2.3 Visualización de figuras geométricas
2.3.2 Reconocimiento de representaciones geométricas
As we have seen before, to sensibly discuss equilibration we need to consider different time-scales and always have to consider a finite precision of equilibration: The system will only equilibrate to some finite precision and only stay equilibrated for a finite but long time (which we expect to diverge with the system size). In the following, we therefore only ask whether equilibration occurs to precision e before some chosen cut-off time T. Both quantities in general depend on the system size and we expect that for a fixed e, the cut-off timeT diverges with the system size.
We can formalize this procedure by introducing regularized quantities. The crucial quantity that governs the dynamics of the chose observable is the distribution of thez∆, which is a discrete distribution for any finite system. We now regularize this quantity into a smooth distribution by convoluting it with a Gaussian of variance1/T, where T is the cut-off time. We hence define the function
zT(λ):=
∑
06=∆∈Gaps(H)
z∆G1/T(λ−∆). (7.18)
Note that we can recover the non-reguralized distribution as
z∞(λ) =
∑
06=∆∈Gaps(H)
A Q UA N T U M O F T H E R M O DY N A M I C S 79
Figure 7.3: System size scal- ing of the regularizedzTdis- tribution. At all system sizes the distributionzTis again evaluated at5000 points in- terpolating the extremal gaps linearly forT ≈ 33. Top: Equilibrating model discussed in Fig. 7.1. With growing sys- tem size, the resulting distri- butionzTspreads its weights more and more evenly leading to the equilibrating behav- ior of the considered system as shown in Fig. 7.1. Bot- tom:non-equilibrating model discussed in Fig. 7.2. The resulting distributionzTcon- centrates most of the weight in two localized peaks which yield the non-equilibrating be- havior shown in Fig. 7.2. (Fig- ure adapted from Ref. [10].)
With this notation we havezT(λ) = [z∞∗ G1/T](λ), where∗denotes the convolution op-
erator. The deviation from equilibrium∆A(t)ρcan then be bounded, using the convolution theorem of Fourier analysis, as
|∆A(t)ρ| ≤ Z zT(λ)eiλtdλ + Z (z∞(λ) − [z∞∗ G1/T](λ))eiλtdλ ≤ Z zT(λ)eiλtdλ +2kAk 1−e −(t/T)2 ≈ |∆AT(t)ρ| +2kAk (t/T)2, t T, (7.20)
with the regularized deviation from equilibrium
∆AT(t)ρ:=
Z
zT(λ)eiλtdλ. (7.21)
The regularized quantity ∆AT(t)ρalways decays to zero on the time-scale of the cut-off
timeT. It is therefore essential that we only consider times much smaller than T once we use this regularized quantity.
Once we have introduced regularized quantities, we can meaningfully compare different system sizes. In particular we can formalize what it means that the distribution of z∆ becomes independent of the system sizeN by saying that there exists a bounded function
λ7→z(λ)such that
lim
T→∞N→∞lim Z
|zT(λ) −z(λ)|dλ=0. (7.22)
We will later see (in section 7.4.1) that if such a function z(λ)exists for a local observ-
able and a local Hamiltonian, then equilibration (in a time independent of the system size) follows.
To see an example of such behaviour, see Figs. 7.1 and 7.3 (top). Fig 7.1 shows the time- evolution of a local observable in an equilibrating system and the corresponding dynamics of thez∆i. Fig 7.3 (top) shows the behaviour ofzT(λ)in the same system for different
system sizes, indicating the emergence of a well-defined functionz(λ).
Conversely, we can also argue that equilibration is expected to fail if λ 7→ z(λ)is not
bounded. We now argue that in this case there will be remaining oscillations with finite amplitude for all times. This can be seen in the following way. For any cut-off timeT and
for any system size we have ∆A(0)ρ=
∑
06=∆∈Gaps(H) z∆= Z zT(λ)dλ. (7.23)Hence,z(λ)needs to have a finite integral, but is unbounded by assumption. The function
will therefore have a finite contribution to the integral from some singularities. Such singu- larities concentrate a finite weight into an arbitrarily small region. As a consequence, they lead to a non-dispersing evolution of thez∆. In the simplest example, assume that these contributions originate from a finite number of δ-distributions. Since z(λ) =z(−λ), they
need to come in pairs of gaps{−∆i,∆i}. In the time-evolution this leads to a contribution
of the form ∆A(t)ρ t→∞ −→
∑
i ricos(∆it), (7.24)withrisome real constants.
An example of a system that seems to show such a behaviour was found in Ref. [212] and its dynamics is illustrated in Figs. 7.2 and Fig. 7.3 (bottom). This example is given by a non-integrable, translational invariant, and local Hamiltonian with a local observable and a pure product-state as initial state. One would hence usually expect that the system equilibrates. In the numerics presented here and in Ref. [212] this does not seem to be the case. However, there is some debate about whether the oscillations indeed persist forever in the infinite system or whether they can be understood to originate from quasi-particles with very long, but finite lifetime [214, 215].
