Lattice Supersymmetry Research Articles

Free fermions beyond Jordan and Wigner

The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such “free fermions in disguise” models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.

Lattice supersymmetry and holography

Over the last twenty years, work based on lattice supersymmetry has generated many new results and insights into the non-perturbative nature of string theory, quantum black holes, and gravity. This endeavor is a broad research program encompassing lattice field theory, supersymmetry, string theory, and quantum gravity. In this volume, we look at a selected subset of the topics covering recent progress in lattice supersymmetry and holography.

Mathcal{N} = 1 Super-Yang-Mills theory on the lattice with twisted mass fermions

Super-Yang-Mills theory (SYM) is a central building block for supersymmetric extensions of the Standard Model of particle physics. Whereas the weakly coupled subsector of the latter can be treated within a perturbative setting, the strongly coupled subsector must be dealt with a non-perturbative approach. Such an approach is provided by the lattice formulation. Unfortunately a lattice regularization breaks supersymmetry and consequently the mass degeneracy within a supermultiplet. In this article we investigate the properties of mathcal{N} = 1 supersymmetric SU(3) Yang-Mills theory with a lattice Wilson Dirac operator with an additional parity mass, similar as in twisted mass lattice QCD. We show that a special 45° twist effectively removes the mass splitting of the chiral partners. Thus, at finite lattice spacing both chiral and supersymmetry are enhanced resulting in an improved continuum extrapolation. Furthermore, we show that for the non-interacting theory at 45° twist discretization errors of order mathcal{O}(a) are suppressed, suggesting that the same happens for the interacting theory as well. As an aside, we demonstrate that the DDαAMG multigrid algorithm accelerates the inversion of the Wilson Dirac operator considerably. On a 163× 32 lattice, speed-up factors of up to 20 are reached if commonly used algorithms are replaced by the DDαAMG.

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.

Lattice SUSY for the DiSSEP at λ2 = 1 (and λ2 = −3)

We investigate whether the dynamical lattice supersymmetry discussed for various Hamiltonians, including one-dimensional quantum spin chains, by (Fendley et al 2003 J. Phys. A 36, 12399424; Yang and Fendley 2004 J. Phys. A 37, 893748; Hagendorf and Fendley 2012 J. Stat. Phys. 146, 112255) and (Hagendorf et al 2013 J. Stat. Phys. 150, 60957; Hagendorf and Liénardy 2017 J. Phys. A 50, 185202; Hagendorf and Liénardy 2018 J. Stat. Mech. 033106) might also exist for the Markov matrices of any one-dimensional exclusion processes, which have the additional constraint of zero column sums by comparison with the spin chains. We find that the DiSSEP (Dissipative Symmetric Simple Exclusion Process), introduced by Crampe et al in (Crampe et al 2016 Phys. Rev. E 94, 032102; Vanicat [arXiv:1708.02440]), provides one such example for suitably chosen parameters. The DiSSEP Markov matrix admits the supersymmetry in these cases because it is conjugate to Ising (λ2 = 1) and Δ = −1/2 XXZ (λ2 = −3) spin chain Hamiltonians which also possess the supersymmetry. The consequences for the spectrum of the DiSSEP Markov matrix are discussed. We also note that the length-changing supersymmetry relation for the DiSSEP Markov matrix ML and the supercharge QL† for L sites, , is reminiscent of a ‘transfer matrix’ symmetry that has been observed in other exclusion processes and discuss the similarity.

Removal of the trace mode in lattice N=4 super Yang-Mills theory

Twisted and orbifold formulations of lattice ${\cal N}=4$ super Yang-Mills theory which possess an exact supersymmetry require a $U(N)=SU(N)\otimes U(1)$ gauge group. In the naive continuum limit, the $U(1)$ modes trivially decouple and play no role in the theory. However, at non-zero lattice spacing they couple to the $SU(N)$ modes and can drive instabilities in the lattice theory. For example, it is well known that the lattice $U(1)$ theory undergoes a phase transition at strong coupling to a chirally broken phase. An improved action that suppresses the fluctuations in the $U(1)$ sector was proposed in arXiv:1505.03135 . Here, we explore a more aggressive approach to the problem by adding a term to the action which can entirely suppress the $U(1)$ mode. The penalty is that the new term breaks the $\mathcal{Q}$-exact lattice supersymmetry. However, we argue that the term is $1/N^2$ suppressed and the existence of a supersymmetric fixed point in the planar limit ensures that any SUSY-violating terms induced in the action possess couplings that also vanish in this limit. We present numerical results on supersymmetric Ward identities consistent with this conclusion.

