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Entropy

Molecules and the Eigenstate Thermalization Hypothesis

David M. Leitner

Published Sep 5, 2018• 16 citations
Publisher

Abstract

We review a theory that predicts the onset of thermalization in a quantum mechanical coupled non-linear oscillator system, which models the vibrational degrees of freedom of a molecule. A system of N non-linear oscillators perturbed by cubic anharmonic interactions exhibits a many-body localization (MBL) transition in the vibrational state space (VSS) of the molecule. This transition can occur at rather high energy in a sizable molecule because the density of states coupled by cubic anharmonic terms scales as N3, in marked contrast to the total density of states, which scales as exp(aN), where a is a constant. The emergence of a MBL transition in the VSS is seen by analysis of a random matrix ensemble that captures the locality of coupling in the VSS, referred to as local random matrix theory (LRMT). Upon introducing higher order anharmonicity, the location of the MBL transition of even a sizable molecule, such as an organic molecule with tens of atoms, still lies at an energy that may exceed the energy to surmount a barrier to reaction, such as a barrier to conformational change. Illustrative calculations are provided, and some recent work on the influence of thermalization on thermal conduction in molecular junctions is also discussed.

DOI
10.3390/e20090673

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•Jan 1, 2008•24 citations
We review a theory that predicts the onset of thermalization in a quantum mechanical coupled non-linear oscillator system, which models the vibrational degrees of freedom of a molecule. A system of N non-linear oscillators perturbed by cubic anharmonic interactions exhibits a many-body localization (MBL) transition in the vibrational state space (VSS) of the molecule. This transition can occur at rather high energy in a sizable molecule because the density of states coupled by cubic anharmonic terms scales as N3, in marked contrast to the total density of states, which scales as exp(aN), where a is a constant. The emergence of a MBL transition in the VSS is seen by analysis of a random matrix ensemble that captures the locality of coupling in the VSS, referred to as local random matrix theory (LRMT). Upon introducing higher order anharmonicity, the location of the MBL transition of even a sizable molecule, such as an organic molecule with tens of atoms, still lies at an energy that may exceed the energy to surmount a barrier to reaction, such as a barrier to conformational change. Illustrative calculations are provided, and some recent work on the influence of thermalization on thermal conduction in molecular junctions is also discussed.
We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the eigenstate thermalization hypothesis (ETH) and the resulting single-eigenstate statistical mechanics. We then focus on a class of systems that fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson-localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can forever locally remember information about their local initial conditions and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL) and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveal dynamically stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and can occur at high energy and low spatial dimensionality, where equilibrium ordering is forbidden.
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators (l∗-bits)({\mathrm{l}}^{*}\text{-bits})(l∗-bits) underlie localization in such systems. In dimensions d>1d>1d>1, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits. Physical consequences of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in d=1d=1d=1 and thermal eigenstates at all parameters in d>1d>1d>1. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of l∗-bits{\mathrm{l}}^{*}\text{-bits}l∗-bits.
The emergence of statistical mechanics from quantum dynamics is conventionally understood through quantum chaos, the Eigenstate Thermalization Hypothesis (ETH), and canonical typicality, which together explain the thermal behavior of local observables in quantum many-body systems. We provide a distinct mechanism in which, for a quantum many-body system under strong random perturbation, the chaotic behavior of the system's eigenvectors and a constraint on the total energy directly generate a canonical ensemble for the entire system. The canonical ensemble occurs as the first moment of the Porter-Thomas distribution of the system's chaotic eigenstates, and the constraint on the total energy sets the temperature in the ensemble. This yields canonically distributed expectation values without invoking ETH, a microcanonical ensemble, or a subsystem-bath partition for local and global observables. We numerically demonstrate the mechanism for the 1D transverse-field Ising model by showing canonically distributed eigenstate projectors and spin-spin correlators with distinctive temperature dependence of the correlation length. An implementation of this thermalization approach on a quantum computer can be utilized for thermal state preparation.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.