7.4.1
A simple argument from harmonic analysis
Let us now argue that equilibration of a local observable in a time independent of the system size is inevitable if the dynamics is generated by a local Hamiltonian andz(λ)exists as a
bounded function. To do this, we make use of recent rigorous results in many-body theory, which show that local observables can only connect energy-eigenstates corresponding to energy-eigenvalues that differ by a small amount. If a local Hamiltonian has eigenstates
|Eii, then [216] hEi|A Ej ≤ kAke−α(|Ei−Ej|−2R). (7.25) Here, α>0 and R>0 are constants independent of the system size and R is proportional to the size of the region on whichA acts.
In the case of a generic, interacting Hamiltonian, neither the gaps∆ nor the eigenvalues Ei are expected to be highly degenerate (potentially after restricting to a super-selection
or symmetry sector). We then see from (7.14) and (7.25) that for large∆, all z∆ have to fall off exponentially, independent of the system size. Hence, alsoz(λ)has to fall off
exponentially for large|λ|. The functionz(λ)therefore has the following properties:
1. It is bounded (by assumption), 2. it has a finite integral,
3. and it falls off exponentially with|λ|.
These three properties together imply that the function is absolutely integrable,R |z(λ)|dλ<
∞. The rest of the argument is then essentially given by the Riemann-Lebesgue Lemma [217]. An absolutely integrable function can always be approximated, to an arbitrary error
δ>0, by a smooth function of compact support gδ:
Z
|gδ(λ) −z(λ)|dλ<δ. (7.26)
Then the Fourier transform ofz(λ), which gives∆A(t)ρin the thermodynamic limit, can
be approximated by the Fourier-transform ˆgδofgδ:
|∆A(t)ρ| ≤ Z gδ(λ)e iλtdλ +δ= |ˆgδ(t)| +δ. (7.27)
A Q UA N T U M O F T H E R M O DY N A M I C S 81
Figure 7.4: Illustration of finite-size effects: Equilibration in theXX chain with Hamiltonian HXX=∑Lj=1(σxjσj+1x +σjyσyj)onL=15 sites with periodic boundary conditions. The top-left plot shows the evolution of the deviation of the instantaneous expectation value from the infinite time average of a σzoperator acting on the first site. The initial state is a charge density wave state, i.e., |Ψi = |1, 0, 1, 0, . . . , 0, 1i, with|0iand|1i denoting the spin up and down state respectively. Up to timet≈3, the system seems to equilibrate to an expectation value such that∆A(t)Ψ≈0.1. Thus, the physically relevant equilibrium value in the thermodynamic limit does not coincide with the time-average in the finite system. The right plot shows a finite-size scaling of the actual expectation valuehA(t)iΨ, indicating that finite-size effects indeed seem to become relevant at about t≈3. The bottom-left panel again shows the contribution to the Fourier transform of zTby plotting the evolution ofzT(λi)in the complex plane at the times marked in the evolution, whereT≈33 and λiinterpolate the between the larges and smallest gap in5000 steps. Again gaps of the size|∆| <10−13are considered to be zero and discarded in order to account for the subtraction of the steady-state value. (Figure adapted from Ref. [10].)
But the Fourier-transform of a compactly supported smooth function decays faster than any power. Therefore, for any fixed precision of equilibration δ, t 7→ ∆A(t)ρequilibrates to
this precision faster than any power. This means that for any δ>0 and k∈N, there exists
a constantCk(δ)such that
|∆A(t)ρ| ≤min kAk,Ck(δ) tk +δ. (7.28)
The constants Ck(δ) can, however, be very large and in fact diverge for δ → 0. This
explains why this bound is not in conflict with equilibration in terms of a slow power-law like t−1/2, as seen, for example, in integrable models (see, for example, [189, 196, 199, 200]).
7.4.2
A brief comment on finite-size effects
The above results concern the dynamics in the thermodynamic limit. In numerical inves- tigations, these dynamics are in principle not accessible exactly, but one has to restrict oneself to finite systems. It is then important to keep in mind how the two settings are related. Suppose an observable equilibrates in a timeteq(w.r.t. some suitable state) to the
value Aeqin the thermodynamic limit. Now suppose that one runs a numerical situation
on a large but finite system (with the corresponding reduced state as initial state). It then follows from Lieb-Robinson bounds that the observable should also evolve to the (approx- imately) the same valueAeqin timeteq. However, in the finite system, this valueAeqdoes
notnecessarily coincide with the equilibrium value in the finite system, which is given by the time-average ofhA(t)iρon the finite system. The difference between these two values will become arbitrarily small when one considers increasingly large systems, but can be observably large in system sizes that are amenable to numerical simulations.
So, while it is true that if a finite system equilibrates, it equilibrates to the time-average (as discussed in section 7.1), this equilibrium value might not be the physically relevant one, if one is interested in what would happen in arbitrarily large systems. For an example of such behaviour, see Fig. 7.4.