the dynamic scaling, we have carefully checked that (in all cases treated herewith) systems of size L = 2 12 = 4096 allow to simulate KZ protocols with a length scale λ up to O(10 2 ).

Unitary KZ dynamics
Before discussing the effects of dissipation, it is instructive to present the outcomes of a typical KZ protocol for the unitary dynamics of the Kitaev quantum wire, without dissipation (u = 0). Figure 1 shows the time behavior of the fixed-time correlation P (x, t) at fixed x/λ [see Eq. (32a)] during a KZ protocol starting from a fixed rescaled initial time τ i < 0 (in the figure, τ i = −10) and running up to positive values of the rescaled time. In this way, the Hamiltonian parameterμ ≡ µ − µ c ofĤ K is slowly changed in time starting from an initial valuē µ i = τ i /λ < 0, whose absolute value decreases with the KZ length scale λ, through the critical pointμ = 0 at t = τ = 0. The various parameters have been rescaled according to Eq. (13b): we set ϕ = 1, κ = 1/2, and plot λ P (x, t) as a function of τ , for increasing values of λ, while keeping the scaling variables x/λ and τ i constant. In Fig. 1 we show results for the rescaled distance x/λ = 1 . Data belonging to one of the two color sets correspond to a given initial Hamiltonian parameterμi < 0, which is kept fixed and equal to eitherμi = −0.1 (black circles) or µi = −0.5 (red squares). The insets in the three panels display rescaled correlations as a function of 1/λ, for both cases ofμi presented in the main frames, at the τ value indicated by the blue arrow.
only; other values of x/λ present analogous behaviors.
Already for values of λ ∼ 10, the curves approach a non trivial scaling behavior, in accordance with the general KZ scaling theory for closed systems (see also the zoom in the upper right inset). The oscillating behavior for τ > 0 is likely due to adiabaticity losses, which are ascribable to the gapless point at τ = 0. The approach to the asymptotic behavior in the limit λ → ∞ is analyzed in the lower inset, for fixed τ = 4.8, where we collected data up to λ = 40. As expected, corrections are suppressed with a power-law behavior that is compatible with O(1/λ). As stated in the previous sections, the dynamic KZ scaling is also expected to be independent of the actual value ofμ i , if this is kept fixed in the dynamic KZ limit. A numerical verification of this conjecture is presented in Fig. 2, for a situation similar to that in Fig. 1 but fixingμ i < 0, rather than τ i < 0. Specifically, we have analyzed the three correlation functions P (x, t), C(x, t), and G(x, t) [see Eqs. (32a)-(32c)] along a KZ protocol where we fixed the initial conditionμ i . Even in this case we can see that, after properly rescaling the various parameters and observables, the curves nicely approach a scaling behavior, which appears to be independent of the choice ofμ i . As previously discussed, the critical point located atμ = 0 prevents the system from remaining in the instantaneous ground state.
A more accurate analysis of the independence of the dynamic scaling functions from the initial conditionμ i is provided in the three insets (each for a different correlation function), where we spotlight the convergence of the rescaled observables with λ → ∞, for a fixed value of τ . In all the three cases we observe that the extrapolated asymptotic value seems to be independent of the two specificμ i analyzed. Notice however that, while for the red data sets (μ i = −0.5) the convergence to the asymptotic behavior appears regular and compatible with a power law ∼ λ −1 , the black data sets (μ i = −0.1) [especially for C(x, t) and G(x, t)] exhibit oscillations in 1/λ (at least up to λ = 40), which should be ascribed to the proximity of the initial ground state atμ i with that at the critical pointμ = 0. We have verified that the above observations hold also for other values of τ and for different initial conditionsμ i (not shown), with a faster convergence for larger values of |μ i |.

Dissipative KZ dynamics
We now turn to a situation where the Hamiltonian Kitaev chainĤ K is coupled to a Markovian bath in the form of either incoherent particle losses, pumping, or dephasing [the three different types of Lindblad operators are reported in Eq. (31) and are supposed to act uniformly over all the sites of the chain]. According to the dynamic KZ scaling framework discussed in Sec. IV, an additional scaling variable γ associated with the dissipation strength u needs to be considered, see Eq. (23). In passing we note that the dissipation parameter u = γ/λ entering the master equation (4) is inversely proportional to λ, therefore it needs to be progressively decreased down to zero when increasing the KZ length scale λ. For the two-point correlations analyzed here, one thus expects the emerging scaling behavior (22b).
