Absence of logarithmic divergence of the entanglement entropies at the phase transitions of a 2D classical hard rod mode
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Absence of logarithmic divergence of the entanglement entropies at the phase transitions of a 2D classical hard rod model Christophe Chatelain 1,a and Andrej Gendiar 2 1 2
Universit´e de Lorraine, CNRS, LPCT, 54000 Nancy, France Institute of Physics, Slovak Academy of Sciences, D´ ubravska´ a cesta 9, 845 11 Bratislava, Slovakia Received 31 January 2020 / Received in final form 14 April 2020 Published online 13 July 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. Entanglement entropy is a powerful tool to detect continuous, discontinuous and even topological phase transitions in quantum as well as classical systems. In this work, von Neumann and Renyi entanglement entropies are studied numerically for classical lattice models in a square geometry. A cut is made from the center of the square to the midpoint of one of its edges, say the right edge. The entanglement entropies measure the entanglement between the left and right halves of the system. As in the strip geometry, von Neumann and Renyi entanglement entropies diverge logarithmically at the transition point while they display a jump for first-order phase transitions. The analysis is extended to a classical model of non-overlapping finite hard rods deposited on a square lattice for which Monte Carlo simulations have shown that, when the hard rods span over 7 or more lattice sites, a nematic phase appears in the phase diagram between two disordered phases. A new Corner Transfer Matrix Renormalization Group algorithm (CTMRG) is introduced to study this model. No logarithmic divergence of entanglement entropies is observed at the phase transitions in the CTMRG calculation discussed here. We therefore infer that the transitions neither can belong to the Ising universality class, as previously assumed in the literature, nor be discontinuous.
1 Introduction The quantum entanglement between the two subsystems A and B of a macroscopic system has attracted a considerable interest in the last decade [1–3]. Besides its purely theoretical interest, the entropy that quantifies this entanglement have found some applications, in particular in the identification of phase boundaries as will be discussed in this work. Denoting ρA = TrB |ψ0 ihψ0 |
(1)
the reduced density matrix of subsystem A in the ground state |ψ0 i of the system, the von Neumann entanglement entropy of the degrees of freedom of A with those of subsystem B is defined as SA = − Tr ρA log ρA
(2)
while the Renyi entropies are Sn = a
1 log Tr ρnA . 1−n
e-mail: [email protected]
(3)
In dimension 1 + 1 and with Open Boundary Conditions, Conformal Field Theory predicts that von Neumann entanglement entropy diverges logarithmically when approaching a critical point [4] SA =
c ξ ln + c0 6 a
(4)
where the correlation length ξ scales with the control parameter δ as ξ ∼ |δ|−ν . The prefactor is proportional to the central charge c which is a universal quantity. At th
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