Nonequilibrium phases and phase transitions

Nonequilibrium phases and phase transitions

Recent experimental progress has brought into existence so-called quantum simulators such as ultracold atoms in optical lattices or trapped ions. These act like an 'analogue' quantum computer by mimicking the behaviour of quantum systems whose properties we are interested in.

In particular, these quantum simulators can be used to study in a controlled way the nonequilibrium dynamics of closed quantum many-body systems. For such nonequilibrium quantum states, the thermodynamic concept of a free energy is not applicable. On the one hand this implies the absence of organizing principles such as the minimization of free energies making a systematic understanding of such states challenging. On the other hand these states promise to show new properties not constrained by the principles of equilibrium thermodynamics.

A natural question is to which extent there are phases of matter without the existence of a free energy? Can these systems exhibit universal behavior independent of microscopic details similar to what happens at equilibrium continuous phase transitions?

We address such questions and develop concepts that allow for a systematic understanding not only for individual problems, but rather for classes of phenomena. This includes the theory of dynamical quantum phase transitions or the construction of dynamical potentials for eigenstate phases. For more details on current and recent research highlights see the collection below.

Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks
Markus Schmitt and Markus Heyl

The efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter constitutes a key challenge for current computational methods. This holds in particular in the regime of two spatial dimensions, whose experimental exploration is currently pursued with strong efforts in quantum simulators. In this work we present a versatile and efficient machine learning inspired approach based on a recently introduced artificial neural network encoding of quantum many-body wave functions. We identify and resolve key challenges for the simulation of time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, we study the dynamics of the paradigmatic two-dimensional transverse-field Ising model, as recently also realized experimentally in systems of Rydberg atoms. Calculating the nonequilibrium real-time evolution across a broad range of parameters, we, for instance, observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached timescales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.

Phys. Rev. Lett. 125, 100503 (2020)

Exceptional points and the topology of quantum many-body spectra
David J. Luitz and Francesco Piazza

We show that in a generic, ergodic quantum many-body system the interactions induce a nontrivial topology for an arbitrarily small non-Hermitian component of the Hamiltonian. This is due to an exponential-in-system-size proliferation of exceptional points which have the Hermitian limit as an accumulation (hyper)surface. The nearestneighbor level repulsion characterizing Hermitian ergodic many-body systems is thus shown to be a projection of a richer phenomenology, where actually all the exponentially many eigenvalues are pairwise connected in a topologically robust fashion via exceptional points.

Phys. Rev. Research 1, 033051 (2019)

Dynamical potentials for nonequilibrium quantum many-body phases
Sthitadhi Roy, Achilleas Lazarides, Markus Heyl, and Roderich Moessner

Out of equilibrium phases of matter exhibiting order in individual eigenstates, such as many-body localized spin glasses and discrete time crystals, can be characterized by inherently dynamical quantities such as spatiotemporal correlation functions. In this paper, we introduce dynamical potentials which act as generating functions for such correlations and capture eigenstate phases and order. These potentials show formal similarities to their equilibrium counterparts, namely thermodynamic potentials. We provide three representative examples: a disordered XXZ chain showing many-body localization, a disordered Ising chain exhibiting spin-glass order, and its periodically-driven cousin exhibiting time-crystalline order.

Phys. Rev. B 97, 205143 (2018)

Odd elasticity
C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, V. Vitelli

A passive solid cannot do work on its surroundings through any quasistatic cycle of deformations. This property places strong constraints on the allowed elastic moduli. In this Article, we show that static elastic moduli altogether absent in passive elasticity can arise from active, non- conservative microscopic interactions. These active moduli enter the antisymmetric (or odd) part of the static elastic modulus tensor and quantify the amount of work extracted along quasistatic strain cycles. In two-dimensional isotropic media, two chiral odd-elastic moduli emerge in addition to the bulk and shear moduli. We discuss microscopic realizations that include networks of Hookean springs augmented with active transverse forces and non-reciprocal active hinges. Using coarse-grained microscopic models, numerical simulations and continuum equations, we uncover phenomena ranging from auxetic behaviour induced by odd moduli to elastic wave propagation in overdamped media enabled by self-sustained active strain cycles. Our work sheds light on the non-Hermitian mechanics of two- and three-dimensional active solids that conserve linear momentum but exhibit a non-reciprocal linear response.