We proceed as in the previous subsection VI A 1 in the absence of dissipation, and first address KZ protocols where the Hamiltonian parameterμ is slowly increased and driven across a critical point (μ = 0), starting from an initial valueμ i < 0 such that the corresponding value of the rescaled time τ i is kept fixed. Results for the correlations P (x, t) and C(x, t), in the presence of either decay, pumping, or dephasing, are shown in Fig. 3 [analogous outcomes have been obtained for G(x, t)-not shown]. Again, we present results for the particular rescaled distance x/λ = 1 and rescaled initial time τ i = −10; analogous scaling behaviors are observed for other values of x/λ and τ i , but, of course, with different asymptotic scaling functions of τ .
The upper and central panels evidence that, after a proper rescaling of the various variables, and in particular fixing the rescaled dissipation rate γ as in Eq. (23), the two observables nicely approach a scaling function with increasing λ. Of course, the latter function depends both on the type of dissipation and on γ. In particular, numerical data show that the dephasing mechanism appears to be more effective in destroying this type of correlations: with increasing γ, the various curves rapidly decay to a very small asymptotic value for λ large (i.e., green curves for γ = 10 in the right panels are hardly distinguishable from zero). On the other hand, for the incoherent decay or pumping, definitely larger values of γ are required to suppress correlations.
The convergence to the asymptotic behavior is analyzed more in depth in the lower panels for a fixed τ , where we explicitly show the dependence of the correlation functions on 1/λ, up to λ = 100. Our data hint at the presence of 1/λ power-law corrections, similarly to what has been observed for the unitary case (compare with Fig. 1).
Even in the presence of dissipation, the dynamic KZ scaling should not depend on the choice of the initialμ i , if this is kept fixed, and thus one expects the scaling behavior reported in Eqs. (24a)-(24b). This has been verified numerically by fixingμ i < 0, as shown in Fig. 4 for P (x, t) and C(x, t), and in Fig. 5 for G(x, t). The displayed data are for a specific value of γ = 0.5 and for two different values ofμ i = −0.1 (black) and −0.5 (red). The various curves in the upper and central panels stand for different values of λ. In all cases we observe that they approach the same asymptotic behavior, irrespective of the choice ofμ i ; the approach becomes faster, when further increasing |μ i | (not shown). Note that, for the incoherent decay, this appears to be much faster than for the other kinds of dissipation (especially for KZ protocols withμ i = −0.1, starting close to the critical pointμ = 0). In contrast, as hinted when commenting the upper and middle right panels of Fig. 3 for fixed τ i and different γ, dephasing seems to be the most disruptive dissipation mechanism: the red curves in the upper and middle right panels of Fig. 4 (where |τ i | is much larger than for the black ones) are rapidly suppressed with λ.
The bottom panels of Figs. 4 and 5, show the convergence to the asymptotic behavior with λ and for a given  Fig. 4, but for the rescaled correlation λ 2 G(x, t). We have computed it the presence of either incoherent decay or pumping, since with dephasing we were only able to compute two-point observables (while densitydensity correlations are a four-point observable) [8].
τ , which is again expected to be power law. Note that the speed with λ at which data for the twoμ i converge to the same value depends on the type of dissipation and observable. In general, we observe that for decay the convergence is much faster than in the other cases. Moreover, in the limit λ → ∞, while with either decay or pumping the correlators go toward a non-zero value, with dephasing the scaling functions (for τ sufficiently larger than zero) are compatible with zero.

B. Dynamic KZ finite-size scaling
We now switch to systems with finite size, and utilize the FSS framework of Sec. IV C to analyze the behavior of the dissipative Kitaev wire undergoing a KZ protocol which crosses the quantum transition point. Results for the three fixed-time correlation functions P (x, t), C(x, t), and G(x, t) are reported in Fig. 6, where we analyze their temporal behavior along a KZ protocol associated with a slow variation of the Hamiltonian parameterμ from negative to positive values, in the presence of incoherent particle losses. Following the FSS scaling behavior of Eq. (26), we kept fixed the ratio x/L, the parameter υ inversely proportional to the speed of the driving [cf. Eq. (19)], the rescaled dissipation strength γ L [cf. Eq. (27)], and the initial rescaled time τ i < 0. Note that, in the FSS framework, the dissipation strength u = γ L /L entering the master equation (4) is inversely proportional to the system size, thus scaling down to zero in the limit L → ∞.
