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Physical review. E

Numerical evidence of hyperscaling violation in wetting transitions of the random-bond Ising model in d=2 dimensions

Ezequiel Vicente Albano, Luciana Melina Luque, Marta L. Trobo, Kurt Binder

Published Feb 1, 2017• 0 citations
Publisher

Abstract

We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent $\ensuremath{\beta}$, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by $\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}$ requires $\ensuremath{\beta}=1/2$ ($\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3$, and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=2$), the thermodynamic scaling establishes that ${\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}$, which in contrast requires $\ensuremath{\beta}=0$ (${\mathrm{\ensuremath{\Delta}}}_{s}=4$), where $\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and ${\mathrm{\ensuremath{\Delta}}}_{s}$ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., $\ensuremath{\beta}=0$) and a direct measurement of the susceptibility critical exponent $\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2$, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.

DOI
10.1103/physreve.95.022801

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We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by \ensuremathγ+2\ensuremathβ=\ensuremathν\ensuremath∥+\ensuremathν\ensuremath⊥\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremathγ+2\ensuremathβ=\ensuremathν\ensuremath∥​+\ensuremathν\ensuremath⊥​ requires \ensuremathβ=1/2\ensuremath{\beta}=1/2\ensuremathβ=1/2 (\ensuremathγ=4,\ensuremathν\ensuremath∥=3\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3\ensuremathγ=4,\ensuremathν\ensuremath∥​=3, and \ensuremathν\ensuremath⊥=2{\ensuremath{\nu}}_{\ensuremath{\perp}}=2\ensuremathν\ensuremath⊥​=2), the thermodynamic scaling establishes that \ensuremathΔs=\ensuremathγ+\ensuremathβ{\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}\ensuremathΔs​=\ensuremathγ+\ensuremathβ, which in contrast requires \ensuremathβ=0\ensuremath{\beta}=0\ensuremathβ=0 (\ensuremathΔs=4{\mathrm{\ensuremath{\Delta}}}_{s}=4\ensuremathΔs​=4), where \ensuremathγ,\ensuremathν\ensuremath∥,\ensuremathν\ensuremath⊥\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremathγ,\ensuremathν\ensuremath∥​,\ensuremathν\ensuremath⊥​, and \ensuremathΔs{\mathrm{\ensuremath{\Delta}}}_{s}\ensuremathΔs​ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., \ensuremathβ=0\ensuremath{\beta}=0\ensuremathβ=0) and a direct measurement of the susceptibility critical exponent \ensuremathγ/\ensuremathν\ensuremath⊥=2.0\ifmmode±\else\textpm\fi0.2\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2\ensuremathγ/\ensuremathν\ensuremath⊥​=2.0\ifmmode±\else\textpm\fi0.2, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.
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In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension D = 6 . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.
The global phase diagram of wetting in the two-dimensional Ising model is obtained through the exact calculation of the surface excess free energy. In addition to a surface field for inducing wetting, a surface-coupling enhancement is also included. The wetting transition (of second order) is critical for any finite ratio of surface coupling J_{s} to bulk coupling J, and becomes of first order in the limit J_{s}/J→∞. However, for J_{s}/J≫1, the critical region is exponentially small and is practically invisible to numerical studies. A distinct preasymptotic regime exists in which the transition displays first-order character. In this regime, surprisingly, the surface susceptibility and surface specific heat develop a divergence and show anomalous scaling with an exponent equal to 3/2.
Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of statistical physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension D=7, i.e., above its upper critical dimension D_{u}=6, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions D>D_{u}, linear length scale L should be replaced in finite-size scaling expressions by the effective scale L_{eff}=L^{D/D_{u}}. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single nonzero exponent $\ensuremath{\theta}$, so that in a spatially isotropic state in ddd spatial dimensions, the specific heat scales with temperature as ${T}^{(d\ensuremath{-}\ensuremath{\theta})/z}$, and the optical conductivity scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where zzz is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013)]. We find that hyperscaling is violated, with $\ensuremath{\theta}=1$ in d=2d=2d=2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d=2d=2d=2.