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arXiv:cond-mat/9802246 · Physical review. B, Condensed matter

Distributions of gaps and end-to-end correlations in random transverse-field Ising spin chains

Daniel S. Fisher, A. P. Young

Published Oct 1, 1998• 92 citations
arXivHTMLPDF

Abstract

A previously introduced real-space renormalization-group treatment of the random transverse-field Ising spin chain is extended to provide detailed information on the distribution of the energy gap and the end-to-end correlation function for long chains with free boundary conditions. Numerical data, using the mapping of the problem to free fermions, are found to be in good agreement with the analytic finite-size scaling predictions.

DOI
10.1103/physrevb.58.9131

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arXiv:cond-mat/9807025•Physical review. B, Condensed matter•Dec 1, 1998•16 citations
A previously introduced real-space renormalization-group treatment of the random transverse-field Ising spin chain is extended to provide detailed information on the distribution of the energy gap and the end-to-end correlation function for long chains with free boundary conditions. Numerical data, using the mapping of the problem to free fermions, are found to be in good agreement with the analytic finite-size scaling predictions.
The end-to-end energy-energy correlations of random transverse field quantum Ising spin chains are computed using a generalization of an asymptotically exact real space renormalization group (RG) previously introduced. Away from the critical point, the average energy-energy correlations decay exponentially with a correlation length that is the same as that of the spin-spin correlations. The typical correlations, however, decay exponentially with a characteristic length, proportional to the square root of the primary correlation length. At the quantum critical point, the average correlations decay subexponentially as $\overline{{C}_{L}}\ensuremath{\sim}{e}^{\ensuremath{-}\mathrm{const}\phantom{\rule{0.2em}{0ex}}{L}^{1∕3}}$, whereas the typical correlations decay faster, as $\ensuremath{\sim}{e}^{\ensuremath{-}K\sqrt{L}}$, with KKK a random variable with a universal distribution. The critical energy-energy correlations behave very similarly to the smallest gap, computed previously; this is explained in terms of the RG flow and the excitation structure of the chain. In order to obtain the energy correlations, an extension of the previously used methods was needed; here, this was carried out via RG transformations that involve a sequence of unitary transformations.
We consider random transverse-field Ising spin chains and study the magnetization and the energy-density profiles by numerically exact calculations in rather large finite systems ( $L\ensuremath{\le}128$). Using different boundary conditions (free, fixed, and mixed) the numerical data collapse to scaling functions, which are very accurately described by simple analytic expressions. The average magnetization profiles satisfy the Fisher--de Gennes scaling conjecture and the corresponding scaling functions are indistinguishable from those predicted by conformal invariance.
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.
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 dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions with which we can obtain numerically exact results for system sizes L as large as 256. The probability distribution of the local imaginary time correlation function $S(\ensuremath{\tau})$ is investigated and found to be simply a function of $\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.$ This scaling behavior implies that the typical correlation function decays algebraically, ${S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},$ where the exponent ${\ensuremath{\alpha}}_{\mathrm{typ}}$ is determined from $P(\ensuremath{\alpha}),$ the distribution of $\ensuremath{\alpha}.$ The precise value for ${\ensuremath{\alpha}}_{\mathrm{typ}}$ depends on exactly how the ``typical'' correlation function is defined. The form of $P(\ensuremath{\alpha})$ for small $\ensuremath{\alpha}$ gives a contribution to the average correlation function ${S}_{\mathrm{av}}(\ensuremath{\tau})$ namely, ${S}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}(\mathrm{ln}\ensuremath{\tau}{)}^{\ensuremath{-}{2x}_{m}},$ where xm{x}_{m}xm​ is the bulk magnetization exponent, which was reported recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.