Supersymmetric Lattice Research Articles

H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta > 0$$\\end{document}. Approaching the critical point from the transient regime, β↘βc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\searrow \\beta _{\ extrm{c}}$$\\end{document}, we show that the expected total time spent at the starting vertex diverges as ∼exp(c/β-βc)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim \\exp (c/\\sqrt{\\beta - \\beta _{\ extrm{c}}})$$\\end{document}. Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values βc<β<βcerg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{\ extrm{c}}< \\beta < \\beta _{\ extrm{c}}^{\ extrm{erg}}$$\\end{document}. In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the H2|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2|2}$$\\end{document}-model.

Entanglement entropy in the ground state of supersymmetric fermion lattice models

In this paper, we study the entanglement entropy for strongly correlated spinless fermions in the ground state of a supersymmetric Hamiltonian on diverse graphs. Then, we use the entanglement entropy to study the graph isomorphism (GI) problem on some non-isomorphic pairs of graphs to distinguish them from each other. The GI problem is considered on pairs of non-isomorphic strongly regular graphs (SRG) and some cases of regular graphs. The SRGs are well-known for being highly symmetric, so that they rarely can be detected by classical and quantum algorithms. However, we show that most of the pairs of non-isomorphic SRGs can be distinguished by utilizing the supersymmetric entanglement entropy. Also, detecting the regular non-isomorphic graphs is often difficult. The obtained result illustrates that the entanglement entropy of the single-particle Hamiltonian cannot detect non-isomorphic pairs of regular graphs, but the entanglement entropy of the two-particle Hamiltonian can.

Kink Dynamics and Quantum Simulation of Supersymmetric Lattice Hamiltonians.

We propose a quantum simulation of a supersymmetric lattice model using atoms trapped in a 1D configuration and interacting through a Rydberg dressed potential. The elementary excitations in the model are kinks or (in a sector with one extra particle) their superpartners-the skinks. The two are connected by supersymmetry and display identical quantum dynamics. We provide an analytical description of the kink (skink) quench dynamics and propose a protocol to prepare and detect these excitations in the quantum simulator. We make a detailed analysis, based on numerical simulation, of the Rydberg atom simulator and show that it accurately tracks the dynamics of the supersymmetric model.

Supersymmetry and multicriticality in a ladder of constrained fermions

Supersymmetric lattice models of constrained fermions are known to feature exotic phenomena such as superfrustration, with an extensive degeneracy of ground states, the nature of which is however generally unknown. Here we address this issue by considering a superfrustrated model, which we deform from the supersymetric point. By numerically studying its two-parameter phase diagram, we reveal a rich phenomenology. The vicinity of the supersymmetric point features period-4 and period-5 density waves which are connected by a floating phase (incommensurate Luttinger liquid) with smoothly varying density. The supersymmetric point emerges as a multicritical point between these three phases. Inside the period-4 phase we report a valence-bond solid type ground state that persists up to the supersymmetric point. Our numerical data for transitions out of density-wave phases are consistent with the Pokrovsky-Talapov universality class. Furthermore, our analysis unveiled a period-3 phase with a boundary determined by a competition between single and two-particle instabilities accompanied by a doubling of the wavevector of the density profiles along a line in the phase diagram.

Lattice mathcal{N} = 4 super Yang-Mills at strong coupling

In this paper we present results from numerical simulations of mathcal{N} = 4 super Yang-Mills for two color gauge theory over a wide range of ’t Hooft coupling 0 < λ ≤ 30 using a supersymmetric lattice action [1]. Numerical study of this lattice theory has been stymied until recently by both sign problems and the occurrence of lattice artifact phases at strong coupling. We have recently developed a new action that appears capable of solving both problems. The resulting action possesses just SU(2) rather than U(2) gauge symmetry. By explicit computations of the fermion Pfaffian we present evidence that the theory possesses no sign problem and exists in a single phase out to arbitrarily strong coupling. Furthermore, preliminary work shows that the logarithm of the supersymmetric Wilson loop varies as the square root of the ’t Hooft coupling λ for large λ in agreement with holographic predictions.

