Eigenstate thermalization in thermal first-order phase transitions

TL;DR

The paper shows that the eigenstate thermalization hypothesis (ETH) must be generalized near thermal first-order phase transitions, by analyzing an all-to-all spin model with two competing mean-field branches that exchange dominance at a microcanonical energy $\varepsilon_\text{cross}$. A semiclassical tunneling framework and exact diagonalization demonstrate two distinct ETH regimes: a multi-branch regime with eigenstates localized to individual branches and a mixed-branch regime with Schrödinger-cat-like inter-branch superpositions; these regimes are separated by an eigenstate phase transition and have distinct dynamical signatures. The results connect the microcanonical description of first-order transitions to eigenstate properties and provide experimentally testable predictions via non-equilibrium dynamics and spectral functions, including a Thouless plateau and bimodal local-observable distributions. The work suggests broader implications for ETH in systems with long-range or power-law interactions and motivates future studies of alternative branch structures and local-model realizations.

Abstract

The eigenstate thermalization hypothesis (ETH) posits how isolated quantum many-body systems thermalize, assuming that individual eigenstates at the same energy density have identical expectation values of local observables in the limit of large systems. While the ETH apparently holds across a wide range of interacting quantum systems, in this work we show that it requires generalization in the presence of thermal first-order phase transitions. We introduce a class of all-to-all spin models, featuring first-order thermal phase transitions that stem from two distinct mean-field solutions (two ``branches'') that exchange dominance in the many-body density of states as the energy is varied. We argue that for energies in the vicinity of the thermal phase transition, eigenstate expectation values do not need to converge to the same thermal value. The system has a regime with coexistence of two classes of eigenstates corresponding to the two branches with distinct expectation values at the same energy density, and another regime with Schrodinger-cat-like eigenstates that are inter-branch superpositions; these two regimes are separated by an eigenstate phase transition. We support our results by semiclassical calculations and an exact diagonalization study of a microscopic spin model, and argue that the structure of eigenstates in the vicinity of thermal first-order phase transitions can be experimentally probed via non-equilibrium dynamics.

Figures (11)

• Figure 1: Schematic picture of the first-order thermal phase transition that we study. (a) Microcanonical mean-field behavior with two distinct solutions, with one solution dominating the many-body density of states $\nu(\varepsilon)$ for energy densities $\varepsilon<\varepsilon_{\rm cross}$, and the second solution becoming dominant for larger energies. This results in a non-convex region in the entropy. (b) Expectations for the ETH, that has two different branches and absence of complete ergodicity for $\varepsilon<\varepsilon_\text{mix}$. The ergodicity is restored for $\varepsilon>\varepsilon_\text{mix}$ by a combination of classical and possibly quantum tunneling. (c) The distribution of local observables in eigenstates for energy densities below $\varepsilon_\text{mix}$ has two species of eigenstates. In contrast, in the mixing regime the eigenstates have bimodal distribution of local observables.
• Figure 2: (a) Contours of constant energy density $\varepsilon$ have a unique local minimum value of $m$ for very negative $\varepsilon$. This minimum gives rise to the branch of mean-field solutions that we label as $m^z<0$. Upon increasing the energy density, a second local minimum of $m$ emerges. This second minimum, labeled as the $m^z>0$ branch, upon further increasing the energy attains a lower value of $m$ compared to the $m^z<0$ branch. (b) Entropy densities of the two branches as a function of $\varepsilon$ have a crossing at energy density $\varepsilon_\text{cross} \approx -0.0877$. (c) Expectation value of $z$-magnetization per spin, $m^z$, shows that the mean-field branch appearing at lower energies has negative $z$-magnetization, while the second branch that appears upon increasing the energy has positive expectation value of $m^z$. Grey line corresponds to the local entropy minimum (local maximum of $m$) that eventually merges with the $m^z<0$ branch at a small positive value of $\varepsilon$ (not shown). Parameters of the LMG-3 model shown here are $h_z=-0.2$ and $h_x = 0.1$.
• Figure 3: Tunneling at the energy density $\varepsilon_\text{cross} = -0.0877$ and $m_\text{cross}=0.586$ corresponding to the branch crossing for $h_z=-0.2$, $h_x=0.1$. The maximal matrix element is attained at a value of $m_\text{t} \approx 0.709 >m_\text{cross}$ shown by the gray dot. Inset shows the constant energy $(\varepsilon=\varepsilon_\text{cross})$ contour in the $\alpha,m$ plane, with the dark gray line indicating the value of $m_\text{t}$ and range of angles where optimal tunneling occurs.
• Figure 4: Phase diagrams (a) value of energy density $\varepsilon_\text{cross}$ and (b) spin size $m_\text{cross}$ at the branch crossing. White regions correspond to parameters where the crossing does not occur. (c) Positive/negative values of the logarithm of the ratio between level spacing and matrix element, correspond to localized and hybridizing phases, respectively. (d) Difference between optimal $m_\text{t}$ where tunneling happens and $m_\text{cross}$ at the crossing shows that the role of within-branch fluctuations is larger at small values of $h_{x,z}$. The red line shows contour where local spin maximum at $\varepsilon_\text{cross}$ is nearly outside of the physically allowed range ($m_\text{max}=0.98$), to the left of red line only combination of quantum and classical tunneling is allowed. To the right of the yellow line tunneling becomes fully classical.
• Figure 5: (a) Expectation value of $z$-magnetization for individual eigenstates for different $N=14, 15, 16$. The mean-field branches are also shown. (b) Distribution of the magnetization for $N=16$ in the 80 eigenstates closest to energy density $\varepsilon =-0.1$ shows the presence of non-bimodal distributions, as well as bimodal distributions with peaks corresponding to the two branches. (c) After separating the bimodal eigenstates each into two parts, as described in the main text, the expectation values of the $z$-magnetization in each part, denoted as $\langle m^z\rangle_E$ clearly show the presence of the two branches for $N=16$. Values of magnetic fields: $h_x=0.1$, $h_z=-0.2$.