being interesting in its own right from the point of view of non-equilibrium dynamics, the first chapter (in particular the first two sections), also provide the necessary background to understand chapters 10 and 11 that deal with thermodynamics in closed quantum systems using Generalized Gibbs ensembles and thermodynamics in the strong coupling regime. I first briefly review some general results about equilibration in complex many-body sys- tems. Then, in section 7.2 I describe in much more detail how this equilibration behaviour can be understood as a mechanism of dephasing. These discussions are illustrated using numerical simulations of well-known models and I also connect the necessary assumptions with rigorous results in mathematical physics. The aim of this chapter is to provide an in- tuitive understanding of the mechanism of equilibration and connect it with general results in many-body theory.
In the next chapter, I will connect the problem of equilibration to a resource theoretic view (although not in the framework of thermal operations): I will provide arguments that show in a quantitative way that it is difficult to prepare a large complex many-body system in a state that does not equilibrate. Furthermore, I will discuss in how far such a resilience to equilibrationcan be seen as a thermodynamic resource with an appropriate corresponding "second law of equilibration".
7.1
Brief review of equilibration and thermalization in closed quantum
systems
When a large many-body system is brought out of equilibrium, physically relevant observ- ables usually quickly come to a rest again. This process, called equilibration, is ubiquitous in nature. A typical example in the context of quantum physics is nowadays studied in
so-called quench-experiments [169–174]. In such an experiment, a quantum system is pre- pared in a stationary state of some Hamiltonian. Then the Hamiltonian is suddenly modi- fied, leading to a situation in which the quantum system is not in equilibrium with respect to the Hamiltonian anymore. After this sudden change of Hamiltonian, called quench, the relaxation dynamics back to equilibrium can be studied. For example, in optical lattices the quench could simply be a sudden change of the lattice-depth.
Despite the fact that equilibration is ubiquitous, it is somewhat surprising from a purely quantum mechanical point of view: After all, a closed system evolves unitarily and hence there are always observables which never approach a stationary value. How can it then be that physically relevant observables do indeed show equilibration? And can one understand and characterize in which kind of systems this behaviour takes place?
This question has in fact already been considered early on by some of the founders of quantum mechanics, von Neumann [29] and Schrödinger [27]. They already observed that such equilibration behaviour should be a very typical behaviour of large interacting systems. Nevertheless, general and rigorous results that prove equilibration behaviour un- der general assumptions mostly only emerged over the last decade [42, 167, 168, 175–191]. These results have been backed up by numerous numerical studies (see the reviews [40, 41] and references therein) and a large body of work on integrable systems [192–202] showing that equilibration is indeed a very generic feature. The revival of the problem of equili- bration can be seen as caused both by new analytical tools that became available leading to rigorous proofs of equilibration behaviour as well as the new possibilities to probe such behaviour experimentally in quench experiments as described above.
In this section, I want to briefly review the simplest general results showing equilibration under general assumptions1. For the remainder of this and the following two sections,
1For a very thorough re-
cent review see Ref. [42]. we consider a large quantum system described by a HamiltonianH = ∑dE
i=1EiPi on a
d-dimensional Hilbert-spaceH . Here, the Ei denote the energy eigenvalues and Pi the
projectors onto the corresponding eigenspaces. When discussing equilibration, I usually have in mind a local, interacting Hamiltonian on a lattice of spins, but the general results in this section do not depend on that. Therefore for now I will not specify the Hamiltonian much further to keep the discussion as general as possible.
