Thermalization and its mechanism for generic isolated quantum systems

TL;DR

The paper investigates how generic isolated quantum systems relax to thermal states and tests the eigenstate thermalization hypothesis (ETH) as the underlying mechanism. Using exact diagonalization of a nonintegrable two-dimensional hard-core boson model, it shows that long-time averages coincide with microcanonical predictions and that the diagonal ensemble reproduces these results due to ETH, while an integrable system does not thermalize. It further argues that a narrow energy distribution is essential for the equivalence of diagonal and thermal ensembles, and demonstrates that thermalization can be understood at the level of individual eigenstates, with implications for foundational statistical mechanics and potential quantum-information applications.

Abstract

Time dynamics of isolated many-body quantum systems has long been an elusive subject. Very recently, however, meaningful experimental studies of the problem have finally become possible, stimulating theoretical interest as well. Progress in this field is perhaps most urgently needed in the foundations of quantum statistical mechanics. This is so because in generic isolated systems, one expects nonequilibrium dynamics on its own to result in thermalization: a relaxation to states where the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable through the time-tested recipe of statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible, in a sense analogous to that in which dynamical chaos makes classical thermalization possible. For example, dynamical chaos itself cannot occur in an isolated quantum system, where time evolution is linear and the spectrum is discrete. Underscoring that new rules could apply in this case, some recent studies even suggested that statistical mechanics may give wrong predictions for the outcomes of relaxation in such systems. Here we demonstrate that an isolated generic quantum many-body system does in fact relax to a state well-described by the standard statistical mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation and that thermalization happens instead at the level of individual eigenstates, as first proposed by J.M. Deutsch and M. Srednicki. A striking consequence of this eigenstate thermalization scenario is that the knowledge of a single many-body eigenstate suffices to compute thermal averages-any eigenstate in the microcanonical energy window will do, as they all give the same result.

Figures (6)

• Figure 1: Relaxation dynamics. a, Two-dimensional lattice on which five hard-core bosons propagate in time. The bosons are initially prepared in the ground state of the sub-lattice in the lower-right corner and released through the indicated link. b, The corresponding relaxation dynamics of the marginal momentum distribution center [$n(k_{x}=0)$] compared with the predictions of the three ensembles. In the microcanonical case, we averaged over all eigenstates whose energies lie within a narrow window (see Supplementary Discussion) $[E_0-\Delta E, E_0+\Delta E]$, where $E_0\equiv\langle \psi(0) | \widehat{H} | \psi(0) \rangle=-5.06 J$, $\Delta E=0.1J$, and $J$ is the hopping parameter. The canonical ensemble temperature is $k_{\textrm{\scriptsize B}}T=1.87 J$, where $k_{\textrm{\scriptsize B}}$ is the Boltzmann constant, so that the ensemble prediction for the energy is $E_0$. c, Full momentum distribution function in the initial state, after relaxation, and in the different ensembles. Here $d$ is the lattice constant and $L_{x}=5$ the lattice width.
• Figure 2: Thermalization in classical vs quantum mechanics. a, In classical mechanics, time evolution constructs the thermal state from an initial state that generally bears no resemblance to the former. b, In quantum mechanics, according to the eigenstate thermalization hypothesis, every eigenstate of the Hamiltonian always implicitly contains a thermal state. The coherence between the eigenstates initially hides it, but time dynamics reveals it through dephasing.
• Figure 3: Eigenstate thermalization hypothesis. a, In our nonintegrable system, the momentum distribution $n(k_x)$ for two typical eigenstates with energies close to $E_0$ is identical to the microcanonical result, in accordance with the ETH. b, Upper panel: $n(k_x=0)$ eigenstate expectation values as a function of the eigenstate energy resemble a smooth curve. Lower panel: the energy distribution $\rho(E)$ of the three ensembles considered in this work. c, Detailed view of $n(k_x=0)$ (left labels) and $|C_\alpha|^2$ (right labels) for 20 eigenstates around $E_0$. d, In the integrable system, $n(k_x)$ for two eigenstates with energies close to $E_0$ and for the microcanonical and diagonal ensembles are very different from each other, i.e., the ETH fails. e, Upper panel: $n(k_x=0)$ eigenstate expectation value considered as a function of the eigenstate energy gives a thick cloud of points rather than resembling a smooth curve. Lower panel: energy distributions in the integrable system are similar to the nonintegrable ones depicted in b. f, Correlation between $n(k_x=0)$ and $|C_\alpha|^2$ for 20 eigenstates around $E_0$. It explains why in d the microcanonical prediction for $n(k_x=0)$ is larger than the diagonal one.
• Figure 4: Temporal vs quantum fluctuations. a, Matrix elements of the observable of interest, $n(k_{x}=0)$, as a function of state indices; the eigenstates of the Hamiltonian are indexed in the order of diminishing overlap with the initial state. The dominance of the diagonal matrix elements is apparent. b, The same time evolution as in Fig. \ref{['fig1']}b with the error bars showing the quantum fluctuations $n(k_{x}=0)\pm \Delta$ with $\Delta=[\langle \widehat{n}^2(k_x=0)\rangle-\langle \widehat{n}(k_x=0)\rangle^2]^{1/2}$, which are clearly much larger than the temporal fluctuations of $n(k_{x}=0)$.
• Figure 5: The lattice for the dynamics. Two-dimensional lattice on which the particles propagate in time. The initial state is the ground state of 5 hard-core bosons confined to the sub-lattice in the lower right-hand corner, and the time evolution starts after the opening of the link indicated by the door symbol.