Critical exponents of one-dimensional models in lattice field theory

Marcel den Nijs

D.I. Kazakov, Alexey A. Vladimirov, O.V. Tarasov

Tohru Koma

We question the validity of the finite-size-correction method based on conformal field theories. It is shown that the method leads to incorrect critical exponents on some of the correlation functions for a one-dimensional free fermion model. The model exhibits the same critical behaviour as in the well-known square-lattice Ising model that is believed to be conformally invariant.

Anatoli G. Izergin, V. E. Korepin

Hiroyuki Shima, Yasunori Sakaniwa

We investigate the dynamic critical exponent of the two-dimensional Ising model defined on a curved surface with constant negative curvature. By using the short-time relaxation method, we find a quantitative alteration of the dynamic exponent from the known value for the planar Ising model. This phenomenon is attributed to the fact that the Ising lattices embedded on negatively curved surfaces act as ones in infinite dimensions, thus yielding the dynamic exponent deduced from mean field theory. We further demonstrate that the static critical exponent for the correlation length exhibits the mean field exponent, which agrees with the existing results obtained from canonical Monte Carlo simulations.

Yan-Wei Dai, Xi Hao Chen, Sam Young Cho +1 more

We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2 XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. As the lattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.