Ergodic Regime Research Articles

Magnetically Tunable Electric Dipolar Interactions of Ultracold Polar Molecules in the Quantum Ergodic Regime.

By leveraging the hyperfine interaction between the rotational and nuclear spin degrees of freedom, we demonstrate extensive magnetic control over the electric dipole moments, electric dipolar interactions, and ac Stark shifts of ground-state alkali-dimer molecules such as KRb(X^{1}Σ^{+}). The control is enabled by narrow avoided crossings and the highly ergodic character of molecular eigenstates at low magnetic fields, offering a general and robust way of continuously tuning the intermolecular electric dipolar interaction for applications in quantum simulation, quantum sensing, and dipolar spinor physics.

Slow dynamics and nonergodicity of the bosonic quantum East model in the semiclassical limit.

We study the unitary dynamics of the bosonic quantum East model, a kinetically constrained lattice model which generalizes the quantum East model to arbitrary occupation per site. We consider the semiclassical limit of large (but finite) site occupancy so that the dynamics is approximated by an evolution equationof the Gross-Pitaevskii kind. This allows us to numerically study in detail system sizes of hundreds of sites. Like in the spin-1/2 case, we find two dynamical phases, an active one of fast thermalization and an inactive one of slow relaxation and the absence of ergodicity on numerically accessible timescales. The location of this apparent ergodic to nonergodic transition coincides with the localization transition of the ground state. We further characterize states which are nonergodic on all timescales in the otherwise ergodic regime.

Quantum information geometry of driven CFTs

Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For Möbius driving by the SL(2, ℝ) subgroup, the BKM metric becomes the hyperbolic metric on the disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via Möbius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.

Ergodicity breaking in rapidly rotating C60 fullerenes.

Ergodicity, the central tenet of statistical mechanics, requires an isolated system to explore all available phase space constrained by energy and symmetry. Mechanisms for violating ergodicity are of interest for probing nonequilibrium matter and protecting quantum coherence in complex systems. Polyatomic molecules have long served as a platform for probing ergodicity breaking in vibrational energy transport. Here, we report the observation of rotational ergodicity breaking in an unprecedentedly large molecule, 12C60, determined from its icosahedral rovibrational fine structure. The ergodicity breaking occurs well below the vibrational ergodicity threshold and exhibits multiple transitions between ergodic and nonergodic regimes with increasing angular momentum. These peculiar dynamics result from the molecule's distinctive combination of symmetry, size, and rigidity, highlighting its relevance to emergent phenomena in mesoscopic quantum systems.

Tracking locality in the time evolution of disordered systems

Using local density correlation functions for a one-dimensional spin system, we introduce a correlation function difference (CFD) which compares correlations on a given site between a full system of size $L$ and its restriction to $\ell<L$ sites. We show that CFD provides useful information on transfer of information in quantum many-body systems by considering the examples of ergodic, Anderson, and many-body localized regimes in disordered XXZ spin chain. In the ergodic phase, we find that the propagation of CFD is asymptotically faster than the spin transport but slower than the ballistic propagation implied by the Lieb-Robinson bound. In contrast, in the localized cases, we unravel an exponentially slow relaxation of CFD. Connections between CFD and other observables detecting non-local correlations in the system are discussed.

Floquet-induced localization in long-range many-body systems

The fate of many-body localization in long-range interacting systems is not fully settled. For instance, the phase boundary between ergodic and many-body localized regimes is still under debate. Here, we use Floquet dynamics which can induce many-body localization in a clean long-range interacting system through spatiotemporal disorder, which are realized by regular operation of random local rotations. The phase diagram has been determined for two types of uniform and nonuniform long-range couplings. Our Floquet mechanism shows more localizing power than conventional static disorder methods as it pushes the phase boundary in favor of the localized phase. Moreover, our comprehensive long-time simulations provide strong support for obtained results based on static analysis.

On the Coexistence of eMBB and URLLC in Multi-Cell Massive MIMO

The non-orthogonal coexistence between the enhanced mobile broadband (eMBB) and the ultra-reliable low-latency communication (URLLC) in the downlink of a multi-cell massive MIMO system is rigorously analyzed in this work. We provide a unified information-theoretic framework blending an infinite-blocklength analysis of the eMBB spectral efficiency (SE) in the ergodic regime with a finite-blocklength analysis of the URLLC error probability relying on the use of mismatched decoding, and of the so-called saddlepoint approximation. Puncturing (PUNC) and superposition coding (SPC) are considered as alternative downlink coexistence strategies to deal with the inter-service interference, under the assumption of only statistical channel state information (CSI) knowledge at the users. eMBB and URLLC performances are then evaluated over different precoding techniques and power control schemes, by accounting for imperfect CSI knowledge at the base stations, pilot-based estimation overhead, pilot contamination, spatially correlated channels, the structure of the radio frame, and the characteristics of the URLLC activation pattern. Simulation results reveal that SPC is, in many operating regimes, superior to PUNC in providing higher SE for the eMBB yet achieving the target reliability for the URLLC with high probability. Moreover, PUNC might cause eMBB service outage in presence of high URLLC traffic loads. However, PUNC turns to be necessary to preserve the URLLC performance in scenarios where the multi-user interference cannot be satisfactorily alleviated.

