Computing quantum phase transitions

This article first gives a concise introduction to quantum phase transitions, emphasizing similarities with and differences to classical thermal transitions. After pointing out the computational challenges posed by quantum phase transitions, a number of successful computational approaches is discussed. The focus is on classical and quantum Monte Carlo methods, with the former being based on the quantum-to classical mapping while the latter directly attack the quantum problem. These methods are illustrated by several examples of quantum phase transitions in clean and disordered systems.Comment: 99 pages, 15 figures, submitted to Reviews in Computational Chemistr

Edge exponents in work statistics out of equilibrium and dynamical phase transitions from scattering theory in one dimensional gapped systems

I discuss the relationship between edge exponents in the statistics of work done, dynamical phase transitions, and the role of different kinds of excitations appearing when a non-equilibrium protocol is performed on a closed, gapped, one-dimensional system. I show that the edge exponent in the probability density function of the work is insensitive to the presence of interactions and can take only one of three values: +1/2, -1/2 and -3/2. It also turns out that there is an interesting interplay between spontaneous symmetry breaking or the presence of bound states and the exponents. For instantaneous global protocols, I find that the presence of the one-particle channel creates dynamical phase transitions in the time evolution.Comment: 5 pages, 2 figures. Revised version published in PR

A Monte Carlo study of surface critical phenomena: The special point

We study the special point in the phase diagram of a semi-infinite system, where the bulk transition is in the three-dimensional Ising universality class. To this end we perform a finite size scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interactions. In order to check for the effect of leading bulk corrections we have also simulated the spin-1/2 Ising model on the simple cubic lattice. We have accurately estimated the surface enhancement coupling at the special point of these models. We find $y_{t_s}=0.718(2)$ and $y_{h_s}=1.6465(6)$ for the surface renormalization group exponents of the special transitions. These results are compared with previous ones obtained by using field theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore we study the behaviour of the surface transition near the special point and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.Comment: 21 pages, 2 figures. figure 1 replaced, various typos correcte

Exponentiation for products of Wilson lines within the generating function approach

We present the generating function approach to the perturbative exponentiation of correlators of a product of Wilson lines and loops. The exponentiated expression is presented in closed form as an algebraic function of correlators of known operators, which can be seen as a generating function for web diagrams. The expression is naturally split onto two parts: the exponentiation kernel, which accumulates all non-trivial information about web diagrams, and the defect of exponentiation, which reconstructs the matrix exponent and is a function of the exponentiation kernel. The detailed comparison of the presented approach with existing approaches to exponentiation is presented as well. We also give examples of calculations within the generating function exponentiation, namely, we consider different configurations of light-like Wilson lines in the multi-gluon-exchange-webs (MGEW) approximation. Within this approximation the corresponding correlators can be calculated exactly at any order of perturbative expansion by only algebraic manipulations. The MGEW approximation shows violation of the dipole formula for infrared singularities at three-loop order.Comment: 33 pages, 5 figures; updated to match journal versio

Stripe-tetragonal phase transition in the 2D Ising model with dipole interactions: Partition-function zeros approach

We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction \delta where the phase characterized by striped configurations of width h=1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between \delta=0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents \nu that are clearly consistent with a single second-order phase transition line, thus excluding such tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.Comment: to appear in Phys. Rev.