Nature Physics 16, 475 (2020)

Many-body dynamical localization in the kicked Bose-Hubbard chain
Michele Fava, Rosario Fazio, and Angelo Russomanno

We provide evidence that a clean kicked Bose-Hubbard model exhibits a many-body dynamically localized phase. This phase shows ergodicity breaking up to the largest sizes we were able to consider. We argue that this property persists in the limit of large size. The Floquet states violate eigenstate thermalization and then the asymptotic value of local observables depends on the initial state and is not thermal. This implies that the system does not generically heat up to infinite temperature, for almost all the initial states. Differently from many-body localization here the entanglement entropy linearly increases in time. This increase corresponds to space- delocalized Floquet states which are nevertheless localized across specific subsectors of the Hilbert space: In this way the system is prevented from randomly exploring all the Hilbert space and does not thermalize.

Phys. Rev. B 101, 064302 (2020)

The Kibble-Zurek mechanism at exceptional points
Balazs Dora, Markus Heyl, Roderich Moessner

Exceptional points (EPs) are ubiquitous in non-Hermitian systems, and represent the complex counterpart of critical points. By driving a system through a critical point at finite rate induces defects, described by the Kibble-Zurek mechanism, which finds applications in diverse fields of physics. Here we generalize this to a ramp across an EP. We find that adiabatic time evolution brings the system into an eigenstate of the final non-Hermitian Hamiltonian and demonstrate that for a variety of drives through an EP, the defect density scales as τ-(d + z)ν/(zν + 1) in terms of the usual critical exponents and 1/τ the speed of the drive. Defect production is suppressed compared to the conventional Hermitian case as the defect state can decay back to the ground state close to the EP. We provide a physical picture for the studied dynamics through a mapping onto a Lindblad master equation with an additionally imposed continuous measurement.

Nat. Commun. 10, 2254 (2019)

Quantum paracrystalline shear modes of the electron liquid
J. Y. Khoo, P.-Y. Chang, F. Pientka, I. Sodemann

Unlike classical fluids, a quantum Fermi liquid can support a long-lived and propagating shear sound wave at arbitrarily small wave vectors and frequencies, reminiscent of the transverse sound in crystals, despite lacking any form of long-range crystalline order. This mode is expected to be present in moderately interacting metals where the quasiparticle mass is renormalized to be more than twice the bare mass in two dimensions (2D), but it has remained undetected because it is hard to excite since it does not involve charge density fluctuations, in contrast to the conventional plasma mode. In this work we propose a strategy to excite and detect this unconventional mode in clean metallic channels. We show that the shear sound is responsible for the appearance of sharp dips in the ac conductance of narrow channels at resonant frequencies matching its dispersion. The liquid resonates while minimizing its dissipation in an analogous fashion to a sliding crystal. Ultraclean 2D materials that can be tuned toward the Wigner crystallization transition such as silicon metal-oxide-semiconductor field-effect transistors, MgZnO/ZnO, p- GaAs, and AlAs quantum wells are promising platforms to experimentally discover the shear sound.

Phys. Rev. B 102, 085437 (2020)

Hierarchy of Relaxation Timescales in Local Random Liouvillians
Kevin Wang, Francesco Piazza, and David J. Luitz

To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size l with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality (n=l), the complex Liouvillian spectrum densely covers a “lemon”-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019)]. However, for local Liouvillians (n<l), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit. Our findings for n locality generalize immediately to the case of spatial locality, introducing further splitting of timescales due to the additional structure.

Phys. Rev. Lett. 124, 100604 (2020)

Temperature Dependence of the Butterfly Effect in a Classical Many-Body System
Thomas Bilitewski, Subhro Bhattacharjee, and Roderich Moessner

We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, μ, and the butterfly speed, vb, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, D, and spin-autocorrelation time, τ. We find that they all exhibit power- law behavior at low temperature, consistent with scaling of the form Dvb2/μ and τ -1T. The vanishing of μ ∼ T 0.48 is parametrically slower than that of the corresponding quantum bound, μ ∼ T, raising interesting questions regarding the semiclassical limit of such spin systems.

Phys. Rev. Lett. 121, 250602 (2018)