Scaling in the critical region: An additional relation among critical parameters

Виктор Степанович Доценко

Zhixuan Li, Shuai Yin, Yu-Rong Shu

We explore the imaginary-time relaxation dynamics near quantum critical points with semi-ordered initial states. Different from the case with homogeneous ordered initial states, in which the order parameter M decays homogeneously as M ∝ τ −β/νz , here M depends on the location x, showing rich scaling behaviors. Similar to the classical relaxation dynamics with an initial domain wall in model A, which describes the purely dissipative dynamics, here as the imaginary time evolves, the domain wall expands into an interfacial region with growing size. In the interfacial region, the local order parameter decays as M ∝ τ −β/νz , with β 1 being an additional dynamic critical exponent. Far away from the interfacial region the local order parameter decays as M ∝ τ −β/νz in the short-time stage, then crosses over to the scaling behavior of M ∝ τ −β/νz when the location x is absorbed in the interfacial region. A full scaling form characterizing these scaling properties is developed. The quantum Ising model in both one and two dimensions are taken as examples to verify the scaling theory. In addition, we find that for the quantum Ising model the scaling function is an analytical function and β 1 is not an independent exponent.

Dominik Schweitzer, Justus Liebig University Giessen

The critical behaviour of a relativistic scalar field theory with Z2 symmetry is investigated near a second-order phase transition. Two sets of dynamic equations are employed, allowing to control conservation laws of both order parameter and energy density. We calculate spectral functions of the order parameter as well as unequal-time correlation functions of the energy momentum tensor at zero and non-vanishing spatial momenta from first-principles classical-statistical lattice simulations in real-time. For both, we investigate general properties and relevant degrees of freedom in distinct regions of the phase diagram. Close to the critical point, we find signatures of dynamic scaling behaviour and calculate the dynamic critical exponent z controlling the divergence of the critical time scale. For both the order parameter spectral function as well as the energy density autocorrelation, we extract universal dynamic scaling functions. Modifying the simulation framework to include dynamically changing temperature and external field allows us to study non-equilibrium phenomena. For the special case of instant quenches to the critical point, we identify universal scaling behaviour controlled by the initial magnetization, and calculate the related additional dynamic critical exponent. We extract the universal non-equilibrium scaling functions for the evolution of both the order parameter as well as the correlation length.

Franz Wegner

Rainer Bischof

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1.

Michael Ellis Fisher, Michael N. Barber