Abstract
Understanding under which conditions physical systems thermalize is a longstanding question in manybody physics. While generic quantum systems thermalize, there are known instances where thermalization is hindered, for example in manybody localized (MBL) systems. Here we introduce a class of stochastic collision models coupling a manybody system out of thermal equilibrium to an external heat bath. We derive upper and lower bounds on the size of the bath required to thermalize the system via such models, under certain assumptions on the Hamiltonian. We use these bounds, expressed in terms of the maxrelative entropy, to characterize the robustness of MBL systems against externallyinduced thermalization. Our bounds are derived within the framework of resource theories using the convex split lemma, a recent tool developed in quantum information. We apply our results to the disordered Heisenberg chain, and numerically study the robustness of its MBL phase in terms of the required bath size.
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Introduction
When pushed out of equilibrium, closed interacting quantum manybody systems generically relax to an equilibrium state, where local subsystems can be described using thermal ensembles that only depend on the energy of the initial state. While this behavior is plausible from the perspective of quantum statistical mechanics^{1,2,3,4,5}, it is far from clear which local properties are responsible for the emergence of thermalization. The discovery of manybody localization (MBL) offers a fresh perspective into this question, as this effect occurs in interacting manybody systems preventing them from actually thermalizing^{6,7}. Examples of systems that are nonthermalizing, like integrable systems, have been known before, but these have been finetuned such that small perturbations restore thermalization. Manybody localization is strikingly different in this respect, as its nonthermalizing behavior appears to be robust to changes in the Hamiltonian^{8}. A related open question, recently considered in several papers, is whether MBL is stable with respect to its own dynamics when small ergodic regions are present^{9,10,11,12,13}, or when the system is in contact with an actual external environment^{14,15,16,17}. This is both a critical question for the experimental realization of systems exhibiting MBL properties, and for its fundamental implications on the process of thermalization in quantum systems.
The main focus of this work is to investigate the robustness of the MBL phase under instances of external dissipative processes. To do so, we introduce a physically realistic class of interaction models, describing the interaction between a manybody system and a finitesized thermal environment. Within these models, the interactions are described in terms of energypreserving stochastic collisions occurring between the system (or regions thereof) and subregions of the bath. During the interactions, the system and bath can be either weakly or strongly coupled. For this class of processes, we are able to derive analytical bounds on the minimum size of the bath required to thermalize the manybody system. We apply these bounds to the setting where the system is in the MBL phase, so as to characterize the robustness of this phase with respect to the coupling with an external environment. It is worth noting, however, that the bounds obtained hold for general manybody systems out of thermal equilibrium.
It is key to the approach taken here—and one of the merits of this work—that in order to arrive at our results, we make use of tools from quantum information theory, tools that might at first seem somewhat alien to the problem at hand, but which turn out to provide a powerful machinery. We demonstrate this by using a technical result known as the convex split lemma^{18,19}, to derive the quantitative bounds on the bath size required for a region of the spinlattice to thermalize. Given that MBL phases are challenging to study theoretically, and most known results are numerical in nature^{20,21,22,23,24,25,26}, our work provides a fresh approach in understanding such phases from an analytical perspective. The convex split lemma has originally been derived in the context of quantum Shannon theory, which is the study of compression and transmission rates of quantum information. Our main contribution is to connect this mathematical result to a class of thermodynamic models that can be used to describe thermalization processes in quantum systems. This gives rise to surprisingly stringent and strong results. Note, however, in the approach taken for thermalizing processes, it is assumed that systems thermalize close to exactly, a requirement that will be softened in future work.
As part of our results, we find that the maxrelative entropy^{27,28} and its smoothed version emerge as operationally significant measures, that quantify the robustness of the MBL phase in a spinlattice. The maxrelative entropy is an element within a family of entropic measures that generalize the Rényi divergences^{29} to the quantum setting. In order to illustrate the practical relevance of our results, we consider a specific system exhibiting the MBL phase, namely the disordered Heisenberg chain. Employing exact diagonalization, we numerically compute the value of the maxrelative entropy as a function of the disorder and of the size of the lattice region that we are interested in thermalizing. Our findings suggest that the MBL phase is robust to thermalization despite being coupled to a finite external bath, under our collision models, indicating that such models allow for a conceptual understanding of the MBL phase stability. This extends the narrative of refs. ^{14,15,16}, which find that MBL is thermalized when coupled to an infinite sized bath. Moreover, our numerical simulations show that the maxrelative entropy signals the transition from the ergodic to the MBL phase.
Results
Thermalization setting
We first set up some basic notation. Given some Hamiltonian H_{S} of a system S with Hilbert space \({{\mathcal{H}}}_{S}\), we define the thermal state with respect to inverse temperature β = 1/(k_{B}T) as the quantum state
In what follows, we model the process of thermalization of S with an external heat bath B. In particular, let H_{S} and H_{B} denote the Hamiltonian of S and B, respectively. If S is initially in a state ρ, then, for fixed β and any ϵ > 0, we say that a global process \({\mathcal{E}}:{\mathcal{B}}\left({{\mathcal{H}}}_{S}\otimes {{\mathcal{H}}}_{B}\right)\to {\mathcal{B}}\left({{\mathcal{H}}}_{S}\otimes {{\mathcal{H}}}_{B}\right)\) ϵthermalizes the system S if
where \({\left\Vert \cdot \right\Vert }_{1}\) is the trace norm. Intuitively, this corresponds to the situation where the process \({\mathcal{E}}\) acts on the compound of the initial state of the system ρ and the bath state τ_{β}(H_{B}), and brings the system state close to its thermal state τ_{β}(H_{S}) while leaving the bath mostly invariant. We write
if Eq. (2) holds. It is worth noting that ϵthermalization requires the global system SB to be close to thermal after the channel \({\mathcal{E}}\) is applied. Monitoring the bath as well is necessary in order to avoid the possibility of trivial thermalization processes in which the nonthermal state ρ is simply swapped into the bath, which would merely move the excitation out of the considered region, rather than describing a physically realistic dissipation process. Thus, our notion of thermalization is different from previously considered ones, where for instance the sole system’s evolution is considered. At the same time, and as mentioned before, it is a rather stringent measure, in that close to full global thermalization is required.