Lattice Supersymmetry and Order-Disorder Coexistence in the Tricritical Ising Model.

We introduce and analyze a quantum spin or Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit but also manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disordered one in the gapped phase. Deforming the model by including explicit chiral symmetry breaking, we find the phases persist up to an unusual chiral phase transition where the supersymmetry becomes exact even on the lattice.

An alternative lattice field theory formulation inspired by lattice supersymmetry

We propose an unconventional formulation of lattice field theories which is quite general, although originally motivated by the quest of exact lattice supersymmetry. Two long standing problems have a solution in this context: 1) Each degree of freedom on the lattice corresponds to 2d degrees of freedom in the continuum, but all these doublers have (in the case of fermions) the same chirality and can be either identified, thus removing the degeneracy, or, in some theories with extended supersymmetry, identified with different members of the same supermultiplet. 2) The derivative operator, defined on the lattice as a suitable periodic function of the lattice momentum, is an addittive and conserved quantity, thus assuring that the Leibniz rule is satisfied. This implies that the product of two fields on the lattice is replaced by a non-local “star product” which is however in general non-associative. Associativity of the “star product” poses strong restrictions on the form of the lattice derivative operator (which becomes the inverse Gudermannian function of the lattice momentum) and has the consequence that the degrees of freedom of the lattice theory and of the continuum theory are in one-to-one correspondence, so that the two theories are eventually equivalent. We can show that the non-local star product of the fields effectively turns into a local one in the continuum limit. Regularization of the ultraviolet divergences on the lattice is not associated to the lattice spacing, which does not act as a regulator, but may be obtained by a one parameter deformation of the lattice derivative, thus preserving the lattice structure even in the limit of infinite momentum cutoff. However this regularization breaks gauge invariance and a gauge invariant regularization within the lattice formulation is still lacking.

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.

Open spin chains with dynamic lattice supersymmetry

The quantum spin 1/2 XXZ chain with anisotropy parameter possesses a dynamic supersymmetry on the lattice. This supersymmetry and a generalisation to higher spin are investigated in the case of open spin chains. A family of non-diagonal boundary interactions that are compatible with the lattice supersymmetry and depend on several parameters is constructed. The cohomology of the corresponding supercharges is explicitly computed as a function of the parameters and the length of the chain. This cohomology is shown to be non-trivial for certain specific values of the parameters that lead to diagonal boundary interactions. For spin 1/2, they were previously studied by Fendley and Yang. The non-trivial cohomology implies that the spin-chain ground states are supersymmetry singlets. Special scalar products involving an arbitrary number of these supersymmetry singlets for chains of different lengths are exactly computed. In the case of spin 1/2, the scalar products are used to determine the logarithmic bipartite fidelity. Its scaling limit is shown to match the predictions of conformal field theory.

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.

General solution of the cyclic Leibniz rule

We study the general solution of the cyclic Leibniz rule (CLR) which was recently proposed as a new approach to the lattice supersymmetry. Introducing some mathematical preliminaries related to the cyclic symmetry, we find the general solution of the 2-body CLR for the naive symmetric difference operator. The main theorems of this paper state that the general solution can be uniquely expressed as (A) a linear combination of the two fundamental solutions with cyclic invariant coefficients, and (B) a linear combination of the minimal solutions with complex coefficients. Moreover, an extension to the general difference operators is also discussed.

Spontaneous supersymmetry breaking in two dimensional lattice super QCD

We report on a non-perturbative study of two dimensional $$ \mathcal{N}=\left(2,2\right) $$ super QCD. Our lattice formulation retains a single exact supersymmetry at non-zero lattice spacing, and contains N f fermions in the fundamental representation of a U(N c ) gauge group. The lattice action we employ contains an additional Fayet-Iliopoulos term which is also invariant under the exact lattice supersymmetry. This work constitutes the first numerical study of this theory which serves as a toy model for understanding some of the issues that are expected to arise in four dimensional super QCD. We present evidence that the exact supersymmetry breaks spontaneously when N f < N c in agreement with theoretical expectations.

