Entanglement entropy and the complex plane of replicas

Abstract

The entanglement entropy of a subsystem $A$ of a quantum system is expressed, in the replica method, through analytic continuation with respect to n of the trace of the n-th power of the reduced density matrix $\trρ_A^n$. We study the analytic properties of this quantity as a function of n in some quantum critical Ising-like models in 1+1 and 2+1 dimensions. Although we find no true singularities for n>0, there is a threshold value of n close to 2 which separates two very different `phases'. The region with larger n is characterized by rapidly convergent Taylor expansions and is very smooth. The region with smaller n has a very rich and varied structure in the complex n plane and is characterized by Taylor coefficients which instead of being monotone decreasing, have a maximum growing with the size of the subsystem. Finite truncations of the Taylor expansion in this region lead to increasingly poor approximations of $\trρ_A^n$. The computation of the entanglement entropy from the knowledge of $\trρ^n_A$ for positive integer n becomes extremely difficult particularly in spatial dimensions larger than one, where one cannot use conformal field theory as a guidance in the extrapolations to n=1.

24 pages, 9 figures. v2: typos corrected, two figures replaced

References in corpus (8)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement spectrum in one-dimensional systems
• Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum
• Topological Entanglement Renyi Entropy and Reduced Density Matrix Structure
• Universal terms for the entanglement entropy in 2+1 dimensions
• Entanglement Entropy in the O(N) model
• Entanglement Entropy and Twist Fields
• Entanglement Entropy in Extended Quantum Systems

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