Fisher's zeros in lattice gauge theory [thesis]

Daping Du
In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models [40, 33] and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes [24] . By reconstructing the density of
more » ... ting the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density [5, 4] . We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) [22] and U(1) models [9] . We also discuss the convergence of the Ferrenberg-Swendsen's method [32] which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. To my wife, Hui-Yun Wu and my family. ii ACKNOWLEDGMENTS First, I would like to thank Professor Yannick Meurice, my Ph.D. thesis advisor, for his tremendous help in my entire graduate career at the University of Iowa. He has been encouraging and helping me in my study and research. He taught me the knowledge as well as the ways of utilizing the knowledge in the research. He is a good friend in addition to a good advisor. The thesis would have not been possible without his generous help, continuous encouragement and meticulous guidance. I am indebted to my colleagues and friends has been helping me with the data generation and partial proof reading of this paper. Yuzhi and Haiyuan has provided me with a lot of helps and suggestions. Alex has written the SU(2) codes which made the numerical calculations possible. Alexei made all the U(1) data available and I benefited from several discussions with him. I have also got a lot of help from my friends Ran Lin, Juan Chen and Nan Chen. I would also like to thank Andreas Kronfeld for his help during my visit at Fermilab which brought me with new understanding on the applications of lattice gauge theory and partially expedited the writing of the paper. I also benefited considerably from the discussions and lectures of Professor Vincent Rodgers, Professor Wayne Polyzou, Professor Craig Pryor, Professor Mary Hall Reno and other respected faculties in the physics department. Finally and importantly, I owe too much to my wife and my family. My wife has been supportive of me ever since we were married. Without her effort in it, I could never make it finished. I am grateful to my parents and my siblings for their encouragement and support. iii ABSTRACT In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models [40, 33] and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes [24] . By reconstructing the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density [5, 4] . We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) [22] and U(1) models [9] . We also discuss the convergence of the Ferrenberg-Swendsen's method [32] which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. The numerical calculations also enable the discussion of the finite volume effects which are important to the weak expansion. We calculate the Fisher's zeros of the SU(2) and U(1) models at various iv volumes using the numerical entropy density functions. We compare different methods of locating the zeros. By the assumption of validity of the saddle point approximation, we find that the roots of the second derivative of the entropy density function have an interesting relation with the actual zeros and may possibly reveal the scaling property of the zeros. Using the analytic approximation of the numerical density of states, we are able to locate the Fisher's zeros of the SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the real axis, which is compatible with the scenario that a crossover instead of a phase transition is expected in the infinite volume limit [56, 16] . In contrast, with the precise determination of the locations of Fisher's zeros for the U(1) model at smaller lattice sizes L = 4, 6 and 8, we show that the imaginary parts of the zeros decrease with a power law of L −3.07 and pinch the real axis at β = 1.01134, which agrees with results using other methods [50] . Preliminary results at larger volumes indicate a first-order transition in the infinite volume limit.
doi:10.17077/etd.bfnqfycu fatcat:umepb547rbe33equlwq6zxeema