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arXiv:2310.14074 · Physical review. B./Physical review. B

Localization renormalization and quantum Hall systems

Bartholomew Andrews, Dominic Reiss, Fenner Harper, Rahul Roy

Published Mar 19, 2024• 4 citations
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Abstract

The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of $\ensuremath{\nu}=0.5$. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result is in accord with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall systems.

DOI
10.1103/physrevb.109.125132

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The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of \ensuremathν=0.5\ensuremath{\nu}=0.5\ensuremathν=0.5. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result is in accord with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall systems.
We discuss the application of a quantum real-space renormalization-group approach to the Anderson-localization problem. States obtained by applying combinations of fixed and free boundary conditions to the system are used as a truncated basis at each iteration. We discuss the problem of intruder states, which occurs when the renormalization-group procedure is used to find states at the center of the band; we also discuss some possible solutions. We present numerical results for the conductance, the localization length, and the scaling function for the Anderson-localization problem in one dimension.
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 construct an effective-field theory for the quantum Hall system which embodies both localization and fractional statistics. The latter involves a Chern-Simons interaction, while the former involves a generalization of conventional localization theory. The theory is invariant under ``complexified'' duality transformations of the conductivities which appear as effective parameters of the model. By exploiting these parameter space symmetries, as well as the conformal symmetry which appears at renormalization-group fixed points, we are able to extract a precise prediction for the whole scaling diagram. It exhibits both fractional and integer phases, the exact location of all fixed points, and universal scaling exponents. With a plausible identification of the universality class of the theory in the replica limit, the value of the critical exponent for the delocalization transition between plateaus in the Hall conductivity is found to be 7/3, in apparent agreement with available scaling experiments.
This dissertation comprises two parts, each of which details an investigation within topological matter.In the first part, we investigate the real space localization properties of static topological insulators in two dimensions. We develop three numerical procedures each of which removes an extensive fraction of the local degrees of freedom. We probe the real space properties of the residual projection operator, showing that the maximally localized length scale ξ supported in the residual Hilbert space diverges as the fraction of states remaining approaches 0. The power-law exponent ν = 1/2 characterizing the singular behavior of ξ appears universal across class A insulators with non-zero Chern number.In the second part, we explore a class of Floquet topological phases in three dimensions. These phases have translational invariance but no other symmetry, and exhibit anomalous transport at a boundary surface. We show that the boundary behavior of such phases falls into equivalence classes up to local 2D unitary evolution. This provides a classification of the 3D bulk, which we conjecture to be complete. We demonstrate that such phases may be generated by exactly solvable “exchange drives” in the bulk. The edge behavior of a general exchange drive in two or three dimensions is shown to derive from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.
We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent ν = 2.73 ± 0.02 that is consistent with that for topologically trivial symplectic systems.
In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\ensuremath{\nu}=1/3$ and 5/25/25/2. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\ensuremath{\nu}=5/2$ for large systems with electron numbers up to Ne=34{N}_{e}=34Ne​=34, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.