Three-dimensional super-Yang-Mills theory on the lattice and dual black branes

In the large-$N$ and strong-coupling limit, maximally supersymmetric SU($N$) Yang--Mills theory in $(2 + 1)$ dimensions is conjectured to be dual to the decoupling limit of a stack of $N$ D$2$-branes, which may be described by IIA supergravity.We study this conjecture in the Euclidean setting using nonperturbative lattice gauge theory calculations.Our supersymmetric lattice construction naturally puts the theory on a skewed Euclidean 3-torus. Taking one cycle to have anti-periodic fermion boundary conditions, the large-torus limit is described by certain Euclidean black holes. We compute the bosonic action---the variation of the partition function---and compare our numerical results to the supergravity prediction as the size of the torus is changed, keeping its shape fixed. Our lattice calculations primarily utilize $N = 8$ with extrapolations to the continuum limit, and our results are consistent with the expected gravity behavior in the appropriate large-torus limit.

Weak-ergodicity-breaking via lattice supersymmetry

We study the spectral properties of D-dimensional N=2 supersymmetric lattice models. We find systematic departures from the eigenstate thermalization hypothesis (ETH) in the form of a degenerate set of ETH-violating supersymmetric (SUSY) doublets, also referred to as many-body scars, that we construct analytically. These states are stable against arbitrary SUSY-preserving perturbations, including inhomogeneous couplings. For the specific case of two-leg ladders, we provide extensive numerical evidence that shows how those states are the only ones violating the ETH, and discuss their robustness to SUSY-violating perturbations. Our work suggests a generic mechanism to stabilize quantum many-body scars in lattice models in arbitrary dimensions.

Characterization of degenerate supersymmetric ground states of the Nicolai supersymmetric fermion lattice model by symmetry breakdown

We study a supersymmetric fermion lattice model defined by Hermann Nicolai. We show that its infinitely many classical supersymmetric ground states are associated to breakdown of hidden local supersymmetries.

Supersymmetry breakdown for an extended version of the Nicolai supersymmetric fermion lattice model

Sannomiya-Katsura-Nakayama have recently studied supersymmetry breakdown for an extension of the Nicolai supersymmetric fermion lattice model. This model called the extended Nicolai model is parameterized by an adjustable constant $g\in \mathbb{R}$ in its defining supercharge, and satisfies ${{\cal{N}}}=2$ supersymmetry. We show that for any non-zero $g$ the extended Nicolai model breaks supersymmetry dynamically, and the energy density of any homogeneous ground state (that breaks supersymmetry) is strictly positive.

Ergodicity Breaking and Localization of the Nicolai Supersymmetric Fermion Lattice Model

We investigate dynamics of the supersymmetric fermion lattice model defined by Hermann Nicolai. We provide its local fermionic constants of motion that exist infinitely many. These generate hidden local supersymmetries that the Nicolai model possesses in addition to its defining dynamical supersymmetry. The existence of such local constants directly implies the breaking ergodicity of the model in the sense of Mazur. At zero temperature, there are infinitely many degenerated classical ground states. We discuss these MBL-like properties. First, we show the delocalization scenario proposed by De Roeck-Huveneers can not naively apply to the Nicolai model at zero temperature despite its disorder-free translation-invariant quantum interaction. Second, we discuss the quantum integrability of the Nicolai model based on the proposal by Caux-Mossel.

Nonperturbative study of dynamical SUSY breaking in N=(2,2) Yang-Mills theory

We examine the possibility of dynamical supersymmetry breaking in two-dimensional $\mathcal{N} = (2, 2)$ supersymmetric Yang-Mills theory. The theory is discretized on a Euclidean spacetime lattice using a supersymmetric lattice action. We compute the vacuum energy of the theory at finite temperature and take the zero temperature limit. Supersymmetry will be spontaneously broken in this theory if the measured ground state energy is non-zero. By performing simulations on a range of lattices up to $96 \times 96$ we are able to perform a careful extrapolation to the continuum limit for a wide range of temperatures. Subsequent extrapolations to the zero temperature limit yield an upper bound on the ground state energy density. We find the energy density to be statistically consistent with zero in agreement with the absence of dynamical supersymmetry breaking in this theory.

One-loop calculations in Supersymmetric Lattice QCD

We study the self energies of all particles which appear in a lattice regularization of supersymmetric QCD (${\cal N}=1$). We compute, perturbatively to one-loop, the relevant two-point Green's functions using both the dimensional and the lattice regularizations. Our lattice formulation employs the Wilson fermion acrion for the gluino and quark fields. The gauge group that we consider is $SU(N_c)$ while the number of colors, $N_c$ and the number of flavors, $N_f$, are kept as generic parameters. We have also searched for relations among the propagators which are computed from our one-loop results. We have obtained analytic expressions for the renormalization functions of the quark field ($Z_\psi$), gluon field ($Z_u$), gluino field ($Z_\lambda$) and squark field ($Z_{A_\pm}$). We present here results from dimensional regularization, relegating to a forthcoming publication our results along with a more complete list of references. Part of the lattice study regards also the renormalization of quark bilinear operators which, unlike the non-supersymmetric case, exhibit a rich pattern of operator mixing at the quantum level.

