Complex Langevin Dynamics and Supersymmetric Quantum Mechanics
CComplex Langevin Dynamics and Supersymmetric Quantum Mechanics
Anosh Joseph ∗ and Arpith Kumar † Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali,Knowledge City, Sector 81, SAS Nagar, Punjab 140306, India (Dated: December 1, 2020)Using complex Langevin method we probe the possibility of dynamical supersymmetry breaking invarious N = 2 supersymmetric quantum mechanical models with complex potentials including theones exhibiting PT symmetry. Traditional Monte Carlo methods based on importance sampling failin these situations. From the simulations, we conclude that complex Langevin method can reliablypredict the absence or presence of dynamical supersymmetry breaking in the these models. ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] N ov CONTENTS
I. Introduction 2II. Supersymmetric Quantum Mechanics 4III. Lattice Regularized Supersymmetric Quantum Mechanics 5A. Lattice Regularization 5B. Theory on a Lattice 6C. Correlation Functions 9D. Ward Identities 10IV. Complex Langevin Simulations 11A. Supersymmetric Anharmonic Oscillator 12B. Double-Well Potential 14C. General Polynomial Potential 15D. Shape Invariant Potential 19E. PT -Symmetric Models 21V. Conclusions 24Acknowledgments 25A. SUSY Invariance of the Action 26B. Bosonic PT -Symmetric Theory 31C. Reliability of Simulations 331. Reliability using Fokker-Planck Operator 332. Decay of the Drift Terms 34References 35 I. INTRODUCTION
A standard tool to investigate numerous nonperturbative features of quantum field theory systems is the MonteCarlo simulations of a lattice regularized version of the field theory path integral. The basic idea behind path integralMonte Carlo is to generate field configurations with a probability weight given by the exponential of the negative ofthe action (in Euclidean spacetime) and then compute the path integral by statistically averaging these importancesampled ensemble of field configurations. However, when the action is complex, for example, when studying QCD atfinite temperature and baryon/quark chemical potential, QCD with a theta term, Chern-Simons gauge theories, orchiral gauge theories, the fermion determinant of the theory can be complex and this feature results in the so-called sign problem (or a phase problem to be more exact). This results in simulation algorithms based on path integralMonte Carlo unreliable.There exist several methods that can handle the sign problem. These include methods such as analytical contin-uation, Taylor series expansion [1], methods based on the complexification of the integration variables such as theLefschetz thimble method [2] and complex Langevin method (CLM) [3–7]. The CLM is a straightforward generaliza-tion of the real Langevin method and it extends the idea of stochastic quantization for ordinary field theoretic systemswith real actions to the cases with complex actions. It overcomes the sign problem by defining a stochastic process, withcomplexified field variables, using Langevin equations for the complex action. The expectation values of observablesin the original path integral are then calculated from an average of the corresponding quantities over this stochasticprocess. See Ref. [7] for a pedagogical review on this method and Ref. [8] for a recent review in the context of thesign problem in quantum many-body physics. See Refs. [9–14] for simulations based on the Lefschetz thimble method.CLM has been used successfully in various models in the recent past [15–24]. There have also been studies of su-persymmetric models based on CLM [25–28]. In Ref. [29] the authors used complex Langevin simulations to observeGross-Witten-Wadia (GWW) [30–32] phase transitions in certain large- N matrix models. In Ref. [33] we looked atcertain classes of zero-dimensional quantum field theories with complex actions using CLM method. In this paper,we continue our investigations to one-dimensional