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arXiv:1204.0621 · Physical Review B

Scaling at chiral quantum critical points in two dimensions

L. Schweitzer, P. Markoš

Published May 14, 2012• 9 citations
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Abstract

We study the localization properties of electrons moving on two-dimensional bipartite lattices in the presence of disorder. The models investigated exhibit a chiral symmetry and belong to the chiral orthogonal (chO), chiral symplectic (chS), or chiral unitary (chU) symmetry class. The disorder is introduced via real random hopping terms for chO and chS, while complex random phases generate the disorder in the chiral unitary model. In the latter case an additional spatially constant, perpendicular magnetic field is also applied. Using a transfer-matrix method, we numerically calculate the smallest Lyapunov exponents that are related to the localization length of the electronic eigenstates. From a finite-size scaling analysis, the logarithmic divergence of the localization length at the quantum critical point at $E=0$ is obtained. We always find for the critical exponent $\ensuremath{\kappa}$, which governs the energy dependence of the divergence, a value close to 2/3. This result differs from the exponent $\ensuremath{\kappa}=1/2$ found previously for a chiral unitary model in the absence of a constant magnetic field. We argue that a strong constant magnetic field changes the exponent $\ensuremath{\kappa}$ within the chiral unitary symmetry class by effectively restoring particle-hole symmetry even though a magnetic-field-induced transition from the ballistic to the diffusive regime cannot be fully excluded.