Having introduced the basic notation and terminology, we now turn to the model used in this work. We consider a spinlattice V, where each site is described by a finitedimensional Hilbert space \({\mathcal{H}}\). The Hamiltonian of the system is composed of local operators, i.e.,
where x is labeling a specific subset of adjacent sites in the lattice and H_{x} is the corresponding Hamiltonian operator whose support is limited to these sites. Within the lattice, we consider a local region R ⊆ V with Hilbert space \({{\mathcal{H}}}_{R}\). We are interested in the stability of the MBL phase with respect to stochastic collisions between the lattice region R and an external thermal bath B. In order to precisely recast this problem in terms of ϵthermalization, we first need to detail our choices for the initial state of the region R, the Hamiltonians H_{B} and H_{R}, and the class of maps \({\mathcal{E}}\) that model the thermalization process.
Given an initial state vector of the lattice \(\left\psi (0)\right\rangle\), if we consider closed evolution generated by the Hamiltonian dynamics, then the global system is always in a pure state. However, if the local subsystems eventually equilibrate, then the equilibrium state is given by partially tracing over the global infinitetime average^{30},
so that the state that describes region R at the time of its first interaction with the bath is \({\omega }_{R}={{\rm{Tr}}}_{{R}^{c}}\left[\omega \right]\), where we trace out the remaining of the lattice R^{c} = V\R. To define a valid Hamiltonian H_{R}, a natural approach is often to disregard interactions between R and the rest of the lattice R^{c}. As such, we consider the Hamiltonian H_{R} = ∑_{x:x⊆R}H_{x}, which includes only terms H_{x} whose support is contained in R, and denote the corresponding thermal state τ_{β}(H_{R}). It is worthwhile to point out that that there is an alternative natural approach to defining thermal states of subsystems, namely \({\hat{\tau }}_{\beta }({H}_{R})={{\rm{Tr}}}_{{R}^{c}}\left[{\tau }_{\beta }({H}_{V})\right]\) the reduced state over the complement R of the thermal state of the full lattice^{31}. These two thermal states are close to each other whenever the interaction terms between R and R^{c} in H_{V} are small. During our later simulations, we check the values of maxrelative entropy using both versions of thermal states, and find that they produced similar values, implying that these different alternatives are actually not too dissimilar from each other for the disordered Heisenberg chain, which is expected, given that interactions are 2local.
There has been a large body of work in thermalization, where a large class of manybody systems can effectively act as their own “environment”^{4,30}, thus local observables tend to equilibrate towards the corresponding thermal values. However, there are exceptions to this, in particular whenever there are nonnegligible interactions between subsystems, of which MBL systems are an example. Nevertheless, most such systems still do equilibrate, and this is the assumption we make throughout the paper, that ω as defined in Eq. (5) exists.
We model the external thermal bath as a collection of n − 1 copies of the region R in thermal equilibrium. More formally, B is a system with Hilbert space \({{\mathcal{H}}}_{B}={{\mathcal{H}}}_{R}^{\otimes n1}\) and Hamiltonian
where the operator \({H}_{R}^{(i)}={{\mathbb{I}}}_{1}\otimes \ldots \otimes {{\mathbb{I}}}_{i1}\otimes {H}_{R}\otimes {{\mathbb{I}}}_{i+1}\otimes \ldots \otimes {{\mathbb{I}}}_{n1}\) only acts nontrivially on the ith subsystem of the bath. With this choice of Hamiltonian, the initial state on B is \({\tau }_{\beta }({H}_{B})={\tau }_{\beta }{({H}_{R})}^{\otimes n1}\). Such a choice for the bath is crucial to make our problem analytically tractable but is also physically relevant for experimental setups, where it is possible to engineer onedimensional systems which are then coupled to a bath per site^{32,33} or a mixture of a system and bath species that interact via contact interactions^{34}. Moreover, we note that for the model of systembath interactions that we introduce below, any state transition that can be realized on R with any heat bath Hamiltonian H_{B} can also be realized, for some n, with a Hamiltonian of the form (6)^{35}. We also refer the interested reader to Supplementary Note 3, for a further discussion on bath choices.
