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arXiv:cond-mat/0611658 · Physical Review B

Emergence of supersymmetry at a critical point of a lattice model

Sung-Sik Lee

Published Aug 6, 2007• 148 citations
arXivHTMLPDF

Abstract

Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing properties have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low-energy limit of a nonsupersymmetric condensed matter system. We propose a $(2+1)$-dimensional lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the $ϵ$ expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.

DOI
10.1103/physrevb.76.075103

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Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing properties have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low-energy limit of a nonsupersymmetric condensed matter system. We propose a (2+1)(2+1)(2+1)-dimensional lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the ϵϵϵ expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.
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.
Supersymmetry, originally proposed in high-energy physics, can emerge as a remarkable low-energy structure in condensed matter systems. While emergent supersymmetry at quantum critical points is widely discussed in models with continuous symmetries, real materials are constrained by microscopic discrete symmetries. To address this, we investigate (2+1)-dimensional Gross-Neveu-Yukawa theories coupling Dirac fermions to a complex order parameter with discrete ZnZ_nZn​ anisotropy. Using the functional renormalization group, we find that for n>3n>3n>3, the anisotropic perturbations are irrelevant at the fixed point, yielding a N=2\mathcal{N}=2N=2 Wess-Zumino supersymmetric critical point. In the ordered phase, this dangerously irrelevant anisotropy gives rise to a second characteristic length scale, ξ′ξ'ξ′, alongside the usual correlation length, ξξξ. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents ν′ν'ν′ and ννν without imposing prior scaling assumptions. For the Z4Z_4Z4​, Z5Z_5Z5​, and Z6Z_6Z6​ models, our results support the scaling relation ν′/ν=1+∣yn∣/pν'/ν= 1+|y_n|/pν′/ν=1+∣yn​∣/p with p=2p=2p=2 in the isotropic framework used here.
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.
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.
The lecture note consists of four parts. In the first part, we review a 2+1 dimensional lattice model which realizes emergent supersymmetry at a quantum critical point. The second part is devoted to a phenomenon called fractionalization where gauge boson and fractionalized particles emerge as low energy excitations as a result of strong interactions between gauge neutral particles. In the third part, we discuss about stability and low energy effective theory of a critical spin liquid state where stringy excitations emerge in a large N limit. In the last part, we discuss about an attempt to come up with a prescription to derive holographic theory for general quantum field theory.
Future quantum computers will enable novel sign-problem-free studies of dynamical phenomena in non-perturbative quantum field theories, including real-time evolution and spontaneous supersymmetry breaking. We are investigating applications of quantum computing to low-dimensional supersymmetric lattice systems that can serve as testbeds for existing and near-future quantum devices. Here we present initial results for the N=1\mathcal{N} = 1N=1 Wess--Zumino model in 1+1~dimensions, building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics. In addition to exploring supersymmetry breaking using the variational quantum eigensolver, we consider the prospects for real-time evolution.