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arXiv:2508.20533 · Quantum Science and Technology

Extensive entanglement between coupled Tomonaga–Luttinger liquids in and out of equilibrium

Taufiq Murtadho, Marek Gluza, Nelly H. Y. Ng

Published Jul 15, 2026• 0 citations

Abstract

Abstract Quantum entanglement exists in nature but is absent in classical physics, hence it fundamentally distinguishes quantum from classical theories. While entanglement is routinely observed for few-body systems, it is significantly more challenging to witness in quantum many-body systems. Here, we theoretically study entanglement between two parallel and spatially separated Tomonaga–Luttinger liquids (TLLs) partitioned along the longitudinal axis. In particular, we focus on 1D Bose gases as a realization of TLLs and investigate two experimentally relevant situations: tunnel-coupled gases at finite temperatures and after coherent splitting. In both scenarios, we analytically calculate the logarithmic negativity and identify a threshold temperature below which the system is entangled. Notably, this threshold temperature is accessible in the current and near-term 1D Bose gas experiments. Furthermore, we investigate the crossover between quantum and classical correlations in the vicinity of the threshold temperature by comparing logarithmic negativity with mutual information. We argue that the initial mutual information established by the coherent splitting is conserved in TLL dynamics, thus preventing certain generalized Gibbs ensembles from being reached during prethermalization. Moreover, both logarithmic negativity and mutual information are found to scale extensively with the subsystem’s length. Although the ground-state entanglement between coupled TLLs has been predicted to be extensive, this setting is largely overlooked compared to other partitions. Our work extends the study of entanglement between coupled TLLs to finite temperatures and out-of-equilibrium regimes, and provides a strategy towards experimental detection of extensive entanglement in quantum many-body systems at finite temperatures.

DOI: 10.1088/2058-9565/ae8b1d

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Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

Abstract The Tomonaga–Luttinger liquid (TLL) theory describes the low-energy excitations of strongly correlated one-dimensional (1D) fermions. In the past years, a number of studies have provided a detailed understanding of this universality class. More recently, theoretical investigations that go beyond the standard low-temperature, linear-response TLL regime have been developed. While these provide a basis for understanding the dynamics of the spin-incoherent Luttinger liquid, there are few experimental investigations in this regime. Here we report the observation of a thermally induced, spin-incoherent Luttinger liquid in a 6Li atomic Fermi gas confined to 1D. We use Bragg spectroscopy to measure the suppression of spin-charge separation and the decay of correlations as the temperature is increased. Our results probe the crossover between the coherent and incoherent regimes of the Luttinger liquid and elucidate the roles of the charge and the spin degrees of freedom in this regime.

We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.

We consider a system of two coupled Tomonaga-Luttinger liquids (TLL's) on parallel chains and study the R\'enyi entanglement entropy

{S}_{n}

between the two chains. Here the entanglement cut is introduced between the chains, not along the perpendicular direction, as has been done in previous studies of one-dimensional systems. The limit $n\ensuremath{\rightarrow}1$ corresponds to the von Neumann entanglement entropy. The system is effectively described by two-component bosonic field theory with different TLL parameters in the symmetric and antisymmetric channels as far as the coupled system remains in a gapless phase. We argue that in this system,

{S}_{n}

is a linear function of the length of the chains (boundary law) followed by a universal subleading constant ${\ensuremath{\gamma}}_{n}$ determined by the ratio of the two TLL parameters. The formulas of ${\ensuremath{\gamma}}_{n}$ for integer $n\ensuremath{\geqslant}2$ are derived using (a) ground-state wave functionals of TLL's and (b) boundary conformal field theory, which lead to the same result. These predictions are checked in a numerical diagonalization analysis of a hard-core bosonic model on a ladder. Although the analytic continuation of ${\ensuremath{\gamma}}_{n}$ to $n\ensuremath{\rightarrow}1$ turns out to be a difficult problem, our numerical result suggests that the subleading constant in the von Neumann entropy is also universal. Our results may provide useful characterization of inherently anisotropic quantum phases such as the sliding Luttinger liquid phase via qualitatively different behaviors of the entanglement entropy with the entanglement partitions along different directions.

The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value ${g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})$ in the quantum critical regime, the TLL phase, and the quasiclassical region, where

p

is the pressure per unit length rescaled by the interaction energy $\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}$ with interaction strength

c

and linear density

n

. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.