It is clear that for a system to equilibrate, there must not be any part of the system that completely decouples from the rest of the system. One way to ensure this is to demand that the Hamiltonian has non-degenerate energy-gaps, meaning that not only are there no degeneracies in the spectrum of the Hamiltonian, but also every difference of energy eigenvaluesEi−Ej is unique. Let us denote byG(∆)the set of pairs of energies whose
difference is∆:
G(∆):=
(Ei, Ej) |Ei−Ej =∆ . (7.1)
Then formally the non-degenerate energy-gaps condition means that every setG(∆)has at most one element as long as∆ 6= 0. Let us, for later convenience, also define the set of non-zero gaps
Gaps(H):=Ei−Ej|i6= j=1, . . . , dE . (7.2)
The condition of non-degenerate energy-gaps implies that it is impossible to find a partition of the Hilbert-spaceH into tensor-factors H1andH2which makes the Hamiltonian non- interacting over this partition, i.e., for which
H=H1⊗1H2+1H1⊗H2. (7.3)
Indeed, if we could find such a partition, then every energy gap of the Hamiltonian H1
would have to have a degeneracy of at least the dimension of the Hilbert-spaceH2, and vice-versa. For the results in the rest of the subsection, I assume for simplicity that the Hamiltonians in question all have non-degenerate energy gaps. While this condition is generically fulfilled (i.e., can be assured by an arbitrarily small random perturbation), it is often too strong. Indeed, much of the following results can be generalized to situations where the degeneracy of energy-gaps is not too large [42, 182]. That the condition of non- degenerate energy-gaps is not always necessary for the equilibration of many physically
A Q UA N T U M O F T H E R M O DY N A M I C S 73
relevant observables can also be seen from the fact that local observables in non-interacting systems, described by local Hamiltonians of free fermions or free bosons, often also show equilibration behaviour [194, 196–199].
Now consider an arbitrary initial state ρ(0), which might in fact be a pure state ρ(0) = |ΨihΨ|. The state evolves unitarily under the dynamics generated byH, evolving along a trajectory ρ: t7→ρ(t). Suppose we consider an arbitrary observableA. If the expectation
value of the observablehAiρ(t) = Tr(Aρ(t))equilibrates, then it has to equilibrate to its time-averageEthhAiρ(t)igiven by Et h hAiρ(t)i= lim T→∞ 1 T Z T 0 hAiρ(t)dt=T→∞lim 1 T Z T 0 Tr(Aρ(t))dt= hAiω(ρ), (7.4)
where we have defined the time-averaged state ω(ρ)given by
ω(ρ) =Et[ρ(t)] = lim T→∞ 1 T Z T 0 ρ(t)dt= dE
∑
j=1 Pjρ(0)Pj. (7.5)Since we consider a finite system, however, perfect equilibration in the sense that ρ(t)
becomes stationary is impossible, due to recurrences [203]. Nevertheless, ρ(t) can be very close to ω(ρ)for most times. In this case the expectation value hAiρ(t) would be
very close to the time-average hAiω(ρ) for most times, possibly with occasional larger deviations for short times. This behaviour is indeed what happens provided that the initial state has overlap with sufficiently many energy-eigenstates.
To quantify this behaviour, let us introduce the effective dimension as a measure for how many energy-eigenstates participate in the initial state. It is defined as [177]
deff(ρ):=deff(ρ(0), H):= 1
∑jTr(Pjρ(0))2 =
1 Tr(ω(ρ)2).
(7.6)
This quantity is typically extremely big. For example, consider a micro-canonical state on a large many-body system, which is evenly distributed (coherently or as a statistical mixture) over an interval of energies [E−δE, E+δE]. For a fixed energy-uncertainty δE, the number of energy-eigenstates that contribute to this initial state will in general
grow exponentially with the system size for local many-body systems. Hence, the effec- tive dimension will also grow exponentially with the system size. At the same time, the uncertainty in terms of the energy-density will go to zero. This is even true if δE grows sub-linearly with the system-size, which would further increase the effective dimension.
For a second example, it has been shown that essentially any state with a finite cor- relation length that is not an eigenstate has an effective dimension that diverges with the system-size [190] (this will be discussed in more detail in section 7.2).
The following theorem then rigorously shows that equilibration indeed happens in the sense discussed above.
Theorem 7.1 (Equilibration on average [204]). Let ρ(t)evolve under an HamiltonianH with non-degenerate energy-gaps. Then for any observableA we have
Et
h
(hAiρ(t)− hAiω(ρ))2i≤ kAk
2
deff(ρ). (7.7)
For large many-body systems and "natural" initial states, we thus see that for most times, the state ρ(t)is practically indistinguishable from ω(ρ)by observables with a finite norm
(bounded by a constant independent of the system size).
