Unified Theory of Local Quantum Many-Body Dynamics: Eigenoperator Thermalization Theorems

Explaining quantum many-body dynamics is a long-held goal of physics. A rigorous operator algebraic theory of dynamics in locally interacting systems in any dimension is provided here in terms of time-dependent equilibrium (Gibbs) ensembles. The theory explains dynamics in closed, open, and time-dependent systems, provided that relevant pseudolocal quantities can be identified, and time-dependent Gibbs ensembles unify wide classes of quantum nonergodic and ergodic systems. The theory is applied to quantum many-body scars, continuous, discrete, and dissipative time crystals, Hilbert space fragmentation, lattice gauge theories, and disorder-free localization, among other cases. Novel pseudolocal classes of operators are introduced in the process: projected-local, which are local only for some states, cryptolocal, whose locality is not manifest in terms of any finite number of local densities, and transient ones, that dictate finite-time relaxation dynamics. An immediate corollary is proving saturation of the Mazur bound for the Drude weight. This proven theory is intuitively the rigorous algebraic counterpart of the weak eigenstate thermalization hypothesis and has deep implications for thermodynamics: Quantum many-body systems “out of equilibrium” are actually always in a time-dependent equilibrium state for any natural initial state. The work opens the possibility of designing novel out-of-equilibrium phases, with the newly identified scarring and fragmentation phase transitions being examples.

Popular Summary

Computing nonequilibrium dynamics of locally interacting quantum many-body systems is a fundamental long-standing challenge in physics with applications ranging from condensed matter physics to quantum information processing. Most of these systems are completely intractable to any analytical methods, and numerical calculations are limited to small systems and short times. In this paper, I unify the dynamics of all such possible systems within a newly developed rigorous theory. I show, counterintuitively, that quantum many-body systems are always in an equilibrium state but with time-dependent chemical potentials, which allows for analytical solutions of long-time dynamics.

The eigenstate thermalization hypothesis (ETH) has been the theoretical cornerstone underpinning nonequilibrium quantum physics for several decades. However, not only has ETH never been proven, but also several models are known counterexamples to its “strong” version. I prove the “weak” version of ETH—a version stating that the time averages of the dynamics are “thermalized,” in a sense—and show that the time averages of observables are given by equilibrium states. Going far beyond ETH, I also prove that the dynamics at finite frequencies is given by generalized local conservation laws that are also at finite frequency. Moreover, the system’s state is always an equilibrium state defined by these conservation laws. This state analytically solves the dynamics of otherwise intractable chaotic models.

This work provides a fundamental rigorous framework for studying nonequilibrium quantum systems with numerous possible research directions, from designing novel quantum error-correcting algorithms to promoting existing powerful theories used in equilibrium physics to the nonequilibrium setting.

Figure 1

An illustration of generic nonequilibrium quantum many-body dynamics. The system starts off in a clustering state (with finite power-law correlations at most) and then proceeds through an Gibbs-like state with transient pseudolocal quantities and corresponding temperatures (or chemical potentials) before thermalizing to a Gibbs ensemble.

Figure 2

An illustration of nonlocality vs different kinds of pseudolocality. The shown subsystem is of arbitrary size $\mathrm{\Lambda }$. First case (nonlocal): $A$ has support on the entire subsystem for all sizes of the subsystem (e.g., a projector to vacuum state). It is not pseudolocal for the infinite-temperature state. Second case (local): $A$ is a simple sum of operators that act only on one site (translated to site $x$). It is pseudolocal for any state with clustering. Third case (projected-local): A local operator (second case) is sandwiched between a projector $P$ (that acts on the entire subsystem), and the clustering state is entirely inside the subspace $P$ projects to. It is pseudolocal for such a state because the state does not “see” the projector. Fourth case (cryptolocal): $A$ contains terms (denoted $R$) that act on the entire subsystem for all size of the subsystem, but they cancel and they are not visible in the size of $A$ for, e.g., the infinite-temperature state.

Figure 3

The growth of $⟨{\tilde{A}}_1{\tilde{A}}_{-1}⟩/V$ with $1/\mu$ and $V$ which can be quite slow, indicating that oscillations can persist beyond the scarring phase for very large systems and times even for finite $\mu$. After some system size that depends on $\mu$, $⟨{\tilde{A}}_1{\tilde{A}}_{-1}⟩/V$ grows exponentially with $V$, and the oscillations are no longer present after that system size. Note that results are the same for $\mu \to -\mu$.

Figure 4

Finite-frequency averages of $J_x^x$. For finite $\mu$, the results are valid at times $1\ll t\ll V$. The magenta line at infinite ${\mu }_0$ agrees with the exact solution (dashed black line) from previous literature [111]. We see that the oscillation amplitudes display an almost discontinuous dependence on system size $V$—they are constant up to some “critical” system size, after which they decay abruptly to 0. In other words, for a fixed system size, the system is effectively in the scarred phase up to some value of the initial chemical potential, after which the oscillations abruptly decay.

Figure 5

The time-averaged value of $S_x^z$ ($S_x^y{S}_{x+1}^y$) are given in (a) [(b)] as a function of the external fields $h$, $d$. We are close to infinite temperature because the chemical potential in the initial state is ${\mu }_0=0$. Hence, the values are small, but there is still a manifest nonlinear dependence.

Figure 6

(a) The fragmentation phase diagram showing the phases where the strings are thermodynamically irrelevant (short-range order) and where they are not. (b) Time average of certain local observables for various values of the initial state parameter [see Eq. (87) for the initial state parameters].