We now turn to our model of the systembath interactions, described via the following master equation,
where ρ_{RB} is the global state on R and B. This equation models a series of collisions, each described by a unitary operator \({U}_{RB}^{(k)}\in {\mathcal{B}}\left({{\mathcal{H}}}_{R}\otimes {{\mathcal{H}}}_{B}\right)\) acting nontrivially on the region R and a subset of bath components, occurring at a given rate \({r}_{k}^{1}\,> \,0\) in time according to a Poissonian distribution (see Supplementary Note 1 for details). We consider elastic collisions, that is, we require that each unitary operator \({U}_{RB}^{(k)}\) conserves the global energy
It is worth noting that the above condition does not imply that the total energy of the lattice is conserved. In fact, this is in general not the case, due to those operators H_{x} in Eq. (4), with support on both R and R^{c}. However, since by assumption these operators have support on at most k adjacent lattice sites, if the region R is sufficiently large one expects these boundary terms to contribute less and less to the total energy of the region. For high temperatures, such notions that boundary terms are negligible have rigorously been established^{31}. Note that Eq. (7) is already in standard Lindblad form, and therefore describes a Markovian dynamical semigroup on RB. It is also important to note, however, that the process happening on the region R is in principle nonMarkovian, since B is modeled here explicitly, and can be a very small heat bath, which retains the memory of the system’s initial state. See Fig. 1 for a graphical illustration of the setup.
In summary, the interaction model is general, in the sense that the unitary operators can act on a single subsystem (thus generating local Hamiltonian dynamics, for example), on two subsystems (these are the standard twobody interactions), or many more subsystems. Longrange interaction terms are also allowed, with the only restriction of Eq. (8)—in the hope that one may work toward relaxation in the future. We are nonetheless neglecting the interaction terms between the region R and R^{c}. While this assumption is critical for allowing us to apply our framework to this problem, we note that there are situations where it is physically relevant—for example, if the interaction between R and B occurs on a much shorter timescale than between R and R^{c}, which is allowed by our class of interactions, as the rates \({\{{r}_{k}\}}_{k}\) can be chosen to be high enough. Moreover, our results on bath size are independent on the systembath coupling strength.
We are finally in a position to define our central measure of robustness to thermalization. Let \({{\mathfrak{E}}}_{n}\) denote the set of quantum channels (i.e., completelypositive tracepreserving maps acting over the quantum states of a system^{36}) on RB that can be generated via the above collision process for a bath of size n − 1. Each channel in this set is realized through a different choice of unitary operations \(\{{U}_{RB}^{(k)}\}\), collision rates \(\{{r}_{k}^{1}\}\), and final time t. Given the region initial state ω_{R}, its Hamiltonian H_{R}, and the bath Hamiltonian H_{B} defined as in Eq. (6), we define n_{ϵ} as the minimum integer such that there exists an element in \({{\mathfrak{E}}}_{{n}_{\epsilon }+1}\) that ϵthermalizes R,
The integer n_{ϵ} then quantifies the smallest size of a thermal environment required to thermalize region R under stochastic collisions, and hence provides a natural measure to quantify the robustness of an MBL system against thermalization.
Upper and lower bounds for thermalization
Our main results are upper and lower bounds on n_{ϵ} that are essentially tight for a wide range of Hamiltonians H_{R}. These bounds are stated using the (smooth) maxrelative entropy, an entropic quantity that has received considerable attention in recent years in quantum information and communication theoretical research^{35,37,38}. Once again, it is worth noting that our results are applicable in general for finitedimensional quantum systems where equilibration occurs, of which MBL is a particularly interesting example. The maxrelative entropy^{27} between two quantum states \(\rho ,\sigma \in {\mathcal{S}}\left({\mathcal{H}}\right)\) such that supp(ρ) ⊆ supp(σ) is defined as
while the smooth maxrelative entropy between the same two quantum states, for some ϵ > 0, is defined as
where B_{ϵ}(ρ) is the ball of radius ϵ around the state ρ with respect to the distance induced by the trace norm.
Upper bound
We first present and discuss the upper bound, which can be easily stated in terms of the quantities just introduced.
Theorem 1 (Upper bound on the size of the bath). For a given Hamiltonian H_{R}, inverse temperature β, and a constant ϵ > 0, we have that
The above theorem provides a quantitative bound on the size of the thermal bath needed to ϵthermalize a lattice region R, when the coupling is mediated by stochastic collisions. For this specific dynamics, the region can be ϵthermalized if the size of the bath (the number of components) is proportional to the exponential of the maxrelative entropy between the state of the region ω_{R} and its thermal state τ_{β}(H_{R}).
Theorem 1 is proven in Supplementary Note 2. Here, we present a sketch of the proof in two steps. In the first step, we show that \({{\mathfrak{E}}}_{n}\) can be connected to socalled random unitary channels^{39}. In the second step, we use this connection to find a particular channel in \({{\mathfrak{E}}}_{n}\) that achieves the upper bound of the above theorem. A central ingredient to the second step is a result known as convex split lemma^{18,19}.
Turning to the first step, recall that a random unitary channel is a map of the form
where \({\left\{{p}_{k}\right\}}_{k}\) is a probability distribution, and \({\{{U}_{k}\}}_{k}\) is a set of unitary operators. For a given number n − 1 of bath subsystems, we define the class of energypreserving random unitary channels \({{\mathfrak{R}}}_{n}\) as those random unitary channels on RB for which each unitary operator \({U}_{k}\in {\mathcal{B}}\left({{\mathcal{H}}}_{R}\otimes {{\mathcal{H}}}_{B}\right)\) commutes with the Hamiltonian of the global system, i. e., \(\left[{U}_{k},{H}_{R}+{H}_{B}\right]=0\). In Supplementary Note 1, we show that for any n ≥ 1, \({{\mathfrak{E}}}_{n}\subseteq {{\mathfrak{R}}}_{n}\), therefore allowing us to analyze any element of \({{\mathfrak{E}}}_{n}\) as a random unitary channel.
