Entropy perspective on the thermal crossover in a fermionic Hubbard chain

The Eigenstate Thermalization Hypothesis (ETH) posits that the reduced density matrix for a subsystem corresponding to an excited eigenstate is "thermal." Here we expound on this hypothesis by asking: for which class of operators, local or non-local, is ETH satisfied? We show that this question is directly related to a seemingly unrelated question: is the Hamiltonian of a system encoded within a single eigenstate? We formulate a strong form of ETH where in the thermodynamic limit, the reduced density matrix of a subsystem corresponding to a pure, finite energy density eigenstate asymptotically becomes equal to the thermal reduced density matrix, as long as the subsystem size is much less than the total system size, irrespective of how large the subsystem is compared to any intrinsic length scale of the system. This allows one to access the properties of the underlying Hamiltonian at arbitrary energy densities/temperatures using just a single eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a non-integrable 1D lattice quantum model with only energy conservation. In addition, we examine the case in which the subsystem size is a finite fraction of the total system size, and find that even in this case, a large class of operators continue to match their canonical expectation values. Specifically, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. We also study, both analytically and numerically, a particle number conserving model at infinite temperature which substantiates our conjectures. PACS numbers:Contents arXiv:1503.00729v2 [cond-mat.str-el]

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“…2) is in agreement with that of the thermal reduced density matrix ρ A,th (β = 0) studied in Ref. 41, consistent with ETH.…”

Section: A Von Neumann and Renyi Entropysupporting

confidence: 90%

Does a Single Eigenstate Encode the Full Hamiltonian?

Garrison¹,

Grover²

2018

Phys. Rev. X

258

291

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“…On the other hand, we observe that for fixed system size the transition temperature T * decreases as the on-site interactions U is reduced re- flecting the enhancement of charge fluctuations in this weakly coupled regime. Finally, we note that we generally find somewhat smaller transition temperatures than recent stochastic series expansion (SSE) simulations 30 of the half-filled Hubbard chain, which can be tracked back to the fact that our DQMC simulations employ a grand-canonical ensemble, while the SSE simulation in Ref. 30 employed a canonical ensemble.…”

Section: B Thermal Crossover Of the Entanglementmentioning

confidence: 92%

“…Finally, we note that we generally find somewhat smaller transition temperatures than recent stochastic series expansion (SSE) simulations 30 of the half-filled Hubbard chain, which can be tracked back to the fact that our DQMC simulations employ a grand-canonical ensemble, while the SSE simulation in Ref. 30 employed a canonical ensemble.…”

Section: B Thermal Crossover Of the Entanglementmentioning

confidence: 92%

Rényi entropies of interacting fermions from determinantal quantum Monte Carlo simulations

Broecker¹

2014

J. Stat. Mech.

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Entanglement measures such as the entanglement entropy have become an indispensable tool to identify the fundamental character of ground states of interacting quantum many-body systems. For systems of interacting spin or bosonic degrees of freedom much recent progress has been made not only in the analytical description of their respective entanglement entropies but also in their numerical classification. Systems of interacting fermionic degrees of freedom, however, have proved to be more difficult to control, in particular with regard to the numerical understanding of their entanglement properties. Here we report a generalization of the replica technique for the calculation of Rényi entropies to the framework of determinantal Quantum Monte Carlo simulations -the numerical method of choice for unbiased, large-scale simulations of interacting fermionic systems. We demonstrate the strength of this approach over a recent alternative proposal based on a decomposition in free fermion Green's functions by studying the entanglement entropy of one-dimensional Hubbard systems both at zero and finite temperatures.

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“…So it is reasonable that the entropy bends down from the linearly increasing line. For a detailed study of properties of thermal Rényi entropy, see [82].…”

Section: Fig 5 (Color Online)mentioning

confidence: 99%

Quantum Joule expansion of one-dimensional systems

2020

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We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time t = 0. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and the canonical ensemble match well for high initial temperatures that correspond to negative effective final temperatures after the expansion. At long times after the barrier is removed, the time-evolved Rényi entropy of subsystems bigger than half can equilibrate to the thermal entropy with exponentially small fluctuations. The time-evolved reduced density matrix at long times can be approximated by a thermal density matrix for small subsystems. Few-body observables, like the momentum distribution function, can be approximated by a thermal expectation of the canonical ensemble with strongly suppressed fluctuations. The negative effective temperatures for finite systems go to nonnegative temperatures in the thermodynamic limit for bosons, but is a true thermodynamic effect for fermions, which is confirmed by finite temperature density matrix renormalization group calculations. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.

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Entropy perspective on the thermal crossover in a fermionic Hubbard chain

Cited by 11 publications

References 88 publications

Does a Single Eigenstate Encode the Full Hamiltonian?

Does a Single Eigenstate Encode the Full Hamiltonian?

Rényi entropies of interacting fermions from determinantal quantum Monte Carlo simulations

Quantum Joule expansion of one-dimensional systems

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