Many-body localization and the area law in two dimensions

We study the high-energy phase diagram of a two-dimensional spin-$\frac{1}{2}$ Heisenberg model on a square lattice in the presence of either quenched or quasiperiodic disorder. The use of large-scale tensor network numerics allows us to compute the bipartite entanglement entropy for systems of up to $60 \times 7$ lattice sites. We provide evidence for the existence of a many-body localized regime for large disorder strength that features an area law in excited states and that violates the eigenstate thermalization hypothesis. From a finite-size analysis, we determine an estimate for the critical disorder strength where the transition to the ergodic regime occurs in the quenched case.

Extreme value statistics of positive recurrent centrally biased random walks

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.

Modeling sample-to-sample fluctuations of the gap ratio in finite disordered spin chains

We study sample-to-sample fluctuations of the gap ratio in the energy spectra in finite disordered spin chains. The chains are described by the random-field Ising model and the Heisenberg model. We show that away from the ergodic-nonergodic crossover, the fluctuations are correctly captured by the Rosenzweig-Porter (RP) model. However, fluctuations in the microscopic models significantly exceed those in the RP model in the vicinity of the crossover. We show that upon introducing an extension to the RP model, one correctly reproduces the fluctuations in all regimes, i.e., in the ergodic and nonergodic regimes as well as at the crossover between them. Finally, we demonstrate how to reduce the sample-to-sample fluctuations in both studied microscopic models.

Long-time memory effects in a localizable central spin problem

We study the properties of the Nakajima–Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e. disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract t → ∞ populations for the qubit from finite time (Jt ⩽ 102) data in the thermalizing phase. This allows us to quantify how the long-time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complement to state-of-the-art numerical methods, for which the long-time dynamics with large baths are impossible to simulate in this phase. Additionally, our numerics on finite baths reveal the possibility for unbounded exponential growth in the memory kernel, a phenomenon rooted in the appearance of exceptional points in the projected Liouvillian governing the reduced dynamics. In small systems amenable to exact numerics, we find that these pathologies may have some correlation with delocalization.

Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains

We analyze the finite-size scaling of the average gap-ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin-chain with quasi-periodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of Berezinskii-Kosterlitz-Thouless (BKT) transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sub-linear drift with the system-size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.

Influence functional of many-body systems: Temporal entanglement and matrix-product state representation

Feynman–Vernon influence functional (IF) was originally introduced to describe the effect of a quantum environment on the dynamics of an open quantum system. We apply the IF approach to describe quantum many-body dynamics in isolated spin systems, viewing the system as an environment for its local subsystems. While the IF can be computed exactly only in certain many-body models, it generally satisfies a self-consistency equation, provided the system, or an ensemble of systems, are translationally invariant. We view the IF as a fictitious wavefunction in the temporal domain, and approximate it using matrix-product states (MPS). This approach is efficient provided the temporal entanglement of the IF is sufficiently low. We illustrate the broad applicability of the IF approach by analyzing several models that exhibit a range of dynamical behaviors, from thermalizing to many-body localized. In particular, we study the non-equilibrium dynamics in the quantum Ising model in both Floquet and Hamiltonian settings. We find that temporal entanglement entropy may be significantly lower compared to the spatial entanglement and analyze the IF in the continuous-time limit. We simulate the thermodynamic-limit evolution of local observables in various regimes, including the relaxation of impurities embedded in an infinite-temperature chain, and the long-lived oscillatory dynamics of the magnetization associated with the confinement of excitations. Furthermore, by incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF of a bond-disordered kicked Ising chain. In this case, we find that the temporal entanglement entropy scales logarithmically with evolution time. The IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems to quantum statistical physics.

From operator statistics to wormholes

For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper we analyse the time dependence of correlation functions of operators. We study them in the ergodic limit as well as their approach to the ergodic limit which is controlled by non-universal massive modes. An effective field theory (EFT) corresponding to the causal symmetry and its breaking describes the ergodic phase. We demonstrate that the resulting Goldstone-mode theory has a topological expansion, analogous to the one described in arXiv:2008.02271 with added operator sources, whose leading non-trivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behaviour of operators in our EFT is seen to result from a combination of RMT-like spectral statistics and Haar averaging over wave-functions. Furthermore we analytically capture the plateau behaviour by taking into account the contribution of a second saddle point. Our main interest are quantum many-body systems with holographic duals and we explicitly establish the validity of the EFT description in the SYK-class of models, starting from their microscopic description. By studying the tower of massive modes above the Goldstone sector we get a detailed understanding of how the ergodic EFT phase is approached and derive the relevant Thouless time scales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby-universes.

