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arXiv:1611.09750 · Journal of Physics Conference Series

Numerical determination of the CFT central charge in the site-diluted Ising model

P. Belov, A A Nazarov, A O Sorokin

Published Nov 1, 2017• 1 citation
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Abstract

We propose a new numerical method to determine the central charge of the conformal field theory models corresponding to the 2D lattice models. In this method, the free energy of the lattice model on the torus is calculated by the Wang-Landau algorithm and then the central charge is obtained from a free energy scaling with respect to the torus radii. The method is applied for determination of the central charge in the site-diluted Ising model.

DOI
10.1088/1742-6596/929/1/012037

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We propose a new numerical method to determine the central charge of the conformal field theory models corresponding to the 2D lattice models. In this method, the free energy of the lattice model on the torus is calculated by the Wang-Landau algorithm and then the central charge is obtained from a free energy scaling with respect to the torus radii. The method is applied for determination of the central charge in the site-diluted Ising model.
We present a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with good precision from a free-energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts, and site-diluted Ising models, and we also discuss an estimation of the conformal weights.
Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly ccc. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.
We present a finite size scaling technique to calculate the central charge and some critical exponents of two-dimensional critical models with a Monte Carlo simulation. We use systems with dimensions $L\ifmmode\times\else\texttimes\fi{}M,$ and focus on the scaling behavior in M/L.M/L.M/L. The finite size scaling relation that we use is the universal expression for the stress tensor on the torus. The stress tensor is the operator that governs the anisotropy of the system, and stems for the theory of conformal invariance. We show that a lattice representation of the stress tensor can easily be constructed, such that its expectation value on the torus can be calculated in a Monte Carlo simulation. In doing so, we observe that the stress tensor turns out to be remarkably insensitive to critical slowing down. We show that the typical simulation time scales with the linear system dimension LLL roughly as L4,{L}^{4},L4, and that this scaling holds for the ordinary Metropolis algorithm as well as for more sophisticated cluster algorithms, such that it is fruitless to invoke the latter. We test the method for the Ising model (with central charge c=c=c=12\frac{1}{2}21​), the Ashkin-Teller model (c=1),(c=1),(c=1), and the F model (also c=1c=1c=1).
Universality classes of non-unitary critical theories in two-dimensions are characterized by the central charge. However, experimental determination of the central charge of a non-unitary critical theory has not been done before because of the intrinsic difficulty that complex parameters usually occur in non-unitary theory, which is not physical. Here we propose to extract the effective central charge of the non-unitary critical point of a two-dimensional lattice model from the quantum coherence measurement of a probe spin which is coupled to the lattice model. A recent discovery shows that quantum coherence of a probe spin which is coupled to a bath is proportional to the partition function of the bath with a complex parameter. Thus the effective central charge of a non-unitary conformal field theory may be extracted from quantum coherence measurement of a probe spin which is coupled to a bath. We have applied the method to the Yang-Lee edge singularity of the two-dimensional Ising model and extracted the effective central charge of the Yang-Lee edge singularity with good precision and tested other predictions of non-unitary conformal field theory. This work paves the way for the first experimental observation of the effective central charge of non-unitary conformal field theory.
We calculate the von Neumann and R\'enyi bipartite entanglement entropy of the O(2)O(2)O(2) model with a chemical potential on a 1+11+11+1-dimensional Euclidean lattice with open and periodic boundary conditions. We show that the Calabrese-Cardy conformal field theory predictions for the leading logarithmic scaling of these entropies are consistent with a central charge c=1c=1c=1. This scaling survives the time-continuum limit and truncations of the microscopic degrees of freedom, modifications which allow us to connect the Lagrangian formulation to quantum Hamiltonians. At half-filling, the forms of the subleading corrections imposed by conformal field theory allow the determination of the central charge with an accuracy better than 2% for moderately sized lattices. We briefly discuss the possibility of estimating the central charge using quantum simulators.
Abstract We compute the entanglement entropy in a two dimensional conformal field theory at finite size and finite temperature in the large central charge limit via the replica trick. We first generalize the known monodromy method for the calculation of conformal blocks on the plane to the torus. Then, we derive a monodromy method for the zero-point conformal blocks of the replica partition function. We explain the differences between the two monodromy methods before applying them to the calculation of the entanglement entropy. We find that the contribution of the vacuum exchange dominates the entanglement entropy for a large class of CFTs, leading to universal results in agreement with holographic predictions from the RT formula. Moreover, we determine in which regime the replica partition function agrees with a correlation function of local twist operators on the torus.