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arXiv:cond-mat/0612420 · Physical Review B

Effect of long-range connections on an infinite randomness fixed point associated with the quantum phase transitions in a transverse Ising model

Amit Dutta, R. Loganayagam

Published Feb 9, 2007• 5 citations
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Abstract

We study the effect of long-range connections on the infinite-randomness fixed point associated with the quantum phase transitions in a transverse Ising model (TIM). The TIM resides on a long-range connected lattice where any two sites at a distance $r$ are connected with a nonrandom ferromagnetic bond with a probability that falls algebraically with the distance between the sites as $1∕{r}^{d+\ensuremath{\sigma}}$. The interplay of the fluctuations due to dilutions together with the quantum fluctuations due to the transverse field leads to an interesting critical behavior. The exponents at the critical fixed point (which is an infinite randomness fixed point) are related to the classical ``long-range'' percolation exponents. The most interesting observation is that the gap exponent $\ensuremath{\psi}$ is exactly obtained for all values of $\ensuremath{\sigma}$ and $d$. Exponents depend on the range parameter $\ensuremath{\sigma}$ and show a crossover to short-range values when $\ensuremath{\sigma}\ensuremath{\ge}2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}$ where ${\ensuremath{\eta}}_{\mathrm{SR}}$ is the anomalous dimension for the conventional percolation problem. Long-range connections are also found to tune the strength of the Griffiths phase.

DOI
10.1103/physrevb.75.052405

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A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field hhh and the dilution ppp, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small hhh and small ppp. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent βββ and the average correlation-length exponent νavν_{\mathrm{av}}νav​ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for ppp smaller than the percolation threshold pcp_cpc​.
We consider the effect of a random longitudinal field on the Ising model in a transverse magnetic field. For spatial dimension d>2d>2d>2, there is at low strength of randomness and transverse field a phase with true long-range order that is destroyed at higher values of the randomness or transverse field. The properties of the quantum phase transition at zero temperature are controlled by a fixed point with no quantum fluctuations. This fixed point also controls the classical finite-temperature phase transition in this model. Many critical properties of the quantum transition are therefore identical to those of the classical transition. In particular, we argue that the dynamical scaling is activated, i.e., the logarithm of the diverging time scale rises as a power of the diverging length scale.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\ensuremath{\alpha}$ is tuned. For $\ensuremath{\alpha}=0$, one recovers the familiar infinite-randomness critical point. For $0<\ensuremath{\alpha}<1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\ensuremath{\alpha}=0$ are absent at any $\ensuremath{\alpha}>0$. When $\ensuremath{\alpha}>1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.