Anomaly and sign problem in N=(2,2) SYM on polyhedra: Numerical analysis

We investigate the two-dimensional $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory on the discretized curved space (polyhedra). We first revisit that the number of supersymmetries of the continuum $\mathcal{N}=(2,2)$ SYM theory on any curved manifold can be enhanced at least to two by introducing an appropriate $U(1)$ gauge background associated with the $U(1)_{V}$ symmetry. We then show that the generalized Sugino model on the discretized curved space, which was proposed in our previous work, can be identified to the discretization of this SUSY enhanced theory, where one of the supersymmetries remains and the other is broken but restored in the continuum limit. We find that the $U(1)_{A}$ anomaly exists also in the discretized theory as a result of an unbalance of the number of the fermions proportional to the Euler characteristics of the polyhedra. We then study this model by using the numerical Monte-Carlo simulation. We propose a novel phase-quench method called "anomaly-phase-quenched approximation" with respect to the $U(1)_A$ anomaly. We numerically show that the Ward-Takahashi (WT) identity associated with the remaining supersymmetry is realized by adopting this approximation. We figure out the relation between the sign (phase) problem and pseudo-zero-modes of the Dirac operator. We also show that the divergent behavior of the scalar one-point function gets milder as the genus of the background increases. These are the first numerical observations for the supersymmetric lattice model on the curved space with generic topologies.

Exact lattice supersymmetry at the quantum level for N = 2 Wess–Zumino models in 1- and 2-dimensions

Supersymmetric lattice Ward–Takahashi identities are investigated perturbatively up to two-loop corrections for super doubler approach of [Formula: see text] lattice Wess–Zumino models in 1- and 2-dimensions. In this approach, notorious chiral fermion doublers are treated as physical particles and momentum conservation is modified in such a way that lattice Leibniz rule is satisfied. The two major difficulties to keep exact lattice supersymmetry are overcome. This formulation defines, however, nonlocal field theory. Nevertheless we confirm that exact supersymmetry on the lattice is realized for all supercharges at the quantum level. Delicate issues of associativity are also discussed.

Supersymmetry on the lattice

We discuss the motivations, difficulties and progress in the study of supersymmetric lattice gauge theories focusing in particular on [Formula: see text] and [Formula: see text] super-Yang–Mills in four dimensions. Brief reviews of the corresponding lattice formalisms are given and current results are presented and discussed. We conclude with a summary of the main aspects of current work and prospects for the future.

Spinon bases in supersymmetric CFTs

We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with or (nonlinear) superconformal symmetry. Generalising the spinon basis of the WZW theories, we introduce supersymmetric spinons , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection of (discrete) momenta. The state-content for given choice of is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the theories. In the simplest case, which is the superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the theory. It includes an operator H2, akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the superconformal CFTs, this finitisation precisely leads to the so-called Mk supersymmetric lattice models with characteristic order-k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.

A euclidean lattice formulation of D = 5 maximally supersymmetric Yang-Mills theory

We construct lattice action for five-dimensional maximally supersymmetric Yang-Mills theory. This supersymmetric lattice formulation can be used to explore the non-perturbative regime of the continuum target theory, which has a known gravitational dual.

Integrable supersymmetric chain without particle conservation

We introduce a new integrable supersymmetric lattice chain which violates fermion conservation and exhibits fermion–hole symmetry. The model displays exponential degeneracy in every eigenstate including the ground state. This degeneracy is expressed in the possibility to create any number of zero modes reminiscent of Cooper pairs.

On Supersymmetric Fermion Lattice Systems

This note provides a C *-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C *-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models.

Quiver gauge theories and integrable lattice models

We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d $\mathcal{N} = 1$ theories known as brane box and brane tilling models, 3d $\mathcal{N} = 2$ and 2d $\mathcal{N} = (2,2)$ theories obtained from them by compactification, and 2d $\mathcal{N} = (0,2)$ theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.