A particular case of physically relevant observables are those that only act on a sub- systemS of the total system. They are completely characterized by the reduced density matrix ρS(t) = TrB(ρ(t)). The following theorem shows that this state is effectively in-
distinguishable from the corresponding reduced density matrix of the time-averaged state TrB(ω(ρ))as measured by the trace-distanceD(·,·).
Theorem 7.2 (Average equilibration of subsystems [177]). Let ρ(t)evolve under an Hamil- tonianH with non-degenerate energy-gaps. Then for any subsystem S we have
Et[D (ρS(t), TrB(ω(ρ)))] ≤
s d2S
deff(ρ). (7.8)
The trace-distance D(ρ, σ) measures the theoretical statistical distinguishability of ρ
and σ in a single shot, i.e., it gives the maximum probability that the two can be correctly distinguished by performing a single measurement.
Given the above comments about the effective dimension, we thus conclude that bounded and local observables in large, complex quantum systems equilibrate in a very stringent sense. Similar theorems can also be shown for more general measurements in terms of POVMs [188]. Furthermore, it can be shown that the fluctuations around equilibrium are slow [179].
As a side remark, note the effective dimension is effectively a measure of the amount of entropy in the probability distribution of energy in a quantum system. Indeed, it can also be expressed as
deff(ρ) =eS2(ω(ρ)), (7.9)
whereS2is the Rényi-2 entropy. The Rényi entropies are defined as
Sα(ρ):=log(d) −Dα(ρk1/d) =
1
1−αlog(Tr(ρ
α)), (7.10)
whereDαare again the Rényi-divergences. From an information theoretic point of view it
is then interesting to ask why it is not the von Neumann entropy of ω(ρ)that appears in
the equilibration bounds, which is a more natural entropy measure. It is thus tempting to try to prove similar equilibration bounds expressed in terms of the von Neumann entropy of ω(ρ). We will see in chapter 8 that this is in fact impossible: There are states which do
not equilibrate, but where the von Neumann entropy of ω(ρ)diverges with the system size.
This also has interesting consequences relating to the role of correlations in the process of equilibration, which we will discuss in chapter 8.
The above results clearly establish that we should expect equilibration to be a generic phenomenon. However, it is also important to note what these results do not show. First, they don’t say anything concrete about the equilibrium state ω(ρ). In particular, they do
not imply that this equilibrium state is well described by a thermal state ofH. Additional assumptions are necessary to formally prove such behaviour [190, 205]. Even in this case, usually referred to as thermalization, it is important to observe that equilibration to a ther- mal state only implies that local subsystems equilibrate to the reduced state of a thermal state, ρS(t) ≈TrB e −βH Zβ ! , (7.11)
but not to the thermal state of the HamiltonianHSof the subsystemS. Indeed, for strongly
interacting systems it is not even clear how one would unambiguously define such a local system Hamiltonian HS (see, however, Ref. [206]). In chapter 11, we will explore the
consequences of this for thermal machines in the strong coupling regime.
Second, the above theorems do not rule out the possibility of having initial states for which relevant observables do not equilibrate. All one needs is an initial state with a small effective dimension that is not diagonal in the energy-eigenbasis. However, as the examples for the effective dimension given above already suggest, it seems difficult to prepare such states, since it would seem to require extremely high control over a large, complex quantum system. In chapter 8, I will provide quantitative arguments which show that it is indeed very difficult to prepare quantum states on complex quantum systems that do not equilibrate.
Third, the above equilibration bounds do not say anything about how long it actually takes to reach equilibrium. While general bounds on such an equilibration time can be de- rived [186], the general bounds diverge exponentially with the size of the total system.
A Q UA N T U M O F T H E R M O DY N A M I C S 75
Moreover, one can construct counter-examples which essentially saturate these bounds [183]. Therefore, it is impossible to prove, without making more concrete physical as- sumptions, that bounded observables equilibrate in finite time also when one takes the thermodynamic limit. In the next sections, we will therefore discuss this problem of equili- bration times in more detail. In particular, I will argue for a simple mechanism that suggests that generic systems in fact equilibrate very rapidly, in a time independent of the total sys- tem size, whereas the precision with which they equilibrates increases with the total system size.