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arXiv:cond-mat/9807025 · Physical review. B, Condensed matter

Dynamical critical properties of the random transverse-field Ising spin chain

Jens Kisker, A. P. Young

Published Dec 1, 1998• 16 citations
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Abstract

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 ${x}_{m}$ 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.

DOI
10.1103/physrevb.58.14397

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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(\ensuremathτ)S(\ensuremath{\tau})S(\ensuremathτ) is investigated and found to be simply a function of \ensuremathα\ensuremath≡\ensuremath−lnS(\ensuremathτ)/ln\ensuremathτ.\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.\ensuremathα\ensuremath≡\ensuremath−lnS(\ensuremathτ)/ln\ensuremathτ. This scaling behavior implies that the typical correlation function decays algebraically, Styp(\ensuremathτ)\ensuremath∼\ensuremathτ\ensuremath−\ensuremathαtyp,{S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},Styp​(\ensuremathτ)\ensuremath∼\ensuremathτ\ensuremath−\ensuremathαtyp​, where the exponent \ensuremathαtyp{\ensuremath{\alpha}}_{\mathrm{typ}}\ensuremathαtyp​ is determined from P(\ensuremathα),P(\ensuremath{\alpha}),P(\ensuremathα), the distribution of \ensuremathα.\ensuremath{\alpha}.\ensuremathα. The precise value for \ensuremathαtyp{\ensuremath{\alpha}}_{\mathrm{typ}}\ensuremathαtyp​ 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 ${x}_{m}$ 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.
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.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems ( L ⩽ 128). The spin-spin autocorrelation function is found to decay proportionally to (log t ) −2 x m and (log t ) −2 x m s in the bulk and on the surface, respectively, with x m and x m s the bulk and surface magnetization exponents, respectively. On the other hand, the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents η e ≈ 2.2 in the bulk and η e s ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at T=0.T=0.T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation C(L)C(L)C(L) for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation C(L)C(L)C(L) decays algebraically as 1/L1/L1/L in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.
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.
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.