models with complex actions. Using complex Langevin dynamicswe study the absence or presence of supersymmetry (SUSY) breaking in supersymmetric quantum mechanics modelswith complex actions. These models also include the interesting case of theories exhibiting PT symmetry.The central theme of stochastic quantization is that expectation values of observables are obtained as equilibriumvalues of a stochastic process. In Langevin dynamics, this is implemented by evolving the system in a fictitious timedirection τ (Langevin time) subject to a stochastic noise. We could think of applying Langevin dynamics when theactions under consideration are complex. In such cases, the field variables become complexified during the Langevinevolution since the gradient of the action, the drift term , is complex. The complex Langevin equation in Eulerdiscretized form reads φ ( τ + ∆ τ ) = φ ( τ ) − ∆ τ (cid:18) δS [ φ ] δφ ( τ ) (cid:19) + √ ∆ τ η ( τ ) , (I.1)where ∆ τ is the Langevin time step, and η ( τ ) is a Gaussian noise satisfying the constraints (cid:104) η ( τ ) (cid:105) = 0 , (cid:104) η ( τ ) η ( τ (cid:48) ) (cid:105) = 2 δ ττ (cid:48) . (I.2)In our simulations, we use real Gaussian stochastic noise to control excursions in the imaginary directions of the fieldconfigurations [34–36].For an arbitrary operator O , we can define a noise averaged expectation value (cid:104)O [ φ ( τ )] (cid:105) η = (cid:90) dφP [ φ ( τ )] O [ φ ] , (I.3)where the probability distribution P [ φ ( τ )] satisfies the Fokker-Planck equation ∂P [ φ ( τ )] ∂τ = δδφ ( τ ) (cid:18) δδφ ( τ ) + δS [ φ ] δφ ( τ ) (cid:19) P [ φ ( τ )] . (I.4)When the action is real, it can be shown that in the limit τ → ∞ , the stationary solution of the Fokker-Planckequation P [ φ ] ∼ exp ( − S [ φ ]) (I.5)will be reached guaranteeing convergence of the Langevin dynamics to the correct equilibrium distribution. When theaction is complex we will end up in a not so easy situation. The drift term will be complex and thus, if we considerLangevin dynamics based on the above equation, we will end up with complexified fields: φ = Re φ + i Im φ . We canstill consider Langevin dynamics with complex probabilities [6, 37–39] but proofs towards convergence to the complexweight, exp( − S ) , will be non-trivial.The plan of this paper is as follows. In Sec. II we briefly introduce the continuum supersymmetric quantummechanics. In Sec. III we discretize the continuum theory on a one-dimensional lattice. There, we compute thepartition function of the lattice regularized action. In Sec. III C we compute the twisted bosonic and fermioniccorrelation functions along with their corresponding Langevin observables. In Sec. III D we define a set of Wardidentities to verify our investigations on SUSY breaking. In Sec. IV we present our simulation results for the presenceor absence of SUSY breaking in various supersymmetric quantum mechanics models using complex Langevin dynamics.We provide our conclusions in Sec. V. In Appendix A we study the invariance of the continuum and lattice-regularizedactions under their respective SUSY transformations. In Appendix. B we extend our analysis of zero-dimensional PT -symmetric scalar field theory [33] to the corresponding bosonic quantum mechanics. In Appendix. C 1 we studya correctness criterion of our simulations using the Fokker-Planck operator. In Appendix. C 2 we study reliability ofour simulations by examining the probability distributions of the magnitude of the drift terms. II. SUPERSYMMETRIC QUANTUM MECHANICS