Publisher’s Note

Feynman-Vernon influence functional (IF) was originally introduced to describe the effect of a quantum environment on the dynamics of an open quantum system. We apply the IF approach to describe quantum many-body dynamics in isolated spin systems, viewing the system as an environment for its local subsystems. While the IF can be computed exactly only in certain many-body models, it generally satisfies a self-consistency equation, provided the system, or an ensemble of systems, are translationally invariant. We view the IF as a fictitious wavefunction in the temporal domain, and approximate it using matrix-product states (MPS). This approach is efficient provided the temporal entanglement of the IF is sufficiently low. We illustrate the versatility of the IF approach by analyzing several models that exhibit a range of dynamical behaviors, from thermalizing to many-body localized. In particular, we study the non-equilibrium dynamics in the quantum Ising model in both Floquet and Hamiltonian settings. We find that temporal entanglement entropy may be significantly lower compared to the spatial entanglement and analyze the IF in the continuous-time limit. We simulate the thermodynamic-limit evolution of local observables in various regimes, including the relaxation of impurities embedded in an infinite-temperature chain, and the long-lived oscillatory dynamics of the magnetization associated with the confinement of excitations. By incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF method. In this case, we find that the temporal entanglement entropy scales logarithmically with evolution time. The IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems theory to quantum statistical physics.

Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.

Closed-form Approximations in Multi-asset Market Making

ABSTRACT A large proportion of market making models derive from the seminal model of Avellaneda and Stoikov. The numerical approximation of the value function and the optimal quotes in these models remains a challenge when the number of assets is large. In this article, we propose closed-form approximations for the value functions of many multi-asset extensions of the Avellaneda–Stoikov model. These approximations or proxies can be used (i) as heuristic evaluation functions, (ii) as initial value functions in reinforcement learning algorithms, and/or (iii) directly to design quoting strategies through a greedy approach. Regarding the latter, our results lead to new and easily interpretable closed-form approximations for the optimal quotes, both in the finite-horizon case and in the asymptotic (ergodic) regime.

Adiabatic landscape and optimal paths in ergodic systems

Whether one is interested in quantum state preparation or in the design of efficient heat engines, adiabatic (reversible) transformations play a pivotal role in minimizing computational complexity and energy losses. Understanding the structure of these transformations and identifying the systems for which such transformations can be performed efficiently and quickly is therefore of primary importance. In this paper we focus on finding optimal paths in the space of couplings controlling the system's Hamiltonian. More specifically, starting from a local Hamiltonian we analyze directions in the space of couplings along which adiabatic transformations can be accurately generated by local operators, which are both realizable in experiments and easy to simulate numerically. We consider a non-integrable 1D Ising model parametrized by two independent couplings, corresponding to longitudinal and transverse magnetic fields. We find regions in the space of couplings characterized by a very strong anisotropy of the variational adiabatic gauge potential (AGP), generating the adiabatic transformations, which allows us to define optimal adiabatic paths. We find that these paths generally terminate at singular points characterized by extensive degeneracies in the energy spectrum, splitting the parameter space into adiabatically disconnected regions. The anisotropy follows from singularities in the AGP, and we identify special robust weakly-thermalizing and non-absorbing many-body "dark" states which are annihilated by the singular part of the AGP and show that their existence extends deep into the ergodic regime.

Many-body localization from dynamical gauge fields

A recent experiment [Nature Physics 10, 1 (2019)] has realized a dynamical gauge system with $\mathbb{Z}_2$ gauge symmetry in a double-well potential. In this work we propose a method to generalize this model from a single double well to a one-dimensional chain. We show that although there is no disordered potential in the original model, the phenomenon of many-body localization can occur. The key ingredient is that different symmetry sectors with different local gauge charges play the role of different disorder configurations, which becomes clear after exactly mapping our model to a transverse Ising model in a random longitudinal field. We show that both the ergodic regime and the many-body localized regime exist in this model from four different metrics, which include level statistics, volume law versus area law of entanglement entropy of eigenstates, quench dynamics of entanglement entropy and physical observables.

Breakdown of ergodicity in disordered U(1) lattice gauge theories

We show how U(1) lattice gauge theories display key signatures of ergodicity breaking in the presence of a random charge background. Contrary to the widely studied case of spin models, in the presence of Coulomb interactions, the spectral properties of such lattice gauge theories are very weakly affected by finite-volume effects. This allows to draw a sharp boundary for the ergodic regime, and thus the breakdown of quantum chaos for sufficiently strong gauge couplings, at the system sizes accessible via exact diagonalization. Our conclusions are independent on the value of a background topological angle, and are contrasted with a gauge theory with truncated Hilbert space, where instead we observe very strong finite-volume effects akin to those observed in spin chains.