supersymmetry and investigate its low-energy properties, paying particular attention to the ground-state degeneracy and low-lying fermionic excitations. First, we establish the existence of a phase with spontaneously broken supersymmetry and a phase transition out of it with the help of variational arguments and the exact ground state. We then develop, based on the superfield formalism, a simple mean-field theory, in which the order parameters detect supersymmetry-breaking, to understand the ground-state phases and low-lying Nambu-Goldstone fermions. At the solvable point ({\em frustration-free point}), the exact ground state of an open chain exhibits large degeneracy of the order of the system size, which is attributed to the existence of a zero-energy domain wall (dubbed kink or skink) separating the topological and trivial states of Majorana fermions. Our results may shed new light on the intriguing ground-state properties of supersymmetric lattice models.

DOI: 10.48550/arxiv.2310.19493

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Supersymmetry breaking and Nambu-Goldstone fermions in interacting Majorana chains

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Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model

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We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges $Q$ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$ . When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large- $g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g>0$ . We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges $Q$ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$. When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-$g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g>0$. We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.

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Supersymmetry in the Standard Sachdev-Ye-Kitaev Model

Jan Behrends, Benjámin Béri

Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.

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Interacting Kitaev Chain with

\mathcal{N}=1

Supersymmetry

\mathcal{N} = 1

\mathcal{N}=1

supersymmetry and investigate its low-energy properties, paying particular attention to the ground-state degeneracy and low-lying fermionic excitations. First, we establish the existence of a phase with spontaneously broken supersymmetry and a phase transition out of it with the help of variational arguments and the exact ground state. We then develop, based on the superfield formalism, a simple mean-field theory, in which the order parameters detect supersymmetry breaking, to understand the ground-state phases and low-lying Nambu-Goldstone fermions. At the solvable point (frustration-free point), the exact ground state of an open chain exhibits large degeneracy of the order of the system size, which is attributed to the existence of a zero-energy domain wall (dubbed kink or skink) separating the topological and trivial states of Majorana fermions. Our results may shed new light on the intriguing ground-state properties of supersymmetric lattice models.

We introduce and study a lattice fermion model in one dimension with explicit

\mathcal{N}=1

supersymmetry (SUSY). The Hamiltonian of the model is defined by the square of a supercharge built from Majorana fermion operators. The model describes interacting Majorana fermions, and its properties depend only on a single parameter

g

. When

g=1

, we find that SUSY is unbroken and the ground states are identical to those of the frustration-free Kitaev chains. We also find a parameter regime in which SUSY is restored in the infinite-volume limit. For sufficiently large

g

, we prove that SUSY is spontaneously broken and the low-lying excitations are gapless, which can be thought of as Nambu-Goldstone fermions. We then show numerically that these gapless modes have cubic dispersion at long wavelengths.

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges

Q

and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter

g

. When the parameter

g

vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-

g

limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when

g>0

. We also rigorously prove that for sufficiently large

g

the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.

We introduce a frustration-free, one-dimensional model of spinless fermions with hopping,

p

-wave superconducting pairing and alternating chemical potentials. The model possesses two exactly degenerate ground states even for finite system sizes. We present analytical results for the strong Majorana zero modes, the phase diagram, and the topological order. Furthermore, we generalize our results to include interactions.

g

, the supersymmetry is spontaneously broken with the ground state energy density scaling as

{g}^{4}

. We also briefly discuss the noninteracting limit of the model, which exhibits a zero-energy flat band and Majorana Chern bands.