In all cases, the rescaled correlations nicely approach a scaling function with increasing L, as predicted by the scaling law in Eq. (26). The finite-size approach to the asymptotic behavior is compatible with a L −1 behavior, as highlighted in the inset of the lower panel of Fig. 6 at fixed τ . We also observe that, for small values of τ , the scaling curves develop complex nonanalytic spikes in τ , whose magnitude and frequency increase with L, similarly to other dynamic situations as after sudden quenches [8,9]; for larger τ dissipation tends to smear those apparent singularities.
To shed light on the effects of the system-bath coupling, in Fig. 7 we have analyzed the correlation C(x, t) for the three different types of dissipation of Eq. (31), and for varying rescaled strength γ L as indicated in the legends. We have also reported the KZ behavior in the unitary case (black curves), to be compared with that in the presence of an environmental interaction (colored curves). Besides the nice convergence to a scaling function for L → ∞, we observe that dephasing appears to be more effective in destroying correlations, since, with increasing γ L , the curves rapidly approach the zero value in time; in contrast, for both incoherent decay and incoherent pumping, definitely larger values of γ L are required to suppress correlations. Moreover, in the presence of pumping, even a tiny amount of dissipation is capable to drive the system far from the equilibrium state in the τ < 0 side (see also the discussion in Sec. VI A 2). We have also numerically verified that, analogously to the dynamic scaling behavior in the infinite-volume limit, the dynamic FSS functions do not depend on the initial Hamiltonian parameterμ i < 0 if this is kept fixed in the dynamic KZ limit [see Eq. (28)]. Figure 8 reports the behavior of the rescaled correlation L P (x, t) as a function of τ , for three different choices ofμ i , and for dissipation provided by incoherent decay at fixed γ L . The various upper panels show convergence with L to a scaling function, which appears to be the same: bottom panels unveil how tiny discrepancies in the temporal behavior, starting from differentμ i , can be suppressed in the large-L limit.

VII. SUMMARY AND CONCLUSIONS
We have investigated the interplay between coherent and dissipative drivings in the dynamics of quantum may-body systems subject to KZ protocols across continuous quantum transitions, starting from the gapped disordered phase, that is when one Hamiltonian parameter is slowly driven across its critical value, for example with a linear dependence on time. Specifically, the issue we have addressed here is whether, and under which con- ditions, open dissipative systems can develop a universal dynamic scaling regime similar to that shown by closed systems at quantum transitions, characterized by asymptotic homogeneous scaling laws. To this purpose we have focused on a class of dissipative mechanisms, whose dynamics can be reliably described through a Lindblad master equation governing the time evolution of the density matrix of the open system. The perturbation arising from the dissipation turns out to be relevant at the quantum transition [8,9]. This implies that open systems cannot develop asymptotic dynamic scaling behaviors controlled by the universality class of the quantum transition when keeping the dissipation decay rate u finite and fixed in the critical limit of the Hamiltonian parameter. This is analogous to the effect of any relevant RG perturbation close to a critical point [7], like the temperature, which makes the system run away from criticality. Nevertheless, we argue that a dynamic KZ scaling limit exists in the presence of a sufficiently weak dissipation. Such a scaling limit, controlled by the universality class of the quantum transition, arises in a regime of weak dissipation. Indeed, it requires a tuning of the dissipative interactions, and in particular of the decay-rate parameter u of the Lindblad master equation describing the evolution of the density matrix, cf. Eq. (4). The decay rate u must decrease as u ∼ t −κ s when increasing the time scale t s of the KZ protocol, where the positive exponent κ = z/(y µ + z) < 1 depends on the dynamic exponent z and the RG dimen-sion y µ of the driving Hamiltonian parameter (usually related to the correlation-length exponent exponent ν by ν = y −1 µ ). The resulting dynamic KZ scaling laws, allowing for the presence of dissipation, provide a unique framework to discuss te interplay between (critical) coherent and dissipative drivings.
The dynamic KZ scaling scenario has been checked within fermionic wires, cf. Eq. (29), in the presence of homogeneous dissipation due to local incoherent pumping, decay and dephasing, which are described by the Lindblad operators reported in Eq. (31). The particularly convenient choice of this model enables to scale its complexity linearly with its size, allowing to simulate the exact dissipative dynamics of systems with thousands of sites; we have thus elected it as a testbed for accurate numerical investigations of the many-body Lindblad master equation. Our numerical analysis ultimately supports the phenomenological dynamic KZ scaling framework addressing the competition between coherent dynamics and dissipation at a continuous quantum transition.