Turning to the second step, we use a stochastic collision model with a simple representation in terms of random unitary channels. Let us first recall that the thermal bath B is described by n − 1 copies of τ_{β}(H_{R}), the Gibbs state of the Hamiltonian H_{R} at inverse temperature β. The collisions occur either between the region R and one subsystem of the bath, or between two bath subsystems. The rate of collisions is uniform, and given by r^{−1} > 0. During a collision involving the ith and jth subsystems of RB, the states of the two colliding components are swapped, so that the interaction is described by the unitary operator \({U}_{\,\text{swap}\,}^{(i,j)}\). The action of this operator over two quantum systems, described by the state vectors \({\left\psi \right\rangle }_{1}\) and \({\left\phi \right\rangle }_{2}\), respectively, is given by \({U}_{{\rm{swap}}}^{(1,2)}{\left\psi \right\rangle }_{1}\otimes {\left\phi \right\rangle }_{2}={\left\phi \right\rangle }_{1}\otimes {\left\psi \right\rangle }_{2}\). For an initial global state \({\rho }_{RB}^{(\,\text{in}\,)}={\omega }_{R}\otimes {\tau }_{\beta }{({H}_{R})}^{\otimes n1}\), the steady state obtained through this process is
where the athermality of the region has been uniformly hidden into the different components of the bath. It is worth noting that, under the stochastic collision model described above, the global system reaches its steadystate exponentially quickly in the collision rate r^{−1}, see Supplementary Note 2 for more details.
The mapping from the initial state of region and bath to the steadystate is achieved by the following random unitary channel
which uniformly swaps each of the bath subsystems with R. Such channels have been studied before in the context of entropy production^{40,41}, see also ref. ^{42} for a similar example. Since all subsystems share the same Hamiltonian H_{R}, it is easy to see that each one of the \({U}_{\,\text{swap}\,}^{(1,i)}\) commutes with the joint Hamiltonian, so that \({\bar{{\mathcal{E}}}}_{n}\in {{\mathfrak{R}}}_{n}\). Finally, we can invoke the convex split lemma, see Supplementary Note 2, which allows us to show that, for any ϵ > 0, the channel \({\bar{{\mathcal{E}}}}_{n}\) can ϵthermalize the region R when the number of subsystems is \(n={\epsilon }^{2}{2}^{{D}_{\text{max}}^{\,}\left({\omega }_{R}  {\tau }_{\beta }({H}_{R})\right)}\).
The collision model presented here already encompasses a wide range of possible and physically realistic thermalization processes. However, before turning to a lower bound, we note that, due to the particularly simple nature of (15), one can use the above construction to upper bound the required size of a bath for other thermalization models as well. For example, by noting that the channel (15) is permutationsymmetric (in the sense that it has permutationinvariant states as its fixed points), it follows that thermalization models with permutationsymmetric dynamics (i.e., those that allow for any permutationsymmetric channel) are also subject to the above upper bound.
Lower bound and optimality results
We now turn to derive a lower bound on n_{ϵ}. This bound is obtained through a further assumption on the Hamiltonian H_{R}, which we call the energy subspace condition (ESC).
Definition 2 (Energy subspace condition). Given a Hamiltonian H_{R}, we say that it fulfills the ESC iff for any \(n\in {\mathbb{N}}\), given the set of energy levels \({\left\{{E}_{k}\right\}}_{k = 1}^{d}\) of the Hamiltonian H_{R}, we have that for any vectors \(m,m^{\prime} \in {{\mathbb{N}}}^{d}\) with the same normalization factor, namely,
Let us here briefly discuss the physical significance of the ESC condition. The ESC entails (but is not equivalent to) that energy levels cannot be exact integer multiples of one another, which also implies full nondegeneracy. Furthermore, note that the ESC is not approximate, in other words, it still holds even if energy levels are very close to each other. Having exact integer multiple energy levels is a very finetuned situation that breaks as soon as randomness is introduced in the Hamiltonian^{43}. Let us for example consider how likely it is for MBL systems to have degenerate energy levels. In the ergodic phase, the level statistics are Wigner–Dyson, and therefore nondegeneracy is enforced by level repulsion. On the other hand, in the strong MBL phase, level statistics are Poissonian, which means that the probability density function is maximum for zeroenergy gaps. Despite this, the probability of exact degeneracy would correspond to a zerovolume integral of the Poissonian distribution (which is bounded from above), and therefore still amounts to zero probability of having degenerate gaps.
It is clear, however, that the ESC is much more stringent than requiring nondegeneracy; If we require Eq. (16) to be satisfied for all \(n\in {\mathbb{N}}\), this implies that the energy levels of the Hamiltonian H_{R} need to be irrational. Nevertheless, one can relax this condition by asking it to hold for all n ≤ N, for some sufficiently large \(N\in {\mathbb{N}}\), for example with the upper bound on n_{ϵ} in Theorem 1. We refer to this as the ESC being satisfied up to N. In the next section, we discuss how a paradigmatic MBL system relates to this condition.
We can now state the following theorem, proved in Supplementary Note 4, on the optimality of the channel associated with the convex split lemma.
Theorem 3 (Optimal thermalization processes). If H_{R} satisfies the ESC and the state ω_{R} is diagonal in the energy eigenbasis, then the channel \({\bar{{\mathcal{E}}}}_{n}\) in Eq. (15) provides the optimal thermalization process, that is, for any \(n\in {\mathbb{N}}\),
Theorem 3 shows that, for Hamiltonians satisfying the ESC, the channel \({\bar{{\mathcal{E}}}}_{n}\) provides the optimal thermalization of R, that is, no other random energypreserving channel acting on the same global system can achieve a smaller value of ϵ in Eq. (2). The above result applies to initial states that are diagonal in the energy eigenbasis; this is in general not the case for the reduced state ω_{R} of the infinitetime average of Eq. (5), since it might have coherence in the eigenbasis of the reduced Hamiltonian H_{R}. For states with coherence, the channel of Eq. (15) is not necessarily optimal anymore, but we can still bound the difference in thermalization achieved by this channel and an optimal one, see Supplementary Note 4 for the proof.
