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 ( $L_A$ ). For sufficiently low energies ( $E$ ) and small subsystems, it is demonstrated numerically that the entanglement entropy $S_A \sim L_A E$ , 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/temperature conformal scaling: $ΔS_A = α\log{(1/E)\sinh{(L_AE)}}$ . 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 (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 ($L_A$). For sufficiently low energies ($E$) and small subsystems, it is demonstrated numerically that the entanglement entropy $S_A \sim L_A E$, 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/temperature conformal scaling: $ΔS_A = α\log{(1/E)\sinh{(L_AE)}}$. 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.

arXiv:2501.03076Jan 6, 20250 citations

Entanglement entropy scaling laws and eigenstate typicality in free fermion systems

Hsin-Hua Lai, Kun Yang

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.

arXiv:1409.1224Physical Review BFeb 23, 201580 citations

Subsystem eigenstate thermalization hypothesis

Anatoly Y. Dymarsky, Nima Lashkari, Hong Liu

Motivated by the qualitative picture of canonical typicality, we propose a refined formulation of the eigenstate thermalization hypothesis (ETH) for chaotic quantum systems. This formulation, which we refer to as subsystem ETH, is in terms of the reduced density matrix of subsystems. This strong form of ETH outlines the set of observables defined within the subsystem for which it guarantees eigenstate thermalization. We discuss the limits when the size of the subsystem is small or comparable to its complement. In the latter case we outline the way to calculate the leading volume-proportional contribution to the von Neumann and Renyi entanglment entropies. Finally, we provide numerical evidence for the proposal in the case of a one-dimensional Ising spin chain.Motivated by the qualitative picture of canonical typicality, we propose a refined formulation of the eigenstate thermalization hypothesis (ETH) for chaotic quantum systems. This formulation, which we refer to as subsystem ETH, is in terms of the reduced density matrix of subsystems. This strong form of ETH outlines the set of observables defined within the subsystem for which it guarantees eigenstate thermalization. We discuss the limits when the size of the subsystem is small or comparable to its complement. In the latter case we outline the way to calculate the leading volume-proportional contribution to the von Neumann and Renyi entanglment entropies. Finally, we provide numerical evidence for the proposal in the case of a one-dimensional Ising spin chain.

arXiv:1611.08764Physical review. EJan 25, 2018164 citations

Generalized free cumulants for quantum chaotic systems

Siddharth Jindal, Pavan Hosur

A bstract The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.A bstract The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.

arXiv:2401.13829Journal of High Energy PhysicsSep 12, 202414 citations

Structure of chaotic eigenstates and their entanglement entropy

Chaitanya Murthy, Mark Srednicki

We consider a chaotic many-body system (i.e., one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density. When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy). This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system. Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover.We consider a chaotic many-body system (i.e., one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density. When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy). This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system. Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover.

arXiv:1906.04295Physical review. EAug 22, 201980 citations

Emergence of a thermal equilibrium in a subsystem of a pure ground state by quantum entanglement

Kazuhiro Seki, Seiji Yunoki

By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters up to 24 sites, we demonstrate that quantum entanglement between subsystems A and B in a pure ground state of a whole system A + B can induce thermal equilibrium in subsystem A. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem A. An effective temperature T A of subsystem A induced by quantum entanglement is not a parameter but can be determined from the entanglement von Neumann entropy S A and the total energy E A of subsystem A calculated for the ground state of the whole system.We show that temperature T A can be derived by minimizing the relative entropy for the reduced density matrix operator of subsystem A and the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem A with respect to the coupling strength between subsystems A and B. Temperature T A is essentially identical to the thermodynamic temperature, for which the entropy and the internal energy evaluated using the canonical ensemble in statistical mechanics for the isolated subsystem A agree numerically with the entanglement entropy S A and the total energy E A of subsystem A. Fidelity calculations ascertain that the reduced density matrix operator of subsystem A for the pure but entangled ground state of the whole system A + B matches, within a maximally 1.5% error in the finite size clusters studied, the thermodynamic density matrix operator of subsystem A at temperature T A , despite that these density-matrix operators are different in general.We also find that temperature T A evaluated from the ground state of the whole system depends insignificantly on the system sizes, which is consistent with the fact that the thermodynamic temperature is an intensive quantity.We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem.We also provide two simple but nontrivial analytical examples of free bosons and free fermions for which the two density-matrix operators are exactly the same if the effective temperature T A is adopted.We furthermore discuss implications and possible applications of our finding.By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters up to 24 sites, we demonstrate that quantum entanglement between subsystems A and B in a pure ground state of a whole system A + B can induce thermal equilibrium in subsystem A. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem A. An effective temperature T A of subsystem A induced by quantum entanglement is not a parameter but can be determined from the entanglement von Neumann entropy S A and the total energy E A of subsystem A calculated for the ground state of the whole system.We show that temperature T A can be derived by minimizing the relative entropy for the reduced density matrix operator of subsystem A and the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem A with respect to the coupling strength between subsystems A and B. Temperature T A is essentially identical to the thermodynamic temperature, for which the entropy and the internal energy evaluated using the canonical ensemble in statistical mechanics for the isolated subsystem A agree numerically with the entanglement entropy S A and the total energy E A of subsystem A. Fidelity calculations ascertain that the reduced density matrix operator of subsystem A for the pure but entangled ground state of the whole system A + B matches, within a maximally 1.5% error in the finite size clusters studied, the thermodynamic density matrix operator of subsystem A at temperature T A , despite that these density-matrix operators are different in general.We also find that temperature T A evaluated from the ground state of the whole system depends insignificantly on the system sizes, which is consistent with the fact that the thermodynamic temperature is an intensive quantity.We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem.We also provide two simple but nontrivial analytical examples of free bosons and free fermions for which the two density-matrix operators are exactly the same if the effective temperature T A is adopted.We furthermore discuss implications and possible applications of our finding.

arXiv:2005.05617Physical Review ResearchOct 16, 20205 citations

L_A

). For sufficiently low energies (

E

) and small subsystems, it is demonstrated numerically that the entanglement entropy

S_A \sim 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/temperature conformal scaling:

ΔS_A = α\log{(1/E)\sinh{(L_AE)}}

. 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.