7 Equilibration in many-body systems
7.3 The basic mechanism
We now discuss equilibration from the point of view of dephasing (see Ref. [213] for a sim- ilar exposition that appeared simultaneously with our manuscript [10]). To understand this basic mechanism behind equilibration, it is useful to re-express the instantaneous deviation from equilibrium∆Aρ(t)in the energy-eigenbasis as
∆Aρ(t) =
∑
Ei6=Ej
Tr PiAPjρ ei(Ej−Ei)t=
∑
06=∆∈Gaps(H)
z∆ei∆t, (7.13)
where we have introduced the complex numbers
z∆=
∑
(Ei,Ej)∈G(∆)
Tr PiAPjρ . (7.14)
SinceA is hermitian and the time-evolution is unitary, the numbers z∆fulfill the relation z−∆=z∆, where the overline denotes complex conjugation.
With this expression at hand, we can now develop a basic understanding of how equi- libration comes about. In a large quantum system, the number of different gaps∆ grows exponentially with the system size. For generic observables and states, we can therefore expect that the number of terms in the sum in∆Aρ(t)is huge. We can visualize this, by drawing a point for eachz∆in the complex plane. We then obtain a cloud of points in the complex plane, each rotating with a different angular velocity∆ on a circle of radius|z∆|
A Q UA N T U M O F T H E R M O DY N A M I C S 77
Figure 7.2: Time evolution in a system failing to equilibrate. We show the exact time evolution of∆A(t)ρ. The model is the transverse field Ising defined as in Fig. 7.1 onL=15 sites, but with parameters J=1, hx =0.5, hz= −1.05. The initial stateΨ is a product state composed of only spin-up states and the observable is a σzoperator in the middle of the chain as used in Ref. [212]. As in Fig. 7.1, the lower plot shows the evolution of the contributions to the Fourier transform of the smoothed distributionzTin the complex plane at the times marked in the evolution. We apply the same scheme as described in the caption of Fig. 7.1, again with5000 interpolation points, T≈33 and treating gaps|∆| <10−13as zero. Initially, the smoothedzTis strongly localized and anisotropic. When evolved in time, we find two distinct and large contributions that revolve around the zero without canceling out one another. These contributions stay approximately in phase and do not disperse strongly as their parts revolve with roughly the same angular velocity. This agrees well with the result shown in Fig. 7.3, which displays the distributionzTin dependence of λ and shows two distinct peaks concentrating most of the weight of the distribution. As a result, the deviation from the steady-state expectation value shows strong oscillations, which only weakly decay. (Figure from Ref. [10].)
(see Fig. 7.1). If the system is initially out of equilibrium,∆A(0) 0, this cloud of points is not distributed isotropically. Since each point moves with a different angular velocity, the cloud of points necessarily starts to spread and distribute more isotropically. But an isotropically distributed cloud of points in the complex plane has a small total absolute value, hence|∆A(t)| 1 and the system equilibrated.
Once the points are spread out approximately isotropically, we can expect that they re- main so for a long time: There are vastly more configurations which remain approximately isotropic for some time under the following time-evolution than configurations which yield a sudden anisotropic distribution again. Nevertheless, there will be rare fluctuations and, importantly, a recurrence time like in any finite system [203], which sometimes bring the system out of equilibrium again. As the system size, and hence number of points, increases, however, we can expect that the fluctuations will become increasingly rare and small and the recurrence time will diverge with the system size, explaining why in large systems, equilibration is essentially perfect.
In finite systems with local interactions, besides the recurrence time there is an ad- ditional time-scale which yields a recurrence-like behaviour and which dominates in any finite-system numerics: information can be transported ballistically and when it reaches the boundaries of the system it is backscattered. This can be seen in∆A(t)as a perturbation on time-scales which increases linearly with the system size.
On the other hand, the timeteqit takes until the system has equilibrated to precision e,
i.e.,|∆Aρ(teq)| ≤e, essentially only depends on the distribution of∆ and the shape of the
distribution of thez∆as a function of∆. If these distributions become roughly independent of the system size as it increases, also the equilibration timeteq should become roughly
independent of the system size as the system size increases. Similarly, we can then expect that for a fixed time larger than teq, the size e of the remaining oscillations of ∆Aρ(t)
should decrease with the system size, while the recurrence-time increases with the system size.
To make these claims clearer, it is useful to consider a very simple toy-model for the mechanism described above. The assumptions that we will make for this toy-model can be motivated from the general results in many-body theory that we will review in the later sections. Let us choose a large number N of gaps∆i uniformly at random from some
interval[−∆max,∆max](this choice will be motivated in sections 7.5.1 and 7.5.3) and for
G1/τ(∆)with variance1/τ ∆max. In the following we assume that we chose a typical
distribution of the gaps∆i. We then normalize thez∆ito ensure∑iz∆i =:∆ ˜A(0)ρis fixed
and of order one (independent ofN)3. Here, we have invented an imaginary observable ˜A
3With this construction,
∑iz∆ican be a complex
number. But its imagi- nary part will be extremely small for largeN, we hence ignore this problem.
and a state ρ that realize the distribution of z∆i. Of course this is only to keep the notation in analogy to the previous and the subsequent sections.
As we increaseN, the time-dependent expectation∆ ˜A(t), given by ∆ ˜A(t)ρ=
∑
i
z∆iei∆it, (7.15)
then approximates to higher and higher precision the Fourier-transform of the Gaussian
G1/τ(∆), since we are essentially sampling the corresponding integral. Thus
∆ ˜A(t)ρ≈
Z
G1/τ(∆)ei∆td∆. (7.16) This approximation will have some error, and for any given error e there will be a maximum timeTN(e)such that this approximation remains valid. For small timest<TN(e)we then
get
∆ ˜A(t)ρ≈e∆ ˜A(0)e−(t/τ)2, t<TN(e). (7.17)
The timeTN(e)increases with increasing N for any fixed e. We can hence identify the
equilibration time-scale as τ and define the equilibration time teq as teq = Cτ for any
constantC>0 which controls the precision of equilibration that we demand.
In more realistic set-ups, the distribution ofz∆also has features on smaller scales than 1/τ, resulting in correspondingly longer equilibration times. However, many of these features can be expected to have small weight, and hence only yield small and slow oscil- lations before they equilibrate. Importantly, while the equilibration times for these features is much longer, they are still largely independent of N as long as the distribution z∆ is essentially independent ofN. In particular, they should not diverge with the system size. The actual problem of explaining equilibration is therefore to explain when and why the distribution ofz∆becomes essentially independent of the system size for large systems.
In the previous discussion, we used a very simplistic model with various implicit as- sumptions. With this intuition we can now start to discuss the problem of equilibration in more detail and develop some formal machinery. We then discuss the various assump- tions that go into the analysis and connect them with known rigorous results in many-body theory.