Theorem 4 (Thermalization bound for coherent states). Fix \(n\in {\mathbb{N}}\), and assume that H_{R} satisfies the ESC. Consider the channel \({{\mathcal{E}}}_{{\rm{opt}}}\in {{\mathfrak{E}}}_{n}\) achieving optimal thermalization
and the decohering channel Δ(⋅) = ∑_{E}Π_{E} ⋅ Π_{E}, where Π_{E} is the eigenprojector onto the energy subspace associated with E. We define the parameter \(\delta ={\left\Vert {\omega }_{R}{{\Delta }}({\omega }_{R})\right\Vert }_{1}\), quantifying the amount of coherence contained in the state of the region. Then, the thermalization achieved by the channel \({\bar{{\mathcal{E}}}}_{n}\) is bounded as
The above theorem provides a quantitative bound on the thermalization achieved by the channel \({\bar{{\mathcal{E}}}}_{n}\) when the input system has coherence in the energy eigenbasis. In the case of MBL systems, the eigenstates of the Hamiltonian are close to product states, see for instance ref. ^{44}, and therefore the reduced state of the infinitetime average ω_{R} is expected to have small and stronglydecaying coherence. Thus, Theorem 4 shows that the stochastic collision model introduced in the previous section is able to effectively thermalize the region of MBL systems. From the two theorems stated above, we can derive the following corollary, providing a lower bound on the size of the thermal bath needed to thermalize a given quantum system.
Corollary 5 (Lower bound to size of the bath). For a given β and ϵ > 0, and some Hamiltonian H_{R} satisfying the ESC, we have
where \(\delta ={\left\Vert {\omega }_{R}{{\Delta }}({\omega }_{R})\right\Vert }_{1}\) and Δ(ω_{R}) is the decohered version of the state ω_{R}.
Note that this lower bound is arising from the stringent model of thermalization of Eq. (2), and that less stringent models will potentially lead to smaller lower bounds. When H_{R} does not satisfy the ESC, it is easy to find counterexamples to the optimality of the channel \(\bar{{\mathcal{E}}}\), as we show in Supplementary Note 4. The idea is that this channel is optimal only when it is able to produce a uniform distribution within each energy subspace of the global system, and this is possible if each subspace is fully characterized by a different frequency of singlesystem eigenvectors, which is exactly given by the ESC. Indeed, when the ESC is maximally violated, i.e., when the system Hamiltonian is completely degenerate, then no bath is required at all. See “Discussion” section for a discussion of this and its relation to known bounds from randomness extraction.
The disordered Heisenberg chain
Our results from the previous section show that, for systems that satisfy the ESC, the maxrelative entropy between the local state of a lattice region and its thermal state provides a natural measure for the robustness of that region to thermalization for a broad family of interactions. This includes manybody systems close to the transition between the ergodic and MBL phase, where both level repulsion and randomness effects favor a lack of exact degeneracies so that it seems reasonable to expect for such systems to satisfy the ESC to sufficient order n. In this section, we use these results to study the robustness of the MBL phase to the thermal noise for a concrete system. Specifically, we consider the disordered Heisenberg chain, a onedimensional spin\(\frac{1}{2}\) lattice system composed of L sites, governed by the Hamiltonian
where \({\sigma }^{x},{\sigma }^{y},{\sigma }^{z}\in {\mathcal{B}}\left({{\mathbb{C}}}^{2}\right)\) are the Pauli operators, Δ is the (dimensionless) disorder strength, and each parameter h_{i} ∈ [ − 1, 1] is drawn uniformly at random. We employ periodic boundary conditions.
It has been demonstrated both theoretically^{45} and experimentally^{7} that this system undergoes a localization transition above the critical disorder strength Δ_{c} ≈ 7. The transition manifests itself in a breakdown of conductance^{7,45}, and a slowdown of entanglement growth after a quench^{23,24}. Moreover, a phenomenological model in terms of quasilocal constants of motions exists which provides an explanation for the nonthermal behavior of the system^{46,47}.
To relate the model to our theoretical results, we note that when the region R is a single qubit (∣R∣ = 1) with a nonzeroenergy gap, then the ESC is always satisfied for all \(n\in {\mathbb{N}}\). For ∣R∣ = 2, we have verified the ESC condition across a large range of disorder strengths Δ ∈ [0.01, 20], up to N = 25 (out of ~2000 realizations, all of them satisfy the ESC); we have additionally considered a small number of realizations for N = 35, for all of which the ESC holds. As ∣R∣ increases, higher values of N become significantly harder to check numerically. However, we have verified that for example ∣R∣ = 4 always satisfies the ESC up to N = 5 (for 2000 generated realizations), while the condition is satisfied with high probability when N = 6 (90% of the realizations).
