All-to-all Spin Models Demonstrate Broken Eigenstate Thermalization Near First-Order Transitions

The behaviour of isolated quantum systems has long been a subject of intense study, with the eigenstate thermalization hypothesis (ETH) offering a compelling explanation for how these systems appear to reach equilibrium. Maksym Serbyn of the Institute of Science and Technology Austria, Alexander Avdoshkin from the Massachusetts Institute of Technology, and Oriana K. Diessel of ITAMP, Harvard-Smithsonian Center for Astrophysics, alongside David A. Huse, challenge this established understanding in new research exploring thermal first-order phase transitions. Their work demonstrates that the ETH requires refinement when applied to systems undergoing such transitions, revealing a more complex picture of how energy distributes within these systems. By investigating all-to-all spin models, the team uncovered regimes where eigenstates exhibit behaviours inconsistent with standard thermalization, including coexistence of distinct expectation values and the emergence of Schrodinger-cat-like superpositions. This research offers crucial insight into the fundamental properties of quantum systems and suggests new avenues for probing these phenomena through non-equilibrium dynamics.

Thermal Phase Transitions Challenge Eigenstate Thermalization Hypothesis

The approach involves detailed analysis of the spin models’ energy landscape and eigenstates, utilising numerical simulations to characterise the behaviour of local observables. Specifically, the team examines how expectation values deviate from the predictions of the standard ETH as the system approaches the phase transition, quantifying these deviations by analysing fluctuations of expectation values across eigenstates and identifying signatures of non-thermal behaviour. This methodology allows for a precise determination of the conditions under which the ETH fails and provides insights into the underlying mechanisms driving the breakdown. A specific contribution of this research is the identification of a novel scaling behaviour of the fluctuations of local observables near the thermal phase transition. The team demonstrates that these fluctuations exhibit a characteristic power-law dependence on system size, differing significantly from the predictions of the ETH, and propose a modified ETH incorporating corrections that account for the presence of the two competing mean-field solutions. These corrections successfully capture the observed behaviour of the system, validating the proposed generalisation of the ETH and providing a framework for understanding thermalisation in systems with first-order phase transitions.

First-Order Transitions in All-to-All Spin Models

The study investigated the eigenstate thermalization hypothesis (ETH) by engineering a novel class of all-to-all spin models exhibiting first-order thermal phase transitions, arising from two competing mean-field solutions, termed “branches”, which alternate in dominance within the energy density. Researchers meticulously constructed these models to challenge the standard ETH, hypothesising that eigenstate expectation values need not converge to a single thermal value near these phase transitions, enabling the observation of regimes where eigenstates fall into two distinct classes, each corresponding to one of the branches and possessing differing expectation values at the same energy density. To characterise the transition between these regimes, the team employed an exact diagonalization study of a microscopic spin model alongside semiclassical calculations. The discrete dataset of eigenstate energies and per-branch weights was smoothed using a Gaussian kernel density estimation (KDE) from the scipy package, with a bandwidth of 0.1, to accurately determine the crossing point between the density of states (DOS) of the two branches.

This crossing point, systematically shifting with increasing magnetic field and system size, provided a precise marker of the phase transition, revealing finite size effects and minor fluctuations across different disorder realisations. Further analysis focused on the diagonal matrix elements of the mx operator, mirroring earlier work on mz, demonstrating that the observed multi-branch and mixed-branch regimes extend to other operators, confirming the generality of the findings. While discrepancies between mean-field predictions and eigenstate values were noted, the overall trend of two distinct branches hybridising around a specific energy remained consistent, ultimately revealing a dominant branch at higher energy densities. This methodological precision, combining exact diagonalization with KDE smoothing and comparative operator analysis, facilitated the identification of Schrodinger-cat-like eigenstates, inter-branch superpositions, and an associated eigenstate phase transition, pioneering a pathway for experimentally probing these complex eigenstate structures through non-equilibrium dynamics.

Eigenstate Splitting at Thermal Phase Transitions

Scientists have demonstrated a modification to the eigenstate thermalization hypothesis (ETH) in quantum systems undergoing thermal first-order phase transitions, centring on all-to-all spin models exhibiting transitions stemming from two distinct mean-field solutions that alternate dominance within the energy density. Experiments revealed that, near these thermal phase transitions, eigenstate expectation values do not necessarily converge to a single thermal value, challenging a core tenet of the standard ETH, and introducing a system where two classes of eigenstates coexist, each corresponding to one of the aforementioned branches and possessing distinct expectation values at the same energy density. The team measured a regime characterized by Schrodinger-cat-like eigenstates, representing superpositions between these two branches, separated from the coexistence regime by an eigenstate phase transition. Semiclassical calculations and exact diagonalization studies supported these findings, providing robust evidence for the altered ETH behaviour, and showing that the microcanonical entropy exhibits non-convexity in the energy range of the transition for systems with long-range interactions, preventing spatial phase separation.

Results demonstrate that the structure of eigenstates in the vicinity of thermal first-order phase transitions can be experimentally probed through non-equilibrium dynamics, opening avenues for future investigation. The study focused on models where equilibrium thermodynamic properties align with a mean-field description, featuring two distinct solutions coexisting at the same energy near the transition, achieved through a generalized Lipkin-Meshkov-Glick (LMG) model comprising spins-1/2 with all-to-all interactions and perturbed by weak random terms. Measurements confirm the existence of a non-monotonic relationship between temperature and energy density in the microcanonical ensemble, a hallmark of inequivalence between canonical and microcanonical ensembles at these transitions, delivering a novel understanding of thermalization in isolated quantum systems and suggesting that the ETH requires generalization when first-order phase transitions are present.

First-Order Transitions Break Eigenstate Thermalisation

This research demonstrates a nuanced understanding of the eigenstate thermalization hypothesis (ETH) by identifying two distinct regimes in quantum systems undergoing thermal first-order phase transitions. The authors reveal that, contrary to the standard ETH, systems near these transitions do not necessarily converge to a single thermal value, instead exhibiting behaviours dependent on the specific energy range, and characterise a ‘multi-branch’ regime where eigenstates display macroscopically distinct properties corresponding to different phases, and a ‘mixed-branch’ regime featuring Schrodinger-cat-like eigenstates representing superpositions of these phases. The significance of this work lies in extending the established framework of ETH to encompass systems with first-order transitions, highlighting a breakdown of conventional thermalization expectations. The identification of an eigenstate phase transition separating these regimes, and the associated exponentially slow dynamics, offers a pathway to experimentally probe the structure of eigenstates via non-equilibrium dynamics, such as quantum quenches. The authors acknowledge that their analysis is based on all-to-all interacting spin models with a specific arrangement of mean-field solutions, suggesting that qualitatively different branch arrangements are also possible and warrant further investigation, potentially including scenarios like false vacuum decay, and that future research could explore the extent to which these findings generalise to.