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arXiv:2608.05968

Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids

Léonce Dupays, Taufiq Murtadho, Bi Hong Tiang, Nelly H. Y. Ng, Paola Ruggiero

quant-phcond-mat.quant-gascond-mat.stat-mechcond-mat.str-elSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations

Abstract

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.

Comments: 45 pages, 8 figures

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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.

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We consider the R\'enyi entanglement entropy of bosonic Tomonaga-Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher-order corrections involve a microscopic length scale, the leading-order scaling depends only on this sole dimensionless parameter which characterizes the low-energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Tomonaga-Luttinger liquid theory for the Lieb-Liniger model of $\ensuremath{\delta}$-function interacting bosons in the one-dimensional spatial continuum.

Abstract We investigate the scaling of the Rényi entanglement entropies for a particle bipartition of interacting spinless fermions in one spatial dimension. In the Tomonaga–Luttinger liquid regime, we calculate the second Rényi entanglement entropy and show that the leading order finite-size scaling is equal to a universal logarithm of the system size plus a non-universal constant. Higher-order corrections decay as power-laws in the system size with exponents that depend only on the Luttinger parameter. We confirm the universality of our results by investigating the one dimensional t − V model of interacting spinless fermions via exact-diagonalization techniques. The resulting sensitivity of the particle partition entanglement to boundary conditions and statistics supports its utility as a probe of quantum liquids.

Luttinger liquids (LLs) arise by coupling left- and right-moving particles through interactions in one dimension. This most natural partitioning of LLs is investigated by the momentum-space entanglement after a quantum quench using analytical and numerical methods. We show that the momentum-space entanglement spectrum of a LL possesses many universal features both in equilibrium and after a quantum quench. The largest entanglement eigenvalue is identical to the Loschmidt echo, i.e., the overlap of the disentangled and final wave functions of the system. The second largest eigenvalue is the overlap of the first excited state of the disentangled system with zero total momentum and the final wave function. The entanglement gap is universal both in equilibrium and after a quantum quench. The momentum-space entanglement entropy is always extensive and saturates fast to a time independent value after the quench, in sharp contrast to a spatial bipartitioning.

The coupling between electronic and lattice degrees of freedom lies at the core of many important properties of solids. Nevertheless, surprisingly little is known about the entanglement between these degrees of freedom. Here, we calculate the entanglement entropy at zero temperature as well as the mutual information and the logarithmic entanglement negativity at finite temperatures between the electrons and the lattice for a one-dimensional chain. The electrons are described within Luttinger-liquid theory. Our results show that the entanglement entropy diverges when one approaches the limit of stability, the so-called Wentzel-Bardeen singularity. We find that the mutual information and the logarithmic entanglement negativity decrease with temperature. The mutual information reaches a finite value in the infinite-temperature limit, which is a consequence of the infinite linear electron spectrum of Luttinger theory. The logarithmic entanglement negativity becomes exactly zero above a certain temperature; that is, the lattice and the electrons become nonentangled above this temperature. If the electron-electron interaction is unscreened or weakly screened, this characteristic temperature diverges with the system size. However, if the interaction is strongly screened, the characteristic temperature is finite and independent of size, indicating a phase transition in the thermodynamic limit.

We study the heating dynamics of a Luttinger liquid, upon suddenly coupling it to a dissipative environment. Within the Lindblad equation, the environment couples to local currents and heats the quantum liquid up to infinite temperatures. The fermionic single particle density matrix, and many other correlators, retain the initial Luttinger liquid correlations in space but decay exponentially in time with a rate that depends on the strength of the interaction. The spectrum of the time evolved density matrix is gapped, which collapses gradually as

-\ln(t)

. The von Neumann entropy crosses over from the early time

-t\ln(t)

behaviour to a

\ln(t)

growth for late times. The early time dynamics is captured numerically by performing simulations on spinless interacting fermions, using several numerically exact methods. Our results establish the validity of bosonization for the early time dynamics of 1D lattice models, and could be observed experimentally in bosonic Luttinger liquids.