Eigenstate thermalization and ensemble equivalence in few-body fermionic systems

Ph. Jacquod

Phys. Rev. E 101, 062141 – Published 25 June 2020

Abstract

We investigate eigenstate thermalization from the point of view of vanishing particle and heat currents between a few-body fermionic Hamiltonian prepared in one of its eigenstates and an external, weakly coupled Fermi-Dirac gas. The latter acts as a thermometric probe, with its temperature and chemical potential set so that there is neither particle nor heat current between the two subsystems. We argue that the probe temperature can be attributed to the few-fermion eigenstate in the sense that (i) it varies smoothly with energy from eigenstate to eigenstate, (ii) it is equal to the temperature obtained from a thermodynamic relation in a wide energy range, (iii) it is independent of details of the coupling between the two systems in a finite parameter range, (iv) it satisfies the transitivity condition underlying the zeroth law of thermodynamics, and (v) it is consistent with Carnot's theorem. For the spinless fermion model considered here, these conditions are essentially independent of the interaction strength. When the latter is weak, however, orbital occupancies in the few-fermion system differ from the Fermi-Dirac distribution so that partial currents from or to the probe will eventually change its state. We find that (vi) above a certain critical interaction strength, orbital occupancies become close to the Fermi-Dirac distribution, leading to a true equilibrium between the few-fermion system and the probe. In that case, the coupling between the Fermi-Dirac gas and few-fermion system does not modify the state of the latter, which justifies our approach a posteriori. From these results, we conjecture that for few-body systems with sufficiently strong interaction, the eigenstate thermalization hypothesis is complemented by ensemble equivalence: individual many-body eigenstates define a microcanonical ensemble that is equivalent to a canonical ensemble with grand canonical orbital occupancies.

Received 13 September 2019
Accepted 3 June 2020

DOI:https://doi.org/10.1103/PhysRevE.101.062141

Physics Subject Headings (PhySH)

Eigenstate thermalization

Statistical Physics & Thermodynamics

Authors & Affiliations

Ph. Jacquod

Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva, Switzerland and School of Engineering, University of Applied Sciences of Western Switzerland HES-SO, CH-1951 Sion, Switzerland

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Issue

Vol. 101, Iss. 6 — June 2020

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Images

Figure 1
Occupation number for the 1001st many-body eigenstate of $H_{sys}$ (3) with $m = 16$ orbitals and $n = 8$ particles, for $U / Δ = 0$ [black histogram, with temperature and chemical potential $(T / Δ, μ / Δ) = (5.41, 0.0141)$ ], 0.1 [red (dark gray), $(T / Δ, μ / Δ) = (5.43, 0.0139)$ ], and 0.2 [green (light gray), $(T / Δ, μ / Δ) = (5.47, 0.0135)$ ]. The black solid line is the Fermi function corresponding to $(T / Δ, μ / Δ) = (5.47, 0.0135)$ . It is slightly unsmooth because the one-body spectrum of (3) is not equidistant.
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Figure 2
Chemical potential (left) and temperature (right) vs many-body eigenstate number $A \in [1, N / 2]$ for a single realization of the two-body randomly interacting fermion model (3) with $m = 16$ orbitals and $n = 8$ fermions and, thus, $N = 12 870$ many-body states. Black dots are for $U / Δ = 0$ and red (gray) dots are for $U / Δ = 0.2$ .
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Figure 3
(a) Many-body density of states. (b) Numerically obtained eigenstate temperature $T$ vs theoretical temperature $T_{th}$ of Eq. (8) for $0 < T_{th} / Δ \leq 50$ . The interval contains about 90% of the positive temperature eigenstates. (c) Same as in (b) for a restricted range $0 < T_{th} / Δ \leq 10$ , showing deviations at low energy due to non-Gaussian tails in the density of states [25]. All panels correspond to a single realization of the two-body randomly interacting fermion model (3) with $m = 16$ orbitals, $n = 8$ fermions, and $N = 12 870$ many-body states. Black dots are for $U = 0$ and red (gray) dots are for $U = 0.2 Δ$ .
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Figure 4
Deviation from detailed balance for the particle current (top panel) and the heat current (bottom panel) as a function of the interaction strength $U / Δ$ for 1000 realizations of the two-body randomly interacting fermion model (3) with $m = 12$ orbitals and $n = 6$ fermions and thus $N = 924$ many-body states. Different curves correspond to different excitations energies above the ground state, starting at $ε = 2 Δ$ and increasing in steps of $δ ε = 2 Δ$ from black to red, green, blue, violet, magenta, and orange (from bottom to top, at $U / Δ = 0.01$ ). The dotted line indicates the arbitrarily chosen threshold $δ I_{A}^{2} / t^{4} = 8 \times 10^{- 3}$ and $δ J_{A}^{2} / t^{4} = 8 \times 10^{- 2}$ used to define critical interaction strengths $U_{c 1}$ and $U_{c 2}$ .
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Figure 5
Critical interaction strength $U_{c}$ vs normalized excitation energy $ε / B$ with the many-body bandwidth $B ≃ n (m - n) Δ$ , for $m = 8$ , $n = 4$ (black circles), $m = 10$ , $n = 5$ (red squares), $m = 12$ , $n = 6$ (green diamonds), $m = 14$ , $n = 7$ (blue triangles, pointing up), and $m = 16$ , $n = 8$ (violet triangles, pointing down). Solid circles give $U_{c 1}$ and empty circles give $U_{c 2}$ (see text).
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