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arXiv:2112.07527 · Physical review. B./Physical review. B

Supersymmetric free fermions and bosons: Locality, symmetry, and topology

Zongping Gong, Robert H. Jonsson, Daniel Malz

Published Feb 24, 2022• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.085423

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•Jan 30, 2013•0 citations
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.
Systems of free fermions are classified by symmetry, space dimensionality, and topological properties described by KKK homology. Those systems belonging to different classes are inequivalent. In contrast, we show that by taking a many-body/Fock-space viewpoint it becomes possible to establish equivalences of topological insulators and superconductors in terms of duality transformations. These mappings connect topologically inequivalent systems of fermions, jumping across entries in existent classification tables, because of the phenomenon of symmetry transmutation by which a symmetry and its dual partner have identical algebraic properties but very different physical interpretations. To constrain our study to established classification tables, we define and characterize mathematically Gaussian dualities as dualities mapping free fermions to free fermions (and interacting to interacting). By introducing a large, flexible class of Gaussian dualities we show that any insulator is dual to a superconductor, and that fermionic edge modes are dual to Majorana edge modes, that is, the Gaussian dualities of this paper preserve the bulk-boundary correspondence. Transmutation of relevant symmetries, particle number, translation, and time reversal is also investigated in detail. As illustrative examples, we show the duality equivalence of the dimerized Peierls chain and the Majorana chain of Kitaev, and a two-dimensional Kekul\'e-type topological insulator, including graphene as a special instance in coupling space, dual to a ppp-wave superconductor. Since our analysis extends to interacting fermion systems, we also briefly discuss some such applications.
In contrast to ordinary symmetries, supersymmetry (SUSY) interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here, we theoretically show that SUSY emerges naturally in condensed matter systems known as topological superconductors. We argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display SUSY when probed at long distances and times. Experimental consequences include 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 SUSY, indicating a deep relation between topological phases and SUSY.
As is well-known in the context of topological insulators and superconductors, short-range-correlated fermionic pure Gaussian states with fundamental symmetries are systematically classified by the periodic table. We revisit this topic from a quantum-information-inspired operational perspective without referring to any Hamiltonians, and apply the formalism to bosonic Gaussian states as well as (both fermionic and bosonic) locality-preserving unitary Gaussian operations. We find that while bosonic Gaussian states are all trivial, there exist nontrivial bosonic Gaussian operations that cannot be continuously deformed into the identity under the locality and symmetry constraint. Moreover, we unveil unexpectedly complicated relations between fermionic Gaussian states and operations, pointing especially out that some of the former can be disentangled by the latter under the same symmetry constraint, while some cannot. In turn, we find that some topological operations are genuinely dynamical, in the sense that they cannot create any topological states from a trivial one, yet they are not connected to the identity. The notions of disentanglability and genuinely dynamical topology can be unified in a general picture of unitary-to-state homomorphism, and apply equally to interacting topological phases and quantum cellular automata.
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.
First proposed in the context of elementary particle physics, supersymmetry is an exotic symmetry that relates bosons and fermions but has not yet been observed in nature. While the traditional expectation is that supersymmetry would be first observed in a high-energy physics experiment, an alternative possibility stressed recently is that supersymmetry could be realized as an emergent symmetry at long wavelengths and low energies in interacting many-particle systems. More specifically, it was suggested that supersymmetry emerges at the quantum phase transition between the gapless Dirac semimetal and a gapped superconductor on the surface of a 3D topological insulator. The authors of this paper provide additional support for the conjecture by performing a three-loop renormalization group calculation and showing that the supersymmetric fixed point in 2+1 dimensions remains stable to that order in perturbation theory.
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.