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Critical exponents of random XX and XY chains: \nExact results via random walks

H. Rieger, R. Juhász, Ferenc Iglói

Published Feb 15, 2000• 2 citations
Publisher

Abstract

\n \nWe study random XY and (dimerized) XX spin-1/2 quantum spin chains\nat their quantum phase transition driven by the anisotropy and\ndimerization, respectively. Using exact expressions for\nmagnetization, correlation functions and energy gap, obtained by the\nfree fermion technique, the critical and off-critical\n(Griffiths-McCoy) singularities are related to persistence\nproperties of random walks. In this way we determine exactly the\ndecay exponents for surface and bulk transverse and longitudinal\ncorrelations, correlation length exponent and dynamical exponent.\n\n\n

DOI
10.1007/s100510050050

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\n \nWe study random XY and (dimerized) XX spin-1/2 quantum spin chains\nat their quantum phase transition driven by the anisotropy and\ndimerization, respectively. Using exact expressions for\nmagnetization, correlation functions and energy gap, obtained by the\nfree fermion technique, the critical and off-critical\n(Griffiths-McCoy) singularities are related to persistence\nproperties of random walks. In this way we determine exactly the\ndecay exponents for surface and bulk transverse and longitudinal\ncorrelations, correlation length exponent and dynamical exponent.\n\n\n
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 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.
Using exact expressions for the persistence probability and for the leading eigenvalue of the Fokker-Planck operator of a random walk in a random environment, we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many exact results are obtained from this correspondence, including the space and time correlations of surviving random walks and the distribution of the gaps of the corresponding Fokker-Planck operator. In turn we derive analytically the dynamical exponent of the random transverse-field Ising spin chain in the Griffiths-McCoy region.
We consider random quantum (tight-binding, XX, and Ising) spin chains in the off-critical region and study their Griffiths-McCoy singularities. These are obtained from the density of states of the low-energy excitations, which is exactly calculated by the Dyson-Schmidt method. In large finite systems, the low-energy excitations are shown to follow the statistics of extremes and their distribution is given by the Fr\'echet form. The relation between the Dyson-Schmidt technique and the strong disorder renormalization group method is also discussed.
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.
The effect of disorder on the quantum phase transitions induced by a transverse field, anisotropy, and dimerization in XY spin chains is investigated. The low-energy behavior near the critical point is described by a Dirac-type equation for which an exact analytic treatment is possible. Results obtained for the dynamical critical exponent, the specific heat, and the transverse susceptibility agree with results recently obtained using a real space renormalization group decimation technique. A nonzero transverse field changes the universality class of the anisotropy transition.