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arXiv:2409.06412 · Physical review. B./Physical review. B

Continuous dimer angles on the silicon surface: Critical properties and the Kibble-Zurek mechanism

Andreas Weitzel, Gernot Schaller, Friedemann Queißer, Ralf Schützhold

Published Dec 12, 2024• 2 citations
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Abstract

Langevin dynamics simulations are used to analyze the static and dynamic properties of an $XY$ model adapted to dimers forming on Si(001) surfaces. The numerics utilize high-performance parallel computation methods on GPUs. The static exponent $\ensuremath{\nu}$ of the symmetry-broken $XY$ model is determined to $\ensuremath{\nu}=1.04$. The dynamic critical exponent $z$ is determined to $z=2.13$ and, together with $\ensuremath{\nu}$, shows the behavior of the Ising universality class. For time-dependent temperatures, we observe frozen domains and compare their size distribution with predictions from Kibble-Zurek theory. We determine a significantly larger quench exponent that shows little dependence on the damping or the symmetry-breaking field.

DOI
10.1103/physrevb.110.245125

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Langevin dynamics simulations are used to analyze the static and dynamic properties of an XYXYXY model adapted to dimers forming on Si(001) surfaces. The numerics utilize high-performance parallel computation methods on GPUs. The static exponent \ensuremathν\ensuremath{\nu}\ensuremathν of the symmetry-broken XYXYXY model is determined to \ensuremathν=1.04\ensuremath{\nu}=1.04\ensuremathν=1.04. The dynamic critical exponent zzz is determined to z=2.13z=2.13z=2.13 and, together with \ensuremathν\ensuremath{\nu}\ensuremathν, shows the behavior of the Ising universality class. For time-dependent temperatures, we observe frozen domains and compare their size distribution with predictions from Kibble-Zurek theory. We determine a significantly larger quench exponent that shows little dependence on the damping or the symmetry-breaking field.
As a simplified description of the nonequilibrium dynamics of buckled dimers on the Si(001) surface, we consider the anisotropic two-dimensional (2D) Ising model and study the freezing of spatial correlations during a cooling quench across the critical point. Depending on the cooling rate, we observe a crossover from one-dimensional (1D) to 2D behavior. For rapid cooling, we find effectively 1D behavior in the strongly coupled direction, for which we provide an exact analytic solution of the nonequilibrium dynamics. For slower cooling rates, we start to see 2D behavior where our numerical simulations show an approach to the usual Kibble-Zurek scaling in 2D.
In context of the search for the QCD critical endpoint in heavy-ion collisions, a deep understanding of the out-of-equilibrium dynamics of the system is necessary to make well-grounded predictions for signatures in final states. To this end, we investigate the dynamic critical behavior of a classical scalar field theory with Z2\mathbb{Z}_2Z2​ symmetry in the dynamic universality class of Model A in two spatial dimensions. The critical dynamics of the system is studied under a linear quench protocol, in which the external symmetry breaking field is changed at a constant rate through the critical point. We discuss the connection to the Kibble-Zurek mechanism and determine universal scaling functions, which fully describe the non-equilibrium evolution of the system near the critical point for all quench rates under consideration. We find that, while the scaling functions are non-trivial, the corresponding scaling exponents are fully determined by static critical exponents and the dynamic critical exponent.
We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLESs) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0+1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0+1)-dimensional and the (1+1)-dimensional YLESs.
The Kibble-Zurek mechanism describes the saturation of critical scaling upon dynamically approaching a phase transition. This is a consequence of the breaking of adiabaticity due to the scale set by the slow drive. By driving the gap parameter, this can be used to determine the leading critical exponents. But this is just the `tip of the iceberg:' Driving more general couplings allows one to activate the entire universal spectrum of critical exponents. Here we establish this phenomenon and its observable phenomenology for the quantum phase transitions in an analytically solvable minimal model and the experimentally relevant transverse XY model. The excitation density is shown to host the sequence of exponents including the subleading ones in the asymptotic scaling behavior by a proper design of the geometry of the driving protocol in the phase diagram. The case of a parallel drive relative to the phase boundary can still lead to the breaking of adiabaticity, and exposes the subleading exponents in the clearest way. Complementarily to disclosing universal information, we extract the restrictions due to the nonuniversal content of the models onto the extent of the subleading scalings regimes.
We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate τQc1τ_{Q}^{c1}τQc1​ above it the KZM scaling begins to appear. Smaller than τQc1τ_Q^{c1}τQc1​, the defect density nnn is a constant independent of the quench rate but depends on the final temperature TfT_fTf​ as n∝LdεTfdνn \propto L^d ε_{T_f} ^{d ν}n∝LdεTf​dν​, the freeze out time t^\hat{t}t^ admits the scaling law t^∝εTf−νz\hat{t} \propto ε_{T_f}^{-νz}t^∝εTf​−νz​ where ddd is the spatial dimension, εTf=(1−Tf/Tc)ε_{T_f}= (1-T_f/T_c)εTf​​=(1−Tf​/Tc​) is the dimensionless reduced temperature, LLL is the sample size, ννν and zzz are spatial and dynamical critical exponents. Quench from TcT_cTc​, the critical rate is determined by the final temperature TfT_fTf​ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.
Kibble-Zurek mechanism relates the domain of nonequilibrium dynamics with the critical properties at equilibrium. It establishes a power law connection between nonequilibrium defects quenched through a continuous phase transition and the quench rate via the scaling exponent. We present a novel numerical scheme to estimate the scaling exponent wherein the notion of defects is mapped to errors, previously introduced to quantify a variety of gapped quantum phases. To demonstrate the versatility of our method we conduct numerical experiments across a broad spectrum of spin-half models hosting local and symmetry protected topological order. Furthermore, an implementation of the quench dynamics featuring a topological phase transition on a digital quantum computer is proposed to quantify the associated criticality. Published by the American Physical Society 2024