Rényi entropies for one-dimensional quantum systems with mixed boundary conditions

Abstract

We present a general method for calculating Rényi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends. In the conformal field theory framework, this computation boils down to the evaluation of the correlation function of one twist field and two boundary condition changing operators in the cyclic orbifold. Exploiting null-vectors of the cyclic orbifold, we derive ordinary differential equations satisfied by these correlation functions. In particular we obtain an explicit expression for the second Rényi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions. In order to compare our results with numerical data for the Ising and three-state Potts critical chains, we also identify and compute the leading finite size corrections.

The operator algebra of cyclic orbifolds

“…conjectured in [51] for all the cases we have checked. We should mention, that this conjectured third order equation has been derived in [18]. Furthermore, the conformal blocks in the z 1 → z 2 channel were found in [18] to correspond to the following twist field fusion rules:…”

“…The main simplification in that case is that such two-point functions are completely fixed by conformal invariance. In contrast, if the system has boundaries, or is at finite temperature with periodic boundary conditions, the CFT computation involves two point functions of twist fields on more complicated Riemann surfaces, namely, the upper half plane or the torus [16][17][18][19][20][21][22][23][24][25]59]. Likewise, if the region A consists of several disjoint intervals, Rényi entropies involve higher-point correlation functions of twist fields, or equivalently the partition function on higher genus Riemann surfaces [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

“…It boils down to the computation of the two-point correlation function of twist fields on the sphere in the Z N orbifold, which is relatively simple from the CFT point of view. In contrast, situations involving boundaries [16][17][18], finite size and finite temperature [19][20][21][22][23][24][25], or simply several intervals [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] are much more complicated and to this day remain mostly unsolved for generic CFTs. The main difficulty in such cases is that the Rényi entropy involves either the two-point function of twist fields on non-trivial surfaces (such as the upperhalf-plane or the torus), or higher-point correlation functions.…”

Entanglement evolution after a global quench across a conformal defect

“…Indeed, this contribution is entirely determined by the boundary CFT, and is related to the boundary entropy of Affleck and Ludwig [10]. Such boundary entropies have been studied extensively both within CFT [7,[11][12][13][14][15] as well as numerically for various critical spin chains in the presence of boundary fields [16][17][18]. Some particular choice of boundary conditions can also lead to the emergence of zero modes.…”

Zero-mode entanglement across a conformal defect

“…Indeed, this contribution is entirely determined by the boundary CFT, and is related to the boundary entropy of Affleck and Ludwig [10]. Such boundary entropies have been studied extensively both within CFT [7,[11][12][13][14][15] as well as numerically for various critical spin chains in the presence of boundary fields [16][17][18].…”