Review of lattice supersymmetry and gauge-gravity duality

We review the status of recent investigations on validating the gauge-gravity duality conjecture through numerical simulations of strongly coupled maximally supersymmetric thermal gauge theories. In the simplest setting, the gauge-gravity duality connects systems of D0-branes and black hole geometries at finite temperature to maximally supersymmetric gauged quantum mechanics at the same temperature. Recent simulations show that nonperturbative gauge theory results give excellent agreement with the quantum gravity predictions, thus proving strong evidence for the validity of the duality conjecture and more insight into quantum black holes and gravity.

Lifting flat directions in lattice supersymmetry

We present a procedure to improve the lattice definition of $\mathcal N = 4$ supersymmetric Yang--Mills theory. The lattice construction necessarily involves U(1) flat directions, and we show how these can be lifted without violating the exact lattice supersymmetry. The basic idea is to modify the equations of motion of an auxiliary field, which determine the moduli space of the system. Applied to numerical calculations, the resulting improved lattice action leads to dramatically reduced violations of supersymmetric Ward identities and much more rapid approach to the continuum limit.

Gauge theory duals of black hole – black string transitions of gravitational theories on a circle

We study the black hole – black string phase transitions of gravitational theories compactified on a circle using the holographic duality conjecture. The gauge theory duals of these theories are maximally supersymmetric and strongly coupled 1 + 1 dimensional SU(N) Yang-Mills theories compactified on a circle, in the large N limit. We perform the strongly coupled finite temperature gauge theory calculations on a lattice, using the recently developed exact lattice supersymmetry methods based on topological twisting and orbifolding. The spatial Polyakov line serves as relevant order parameter of the confinement – deconfinement phase transitions in the gauge theory duals.

Two-loop Wess-Zumino model with exact supersymmetry on the lattice

We consider a lattice formulation of the four dimensional N=1 Wess-Zumino model in terms of the Ginsparg-Wilson relation. This formulation has an exact supersymmetry on the lattice. The lattice action is invariant under a deformed supersymmetric transformation which is non-linear in the scalar fields and it is determined by an iterative procedure in the coupling constant to all orders in perturbation theory. We also show that the corresponding Ward-Takahashi identity is satisfied at fixed lattice spacing. The calculation is performed in lattice perturbation theory up to order $g^3$ (two-loop) and the Ward-Takahashi identity (containing 110 connected non-tadpole Feynman diagrams) is satisfied at fixed lattice spacing thanks to this exact lattice supersymmetry.

A criterion for lattice supersymmetry: cyclic Leibniz rule

It is old folklore that the violation of Leibniz rule on a lattice is an obstruction for constructing a lattice supersymmetric model. While it is still true for full supersymmetry, we show that a slightly modified form of the Leibniz rule, which we call cyclic Leibniz rule (CLR), is actually a criterion for the existence of partial lattice supersymmetry. In one dimension, we find sets of lattice difference operator and field multiplication smeared over lattice which satisfy the CLR under some natural assumptions such as translational invariance and locality. Thereby we construct a model of supersymmetric lattice quantum mechanics without spoiling locality. The CLR relation is coincident with the condition that the vanishing of the so-called surface term in the construction by lattice Nicolai map. We can construct superfield formalism with arbitrary superpotential. This also enables us to apply safely a localization technique to our model, because the kinetic term and the interaction terms of our model are independently invariant under the supersymmetry transformation. A preliminary attempt in finding a solution for the higher dimensional case is also discussed.

Spin Chains with Dynamical Lattice Supersymmetry

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension admits a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.

Species doublers as super multiplets in lattice supersymmetry: chiral conditions of Wess-Zumino model for D = N = 2

We propose an algebraic lattice supersymmetry formulation which has an exact supersymmetry on the lattice. We show how lattice version of chiral conditions can be imposed to satisfy an exact lattice supersymmetry algebra. The species doublers of chiral fermions and the corresponding bosonic counterparts can be accommodated to fit into chiral supermultiplets of lattice supersymmetry and thus lattice chiral fermion problem does not appear. We explicitly show how N=2 Wess-Zumino model in one and two dimensions can be formulated to keep exact supersymmetry for all super charges on the lattice. The momentum representation of N=2 lattice chiral sypersymmetry algebra has lattice periodicity and thus momentum conservation should be modified to a lattice version of sine momentum conservation, which generates nonlocal interactions and leads to a loss of lattice translational invariance. It is shown that the nonlocality is mild and the translational invariance is recovered in the continuum limit. In the coordinate representation a new type of product is defined and the difference operator satisfies Leibnitz rule and an exact lattice supersymmetry is realized on this product.