Exact entanglement characterization of the Rokhsar-Kivelson quantum hard-square lattice gas: Rényi entropies, integrable entanglement spectra, and the extension beyond two-leg ladders

The Rokhsar-Kivelson (RK) quantum hard-square lattice gas has an exact ground state given by a weighted superposition of classical nearest-neighbor-exclusion particle configurations. On two-leg ladders, its entanglement Hamiltonian is the logarithm of the transfer matrix of the corresponding two-dimensional classical hard-square or hard-hexagon model, so the entanglement spectrum is critical exactly at the classical transition, realizing the c = 1/2 and c = 4/5 minimal conformal field theories ('holographic minimal models'); for the triangular ladder the entanglement Hamiltonian is integrable at all activities through Baxter's commuting-family construction. The 2012 source analysis, however, computed only the von Neumann entropy and numerically diagonalized entanglement spectra, and only for two legs. The open problem is to complete this exact characterization: derive exact analytic Rényi entropies from the transfer-matrix identification, extract closed-form finite-size spectra and operator content from the commuting family including away from criticality, and extend the entanglement analysis to N-leg ladders with N > 2 and to genuinely two-dimensional lattices, where the Gram-matrix reduction to a one-dimensional transfer matrix fails and new methods are required.

Background

A Rokhsar-Kivelson (RK) state is an exactly solvable quantum ground state written as a weighted superposition of classical lattice configurations, with amplitudes given by the square root of the Boltzmann weights of a classical statistical-mechanical model. The model studied here is a quantum lattice gas of hard-core particles (at most one particle per site) with nearest-neighbor exclusion, meaning two adjacent sites cannot both be occupied. Its parent Hamiltonian is a sum of positive-semidefinite local terms $h_i^\dagger(z)h_i(z)$, so the exact zero-energy ground state is

$$|z\rangle = \Xi(z)^{-1/2}\sum_C z^{n_C/2}|C\rangle$$

where $C$ runs over all allowed particle configurations, $n_C$ counts the particles in $C$, and the normalization $\Xi(z)=\sum_C z^{n_C}$ is exactly the partition function of the classical hard-square lattice gas at activity (fugacity) $z$. The entanglement properties of the quantum state are therefore controlled by this classical model. On the square lattice the classical hard-square model undergoes an order-disorder transition at $z_c\simeq 3.796$ in the two-dimensional Ising universality class and is not exactly solved, while the triangular-lattice version (hard hexagons) was solved exactly by Baxter and transitions at $z_c=(11+5\sqrt{5})/2$ in the three-state Potts universality class.

Tanaka, Tamura and Katsura (2012) placed this RK state on two-leg ladders (two coupled chains, periodic along the ladder) and studied the entanglement between the two legs. Writing the unnormalized state with the transfer matrix $T(z)$ of the corresponding two-dimensional classical model, the reduced density matrix of one leg has the same nonzero eigenvalues as the Gram matrix (overlap matrix) $M = T^T T/\Xi(z)$. The entanglement Hamiltonian $H_E := -\ln M$ is thus the logarithm of a classical transfer matrix: for the square ladder $T$ is symmetric, so $M = T^2/\Xi(z)$ with $T^2$ a two-row transfer matrix of the classical hard-square model, while for the triangular ladder $T^T T$ is the transfer matrix of a hard-hexagon-type model. Consequently the entanglement spectrum (ES) - the levels $\lambda_\alpha = -\ln p_\alpha$ built from the eigenvalues $p_\alpha$ of the reduced density matrix - is critical exactly when the corresponding classical model is critical. At $z=z_c$ the ES of the square (triangular) ladder is described by the minimal conformal field theory (CFT) with central charge $c=1/2$ (the Ising CFT) and $c=4/5$ (the three-state Potts CFT), respectively; here a minimal CFT is one of the unitary $c<1$ Virasoro theories, and the central charge $c$ fixes the universal finite-size structure of a critical one-dimensional spectrum. The authors call these critical entanglement spectra holographic minimal models, by analogy with the earlier correspondence established for the two-dimensional valence-bond-solid (AKLT) state. For the triangular ladder they further proved, by embedding the Gram matrix in Baxter's commuting-family construction for a generalized hard-square model with diagonal interactions, that $H_E$ is integrable at arbitrary $z$, in the sense of belonging to a one-parameter family of mutually commuting transfer matrices - even though the underlying quantum lattice gas is not integrable. The ground state also resembles that of a Rydberg lattice gas, which motivates but is not part of the present question.

What the 2012 analysis left undone is threefold. First, the von Neumann entropy and the ES were obtained by exact numerical diagonalization at finite length (level structure up to $L=18$); the Rényi entropies $S_n = (1/(1-n))\ln \mathrm{Tr}\,\rho^n$ - a one-parameter family of entanglement measures - were not computed, although the authors noted that the entanglement-Hamiltonian-to-transfer-matrix connection, e.g. through Suzuki's virtual-space transfer-matrix method, should make them accessible. Second, no closed-form spectral data (finite-length eigenvalues, exact scaling dimensions, operator content, or the spectrum away from criticality) were extracted from the commuting family, whose potential for exact analytic results was only noted. Third, the whole construction is special to two legs: for $N$-leg ladders with $N>2$ the Gram matrix can no longer be written in terms of a one-dimensional transfer matrix, so the analysis of entanglement spectra and entropies on wider ladders and on genuinely two-dimensional lattices requires methods that do not yet exist for this class of non-determinantal RK states.

Problem Statement

For the quantum hard-square lattice gas with Rokhsar-Kivelson ground state $|z\rangle = \Xi(z)^{-1/2}\sum_C z^{n_C/2}|C\rangle$, whose two-leg-ladder entanglement Hamiltonian $H_E = -\ln(T^T T/\Xi(z))$ realizes the $c=1/2$ and $c=4/5$ holographic minimal models at the classical critical activities, complete the exact entanglement characterization that the 2012 analysis left unfinished:

1. Exact entanglement data for the two-leg ladders. Derive exact analytic expressions for the Rényi entropies $S_n$ of one leg of the square and triangular ladders as functions of the activity $z$ and ladder length $L$, exploiting the identification of the Gram matrix with the (two-row) transfer matrices of the classical hard-square and hard-hexagon models, including the thermodynamic-limit behavior and finite-size corrections, both at and away from the critical activity. Extract closed-form spectral data for the triangular-ladder entanglement Hamiltonian from the commuting transfer-matrix family constructed in the source - finite-$L$ eigenvalues, exact scaling dimensions and operator content at $z = z_c$, and the spectrum away from criticality - replacing the numerical diagonalization used there.

2. Beyond two legs. Develop the analysis of entanglement spectra and entropies of the same RK quantum lattice gas on $N$-leg ladders with $N > 2$ and on genuinely two-dimensional lattices, where the Gram matrix can no longer be written in terms of a one-dimensional transfer matrix alone, and determine the structure of the resulting entanglement Hamiltonians - whether their spectra are critical or gapped as functions of $z$, their operator content where critical, and the scaling of the entropies - in particular establishing whether the holographic minimal-model spectra of the two-leg case survive beyond two legs.

These two items enumerate independent open cores - the exact analytic Rényi entropies, the closed-form spectral data from the commuting family, and the extension beyond two legs - any one of which constitutes a genuine partial resolution on its own, while the exact entanglement characterization is complete only when all of them have been obtained.

References

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