Eigenstate Thermalization Hypothesis

This project presents numerical evidence supporting the Eigenstate Thermalization Hypothesis (ETH) in the non-integrable Ising spin chain and illustrates some of its main consequences. It supports my undergraduate research on irreversibility in quantum mechanics, advised by Prof. Renato Angelo at UFPR.

What is ETH?

A brief summary follows. For further details, see the references below.

The problem of thermalization

At first glance, nothing in quantum theory explains how thermalization occurs. Given a Hamiltonian eigenbasis ${|n\rangle}$ with eigenvalues ${E_n}$ and an initial state $|\psi\rangle=\sum_n C_n |n\rangle$, quantum mechanics tells us that

$$\langle O \rangle (t) = \langle \psi(t)|O|\psi(t)\rangle = \sum_n |C_n|^2O_{nn}+\sum_{n,\ m \neq n}C_m^* C_n O_{mn} e^{i(E_m-E_n)t},$$

where $O_{mn}=\langle m|O|n\rangle$. Averaged over long times, this tends to

$$\overline{\langle O \rangle}=\lim_{t_0 \to \infty}\frac{1}{t_0}\int_0 ^{t_0}\langle O \rangle(t)\ dt=\sum_n |C_n|^2O_{nn}.$$

However, statistical mechanics states that the expected value of an observable must equal that predicted by the microcanonical ensemble, that is,

$$\langle O \rangle_{\textrm{micro}}(E) = \frac{1}{N_{E,\ \Delta E}} \sum_{\substack{n \ |E - E_n| < \Delta E}}O_{nn},$$

i.e., all eigenstates inside the system's energy window are equally weighted. So how is it that the long-time average of an observable equals the microcanonical prediction?

ETH

ETH proposes that, in fact, for a local observable each eigenstate is already approximately thermal. That is,

$$\langle n|O|n\rangle=O_{nn}\approx \langle O \rangle_{\textrm{micro}}(E_n),$$

i.e., the diagonal matrix elements (in the Hamiltonian eigenbasis) already agree with the microcanonical prediction. According to Rigol et al. [1], coherence initially "hides" this eigenstate thermalization, but time reveals it through dephasing. ETH can be seen as an ansatz [2], [3] for the observable's matrix elements in the Hamiltonian eigenbasis:

$$O_{mn}=\langle O \rangle_{\mathrm{micro}}(\overline{E}) \delta_{mn} + e^{-S(\overline{E})/2} f_O(\overline{E}, \omega) R_{mn},$$

where

• $\overline{E}=(E_m +E_n)/2$;
• $S(\cdot)$ is the thermodynamic entropy;
• $\omega=E_n-E_m$;
• $f_O(\cdot)$ is a smooth function of its arguments;
• $R_{mn}$ is a random number with zero mean and unit variance.

It can be shown (see Ref. [3]) that this ansatz leads to the long-time average of the temporal fluctuations decaying exponentially with system size, that is,

$$\sigma_O ^2 \equiv \lim_{t_0 \to \infty}\frac{1}{t_0} \int _0 ^{t_0}\left[\langle O \rangle(t) - \overline{\langle O \rangle}\right]^2 dt \propto e^{-S(\overline{E})}.$$

Implementation

The Hamiltonian used for $L$ spins is

$$H = -J\sum _{i=1} ^{L-1} \sigma^z_i \sigma^z _{i+1} - \sum _{i=1} ^L \left(h\sigma^x_i+ g\sigma^z_i \right),$$

with the conventionally non-integrable parameters $J=1.0$, $g=0.5$, and $h=1.05$. Taking the observable to be

$$O=\frac{1}{L}\sum_{i=1}^{L}\sigma^x_i,$$

i.e., the mean magnetization of the system. The distribution of diagonal elements in the Hamiltonian eigenbasis, as a function of energy, is shown below.

The diagonal elements already cluster around the microcanonical mean with a small fluctuation, in accordance with the ETH ansatz. This fluctuation was computed at the center of the spectrum (standard deviation of the diagonal elements in that region) as a function of system size, and it decays exponentially with system size, as shown below.

The ansatz also makes a prediction about the off-diagonal elements: they must decrease exponentially with system size. The heatmap below shows the diagonal elements in contrast to the off-diagonal ones; the latter remain small compared to the former, in agreement with the ansatz.

Finally, the time evolution of the system is examined below. The figure shows the expectation value of the observable (top) and the entanglement entropy (bottom). The expectation value relaxes to nearly zero and remains there, with small fluctuations, after a transient time. The entanglement entropy (the von Neumann entropy of the reduced state) is also shown: it evidences that the system behaves as a thermal bath for itself. Even though the von Neumann entropy of a pure state is invariant under unitary evolution, that of a reduced state is not. Once you trace out part of the system, information about that subsystem scrambles into the rest of the system, increasing the reduced state's entropy — in this case, tracking only the first spin's magnetization while tracing out the rest.

How to run

This project uses Julia's built-in package manager to keep the environment reproducible.

1. Install Julia.
2. Clone this repository and enter its folder.
3. Instantiate the project environment:

   julia --project=. -e 'using Pkg; Pkg.instantiate()'

4. Run the main script:

   julia --project=. scripts/plots.jl

Project Structure

• src/
  • hamiltonians.jl # Hamiltonian construction and diagonalization
  • observables.jl # observable computation and statistics
• scripts/
  • plots.jl # runs the simulation and generates figures
• figures/ # figures used in this README
• Project.toml
• Manifest.toml

References

1. Rigol, M., Dunjko, V., & Olshanii, M. (2008). Thermalization and its mechanism for generic isolated quantum systems. Nature, 452(7189), 854–858.
2. Srednicki, M. (1999). The approach to thermal equilibrium in quantized chaotic systems. J. Phys. A: Math. Gen. 32, 1163.
3. D'Alessio, L., Kafri, Y., Polkovnikov, A., & Rigol, M. (2016). From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Advances in Physics, 65(3), 239-362.

Author

Gabriel Boldo Martins — undergraduate research on irreversibility in quantum mechanics, advised by Prof. Renato Angelo, UFPR.