We believe that, in the near future, it will be also possible to address and verify this scenario through suitably engineered experiments with ultracold atoms or cavity-QED technology aimed at realizing and controlling driven-dissipative quantum many-body systems (see, e.g., Ref [64]).
It would be tempting to investigate and carefully verify our dynamic KZ scaling in other quantum dissipative systems, such as Ising-like quantum spin models. To that purpose, given the difficulties in finding a numerical solution to the Lindblad master equation for a generic manybody problem (4), a FSS framework should be adopted as the primary setting, due to the relatively small system sizes that could be reached and the substantial impossibility to address infinite-volume systems (this would be the case, e.g., for the standard quantum Ising chain, with realistic local dissipation related to the spin operators). In that respect, an interesting issue would be to extend the dynamic KZ scaling to protocols across first-order quantum transitions (e.g. in the quantum Ising chain in a transverse plus longitudinal field). The exponentially closing gap between the two lowest states of the ordered phase might be relevant and new features may become apparent already for systems with O(10) spins, such as the sensitivity on the type of boundary conditions [57,65].
Indeed, it can be easily shown that, neglecting boundary terms, the above transformation mapsĤ K of Eq. (29) into the XY chain (δ = 0): (A2) In particular, for t = δ = 1, the corresponding spin model coincides with the quantum Ising chain with g = −µ/2. It is however crucial to stress that the boundary conditions play an important role in this mapping. As a matter of fact, the non-local Jordan-Wigner transformation of the Ising chain with periodic or antiperiodic boundary conditions does not map into the fermionic model (29) with periodic or antiperiodic boundary conditions. Indeed further considerations apply [66,67], leading to a less straightforward correspondence, which also depends on the parity of the particle number eigenvalue (see below). Therefore, although the bulk behaviors of the above models in the infinite-volume limit (and thus their phase diagram) are analogous, the resulting FSS functions are different, since they subtly depend on the choice of the boundary conditions. Even more, the Kitaev quantum wire with antiperiodic boundary conditions, explicitly studied in this paper, turns out to be gapped in both of the phases separated by the quantum transition at µ c = −2. Indeed, the energy difference ∆ of the two lowest states is given by Therefore, the Kitaev quantum wire studied here does not exhibit the lowest-state degeneracy of the ordered phase of the quantum Ising chain (i.e., the exponential suppression of the gap with increasing L). The reason for that substantial discrepancy resides in the fact that the Hilbert space of the Kitaev quantum wire with antiperiodic boundary conditions alone is restricted with respect to that of the quantum Ising chain, so that it is not possible to restore the competition between the two vacua belonging to the symmetric/antisymmetric sectors of the Ising model [42,56,66].
The ultimate reason why we prefer to stick with the Kitaev quantum wire is twofold: i) the dissipation that we consider in this paper is more naturally defined for Fermi lattice gases; ii) the dissipative fermionic decay/pumping mechanisms cannot be mapped into simple spin operators, due to the presence of a nonlocal string operator in the transformation (A1). In this respect, simulating a conventional quantum Ising chain with local dissipation in the form of spin losses (L l,j =σ − j ) or pumping (L p,j =σ + j ) would prevent one from exploiting the particularly simple solvability of the Kitaev model with a polynomial amount of resources, due to the appearance of Jordan-Wigner strings when mapping the termL j ρL † j of the Lindblad master equation in fermionic language [29].
In light of this, it is finally worth mentioning that, although we have only shown numerical results for KZ protocols where the Hamiltonian parameter µ is linearly driven in time from an initial value µ i < µ c to a final value µ f > µ c , there is no reason to expect qualitative differences when reverting the protocol, i.e., starting from µ i > µ c and ending into µ f < µ c . The reason resides in the fact that the Kitaev chain with antiperiodic boundary conditions is gapped in both phases on the left and on the right of the quantum transition point µ c , and thus the evolution arising from slow changes of µ is essentially adiabatic, from anyμ i far from criticality to the relevant scaling interval around µ c . The situation may change for the above mentioned quantum Ising chain, since one of the two phases is ordered and presents a double degeneracy.