DOI
10.1103/physrevb.85.195424

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We study the localization properties of electrons moving on two-dimensional bipartite lattices in the presence of disorder. The models investigated exhibit a chiral symmetry and belong to the chiral orthogonal (chO), chiral symplectic (chS), or chiral unitary (chU) symmetry class. The disorder is introduced via real random hopping terms for chO and chS, while complex random phases generate the disorder in the chiral unitary model. In the latter case an additional spatially constant, perpendicular magnetic field is also applied. Using a transfer-matrix method, we numerically calculate the smallest Lyapunov exponents that are related to the localization length of the electronic eigenstates. From a finite-size scaling analysis, the logarithmic divergence of the localization length at the quantum critical point at E=0E=0E=0 is obtained. We always find for the critical exponent \ensuremathκ\ensuremath{\kappa}\ensuremathκ, which governs the energy dependence of the divergence, a value close to 2/3. This result differs from the exponent \ensuremathκ=1/2\ensuremath{\kappa}=1/2\ensuremathκ=1/2 found previously for a chiral unitary model in the absence of a constant magnetic field. We argue that a strong constant magnetic field changes the exponent \ensuremathκ\ensuremath{\kappa}\ensuremathκ within the chiral unitary symmetry class by effectively restoring particle-hole symmetry even though a magnetic-field-induced transition from the ballistic to the diffusive regime cannot be fully excluded.
Gade [Nucl. Phys. B 398,398,398, 499 (1993)] has shown that the density of states for a particle hopping on a two-dimensional bipartite lattice in the presence of weak disorder and in the absence of time-reversal symmetry (chiral unitary universality class) is anomalous in the vicinity of the band center $\ensuremath{\varepsilon}=0$ whenever the disorder preserves the sublattice symmetry. More precisely, using a nonlinear $\ensuremath{\sigma}$ model that encodes the sublattice (chiral) symmetry and the absence of time-reversal symmetry she argues that the disorder average density of states diverges as $|\ensuremath{\varepsilon}{|}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}c|\mathrm{ln}\ensuremath{\varepsilon}{|}^{\ensuremath{\kappa}})$ with c some nonuniversal positive constant and $\ensuremath{\kappa}=1/2$ a universal exponent. Her analysis has been extended to the case when time-reversal symmetry is present (chiral orthogonal universality class) for which the same exponent $\ensuremath{\kappa}=1/2$ was predicted. Motrunich et al. [Phys. Rev. B 65,65,65, 064206 (2002)] have argued that the exponent $\ensuremath{\kappa}=1/2$ does not apply to the density of states in the chiral orthogonal universality class. They predict that $\ensuremath{\kappa}=2/3$ instead. We confirm the analysis of Motrunich et al. within a field theory for two flavors of Dirac fermions subjected to two types of weak uncorrelated random potentials: a purely imaginary vector potential and a complex valued mass potential. This model is the naive continuum limit of a model describing a particle hopping on a square lattice in the background of a $\ensuremath{\pi}$-flux phase and subjected to weak disorder that preserves the sublattice symmetry and time-reversal invariance. By commonly held universality arguments, this model is believed to belong to the chiral orthogonal universality class. Our calculation relies in an essential way on the existence of infinitely many local composite operators with negative anomalous scaling dimensions.
The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.
Anderson localization is studied for two flavors of massless Dirac fermions in two-dimensional space perturbed by static disorder that is invariant under a chiral symmetry (chS) and a time-reversal symmetry (TRS) operation which, when squared, is equal either to plus or minus the identity. The former TRS (symmetry class BDI) can, for example, be realized when the Dirac fermions emerge from spinless fermions hopping on a two-dimensional lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\ensuremath{\pi}$ flux per plaquette. The latter TRS is realized by the surface states of three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulators in symmetry class CII. In the phase diagram parametrized by the disorder strengths, there is an infrared stable line of critical points for both symmetry classes BDI and CII. Here we discuss a ``global phase diagram'' in which disordered Dirac fermion systems in all three chiral symmetry classes, AIII, CII, and BDI, occur in four quadrants, sharing one corner which represents the clean Dirac fermion limit. This phase diagram also includes symmetry classes AII [e.g., appearing at the surface of a disordered three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulator in the spin-orbit (symplectic) symmetry class] and D (e.g., the random bond Ising model in two dimensions) as boundaries separating regions of the phase diagram belonging to the three chS classes AIII, BDI, and CII. Moreover, we argue that physics of Anderson localization in the CII phase can be presented in terms of a non-linear-$\ensuremath{\sigma}$ model (NL$\ensuremath{\sigma}$M) with a Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NL$\ensuremath{\sigma}$M description of disordered fermionic models in all ten symmetry classes relevant to Anderson localization in two spatial dimensions.
The scaling behavior of the current-voltage characteristics of chiral and gauge-glass models of disordered superconductors, are studied numerically, in two dimensions. For both models, the linear resistance is nonzero at finite temperatures and the scaling analysis of the nonlinear resistivity is consistent with a phase transition at T=0T=0T=0 temperature characterized by a diverging correlation length $\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}{\ensuremath{\nu}}_{T}}$ and thermal critical exponent ${\ensuremath{\nu}}_{T}.$ The values of ${\ensuremath{\nu}}_{T},$ however, are found to be different for the chiral and gauge-glass models, suggesting different universality classes, in contrast to the result obtained recently in three dimensions.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We determine the dynamical critical exponent z appearing at the Bose glass to superfluid transition in two dimensions by performing large scale numerical studies of two microscopically different quantum models within the universality class: The hard-core boson model and the quantum rotor (soft core) model, both subject to strong on-site disorder. By performing many simulations at different system size L and inverse temperature β close to the quantum critical point, the position of the critical point and the critical exponents, z, ν, and η can be determined independently of any implicit assumptions of the numerical value of z, in contrast to most prior studies. This is done by a careful scaling analysis close to the critical point with a particular focus on the temperature dependence of the scaling functions. For the hard-core boson model we find z=1.88(8), ν=0.99(3), and η=-0.16(8) with a critical field of h(c)=4.79(3), while for the quantum rotor model we find z=1.99(5), ν=1.00(2), and η=-0.3(1) with a critical hopping parameter of t(c)=0.0760(5). In both cases do we find a correlation length exponent consistent with ν=1, saturating the bound ν≥2/d as well as a value of z significantly larger than previous studies, and for the quantum rotor model consistent with z=d.