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arXiv:0803.2617 · Physical Review B

Griffiths-McCoy singularities in random quantum spin chains: Exact results

Ferenc Iglói, István A. Kovács

Published Apr 8, 2008• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.77.144203

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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.
In random quantum magnets, like the random transverse Ising chain, low energy excitations are localized in rare regions and there are only weak correlations between them. It is an open question whether these correlations are relevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and---for shorter chains---by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Considering the finite size corrections, the two numerical methods give very similar results, but these differ from the correction term for uncorrelated random variables, indicating that the weak correlations between low-energy excitations in random quantum magnets are relevant.
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 consider interacting many-particle systems with quenched disorder having strong Griffiths singularities, which are characterized by the dynamical exponent, zzz, such as random quantum systems and exclusion processes. In several d=1d=1d=1 and d=2d=2d=2 dimensional problems we have calculated the inverse time scales, ${\ensuremath{\tau}}^{\ensuremath{-}1}$, in finite samples of linear size, LLL, either exactly or numerically. In all cases, having a discrete symmetry, the distribution function, $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$, is found to depend on the variable, $u={\ensuremath{\tau}}^{\ensuremath{-}1}{L}^{z}$, and to be universal given by the limit distribution of extremes of independent and identically distributed random numbers. This finding is explained in the framework of a strong disorder renormalization group approach when, after fast degrees of freedom are decimated out, the system is transformed into a set of noninteracting localized excitations. The Fr\'echet distribution of $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$ is expected to hold for all random systems having a strong disorder fixed point, in which the Griffiths singularities are dominated by disorder fluctuations.
\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.