ABSTRACT The entanglement spectrum serves as a powerful tool for probing the structure and dynamics of quantum many‐body systems, revealing key information about symmetry, topology, and excitations. While the entanglement entropy (EE) of ground states typically follows an area law, highly excited states obey a volume law, leading to a striking contrast in their scaling behavior. In this paper, we investigate the crossover between these two regimes, focusing on the role of quasi‐particles (QPs) in mediating this transition. By analyzing the energy dependence of EE in various many‐body systems, we explore how the presence of long‐lived QPs influences the entanglement structure of excited states. We present numerical results for spinless fermions, a spin chain near a many‐body localization transition, and the Sachdev‐Ye‐Kitaev (SYK) model, which lacks a conventional QP description. Our findings are complemented by a theoretical model based on Fermi liquid theory, providing insight into the interaction‐dependent scaling of EE and its consistency with numerical simulations. We find that a hallmark of QPs is a linear dependence of the EE on energy, which breaks down at high energies and in the limit of strong interaction. The slope of this linear dependence reflects the QP weight, which decreases with interaction strength.
ABSTRACT The entanglement spectrum serves as a powerful tool for probing the structure and dynamics of quantum many‐body systems, revealing key information about symmetry, topology, and excitations. While the entanglement entropy (EE) of ground states typically follows an area law, highly excited states obey a volume law, leading to a striking contrast in their scaling behavior. In this paper, we investigate the crossover between these two regimes, focusing on the role of quasi‐particles (QPs) in mediating this transition. By analyzing the energy dependence of EE in various many‐body systems, we explore how the presence of long‐lived QPs influences the entanglement structure of excited states. We present numerical results for spinless fermions, a spin chain near a many‐body localization transition, and the Sachdev‐Ye‐Kitaev (SYK) model, which lacks a conventional QP description. Our findings are complemented by a theoretical model based on Fermi liquid theory, providing insight into the interaction‐dependent scaling of EE and its consistency with numerical simulations. We find that a hallmark of QPs is a linear dependence of the EE on energy, which breaks down at high energies and in the limit of strong interaction. The slope of this linear dependence reflects the QP weight, which decreases with interaction strength.
We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law $\ensuremath{\alpha}$ is varied. We ask if a common ${\ensuremath{\alpha}}_{c}$ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as $\ensuremath{\alpha}$ approaches zero. These general features are expected to persist on turning on interactions.
For the typical quantum many-body systems that obey the eigenstate thermalization hypothesis (ETH), we argue that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function. The ETH implies that the crossover functions can be deduced from subsystem entropies of thermal ensembles and have universal properties. These functions capture the full crossover from the ground-state entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For critical one-dimensional systems, a universal scaling function follows from conformal field theory and can be adapted for nonlinear dispersions. We use it to also deduce the crossover scaling function for Fermi liquids in d>1 dimensions. The analytical results are complemented by numerics for large noninteracting systems of fermions in d≤3 dimensions and have also been confirmed for bosonic systems and nonintegrable spin chains.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent α. For α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size L is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2. For α>3/2, the EE is in the area-law phase, namely, no scaling with L, for any monitoring strength. For 1<α≲3/2, we identify an α-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
law is enhanced by a logarithmic factor. In Ref. [1], we use multi-dimensional bosonization to provide a simple derivation of this result, and show that the logarithimic factor has a 1D origin. More importantly the bosonization technique allows us to take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids. The central result of our work is that Fermi liquid interactions do not alter the leading scaling behavior of the entanglement entropy, and the logarithmic enhancement of area law is a robust property of the Fermi liquid phase. In sharp contrast to the fermioic systems with Fermi surfaces, quantum critical (or gapless) bosonic systems do not violate the area law above 1D (except for the case discussed below). The fundamental difference lies in the fact that gapless excitations live near a single point (usually origin of momentum space) in such bosonic systems, while they live around an (extended) Fermi surface in Fermi liquids. In Ref. [2], we studied entanglement properties of some specific examples of the so called Bose metal states, in which bosons neither condense (and become a superfluid) nor localize (and insulate) at T=0. The system supports gapless excitations around “Bose surfaces,” instead of isolated points in momentum space. We showed that similar to free Fermi gas and Fermi liquids, these states violate the entanglement area law in a logarithmic fashion. Compared to ground states, much less is known concretely about entanglement in (highly) excited states. Going back to free fermion systems, in Ref. [3] we show that there exists a duality relation between ground and excited states, and the area law obeyed by ground state turns into a volume law for excited states, something widely expected but hard to prove. Most importantly, we find in appropriate limits the reduced density matrix of a subsystem takes the form of thermal density matrix, providing an explicit example of the eigenstate thermalization hypothesis. Our work [3] explicitly demonstrates how statistical physics emerges from entanglement in a single eigenstate.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of fermions interacting with q-body random couplings. For q=2, it describes free particles and is nonchaotic in the many-body sense, while for q>2 it is strongly interacting and exhibits many-body chaos. In this work we study the entanglement entropy (EE) of the SYKq models for a bipartition of N real or complex fermions into subsystems containing 2m real/m complex fermions and $N\ensuremath{-}2m$/$N\ensuremath{-}m$ fermions in the remainder. For the free model SYK2, we obtain an analytic expression for the EE, derived from the $\ensuremath{\beta}$-Jacobi random matrix ensemble. Furthermore, we use the replica trick and path-integral formalism to show that the EE is maximal when one subsystem is small, i.e., $m\ensuremath{\ll}N$, for arbitrary q. We also demonstrate that the EE for the SYK4 model is noticeably smaller than the Page value when the two subsystems are comparable in size, i.e., m/N is O(1). Finally, we explore the EE for a model with both SYK2 and SYK4 interactions and find a crossover from SYK2 (low-temperature) to SYK4 (high-temperature) behavior as we vary energy.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.