Entanglement negativity in extended systems: A field theoretical approach (1210.5359v2)

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.

• The paper introduces a systematic replica trick approach to calculate entanglement negativity in one-dimensional quantum field theories.
• It derives analytical expressions for negativity in adjacent and disjoint intervals, emphasizing universal scaling features and central charge effects.
• The framework is numerically validated using a 1D harmonic chain model, underscoring its practical applicability to extended quantum systems.

Entanglement Negativity in Quantum Field Theory: Methodology and Implications

This paper, authored by Pasquale Calabrese, John Cardy, and Erik Tonni, presents a field-theoretical framework to compute entanglement negativity in extended quantum systems, particularly focusing on one-dimensional (1D) quantum field theories (QFTs). Entanglement negativity is a prominent measure of entanglement for bipartite quantum states, especially important for mixed states where traditional measures like entanglement entropy become inadequate. The authors adopt a systematic approach to derive the negativity, employing a replica trick and path integral formalism.

Key Concepts and Methodology

1. Replica Method for Negativity: The authors extend the replica trick approach, which is traditionally used to calculate the entanglement entropy, to determine entanglement negativity. The analytical continuation of the traces of even powers of the partial transpose of the density matrix plays a critical role. The entanglement negativity is defined as the logarithmic negativity and is expressed as:

$$\mathcal{E} = \lim_{n_e \to 1} \ln \text{Tr}(\rho_A^{T_2})^{n_e}$$

where $n_e$ denotes even integer replicas and $\rho_A^{T_2}$ is the partial transpose of the reduced density matrix.

1. Application to Conformal Field Theories (CFTs): The authors analyze several cases where the theoretical predictions can be verified against exact results. They find that for a single interval in an infinite system, the negativity matches the R\'enyi entropy of order 1/2. For adjacent intervals, negativity is expressed as $$\mathcal{E} \sim (c/4)\ln(\ell_1\ell_2/(\ell_1+\ell_2))$$, highlighting the central charge $c$ of the theory.
2. Disjoint Intervals: For disjoint intervals, the negativity exhibits dependence on the harmonic ratio of the intervals' endpoints, a feature reflecting scale invariance. The analysis involves more complex functions dependent on the specific operator content of CFTs, and extensions to these results are provided for systems with free compactified bosons and non-compactified bosons.
3. Numerical Verification: The theoretical predictions are numerically verified using a 1D harmonic chain model, under both periodic and open boundary conditions, showing remarkable agreement between theory and numerical simulation.

Implications and Future Directions

• Theoretical Implications: The paper provides a robust framework for examining tripartite entanglement in quantum systems, revealing, for instance, universal scaling features near criticality and connections between logarithmic negativity and symmetries in conformal systems.
• Computational Methods: Numerically, the paper's approach facilitates the computation of entanglement measures in large systems, serving as a benchmark to paper entanglement dynamics in lattice models.
• Extensions and Applications: The paper hints at extensions to finite-temperature systems and higher-dimensional theories. Moreover, the framework could be applied to analyze entanglement in models with varying degrees of freedom, such as those in the Luttinger liquid universality class.

Overall, this work lays a comprehensive foundation for understanding entanglement in non-trivial quantum states and sets the stage for more sophisticated investigations into the role of entanglement in quantum statistical mechanics and quantum information science. Future studies could focus on uncovering further connections to holographic entanglement entropy and topological phases of matter, potentially uncovering deeper insights into the structure of quantum correlations.