On the boundary Ising model with disorder operators

G. M. T. Watts

We extend the well-known method of calculating bulk correlation functions of the conformal Ising model via bosonisation to situations with boundaries. Oshikawa and Affleck have found the boundary states of two decoupled Ising models in terms of the orbifold of a single free boson compactified on a circle of radius r=1; we adapt their results to include disorder operators. Using these boundary states we calculate the expectation value of a single disorder field on a cylinder with free boundary conditions and show that in the appropriate limits we recover the standard and frustrated partition functions. We also show how to calculate Ising correlation functions on the upper half plane. 1

Subinay Dasgupta, Dietrich Stauffer, V. Dohm

Erich Eisenriegler

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles 〈σ(r_{1})〉 and 〈ε(r_{2})〉, respectively. Does increasing the order at r_{1} always decrease the disorder at r_{2}? Does increasing the disorder at r_{2} always decrease the order at r_{1}? The answer to these questions is contained in the cumulant response function 〈σ(r_{1})ε(r_{2})〉^{(cum)}. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of σ(r_{1})ε(r_{2}) and exact results for the Ising model, we analyze 〈σ(r_{1})ε(r_{2})〉^{(cum)} in two-dimensional critical systems defined on the x-y plane with mixed + and - boundary conditions. Particularly interesting behavior is found when either of the operators σ or ε is located on a "zero line" in the x-y plane, along which 〈σ(r)〉 vanishes. Results for half-plane, triangular, and rectangular geometries are presented.

Abhishek Samlodia, Simon Catterall, Alexander F. Kemper +2 more

We study the quantum Ising model on (2+1)-dimensional anti-de Sitter space using Matrix Product States (MPS) and Matrix Product Operators (MPOs). We explore the bulk phase diagram of the theory on regular tessellations of hyperbolic space with coordination number seven and find disordered and ordered phases separated by a phase transition. We find that the boundary-boundary spin correlation function exhibits power law scaling deep in the disordered phase of the Ising model consistent with holography. At the critical point, we find the boundary entanglement entropy scales logarithmically with subsystem size but away from this, we see a linear scaling. In comparison, the full system exhibits a volume law scaling, which is expected in chaotic and/or highly connected systems. We also measure Out of time Ordered Correlators (OTOCs) to explore the scrambling behavior of the theory.

Marc Sigelle D

N. Sh. Izmailian