In our simulation, we choose as initial state vector \(\left{{\Psi }}(0)\right\rangle\) a variation of the Néel state with support on the totalmagnetization sectors M = ±1, 0. Our choice is motivated by the fact that this state, due to its increased overlap over different symmetric subspaces of the Hamiltonian, thermalizes more easily during the ergodic phase. For each random realization, we numerically compute the infinitetime average of \(\left{{\Psi }}\right\rangle\) as defined in Eq. (5), using exact diagonalization. We then trace out part of the lattice so as to obtain the state ω_{R}, describing the infinitetime averaged state reduced to the region R. Notice that in the ergodic phase, when the disorder strength Δ < Δ_{c}, this state is expected to be close to thermal, with a temperature which depends on the energy of the initial state of the lattice. However, when the disorder strength Δ passes its critical value, the state ω_{R} is not thermal anymore^{7}.
To numerically compute the maxrelative entropy for this system, we use the Gibbs state of the reduced Hamiltonian H_{R}, obtained from the Hamiltonian in Eq. (21) by only considering terms with full support on the region R. The inverse temperature β is obtained by constructing the global Gibbs state of the lattice and requiring its energy to equal to the one of the initial state vector \(\left{{\Psi }}(0)\right\rangle\). We compute \({D}_{\,\text{max}\,}^{\,}({\omega }_{R}  {\tau }_{\beta }({H}_{R}))\) for different disorder strengths Δ, and different sizes of the region R (Fig. 2).
We find that in the ergodic phase the state is approximately thermal, and the maxrelative entropy remains almost constant as ∣R∣ increases. For big enough sizes of the region, the maxrelative entropy starts increasing even in the ergodic case. However, this effect is due to the finite size of the lattice in our simulation, and it can be mitigated by increasing the number of lattice sites (at the expense of a higher computational cost). As Δ approaches the critical value, we find that the maxrelative entropy scales linearly in the region size ∣R∣, with a linear coefficient which increases with the disorder strength (Fig. 2a). As a result, the size of the external thermal bath n_{ϵ} scales exponentially in the region size, due to the bounds we have obtained in the previous section. This exponential scaling in the size of the bath suggests robustness of the MBL phase with respect to the dynamics given by Eq. (7), since the relative size of the bath n_{ϵ}/∣R∣ needs to diverge as ∣R∣ tends to infinity. In other words, for the MBL phase to be destroyed one needs, under the interaction models we consider, an exponentially vast amount of thermal noise. It is worth noting that our characterization of the robustness of the MBL phase to thermal noise is distinctly different from others found in the literature^{14,15,16,17}. Indeed, we couple the system with a finitesized thermal bath, and we quantify the robustness in terms of its size. Furthermore, our notion of thermalization accounts for the evolution of both the system and bath, rather than focusing on the system only. Other works instead consider infinite thermal reservoirs and quantified the robustness as a function, for instance, of the coupling between system and environment. A promising experimental realization is recent optical lattice experiments^{32,33,34}. However, to connect to our findings one would need full state tomography on both system and bath which so far is out of reach for these platforms.
We additionally study the first derivative of the maxrelative entropy with respect to the region size ∣R∣, as a function of the disorder strength, shown in Fig. 2b. We find that, during the ergodic phase, the derivative remains constant and small. As Δ approaches the critical value, the derivative increases, and for Δ ≫ Δ_{c} the derivative becomes constant again. Thus, we find that the derivative of the maxrelative entropy with respect to the region size is an order parameter for the MBL phase transition. We then use this order parameter to estimate the critical value \({{{\Delta }}}_{c}^{(L)}\) for the finitelength spin chain we are considering, obtaining a value of ~4.5 for L = 15 sites. While the critical value for infinitelength spin chains is considered to be Δ_{c} ≈ 7, we find that our value, which we stress is obtained for a finite number of sites, seems to be in good accord with known results found in the literature using other measures^{48,49}.
Discussion
We show that mathematical results originally developed to study quantum information processing may find their applications in manybody physics as well, in particular for the study of MBL in this paper. We demonstrate this by applying the recently developed convex split lemma technique, to derive upper and lower bounds for the size of the external thermal bath required to thermalize an MBL system. The class of interaction models between the lattice and the thermal bath is described by the master equation (7) and consists of stochastic energypreserving collisions between the system and bath components. The bounds we obtain depend on the maxrelative entropy between the state we aim to thermalize and its thermal state.
We make use of these analytic results to study a specific and at the same time much ubiquitous system exhibiting MBL features, known as the disordered Heisenberg chain. We show that the MBL phase in this system is in fact robust with respect to the thermalization processes considered here and that the derivative of the maxrelative entropy with respect to region size serves as an approximate order parameter of the ergodic to MBL transition. We emphasize that this is not in contradiction with previous results, where a breakdown of localization was reported^{14,15,16}, as the size of the baths considered in these works was unbounded. Resourcetheoretic frameworks offer another potentially useful approach for studying thermalization with infinitedimensional baths; the framework of elementary thermal operations^{50} which involves a single bosonic bath that is coupled only with two levels of the system of interest. One may then study the resources (the number of bosonic baths with different frequencies) required to achieve thermalization. Also, and more technically, it would be interesting to study the extent to which both the ESC condition and the requirement of exact commutation in our framework can be relaxed to only approximately hold true and how this, in turn, affects the lower bound of Corollary 5 (Eq. 20). These questions we leave to be studied in future work.
The success of our application implies that, potentially, other informationtheoretic tools could be employed to study the thermalization of MBL systems—and nonequilibrium dynamics of manybody systems in more generality, for that matter. For instance, results in randomness extraction^{51} might be useful to provide new bounds. In randomness extraction, a weakly random source is converted into an approximately uniform distribution, with the use of seed (a small, uniformly distributed auxiliary system). In analogy, thermalization requires a nonthermal state to be mapped into an almost thermal state, with the help of an external bath (the seed). Thus, it seems possible that results from randomness extraction might be modified to study this setting and to obtain bounds on the thermal seed.
