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arXiv:1712.07467 · Physical review. B./Physical review. B

Transverse spin correlations of the random transverse-field Ising model

Ferenc Iglói, István A. Kovács

Published Mar 21, 2018• 4 citations
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Abstract

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 $d=1,2$, 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 $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.

DOI
10.1103/physrevb.97.094205

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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∼r\ensuremath−\ensuremathηt\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}\ensuremath∼r\ensuremath−\ensuremathηt​ is found, with a decay exponent being approximately \ensuremathηt\ensuremath≈2+2d{\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d\ensuremathηt​\ensuremath≈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.
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.
We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First, we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for the arbitrary size of the block. For the model with three-spin couplings, we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus, this model represents a new infinite disorder universality class.
Abstract Motivated by experimental results on compounds like LiHo x Y 1 − x F 4 , we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalisation group method. In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behaviour is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to a set of disordered fixed-points, in which the disorder is either due to random quantum fluctuations, or due to classical random-field effects. Between the two regimes there is a smooth cross-over, which becomes sharp at the infinite disorder fixed-point. This local separatrix defines the relevant scaling direction, where the correlation-length is shown to diverge with an exponent ν h ≈ 1 .
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 study critical behavior of the diluted two-dimensional Ising model in the presence of disorder correlations which decay algebraically with distance as $\ensuremath{\sim}{r}^{\ensuremath{-}a}$. Mapping the problem onto two-dimensional Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a nontrivial fixed point which is stable for 0.995<a<20.995<a<20.995<a<2 and is characterized by the correlation length exponent $\ensuremath{\nu}=2/a+O({(2\ensuremath{-}a)}^{3})$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent ${\ensuremath{\eta}}_{2}=1/2\ensuremath{-}(2\ensuremath{-}a)/4+O({(2\ensuremath{-}a)}^{2})$.
Motivated by the compound LiHoxY1-xF4, we consider the Ising chain with random couplings and in the presence of simultaneous random transverse and longitudinal fields, and study its low-energy properties at zero temperature by the strong disorder renormalization group approach. In the absence of longitudinal fields, the system exhibits a quantum-ordered and a quantum-disordered phase separated by a critical point of infinite disorder. When the longitudinal random field is switched on, the ordered phase vanishes and the trajectories of the renormalization group are attracted to two disordered fixed points: one is characteristic of the classical random field Ising chain, the other describes the quantum disordered phase. The two disordered phases are separated by a separatrix that starts at the infinite disorder fixed point and near which there are strong quantum fluctuations.