We find the exact finite size and lattice corrections, to the partition function
Z, of a continuous spin Gaussian model on a toroidal triangular lattice, with periods
L0 and
L1. The spins carry a representation of the fundamental group of the torus labeled by
u0 and
u1 and a mass
m. Summing ${Z}^{\ensuremath{-}1/2}$ over
ui gives the corresponding result for the Ising model. The limits $m\ensuremath{\rightarrow}0$ and ${u}_{i}\ensuremath{\rightarrow}0$ do not commute. With
m=0 and
ui nonzero the model exhibits a vortex critical phase. In the continuum limit, for arbitrary
m, the finite size corrections to $\ensuremath{-}\mathrm{ln}Z$ are modular invariant. In the limit ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ the ``cylinder charge''
c(u0,m2L02) ranges nonmonotonically from $2[{1+6u}_{0}({u}_{0}\ensuremath{-}1)]$ for
m=0 to zero for $m\ensuremath{\rightarrow}\ensuremath{\infty}$.
The critical-point anomaly of a plane square $m\ifmmode\times\else\texttimes\fi{}n$ Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ with $\ensuremath{\xi}=\frac{m}{n}$ fixed. Among other results, it is shown that for fixed $\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, the specific heat per spin of a large lattice is given by $\frac{{C}_{\mathrm{mn}}(T)}{{k}_{\mathrm{B}}\mathrm{mn}}={A}_{0}\mathrm{ln}n+B(\ensuremath{\tau}, \ensuremath{\xi})+{B}_{1}(\ensuremath{\tau})\frac{(\mathrm{ln}n)}{n}+\frac{{B}_{2}(\ensuremath{\tau}, \ensuremath{\xi})}{n}+O[\frac{{(\mathrm{ln}n)}^{3}}{{n}^{2}}],$ where explicit expressions can be given for
A0 and for the functions
B,
B1, and
B2. It follows that the specific-heat peak of the finite lattice is rounded on a scale $\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$, while the maximum in
Cmn(T) is displaced from
Tc by $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$. For ${\ensuremath{\xi}}_{0}>\ensuremath{\xi}>{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, where ${\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}$, the maximum lies above
Tc; but for $\ensuremath{\xi}>{\ensuremath{\xi}}_{0}$ or $\ensuremath{\xi}<{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the maximum is depressed below
Tc; when $\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the relative shift in the maximum from
Tc is only of order
n2(lnn). Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.
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.
Abstract We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with c > 1. This formula is valid when one or more of the operators has large dimension or — in the presence of a twist gap — has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT2 structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT2, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.
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
c. 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.