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arXiv:2211.11376 · Physical review. B./Physical review. B

Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge

Yanan Sun, Jin Lyu, Jian-Ping Lv

Published Nov 28, 2022• 8 citations
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Abstract

There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents ${y}_{t}=0.58(1)$ and ${y}_{h}=1.690(1)$, respectively, which are close to recent estimates from models with discrete spin variables. The existence of extraordinary-log universality is evidenced by the critical exponent $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}=0.58(2)$ from two-point correlation and the renormalization-group parameter $\ensuremath{\alpha}=0.28(1)$ from superfluid stiffness, which obey the scaling relation of extraordinary-log critical theory and recover the logarithmic finite-size scaling of critical superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results bridge recent observations of surface critical behavior in the classical statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701 (2021); Hu et al., Phys. Rev. Lett. 127, 120603 (2021); Parisen Toldin et al., Phys. Rev. Lett. 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun $et$ $al.$, arXiv:2205.00878 (2022)].

DOI
10.1103/physrevb.106.174516

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There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents yt=0.58(1){y}_{t}=0.58(1)yt​=0.58(1) and yh=1.690(1){y}_{h}=1.690(1)yh​=1.690(1), respectively, which are close to recent estimates from models with discrete spin variables. The existence of extraordinary-log universality is evidenced by the critical exponent q\ifmmode^\elseˆ\fi=0.58(2)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}=0.58(2)q\ifmmode^\elseˆ\fi​=0.58(2) from two-point correlation and the renormalization-group parameter \ensuremathα=0.28(1)\ensuremath{\alpha}=0.28(1)\ensuremathα=0.28(1) from superfluid stiffness, which obey the scaling relation of extraordinary-log critical theory and recover the logarithmic finite-size scaling of critical superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results bridge recent observations of surface critical behavior in the classical statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701 (2021); Hu et al., Phys. Rev. Lett. 127, 120603 (2021); Parisen Toldin et al., Phys. Rev. Lett. 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun etetet al.al.al., arXiv:2205.00878 (2022)].
The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.
We demonstrate the emergence of extraordinary-log criticality in a quantum many-body system by coupling a spin-1/21/21/2 Heisenberg chain to a two-dimensional quantum critical Heisenberg bilayer. Under the quantum-to-classical mapping, the chain acts as a plane defect in a three-dimensional critical environment for which the extraordinary-log universality class has recently been predicted. Using large-scale sign-problem-free quantum Monte Carlo simulations, we directly probe the renormalization-group flow of the effective defect coupling through the spin stiffness. We observe the characteristic logarithmic growth with linear system size LLL, corresponding to a stable weak-coupling fixed point and a running coupling g(L)∼1/log⁡Lg(L)\sim 1/\log Lg(L)∼1/logL. Furthermore, the extracted step-scaling function quantitatively follows the universal renormalization-group flow predicted for the extraordinary-log universality class, including the leading and subleading universal coefficients. Our results establish extraordinary-log criticality in a genuine quantum many-body setting and show that its universal behavior persists despite the topological θθθ-term present in the low-energy field theory of the isolated spin chain.
Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance rrr as $g(r)\ensuremath{\sim}{r}^{2\ensuremath{-}d\ensuremath{-}\ensuremath{\eta}}$, with ddd the spatial dimension and $\ensuremath{\eta}$ 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 $\ensuremath{\alpha}$ 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.
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.
We investigate the zero-temperature superfluid to insulator transitions in a diluted two-dimensional quantum rotor model with particle-hole symmetry. We map the Hamiltonian onto a classical (2+1)(2+1)(2+1)-dimensional XYXYXY model with columnar disorder which we analyze by means of large-scale Monte Carlo simulations. For dilutions below the lattice percolation threshold, the system undergoes a generic superfluid-Mott glass transition. In contrast to other quantum phase transitions in disordered systems, its critical behavior is of conventional power-law type with universal (dilution-independent) critical exponents $z=1.52(3), \ensuremath{\nu}=1.16(5), \ensuremath{\beta}/\ensuremath{\nu}=0.48(2), \ensuremath{\gamma}/\ensuremath{\nu}=2.52(4)$, and $\ensuremath{\eta}=\ensuremath{-}0.52(4)$. These values agree with and improve upon earlier Monte Carlo results [Phys. Rev. Lett. 92, 015703 (2004)] while (partially) excluding other findings in the literature. As a further test of universality, we also consider a soft-spin version of the classical Hamiltonian. In addition, we study the percolation quantum phase transition across the lattice percolation threshold; its critical behavior is governed by the lattice percolation exponents in agreement with recent theoretical predictions. We relate our results to a general classification of phase transitions in disordered systems, and we briefly discuss experiments.
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent q=0.60(2)q=0.60(2)q=0.60(2), however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the Z6{Z}_{6}Z6​ symmetry-breaking phase, with critical exponent ys=0.41(2){y}_{\mathrm{s}}=0.41(2)ys​=0.41(2). The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent q=0.37(1)q=0.37(1)q=0.37(1); however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.