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arXiv:1008.3723 · Physical Review Letters

Localization Properties of Random-Mass Dirac Fermions from Real-Space Renormalization Group

V. V. Mkhitaryan, M. É. Raǐkh

Published Jun 22, 2011• 4 citations
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Abstract

Localization properties of random-mass Dirac fermions for a realization of mass disorder, commonly referred to as the Cho-Fisher model, are studied on the D-class chiral network. We show that a simple renormalization group (RG) description captures accurately a rich phase diagram: thermal metal and two insulators with quantized σ(xy), as well as transitions (including critical exponents) between them. Our main finding is that, even with small transmission of nodes, the RG block exhibits a sizable portion of perfect resonances. Delocalization occurs by proliferation of these resonances to larger scales. Evolution of the thermal conductance distribution towards a metallic fixed point is synchronized with evolution of signs of transmission coefficients, so that delocalization is accompanied with sign percolation.

DOI
10.1103/physrevlett.106.256803

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Localization properties of random-mass Dirac fermions for a realization of mass disorder, commonly referred to as the Cho-Fisher model, are studied on the D-class chiral network. We show that a simple renormalization group (RG) description captures accurately a rich phase diagram: thermal metal and two insulators with quantized σ(xy), as well as transitions (including critical exponents) between them. Our main finding is that, even with small transmission of nodes, the RG block exhibits a sizable portion of perfect resonances. Delocalization occurs by proliferation of these resonances to larger scales. Evolution of the thermal conductance distribution towards a metallic fixed point is synchronized with evolution of signs of transmission coefficients, so that delocalization is accompanied with sign percolation.
Two-dimensional (2D) disordered superconductor (SC) in class D exhibits a disorder-induced quantum multicritical phenomenon among diffusive thermal metal (DTM), topological superconductor (TS), and conventional localized (AI) phases. To characterize the quantum tricritical point where these three phases meet, we carry out a two-loop renormalization group (RG) analysis for 2D Dirac fermion with random mass in terms of the $\ensuremath{\epsilon}$-expansion in the spatial dimension $d=2\ensuremath{-}\ensuremath{\epsilon}$. In two dimensions $(\ensuremath{\epsilon}=0)$, the random mass is marginally irrelevant around a clean-limit fixed point of the gapless Dirac fermion, while there exists an IR unstable fixed point at finite disorder strength that corresponds to the tricritical point. The critical exponent, dynamical exponent, and scaling dimension of the (uniform) mass term are evaluated around the tricritical point by the two-loop RG analysis. Using a mapping between an effective theory for the 2D random-mass Dirac fermion and the (1+1)(1+1)(1+1)-dimensional Gross-Neveu model, we further deduce the four-loop evaluation of the critical exponent, and the scaling dimension of the uniform mass around the tricritical point. Both the two-loop and four-loop results suggest that criticalities of a AI-DTM transition line as well as TS-DTM transition line are controlled by other saddle-point fixed point(s) at finite uniform mass.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
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.
Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length $\ensuremath{\zeta}$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.
Exploring and understanding topological phases in systems with strong distributed disorder requires developing fundamentally new approaches to replace traditional tools such as topological band theory. Here, we present a general real-space renormalization group (RG) approach for scattering models, which is capable of dealing with strong distributed disorder without relying on the renormalization of Hamiltonians or wave functions. Such scheme, based on a block-scattering transformation combined with a replica strategy, is applied for a comprehensive study of strongly disordered unitary scattering networks with localized bulk states, uncovering a connection between topological physics and critical behavior. Our RG scheme leads to topological flow diagrams that unveil how the microscopic competition between reflection and non-reciprocity leads to the large-scale emergence of macroscopic scattering attractors, corresponding to trivial and topological insulators. Our findings are confirmed by a scaling analysis of the localization length (LL) and critical exponents, and experimentally validated. The results not only shed light on the fundamental understanding of topological phase transitions and scaling properties in strongly disordered regimes, but also pave the way for practical applications in modern topological condensed-matter and photonics, where disorder may be seen as a useful design degree of freedom, and no longer as a hindrance.
We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically and is mathematically equivalent to a domain coarsening model that has been previously solved. The critical fixed-point distribution and critical exponents (that satisfy the Chayes inequality) are thus obtained analytically or to numerical precision. This reproduces some, but not all, of the qualitative features of the MBL phase transition that are indicated by previous numerical work and approximate RG studies: our RG might serve as a ``zeroth-order'' approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.