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arXiv:0909.4442 · Physical Review B

Critical behavior and entanglement of the random transverse-field Ising model between one and two dimensions

István A. Kovács, Ferenc Iglói

Published Dec 17, 2009• 46 citations
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Abstract

We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, $w\ensuremath{\le}20$, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite $w$ are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, $\ensuremath{\nu}(2d)$. We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the $w$ dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: $\ensuremath{\nu}(2d)=1.25(8)$, $x(2d)=0.996(15)$, and $\ensuremath{\psi}(2d)=0.51(3)$.

DOI
10.1103/physrevb.80.214416

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We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, w\ensuremath≤20w\ensuremath{\le}20w\ensuremath≤20, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite www are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, \ensuremathν(2d)\ensuremath{\nu}(2d)\ensuremathν(2d). We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the www dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: \ensuremathν(2d)=1.25(8)\ensuremath{\nu}(2d)=1.25(8)\ensuremathν(2d)=1.25(8), x(2d)=0.996(15)x(2d)=0.996(15)x(2d)=0.996(15), and \ensuremathψ(2d)=0.51(3)\ensuremath{\psi}(2d)=0.51(3)\ensuremathψ(2d)=0.51(3).
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.
Using a very efficient numerical algorithm of the strong disorder renormalization group method, we have extended the investigations about the critical behavior of the random transverse-field Ising model in three and four dimensions, as well as for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs, which represent infinite dimensional lattices. In all studied cases, an infinite disorder quantum critical point is identified, which ensures that the applied method is asymptotically correct and the calculated critical exponents tend to the exact values for large scales. We have found that the critical exponents are independent of the form of (ferromagnetic) disorder and they vary smoothly with the dimensionality.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.
The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as ${\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)$, together with critical exponents such as $\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)$, and z=3.3(3)z=3.3(3)z=3.3(3) or $\ensuremath{\psi}=0.50(3)$. We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.