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arXiv:1610.07627 · Physical review. B./Physical review. B

Supersymmetry in the fractional quantum Hall regime

Eran Sagi, Raul A. Santos

Published May 31, 2017• 16 citations
arXivHTMLPDF

Abstract

Supersymmetry (SUSY) is a symmetry transforming bosons to fermions and vice versa. Indications of its existence have been extensively sought after in high-energy experiments. However, signatures of SUSY have yet to be detected. In this paper we study a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form $\ensuremath{\nu}=\frac{k}{k+2}$, where $k$ is an integer. As we show explicitly, this strongly interacting state exhibits an $\mathcal{N}=2$ SUSY. This allows us to use a topological invariant---the Witten index---defined specifically for supersymmetric theories, to count the difference between the number of bosonic and fermionic zero modes in a circular edge. In this system, we argue that the edge hosts $k+1$ protected zero modes. We further discuss the stability of SUSY with respect to generic perturbations and find that much of the above results remain unchanged. In particular, these results directly apply to the well-established $\ensuremath{\nu}=1/3$ Laughlin state, in which case SUSY is a robust property of the edge theory. These results unveil a hidden topological structure on the long-studied Read-Rezayi states.

DOI
10.1103/physrevb.95.205144

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Supersymmetry (SUSY) is a symmetry transforming bosons to fermions and vice versa. Indications of its existence have been extensively sought after in high-energy experiments. However, signatures of SUSY have yet to be detected. In this paper we study a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form \ensuremathν=kk+2\ensuremath{\nu}=\frac{k}{k+2}\ensuremathν=k+2k​, where kkk is an integer. As we show explicitly, this strongly interacting state exhibits an N=2\mathcal{N}=2N=2 SUSY. This allows us to use a topological invariant---the Witten index---defined specifically for supersymmetric theories, to count the difference between the number of bosonic and fermionic zero modes in a circular edge. In this system, we argue that the edge hosts k+1k+1k+1 protected zero modes. We further discuss the stability of SUSY with respect to generic perturbations and find that much of the above results remain unchanged. In particular, these results directly apply to the well-established \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 Laughlin state, in which case SUSY is a robust property of the edge theory. These results unveil a hidden topological structure on the long-studied Read-Rezayi states.
The Moore-Read state, one of the leading candidates for describing the fractional quantum Hall effect at filling factor ν=5/2, is a paradigmatic p-wave superconductor with non-Abelian topological order. Among its many exotic properties, the state hosts two collective modes: a bosonic density wave and a neutral fermion mode that arises from an unpaired electron in the condensate. It has recently been proposed that the descriptions of the two modes can be unified by postulating supersymmetry (SUSY) that relates them in the long-wavelength limit. Here we extend the SUSY description to construct wave functions of the two modes on closed surfaces, such as the sphere and torus, and we test the resulting states in large-scale numerical simulations. We demonstrate the equivalence in the long-wavelength limit between SUSY wave functions and previous descriptions of collective modes based on the Girvin-MacDonald-Platzman ansatz, Jack polynomials, and bipartite composite fermions. Leveraging the first-quantized form of the SUSY wave functions, we study their energies using the Monte Carlo method and show that realistic ν=5/2 systems are close to the putative SUSY point, where the two collective modes become degenerate in energy.
The neutral fermionic edge mode is essential to the non-Abelian topological property and its experimental detection in Zk{Z}_{k}Zk​ fractional quantum Hall (FQH) state for k>1k>1k>1. Usually, the identification of the edge modes in a finite size system is difficult, especially near the region of the edge reconstruction, due to mixing with the bosonic edge mode and the bulk states as well. We study the edge-mode excitations of the Moore-Read (MR) and Read-Rezayi (RR) states by using Jack polynomials in the truncated subspace. It is found that the electron density, as a detector, has marked different behaviors between the bosonic and fermionic edge modes. As an application, it helps us to identify them near the edge reconstruction and in the RR edge spectrum. On the other hand, we systematically study the edge excitations for the RR state, extrapolating the edge velocities and their related coherence length and temperature in the interferometer experiments.
In contrast to ordinary symmetries, supersymmetry interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here we theoretically show that supersymmetry emerges naturally in topological superconductors, which are well-known condensed matter systems. Specifically, we argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display supersymmetry when probed at long distances and times. Supersymmetry entails several experimental consequences for these systems, such as, exact relations between quantities measured in disparate experiments, and in some cases, exact knowledge of the universal critical exponents. The topological surface states themselves may be interpreted as arising from spontaneously broken supersymmetry, indicating a deep relation between topological phases and SUSY. We discuss prospects for experimental realization in films of superfluid He3_33​-B.
Proposed as a fundamental symmetry describing our Universe, spacetime supersymmetry (SUSY) has not been discovered yet in nature. Nonetheless, it has been predicted that SUSY may emerge in low-energy physics of quantum materials such as topological superconductors and Weyl semimetals. Here, by performing state-of-the-art sign-problem-free quantum Monte Carlo simulations of an interacting two-dimensional topological superconductor, we show convincing evidence that the N=1 SUSY emerges at its edge quantum critical point (EQCP) while its bulk remains gapped and topologically nontrivial. Remarkably, near the EQCP, we find that the edge Majorana fermion acquires a mass that is identical with that of its bosonic superpartner. To the best of our knowledge, this is the first observation that fermions and bosons have equal dynamically generated masses, a hallmark of emergent SUSY. We further discuss experimental signatures of such EQCP and associated SUSY.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE.An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality.Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus.Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information.We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.
Supersymmetry (SUSY), originally proposed in particle physics, refers to a dual relation that connects fermionic and bosonic degrees of freedom in a system. Recently, there has been considerable interest in applying the idea of SUSY to topological phases, motivated by the attempt to gain insights from the fermion side into the boson side and vice versa. We present a systematic study of this construction when applied to band topology in noninteracting systems. First, on top of the conventional tenfold way, we find that topological insulators and superconductors are divided into three classes depending on whether the supercharge can be local and symmetric, must break a symmetry to preserve locality, or needs to break locality. Second, we resolve the apparent paradox between the nontriviality of free fermions and the triviality of free bosons by noting that the topological information is encoded in the identification map. We also discuss how to understand a recently revealed SUSY entanglement duality in this context. These findings are illustrated by prototypical examples. In this paper, we shed light on band topology from the perspective of SUSY.