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arXiv:1111.3468 · Journal of Statistical Mechanics Theory and Experiment

Random transverse field Ising model in dimensionsd= 2, 3: infinite disorder scaling via a non-linear transfer approach

Cécile Monthus, Thomas Garel

Published Jan 16, 2012• 13 citations
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Abstract

The 'cavity-mean-field' approximation developed for the random transverse field Ising model on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) has been found to reproduce the known exact result for the surface magnetization in d = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the present paper, we propose to extend these ideas in finite dimensions d > 1 via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations corresponds to the transfer matrix for a directed polymer in a random medium of transverse dimension D = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011 arXiv: 1110.3145 ). We present numerical results of the non-linear transfer approach in dimensions d = 2 and 3. In both cases, we find that the critical point is governed by infinite disorder scaling. In particular, exactly at criticality, the one-point surface magnetization scales as , where ω c ( d ) coincides with the droplet exponent ω DP ( D = d − 1) of the corresponding directed polymer model, with ω c ( d = 2) = 1/3 and . The distribution P ( v ) of the positive random variable v of order O(1) presents a power-law singularity near the origin with a ( d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-law decay with x s = ω c (1 + a ) independently of the order k .

DOI
10.1088/1742-5468/2012/01/p01008

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The 'cavity-mean-field' approximation developed for the random transverse field Ising model on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) has been found to reproduce the known exact result for the surface magnetization in d = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the present paper, we propose to extend these ideas in finite dimensions d > 1 via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations corresponds to the transfer matrix for a directed polymer in a random medium of transverse dimension D = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011 arXiv: 1110.3145 ). We present numerical results of the non-linear transfer approach in dimensions d = 2 and 3. In both cases, we find that the critical point is governed by infinite disorder scaling. In particular, exactly at criticality, the one-point surface magnetization scales as , where ω c ( d ) coincides with the droplet exponent ω DP ( D = d − 1) of the corresponding directed polymer model, with ω c ( d = 2) = 1/3 and . The distribution P ( v ) of the positive random variable v of order O(1) presents a power-law singularity near the origin with a ( d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-law decay with x s = ω c (1 + a ) independently of the order k .
For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
To avoid the complicated topology of surviving clusters induced by standard strong disorder RG in dimension d > 1, we introduce a modified procedure called 'boundary strong disorder RG' where the order of decimations is chosen a priori . We apply this modified procedure numerically to the random transverse field Ising model in dimension d = 2. We find that the location of the critical point, the activated exponent ψ ≃ 0.5 of the infinite-disorder scaling, and the finite-size correlation exponent ν FS ≃ 1.3 are compatible with the values obtained previously using standard strong disorder RG. Our conclusion is thus that strong disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze the RG flows within the two phases in more detail, to show explicitly the presence of various correlation length exponents: we measure the typical correlation exponent ν typ ≃ 0.64 for the disordered phase (this value is very close to the correlation exponent of the pure two-dimensional quantum Ising model), and the typical exponent ν h ≃ 1 for the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at infinite-disorder critical points. We also measure, within the disordered phase, the fluctuation exponent ω ≃ 0.35 which is compatible with the directed polymer exponent in (1 + 1) dimensions.
Strong disorder renormalization for the random transverse field Ising model leads to a complicated topology of surviving clusters as soon as d > 1. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can be obtained. Here we introduce a modified procedure called 'boundary strong disorder renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows one to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling J and random transverse fields with support [ h min , h max ]. We find the following picture: (i) for J > h max , only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster; (ii) for J < h min , only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a directed polymer model in a random medium; (iii) for h min < J < h max , both sites and bonds can be decimated, and the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified directed polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types: (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent ν typ = 1; (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears, with a correlation length exponent ν ≃ 0.75.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
We consider the critical and off-critical properties at the boundary of the random transverse-field Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance l from the surface, deviates from its uniform bulk value by terms of order ${l}^{\ensuremath{-}\ensuremath{\kappa}}$ with an amplitude A. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, $\ensuremath{\kappa}<1/2,$ the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of A. In the marginal situation $\ensuremath{\kappa}=1/2,$ the average surface magnetization decays as a power law with a continuously varying, A-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions.
The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.