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arXiv:cond-mat/9709260 · Physical review. B, Condensed matter

Random transverse Ising spin chain and random walks

Ferenc Iglói, Heiko Rieger

Published May 1, 1998• 105 citations
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Abstract

We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low-energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.

DOI
10.1103/physrevb.57.11404

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We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low-energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.
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.
We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11 404 (1998)] to new quantities, such as the nonlinear susceptibility, higher excitations, and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)1/z]av=1[{(J/h)}^{1/z}{]}_{\mathrm{av}}=1[(J/h)1/z]av​=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as $[G{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/z}$, the average energy-density autocorrelations decay with another exponent as $[{G}^{e}{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2\ensuremath{-}1/z}$.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
We study numerically the paramagnetic phase of the spin-1/21/21/2 random transverse-field Ising chain, using a mapping to noninteracting fermions. We extend our earlier work, Phys. Rev. B 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are Griffiths-McCoy singularities in the paramagnetic phase described by a continuously varying exponent $z(\ensuremath{\delta}),$ where \ensuremath{\delta} measures the deviation from criticality. There are some discrepancies between the values of $z(\ensuremath{\delta})$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely, ${\ensuremath{\tau}}^{\ensuremath{-}1/z(\ensuremath{\delta})},$ where \ensuremath{\tau} is imaginary time. However, the typical value decays with a stretched exponential behavior, $\mathrm{exp}(\ensuremath{-}c{\ensuremath{\tau}}^{1/\ensuremath{\mu}}),$ where \ensuremath{\mu} may be related to $z(\ensuremath{\delta}).$ We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.
We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities, and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random-walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confirmed by numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L< 512)(L<~512)(L< 512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution shows multiscaling character. In the Griffiths phase, which is an extended part of the off-critical region, average autocorrelations have a power-law form with a nonuniversal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.
The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.