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arXiv:2505.06689 · Physical Review Letters

Critical States of Fermions with Z 2 Flux Disorder

Hiranmay Das, Naba P. Nayak, Soumya Bera, Vijay B. Shenoy

Published Nov 21, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/wsqw-2xh4

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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.
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=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 $801\ifmmode\times\else\texttimes\fi{}801)$ 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.
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.
Abstract. We study the superfluid properties of attractively interacting fermions hopping in a family of 2D and 3D lattices in the presence of synthetic gauge fields having pi-flux per plaquette. The reason for such a choice is that the pi-flux cubic lattice displays Dirac points and that decreasing the hopping coefficient in a spatial direction (say, tz) these Dirac points are unaltered: it is then possible to study the 3D-2D interpolation towards the pi-flux square lattice. We also consider the lattice configuration providing the continuous interpolation between the 2D pi-flux square lattice and the honeycomb geometry. We investigate by a mean field analysis the effects of interaction and dimensionality on the superfluid gap, chemical potential and critical temperature, showing that these quantities continuously vary along the patterns of interpolation. In the two-dimensional cases at zero temperature and at half filling there is a quantum phase transition occurring at a critical (negative) interaction Uc presenting a linear critical exponent for the gap as a function of |U − Uc|. We show that in three dimensions this quantum phase transition is again retrieved, pointing out that the critical exponent for the gap changes from 1 to 1/2 for each finite value of tz. ar X iv
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.
What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers that specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations—defects of crystalline symmetries—generally lead to a certain “emanant” quantized magnetic flux in the continuum. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are ( 2 + 1 ) D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and antiparticles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects, and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single ( 2 + 1 ) D Dirac cone in symmetry class AII is incompatible with a commuting C M rotational symmetry with ( C M ) M = + 1 .
We study the superfluid properties of attractively interacting fermions hopping in a family of 2D and 3D lattices in the presence of synthetic gauge fields having π-flux per plaquette. The reason for such a choice is that the π-flux cubic lattice displays Dirac points and that decreasing the hopping coefficient in a spatial direction (say, t z ), these Dirac points are unaltered: it is then possible to study the 3D–2D interpolation towards the π-flux square lattice. We also consider the lattice configuration providing the continuous interpolation between the 2 D π-flux square lattice and the honeycomb geometry. We investigate by a mean-field analysis the effects of interaction and dimensionality on the superfluid gap, chemical potential and critical temperature, showing that these quantities continuously vary along the patterns of interpolation. In the two-dimensional cases at zero temperature and half-filling, there is a quantum phase transition occurring at a critical (negative) interaction U c presenting a linear critical exponent for the gap as a function of | U − U c |. We show that in three dimensions, this quantum phase transition is again retrieved, pointing out that the critical exponent for the gap changes from 1 to 1/2 for each finite value of t z .