We demonstrate that the entanglement entropy area law for free fermion ground states and the corresponding volume law for highly excited states are related by a position-momentum duality, and thus are of the same origin. For a typical excited state in the thermodynamic limit, we further show that the reduced density matrix of a subsystem approaches the thermal density matrix, provided the subsystem's linear size is small compared to that of the whole system in all directions, a property we dub eigenstate typicality. This provides an explicit example of thermalization via entanglement, and reveals how statistical physics emerges from a single eigenstate by tracing out a large number of degrees of freedom.
We demonstrate that the entanglement entropy area law for free fermion ground states and the corresponding volume law for highly excited states are related by a position-momentum duality, and thus are of the same origin. For a typical excited state in the thermodynamic limit, we further show that the reduced density matrix of a subsystem approaches the thermal density matrix, provided the subsystem's linear size is small compared to that of the whole system in all directions, a property we dub eigenstate typicality. This provides an explicit example of thermalization via entanglement, and reveals how statistical physics emerges from a single eigenstate by tracing out a large number of degrees of freedom.
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.
We consider free fermion systems in arbitrary dimensions and represent the occupation pattern of each eigenstate as a classical binary string. We find that the Kolmogorov complexity of the string correctly captures the scaling behavior of its entanglement entropy (EE). In particular, the logarithmically enhanced area law for EE in the ground state and the volume law for EE in typical highly excited states are reproduced. Since our approach does not require bipartitioning the system, it allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We reveal that the fraction of atypical eigenstates which do not thermalize in the free fermion system vanishes exponentially in the thermodynamic limit. Our results illustrate explicitly the connection between complexity and EE of individual pure states in quantum systems.
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.
It is known that the long-range quantum entanglement exhibited in free fermion systems is sufficient to "thermalize" a small subsystem in that the subsystem reduced density matrix computed from a typical excited eigenstate of the combined system is approximately thermal. Remarkably, fermions without any interactions are thus thought to satisfy the Eigenstate Thermalization Hypothesis (ETH). We explore this hypothesis when the fermion subsystem is only minimally coupled to a quantum reservoir (in the form of another fermion system) through a quantum point contact (QPC). The entanglement entropy of two 2-d free fermion systems connected by one or more quantum point contacts (QPC) is examined at finite energy and in the ground state. When the combined system is in a typical excited state, it is shown that the entanglement entropy of a subsystem connected by a small number of QPCs is sub-extensive, scaling as the linear size of the subsystem (LA). For sufficiently low energies (E) and small subsystems, it is demonstrated numerically that the entanglement entropy SA∼LAE, what one would expect for the thermodynamics of a one-dimensional system. In this limit, we suggest that the entropy carried by each additional QPC is quantized using the one-dimensional finite size/temperature conformal scaling: ΔSA=αlog(1/E)sinh(LAE). The sub-extensive entropy in the case of a small number of QPCs should be contrasted with the expectation for both classical, ergodic systems and quantum chaotic systems wherein a restricted geometry might affect the equilibrium relaxation times, but not the equilibrium properties themselves, such as extensive entropy and heat capacity.
It is known that the long-range quantum entanglement exhibited in free fermion systems is sufficient to "thermalize" a small subsystem in that the subsystem reduced density matrix computed from a typical excited eigenstate of the combined system is approximately thermal. Remarkably, fermions without any interactions are thus thought to satisfy the eigenstate thermalization hypothesis. We explore this hypothesis when the fermion subsystem is only minimally coupled to a quantum reservoir (in the form of another fermion system) through a quantum point contact (QPC). The entanglement entropy of two free fermion systems connected by one or more QPCs is examined at finite energy and in the ground state. When the combined system is in a typical excited state, it is shown that the entanglement entropy of a subsystem connected by a small number of QPCs is subextensive, scaling as the linear size of the subsystem (L_{A}). For sufficiently low energies (E) and small subsystems, it is demonstrated numerically that the entanglement entropy S_{A}∼L_{A}E, what one would expect for the thermodynamics of a one-dimensional system. In this limit, we suggest that the entropy carried by each additional QPC is quantized using the one-dimensional finite size and temperature conformal scaling: ΔS_{A}=αlog(1/E)sinh(L_{A}E). The subextensive entropy in the case of a small number of QPCs should be contrasted with the expectation for both classical, ergodic systems and quantum chaotic systems, wherein a restricted geometry might affect the equilibrium relaxation times but not the equilibrium properties themselves, such as extensive entropy and heat capacity.
We consider a quantum system of large size N and its subsystem of size L, assuming that N is much larger than L, which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page’s formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.