Entanglement entropy and conformal field theory (0905.4013v2)

Abstract: We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.

• The paper presents the effective computation of entanglement entropy in 1+1 systems using the replica trick and conformal field theory methods.
• It demonstrates the universal scaling law S_A = (c/3) ln(ℓ/a) and explores finite-size, thermal, and boundary effects on quantum correlations.
• The study extends its analysis to non-equilibrium quantum quenches, linking the entanglement spectrum to practical tensor network simulations.

Entanglement Entropy and Conformal Field Theory

The paper by Pasquale Calabrese and John Cardy provides a comprehensive review of the application of conformal field theory (CFT) to calculate entanglement entropy in 1+1 dimensional systems. Entanglement entropy, a measure of quantum correlations between different parts of a system, plays a crucial role in understanding quantum many-body systems. Conformal field theory, known for its ability to describe critical phenomena in various physics domains, emerges as a robust tool for evaluating this entropy in systems at critical points.

Summary of Key Concepts and Results

The authors begin by introducing the concept of entanglement entropy for a subsystem $A$ of a quantum system. The entanglement entropy is defined as the von Neumann entropy of the reduced density matrix of the subsystem, which provides insights into the quantum correlations between $A$ and its complement. The efficacious computation of entanglement entropy becomes achievable due to its universal scaling properties at one-dimensional conformal critical points, captured by the formula:

$S_A = \frac{c}{3} \ln \frac{\ell}{a} + c'_1$

where $c$ denotes the central charge of the underlying CFT, $\ell$ is the length of the subsystem $A$, and $a$ is a short-distance cutoff.

Entanglement in Various Geometries and Settings

1. Single Interval in Infinite Systems: The paper meticulously derives the entanglement entropy for a single interval in an infinite system. Essential to this analysis is the use of the "replica trick", which involves constructing an $n$th power of the density matrix and considering Riemann surfaces with branch cuts to account for the entanglement across intervals.
2. Finite Size and Finite Temperature: The mapping techniques from CFT are applied to finite systems and systems at non-zero temperatures, demonstrating that entanglement entropy exhibits crossovers from quantum to thermal behavior. Such analyses show that entanglement entropy can provide more than just a measure of quantum correlations; it can also signify a universal scaling form akin to a thermal entropy in the proper regimes.
3. Systems with Boundaries: The entanglement entropy in systems with boundaries, where a fascinating combination of bulk and boundary effects are manifest, leads to insights about "boundary entropy" as defined by Affleck and Ludwig. Such calculations reveal details of how the constraints influence quantum correlations.
4. Quenched and Non-Equilibrium Systems: Beyond equilibrium states, the paper extends the CFT formalism to the paper of quantum quenches—situations where a system is suddenly taken out of equilibrium. Here, entanglement entropy dynamics provide a powerful diagnostic tool for understanding how correlations spread in quantum systems.

Challenges and Future Directions

The initial misstep in analyzing multiple disjoint intervals using uniformization is corrected through a detailed investigation of higher genus Riemann surfaces. The general case involves intricate correlation functions and further development of these methodologies for practical systems remains an active area of research.

The paper concludes with an exploration of the "entanglement spectrum", revealing that the eigenvalue distribution of the reduced density matrix, not just its entropy, is fundamentally important. This perspective links with effective simulations of quantum systems, particularly within tensor network frameworks like MPS and MERA.

In summary, this paper is a cornerstone in the paper of quantum entanglement through CFT, laying out systematic methods for a variety of configurations and opening channels for future exploration in higher dimensions, more complex settings, and further connections with quantum computational paradigms.