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arXiv:cond-mat/9603169 · Physical review. B, Condensed matter

Disordered critical wave functions in random-bond models in two dimensions: Random-lattice fermions at E=0 without doubling

Yasuhiro Hatsugai, Xiao-Gang Wen, Mahito Kohmoto

Published Jul 15, 1997• 71 citations
arXivHTMLPDF

Abstract

Random-bond Hamiltonians of the $\ensuremath{\pi}$ flux state on the square lattice are investigated. It has a special symmetry and all states are paired except the ones with zero energy. Because of this, there are always zero modes. The states near $E=0$ are described by massless Dirac fermions. For the zero mode, we can construct a random lattice fermion without a doubling and quite large systems (up to $801\ifmmode\times\else\texttimes\fi{}801)$ are treated numerically. We clearly demonstrate that the zero mode is given by a critical wave function. Its multifractal behavior is also compared with the effective field theory.

DOI
10.1103/physrevb.56.1061

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Random-bond Hamiltonians of the \ensuremathπ\ensuremath{\pi}\ensuremathπ flux state on the square lattice are investigated. It has a special symmetry and all states are paired except the ones with zero energy. Because of this, there are always zero modes. The states near E=0E=0E=0 are described by massless Dirac fermions. For the zero mode, we can construct a random lattice fermion without a doubling and quite large systems (up to 801\ifmmode×\else\texttimes\fi801)801\ifmmode\times\else\texttimes\fi{}801)801\ifmmode×\else\texttimes\fi801) are treated numerically. We clearly demonstrate that the zero mode is given by a critical wave function. Its multifractal behavior is also compared with the effective field theory.
Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\ensuremath{\pi}$-flux model with random hopping. It preserves a symmetry denoted by ${H,\ensuremath{\gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\ensuremath{\sim}{E}^{\ensuremath{\alpha}}$ and the exponent $\ensuremath{\alpha}$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at E=0E\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0E=0.
We investigate the physics of fermions on a square lattice with π flux, subjected to disordered random Z_{2} gauge fields that arise from flux defects, i.e., plaquettes with zero flux. At half filling, where the system possesses BDI symmetry, we show that a new class of critical states is realized, with the states at zero energy showing a multifractal character that depends on the flux defect concentration c and local correlation between defects. For any concentration of flux defects, we find a nonfreezing multifractal spectrum with a tendency toward termination. We characterize this class of critical states by uncovering a robust relation between the conductivity and the Lyapunov exponent, which is satisfied by the states irrespective of the concentration or the local correlations between the flux defects. We demonstrate that renormalization group methods, based on perturbing the Dirac point, fail to capture this new class of critical states. This Letter not only offers new challenges to the theory of disordered systems in the chiral classes, but is also likely to be useful in understanding a variety of problems where fermions interact with discrete gauge fields.
We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wave functions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavors, are obtained by deriving from conformal field theory an effective Gaussian model for the wave function amplitudes and mapping it to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wave functions remain delocalized in the presence of strong disorder.
The random flux model (RFM) on a square lattice with random flux $\ensuremath{\Phi}\ensuremath{\in}(\ensuremath{-}p\ensuremath{\pi},p\ensuremath{\pi}),0<p<~1$ per plaquette has been studied by analyisis of high-precision transfer-matrix calculations, as well as the exact numerical diagonalization in the vicinity and at the band center. Interpreting thus obtained results using the recent findings for the Anderson bond-disordered (ABD) model from Phys. Rev. B 62, 12 775 (2000), the importance of chiral symmetry is further emphasized and localization properties explained in terms of the scaling associated with the central two states of the spectrum. The results support the existence of two critical states at the band center of the infinite system. These states exhibit a different kind of critical behavior, named anomalous, allowing one to calculate the critical exponent which has been done for p=0.5p=0.5p=0.5 and 1.1.1. Some differences between the ABD and RFM models are also discussed, the most important one being a broken symmetry at the critical point of the former.
A disordered fermion model defined on a two-dimensional square lattice with $\ensuremath{\pi}$ flux is studied. Time-reversal symmetry and one of four possible chiral symmetries are imposed. The continuum limit of the model near zero energy yields Dirac fermions with chiral symmetry. It is shown that one of them has $\mathrm{Sp}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{Sp}(n)$ symmetry whereas others have U(2n)(2n)(2n) symmetry, both of which are broken to Sp(n),(n),(n), and the fluctuation around a saddle point is described, respectively, by the Sp(n)2{(n)}_{2}(n)2​ Wess-Zumino-Witten model and U(2n)(2n)(2n)/Sp(n)(n)(n) nonlinear sigma model.
We calculate nonperturbatively the multifractal scaling exponents of the critical wave function for two dimensional Dirac fermions in the presence of a random magnetic field. We do so by arguing that the multifractal scaling exponents can be expressed in terms of the free energy of random directed polymers on a Cayley tree. We find a weak-strong disorder transition for the multifractal scaling exponents of the wave function that is parallel to the freezing or glassy transition of the random polymer model.