Entanglement negativity in extended systems: A field theoretical approach

TL;DR

This work develops a systematic field-theoretical framework to compute entanglement negativity in the ground state of 1D quantum field theories via a replica-imaginary-time path-integral approach. By translating Tr(ρ_A^{T2})^n into twist-field correlators and exploiting conformal invariance, it derives universal results for adjacent and disjoint interval configurations, and provides finite-size mappings and boundary generalizations. The authors validate the CFT predictions through extensive numerical checks in harmonic chains with various boundary conditions, revealing precise agreement and informative finite-size corrections. They also explore the impact of mass (scaling toward massive theories) and provide analytic structures for both non-compact and compact bosons, highlighting parity effects in the replica continuation. The framework lays groundwork for future extensions to finite temperature and more complex lattice systems while delivering concrete, testable predictions for universal entanglement properties in 1D quantum systems.

Abstract

We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose rho_A^{T_2} of the reduced density matrix of a subsystem A=A_1 U A_2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces Tr (rho_A^{T_2})^n are obtained. The logarithmic negativity E= log||rho_A^{T_2}|| is then the continuation to n->1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths L1, L2 in an infinite system, we derive the result E\sim(c/4) ln(L1 L2/(L1+L2)), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end-points and so is manifestly scale invariant. We explicitly calculate the scale-invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n->1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.

Figures (14)

• Figure 1: The three main configurations of 1D systems we consider. Top: the entanglement between two disjoint intervals $A_1$ and $A_2$ embedded in the ground-state of a larger system formed by the union of $A_1$, $A_2$ and the remainder $B$. The whole system can be either finite or infinite. Middle: The entanglement between two adjacent intervals in a larger system. Bottom: The entanglement between two adjacent intervals which form the full system ($B\to\emptyset$). In all cases we denote $A=A_1\cup A_2$.
• Figure 2: Top: The reduced density matrix $\rho_A$ o f two disjoint intervals. Middle: Partial transpose with respect to the second interval $\rho_A^{T_{2}}$. Bottom: Reversed partial transpose $\rho_A^{C_{2}}=C \rho_A^{T_{2}}C$, where $C$ reverses the order of the row and column indices.
• Figure 3: The path integral representation of ${\rm Tr}\rho_A^n$ gives a $n$-sheeted Riemann surface ${\cal R}_n$ depicted here for $n=3$ and $A=[u_1,v_1]\cup [u_2,v_2]$.
• Figure 4: Path integral representation of ${\rm Tr}(\rho_A^{T_{2}})^n={\rm Tr}(\rho_A^{C_{2}})^n$ for $n=3$.
• Figure 5: Cut of the Riemann surface defined by $\langle {\cal T}^2_{n}(u_2) \overline{\cal T}^2_{n}(v_2)\rangle$. Left: A typical example for $n$ even ($n=4$ in the figure) showing how the surface decouples in two independent parts, each with half of the sheets. Right: A typical example for $n$ odd ($n=5$ in the figure) showing that the net effect is just a reshuffling of the sheet numeration going from $(1,2,3,4,5)$ to $(1,3,5,2,4)$.