It has been shown that excited states of onedimensional MBL systems are wellapproximated by matrix product states (MPS) with a low bond dimension^{44,52} if the system features an information mobility gap. These states have several interesting properties, and in particular, they feature an area law for the entanglement entropy which is logarithmic in the bond dimension^{53}. Since our result is based on a particular entropic quantity, it might be possible to use the properties of MPS to derive a fully analytical bound on the robustness of these systems with respect to thermal noise. It is the hope that our work stimulates further crossfertilization between the fields of quantum thermodynamics and the study of quantum manybody systems out of equilibrium.
Data availability
The data sets generated during the numerical simulation are available from the corresponding author on reasonable request.
Code availability
The code used for the numerical simulations is written in Python, and it is available from the corresponding author on reasonable request.
References
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888 (1994).
Popescu, S., Short, A. J. & Winter, A. Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754–758 (2006).
Eisert, J., Friesdorf, M. & Gogolin, C. Quantum manybody systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).
Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863–883 (2011).
Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).
Schreiber, M. et al. Observation of manybody localization of interacting fermions in a quasirandom optical lattice. Science 349, 842 (2015).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Manybody localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Luitz, D. J., Huveneers, F. & Roeck, W. D. How a small quantum bath can thermalize long localized chains. Phys. Rev. Lett. 119, 150602 (2017).
Ponte, P., Laumann, C. R., Huse, D. A. & Chandran, A. Thermal inclusions: How one spin can destroy a manybody localized phase. Phil. Trans. R. Soc. A 375, 20160428 (2017).
Hetterich, D., Serbyn, M., Domínguez, F., Pollmann, F. & Trauzettel, B. Noninteracting central site model: localization and logarithmic entanglement growth. Phys. Rev. B 96, 104203 (2017).
Barišić, O. S., Prelovšek, P., Metavitsiadis, A. & Zotos, X. Incoherent transport induced by a single static impurity in a Heisenberg chain. Phys. Rev. B 80, 125118 (2009).
Goihl, M., Eisert, J. & Krumnow, C. Exploration of the stability of manybody localized systems in the presence of a small bath. Phys. Rev. B 99, 195145 (2019).
Nandkishore, R., Gopalakrishnan, S. & Huse, D. A. Spectral features of a manybodylocalized system weakly coupled to a bath. Phys. Rev. B 90, 064203 (2014).
Fischer, M. H., Maksymenko, M. & Altman, E. Dynamics of a manybodylocalized system coupled to a bath. Phys. Rev. Lett. 116, 160401 (2016).
Levi, E., Heyl, M., Lesanovsky, I. & Garrahan, J. P. Robustness of manybody localization in the presence of dissipation. Phys. Rev. Lett. 116, 237203 (2016).
Johri, S., Nandkishore, R. & Bhatt, R. N. Manybody localization in imperfectly isolated quantum systems. Phys. Rev. Lett. 114, 117401 (2015).
Anshu, A., Devabathini, V. K. & Jain, R. Quantum communication using coherent rejection sampling. Phys. Rev. Lett. 119, 120506 (2017).
Anshu, A., Hsieh, M.H. & Jain, R. Quantifying resources in general resource theory with catalysts. Phys. Rev. Lett. 121, 190504 (2018).
Luitz, D. J., Laflorencie, N. & Alet, F. Manybody localisation edge in the randomfield Heisenberg chain. Phys. Rev. B 91, 081103 (2015).
Devakul, T., Khemani, V., Pollmann, F., Huse, D. & Sondhi, S. Obtaining highly excited eigenstates of the localized xx chain via dmrgx. Phil. Trans. R. Soc. Lond. A 375, 2108 (2017).
Yu, X., Pekker, D. & Clark, B. K. Finding matrix product state representations of highly excited eigenstates of manybody localized hamiltonians. Phys. Rev. Lett. 118, 017201 (2017).
Žnidarič, M., Prosen, T. & Prelovšek, P. Manybody localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008).
Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of manybody localization. Phys. Rev. Lett. 109, 017202 (2012).
Wahl, T. B., Pal, A. & Simon, S. H. Signatures of the manybody localized regime in two dimensions. Nature Phys. 15, 164–169 (2019).
Kshetrimayum, A., Goihl, M. & Eisert, J. Time evolution of manybody localized systems in two spatial dimensions. Phys. Rev. B 102, 235132 (2020).
Datta, N. Min and maxrelative entropies and a new entanglement monotone. IEEE Trans. Inf. Theory 55, 2816–2826 (2009).
Tomamichel, M. Quantum Information Processing with Finite Resources  Mathematical Foundations (Springer, Cham, 2016) .
Erven, T. V. & Harremos, P. Rényi divergence and KullbackLeibler divergence. IEEE Trans. Inf. Theory. 60, 3797–3820 (2014).
Gogolin, C. & Eisert, J. Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 79, 056001 (2016).
Kliesch, M., Gogolin, C., Kastoryano, M. J., Riera, A. & Eisert, J. Locality of temperature. Phys. Rev. X 4, 031019 (2014).
Bordia, P. et al. Coupling identical onedimensional manybody localized systems. Phys. Rev. Lett. 116, 140401 (2016).
Bordia, P. et al. Probing slow relaxation and manybody localization in twodimensional quasiperiodic systems. Phys. Rev. X 7, 041047 (2017).
