A. Denbleyker

Fisher's Zeros as the Boundary of Renormalization Group Flows in Complex Coupling Spaces

We propose new methods to extend the renormalization group transformation to complex coupling spaces. We argue that Fishers zeros are located at the boundary of the complex basin of attraction of infrared fixed points. We support this picture with numerical calculations at finite volume for two-dimensional O(N) models in the large-N limit and the hierarchical Ising model. We present numerical evidence that, as the volume increases, the Fishers zeros of four-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action stabilize at a distance larger than 0.15 from the real axis in the complex β=4/g{2} plane. We discuss the implications for proofs of confinement and searches for nontrivial infrared fixed points in models beyond the standard model.

Tensor renormalization group study of classical XY model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.

Controlling sign problems in spin models using tensor renormalization

We consider the sign problem for classical spin models at complex beta = 1/g(0)(2) on L x L lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im beta than the reweighting Monte Carlo method. For the Ising model with complex beta we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the O(2) model on L x L lattices when the number of states D-s increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the O(2) model with a chemical potential is briefly discussed.

Series expansions of the density of states in SU(2) lattice gauge theory.

We calculate numerically the density of states n(S) for SU(2) lattice gauge theory on L{sup 4} lattices [S is the Wilsons action and n(S) measures the relative number of ways S can be obtained]. Small volume dependences are resolved for small values of S. We compare ln(n(S)) with weak and strong coupling expansions. Intermediate order expansions show a good overlap for values of S corresponding to the crossover. We relate the convergence of these expansions to those of the average plaquette. We show that, when known logarithmic singularities are subtracted from ln(n(S)), expansions in Legendre polynomials appear to converge and could be suitable to determine the Fishers zeros of the partition function.

Fisher's zeros of quasi-Gaussian densities of states

We discuss apparent paradoxes regarding the location of the zeros of the partition function in the complex {beta} plane (Fishers zeros) of a pure SU(2) lattice gauge theory in 4 dimensions. We propose a new criterion to draw the region of the complex {beta} plane where reweighting methods can be trusted when the density of states is almost but not exactly Gaussian. We propose new methods to infer the existence of zeros outside of this region. We demonstrate the reliability of these proposals with quasi-Gaussian Monte Carlo distributions where the locations of the zeros can be calculated by independent numerical methods. The results are presented in such a way that the methods can be applied for general lattice models. Applications to specific lattice models will be discussed in a separate publication.

Dyson's Instability in Lattice Gauge Theory

We discuss Dysons argument that the vacuum is unstable under a change g^2 -> - g^2, in the context of lattice gauge theory. For compact gauge groups, the partition function is well defined at negative g^2, but the average plaquette P has a discontinuity when g^2 changes sign. This reflects a change of vacuum rather than a loss of vacuum. In addition, P has poles in the complex g^2 plane, located at the complex zeros of the partition function (Fishers zeros). We discuss the relevance of these singularities for lattice perturbation theory. We present new methods to locate Fishers zeros using numerical values for the density of state in SU(2) and U(1) pure gauge theory. We briefly discuss similar issues for O(N) nonlinear sigma models where the local integrals are also over compact spaces.

Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models

We show that the Tensor Renormalization Group (TRG) method can be applied to O(N) spin models, principal chiral models and pure gauge theories (Z2, U(1) and SU(2)) on (hyper) cubic lattices. We explain that contrarily to some common belief, it is very difficult to write compact formulas expressing the blockspinning of lattice models. We show that in contrast to other approaches, the TRG formulation allows us to write exact blocking formulas with numerically controllable truncations. The basic reason is that the TRG blocking separates neatly the degrees of freedom inside the block and which are integrated over, from those kept to communicate with the neighboring blocks. We argue that the TRG is a method that can handle large volumes, which is crucial to approach quasi-conformal systems. The method can also get rid of some sign problems. We discuss recent results regarding the critical properties of the 2D O(2) nonlinear sigma model with complex beta and chemical potential. As some of these results appeared in a recently published paper (PRD 88, 056005) and two recent preprints (arXiv:1309.4963 and arXiv:1309.6623), these proceedings rather emphasize the conceptual aspects of our ongoing effort.

Fisher's zeros, complex RG flows and confinement in LGT models.

The zeros of the partition function in the complex beta plane (Fishers zeros) play an important role in our understanding of phase transitions and RG flows. Recently, we argued that they act as gates or separatrices for complex RG flows. Using histogram reweighting to construct the density of states, we calculate the Fishers zeros for pure gauge SU(2) and U(1) on L^4 lattices. For SU(2), these zeros appear to move almost horizontally when the volume increases. They stay away from the real axis which indicates a confining theory at zero temperature. We discuss the effect of an adjoint term on these results. In contrast, using recent multicanonical simulations for the U(1) model for L up to 8 we find that the zeros pinch the real axis near beta =1.0113. Preliminary results concerning U(1) at larger volumes, SU(3) with 3 light flavors and plans to delimit the boundary of the conformal window are briefly discussed.

Volume dependence of Fisher's zeros

We study the location of the partition function zeros in the complex β plane (Fisher’s Zeros) for SU(2) lattice gauge theory on L 4 lattices. We discuss recent attempts to locate complex zeros for L = 4 and 6. We compare results obtained using various polynomial approximations of the logarithm of the density of states and a straightforward MC reweighting. We conclude that the method based on a combination of discrete Chebyshev orthogonality and patching plaquette distributions at different β provides the more reliable estimates.

Finite size scaling and universality in SU(2) at finite temperature

We study the 4-th Binder cumulant on