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arXiv:2104.05152 · Physical Review Letters

Extraordinary-Log Surface Phase Transition in the Three-Dimensional XY Model

Minghui Hu, Youjin Deng, Jian-Ping Lv

Published Sep 16, 2021• 40 citations
arXivHTMLPDF

Abstract

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance $r$ as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with $d$ the spatial dimension and $\ensuremath{\eta}$ the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the $\mathrm{O}(N)$ system. In this logarithmic universality, $g(r)$ decays in a power of logarithmic distance as $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, dramatically different from the standard scenario. We explore the three-dimensional $XY$ model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with $L$ as $g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}}$ as well as the $r$-dependent term $g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}$, with ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1$. The critical exponent ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}$, characterizing the height of the plateau, obeys the scaling relation ${\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha})$ with the RG parameter $\ensuremath{\alpha}$ of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.

DOI
10.1103/physrevlett.127.120603

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Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance rrr as g(r)\ensuremath∼r2\ensuremath−d\ensuremath−\ensuremathηg(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}g(r)\ensuremath∼r2\ensuremath−d\ensuremath−\ensuremathη, with ddd the spatial dimension and \ensuremathη\ensuremath{\eta}\ensuremathη the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the O(N)\mathrm{O}(N)O(N) system. In this logarithmic universality, g(r)g(r)g(r) decays in a power of logarithmic distance as g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}, dramatically different from the standard scenario. We explore the three-dimensional XYXYXY model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of g(r,L)g(r,L)g(r,L) has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with LLL as g(L)\ensuremath{\sim}(\mathrm{ln}L{)}^{\ensuremath{-}{\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}} as well as the rrr-dependent term g(r)\ensuremath{\sim}(\mathrm{ln}r{)}^{\ensuremath{-}\stackrel{^}{\ensuremath{\eta}}}, with {\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}\ensuremath{\approx}\stackrel{^}{\ensuremath{\eta}}\ensuremath{-}1. The critical exponent {\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}, characterizing the height of the plateau, obeys the scaling relation {\stackrel{^}{\ensuremath{\eta}}}^{\ensuremath{'}}=(N\ensuremath{-}1)/(2\ensuremath{\pi}\ensuremath{\alpha}) with the RG parameter \ensuremathα\ensuremath{\alpha}\ensuremathα of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.
Recent advances in boundary critical phenomena have led to the discovery of a new surface universality class in the three-dimensional O ( N ) model. The newly found “extraordinary-log” phase can be realized on a two-dimensional surface for N < N c , with N c > 3 , and on a plane defect embedded into a three-dimensional system, for any N . One of the key features of the extraordinary-log phase is the presence of logarithmic violations of standard finite-size scaling. In this work we study finite-size scaling in the extraordinary-log universality class by means of Monte Carlo simulations of an improved lattice model. We simulate the model with open boundary conditions, realizing the extraordinary-log phase on the surface for N = 2 , 3 , as well as with fully periodic boundary conditions and in the presence of a plane defect for N = 2 , 3 , 4 . In line with theory predictions, renormalization-group invariant observables studied here exhibit a logarithmic dependence on the size of the system. We numerically access not only the leading term in the β function governing these logarithmic violations, but also the subleading term, which controls the evolution of the boundary phase diagram as a function of N .
The recent discovery of the extraordinary-log (E-Log) criticality is a celebrated achievement in modern critical theory and calls for generalization. Using large-scale Monte Carlo simulations, we study the critical phenomena of plane defects in three- and four-dimensional O(n) critical systems. In three dimensions, we provide the first numerical proof for the E-Log criticality of plane defects. In particular, for n=2, the critical exponent q[over ^] of two-point correlation and the renormalization-group parameter α of helicity modulus conform to the scaling relation q[over ^]=(n-1)/(2πα), whereas the results for n≥3 violate this scaling relation. In four dimensions, it is strikingly found that the E-Log criticality also emerges in the plane defect. These findings have numerous potential realizations and would boost the ongoing advancement of conformal field theory.
We investigate nonequilibrium driven dynamics across the special surface phase transition in the three-dimensional classical Heisenberg model with open boundaries, where tuning the surface coupling gives access to an extraordinary-log boundary critical state characterized by logarithmic, rather than power-law, decay of correlations. Using Monte Carlo simulations, we realize four driving protocols: temperature heating and cooling across the special transition, and surface-coupling ramps from the ordinary and extraordinary-log critical states into the special point. For temperature-driven protocols, the surface order parameter obeys a generalization of the finite-time scaling (FTS) and the Kibble-Zurek mechanism. The central finding emerges when the system is driven from the extraordinary-log critical state: the large-rate scaling relation acquires a logarithmic correction and takes the novel form Ms2∝R(1+ηs)/rs[log⁡(LR1/ηs)]−qM^{2}_{s}\propto R^{(1+η_{s})/r_{s}}[\log(LR^{1/η_{s}})]^{-q}Ms2​∝R(1+ηs​)/rs​[log(LR1/ηs​)]−q , where RRR is the driving rate, LLL the system size, ηsη_{s}ηs​ the surface anomalous dimension, rsr_{s}rs​ the scaling dimension of RRR, and qqq the exponent governing the logarithmic boundary criticality. We demonstrate that this form follows from the general FTS framework by incorporating the logarithmic initial-state memory, and we achieve excellent data collapse over a wide range of system sizes and driving rates. Our results establish that extraordinary-log initial states alter nonequilibrium critical scaling, extending boundary FTS beyond conventional power-law initial conditions.
Abstract Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (dc = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows r2−dcr^{2-d_c}r2−dc​ governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as L−dc/2(lnL)p^L^{-d_c/2}({\rm ln}L)^{\hat{p}}L−dc​/2(lnL)p^​ with p^=1/2\hat{p}=1/2p^​=1/2 a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.