RubioAbadal, A. et al. Manybody delocalization in the presence of a quantum bath. Phys. Rev. X 9, 041014 (2019).
Horodecki, M. & Oppenheim, J. Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2010).
Konig, R., Renner, R. & Schaffner, C. The operational meaning of minand maxentropy. IEEE Trans. Inf. Theor. 55, 4337–4347 (2009).
Bu, K., Singh, U., Fei, S.M., Pati, A. K. & Wu, J. Maximum relative entropy of coherence: an operational coherence measure. Phys. Rev. Lett. 119, 150405 (2017).
Audenaert, K. M. R. & Scheel, S. On random unitary channels. New J. Phys. 10, 023011 (2008).
Diósi, L., Feldmann, T. & Kosloff, R. On the exact identity between thermodynamic and informatic entropies in a unitary model of friction. Int. J. Quant. Inf. 4, 99–104 (2006).
Csiszár, I., Hiai, F. & Petz, D. Limit relation for quantum entropy and channel capacity per unit cost. J. Math. Phys. 48, 092102 (2007).
Scarani, V., Ziman, M., Štelmachovič, P., Gisin, N. & Bužek, V. Thermalizing quantum machines: dissipation and entanglement. Phys. Rev. Lett. 88, 097905 (2002).
Linden, N., Popescu, S., Short, A. J. & Winter, A. Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009).
Friesdorf, M., Werner, A. H., Brown, W., Scholz, V. B. & Eisert, J. Manybody localization implies that eigenvectors are matrixproduct states. Phys. Rev. Lett. 114, 170505 (2015).
Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metalinsulator transition in a weakly interacting manyelectron system with localized singleparticle states. Ann. Phys. 321, 1126 (2006).
Serbyn, M., Papić, Z. & Abanin, D. A. Local conservation laws and the structure of the manybody localized states. Phys. Rev. Lett. 111, 127201 (2013).
Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully manybodylocalized systems. Phys. Rev. B 90, 174202 (2014).
Luitz, D. J., Laflorencie, N. & Alet, F. Manybody localization edge in the randomfield Heisenberg chain. Phys. Rev. B 91, 081103 (2015).
Gray, J., Bose, S. & Bayat, A. Manybody localization transition: Schmidt gap, entanglement length, and scaling. Phys. Rev. B 97, 201105 (2018).
Lostaglio, M., Alhambra, Á. M. & Perry, C. Elementary thermal operations. Quantum 2, 52 (2018).
Trevisan, L. & Vadhan, S. Extracting randomness from samplable distributions. In Proc. 41st Annual Symposium on Foundations of Computer Science 32–42 (2000).
Bauer, B. & Nayak, C. Area laws in a manybody localized state and its implications for topological order. J. Stat. Mech. Theory Exp. 2013, P09005 (2013).
Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010).
Janzing, D., Wocjan, P., Zeier, R., Geiss, R. & Beth, T. Thermodynamic cost of reliability and low temperatures: tightening Landauer’s principle and the second law. Int. J. Theor. Phys. 39, 2717–2753 (2000).
Brandão, F. G. S. L., Horodecki, M., Oppenheim, J., Renes, J. M. & Spekkens, R. W. Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett. 111, 250404 (2013).
Brandao, F. G. S. L., Horodecki, M., Ng, N., Oppenheim, J. & Wehner, S. The second laws of quantum thermodynamics. PNAS 112, 3275–3279 (2015).
Alhambra, Á. M., Masanes, L., Oppenheim, J. & Perry, C. Fluctuating work: From quantum thermodynamical identities to a second law equality. Phys. Rev. X 6, 041017 (2016).
Masanes, L. & Oppenheim, J. A general derivation and quantification of the third law of thermodynamics. Nat. Commun. 8, 14538 (2017).
Wilming, H. & Gallego, R. Third law of thermodynamics as a single inequality. Phys. Rev. X 7, 041033 (2017).
Woods, M. P., Ng, N. H. Y. & Wehner, S. The maximum efficiency of nano heat engines depends on more than temperature. Quantum 3, 177 (2019).
Halpern, N. Y., White, C. D., Gopalakrishnan, S. & Refael, G. Quantum engine based on manybody localization. Phys. Rev. B 99, 024203 (2019).
Alhambra, Á. M. & Wilming, H. Revivals imply quantum manybody scars. Phys. Rev. B 101, 205107 (2020).
Acknowledgements
We would like to thank Álvaro Martin Alhambra, Johnnie Gray, Volkher Scholz, and Henrik Wilming for helpful discussions. C.S. is funded by EPSRC. N.H.Y.N. is funded by the Alexander von Humboldt foundation. P.B. acknowledges funding from the Templeton Foundation. J.E. has been supported by the DFG (FOR 2724, CRC 183), the FQXi, and the Templeton Foundation. We acknowledge the Freie Universität Berlin for covering the costs of offsetting the emission generated by this research. In the supplemental material, we summarize the climate expenses of this work.
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C.S., P.B., and N.H.Y.N. have formulated the proofs, M.G. has written the code and produced the numerics, and all authors C.S, M.G., P.B., J.E., and N.H.Y.N. have contributed to research design and the writing of the paper.
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Sparaciari, C., Goihl, M., Boes, P. et al. Bounding the resources for thermalizing manybody localized systems. Commun Phys 4, 3 (2021). https://doi.org/10.1038/s42005020005031
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DOI: https://doi.org/10.1038/s42005020005031
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