Generalized Gibbs Ensemble from Eigenstate Entanglement Hamiltonian

TL;DR

This work establishes a framework to derive a Generalized Gibbs Ensemble (GGE) from the entanglement structure of a larger auxiliary system. By constructing entanglement Hamiltonians of eigenstates that share common modes with the target system, the authors extract conserved bilinear operators and assemble a GGE that can be non-Abelian when degeneracies are present. They prove exact matching of long-time averages for fermionic bilinears with a non-Abelian GGE in 1D free-fermion lattices and show numerical evidence that this NA GGE outperforms the Abelian version for both fermionic and hardcore-boson observables. The approach offers a new numerical vantage point on quantum integrability and may generalize to interacting systems through careful auxiliary-system design.

Abstract

Relaxed quantum systems with conservation laws are believed to be approximated by the Generalized Gibbs Ensemble (GGE), which incorporates the constraints of certain conserved quantities serving as integrals of motion. By drawing an analogy between eigenstate reduced density matrix and GGE, we conjecture that a natural set of conserved quantities for GGE can emerge from the reduced density matrices of properly chosen eigenstates by the entanglement Hamiltonian superdensity matrix (EHSM) framework, and we demonstrate this explicitly for models mappable to free fermions. The framework proposes that such conserved quantities are linear superpositions of eigenstate entanglement Hamiltonians of a larger auxiliary system, where the eigenstates are Fock states occupying what we call the common eigenmodes, which remain eigenmodes when truncated within the physical subsystem. For 1D homogeneous free fermions with (anti-)periodic boundary conditions, which maps to 1D hardcore bosons with nearest neighbor hoppings, these conserved quantities lead to a non-Abelian GGE, which predicts the relaxation of both fermion and boson bilinears more accurately than the conventional Abelian GGE. Generalization of this framework may provide novel numerical insights for quantum integrability.

Figures (8)

• Figure 1: (a),(c) show the auxiliary system $A\cup B$ for homogeneous system $S$ in (b),(d) with OBC and PBC, respectively. Blue dashed curves (in (a),(c)) denote common eigenmodes in $A\cup B$ and their accordance in system $S$ (in (b),(d)). Red dashed curve in (a) is an eigenmode of $A\cup B$ which is not a common eigenmode.
• Figure 2: EHSM weights $p_{A,n}$ in descending order for 1D homogeneous free fermion chain with (a) OBC, (c) PBC, (e) A-PBC, in which $L=10,15,20$ and $L=20L_A$ respectively. (b) Matrix elements $\kappa_{Q,kk'}^{(n)}$ of a typical EHSM eigen-operator $Q_A^{(n)}\simeq Q_S^{(n)}=\sum_{k,k'}\kappa_{Q,kk'}^{(n)}f^\dag_k f_{k'}$ for OBC, with $k=\frac{\pi m}{L_S+1}$ in \ref{['eq:OBC-phiS']}. The horizontal and vertical labels are $1\le m\le L_S$. (d) and (f): Matrix elements $\kappa_{Q,kk'}^{(n)}$ of an EHSM eigen-operator $Q_A^{(n)}\simeq Q_S^{(n)}=\sum_{k,k'}\kappa_{Q,kk'}^{(n)}f^\dag_k f_{k'}$ for (d) PBC and (f) A-PBC, with $k=\frac{2\pi m}{L_S}$ in \ref{['eq:PBC-phiS']}. The horizontal and vertical labels are $-\frac{L_S}{2}< m\le \frac{L_S}{2}$.
• Figure 3: The relaxed expectation values of hardcore boson bilinears of the initial state adopted in rigol_2007 (see description below \ref{['eq-2p-fb']}) with PBC, compared with ensemble averages from the Abelian and non-Abelian GGE. (a) The results for $\braket{b_k^\dagger b_k}$. (b) The results for $\text{Im}\langle b_{k}^\dag b_{-k}\rangle=-\frac{i}{2}\braket{b_k^\dagger b_{-k} - h.c.}$.
• Figure 4: The mirror extension method of designing the auxiliary system for generic free fermion models. The blue (red) dashed curve represents an anti-symmetric (symmetric) eigenmode that is (is not) a common eigenmode.
• Figure S1: Relaxed values of the real part of $\braket{b_k^\dag b_{-k}}$ predicted by the Abelian GGE and non-Abelian GGE, compared with the results from time-evolution calculations.