In this section we introduce the action S [ φ, ψ, ψ ] for supersymmetric quantum mechanics with a general superpo-tential W ( φ ) . The degrees of freedom are a scalar field φ and two fermions ψ and ψ . The partition function in pathintegral formalism takes the form Z = (cid:90) D φ D ψ D ψ e − S [ φ,ψ,ψ ] . (II.1)We take the action to be an integral over a compactified time circle of circumference β in Euclidean spacetime. Inour case it has the form S [ φ, ψ, ψ ] = (cid:90) β dτ (cid:20) B ( τ ) + i B (cid:18) ∂∂τ φ ( τ ) + ∂∂φ W ( φ ( τ )) (cid:19) + ψ ( τ ) (cid:18) ∂∂τ + ∂ ∂φ W ( φ ( τ )) (cid:19) ψ ( τ ) (cid:21) , (II.2)where B is an auxiliary field. In the above expression the derivatives with respect to τ and φ , are denoted by dot andprime, respectively.The action given in Eq. (II.2) is invariant under N = 2 supersymmetry. There are two independent supercharges: Q and Q . The SUSY transformations on the fields have the form Q φ = ψ, (II.3a) Q ψ = 0 , (II.3b) Q ψ = − i B , (II.3c) QB = 0 , (II.3d)and Q φ = − ψ, (II.4a) Q ¯ ψ = 0 , (II.4b) Q ψ = − i B + 2 ˙ φ, (II.4c) QB = 2 i ˙ ψ. (II.4d)The supercharges satisfy the algebra {Q , Q} = 0 , {Q , Q} = 0 , {Q , Q} = 2 ∂ τ . (II.5)We also note that the action can be expressed in Q - and QQ -exact forms. That is, S = Q (cid:90) β dτ ψ (cid:26) i B − (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19)(cid:27) , (II.6) = QQ (cid:90) β dτ (cid:18) ψψ + W ( φ ) (cid:19) . (II.7)The auxiliary field B was introduced for off-shell completion of the SUSY algebra. It also acts as a crucial observableto probe SUSY breaking. It is possible to integrate out this field using its equation of motion B = − i (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) (II.8)to get the on-shell form of the action S = (cid:90) β dτ (cid:34) (cid:26)(cid:18) ∂φ∂τ (cid:19) + W (cid:48) ( φ ) (cid:27) + ψ (cid:26) ∂∂τ + W (cid:48)(cid:48) ( φ ) (cid:27) ψ (cid:35) . (II.9)Upon using the Leibniz integral rule, and discarding the resultant total derivative term, the action takes the form S = (cid:90) β dτ (cid:34) (cid:40)(cid:18) ∂φ∂τ (cid:19) + [ W (cid:48) ( φ )] (cid:41) + ψ (cid:26) ∂∂τ + W (cid:48)(cid:48) ( φ ) (cid:27) ψ (cid:35) . (II.10)In the above expression, the total derivative term we omitted was ( ∂φ/∂τ ) W (cid:48) ( φ ) . We note that such an omission isonly possible in the continuum theory. When we discretize the theory on a lattice, this term is non-vanishing, and itspresence is crucial to ensure the Q -exact lattice supersymmetry. Thus, in our lattice analysis we will use Eq. (II.9)as the continuum theory. III. LATTICE REGULARIZED SUPERSYMMETRIC QUANTUM MECHANICSA. Lattice Regularization
In this section, we discretize the action given in Eq. (II.9) on a one-dimensional lattice. Let us take the lattice tobe Λ , having T number of equally spaced sites with lattice spacing a . The integral and continuum derivatives arereplaced by a Riemann sum a Σ and a lattice difference operator ∇ , respectively. The physical extent of the lattice isdefined as β ≡ T a .There are several ways to implement the lattice discretization of the theory. We will choose the prescription inwhich the derivatives appearing in the action take the form of a symmetric difference operator ∇ Sij = 12 (cid:0) ∇ + ij + ∇ − ij (cid:1) , (III.1)where ∇ + ij = 1 a ( δ i +1 ,j − δ i,j ) −→ ∇ + ij f j = 1 a ( f i +1 − f i ) , (III.2) ∇ − ij = 1 a ( δ i,j − δ i − ,j ) −→ ∇ − ij f j = 1 a ( f i − f i − ) , (III.3)are the forward and backward difference operators, respectively. However, it is known that the symmetric derivativeleads to the so-called fermion doubling problem and this in turn leads to a non-supersymmetric lattice theory. Wecan use the Wilson discretization prescription to decouple these extra fermion modes from the system. The differenceoperator is modified as ∇ Wij ( r ) = ∇ Sij − ra (cid:3) ij , (III.4)where (cid:3) ij = ∇ + ik ∇ − kj is the usual lattice Laplacian and the Wilson parameter r ∈ [ − , / { } [40]. For one-dimensional derivatives it turnsout that the standard choice of r = ± yields ∇ Wij ( ±
1) = ∇ ∓ ij , (III.5)thereby suggesting that the doubling problem can be resolved by simply using forward or backward difference op-erator. The reason being that for any choice of the lattice difference operator, the theories can be made manifestlysupersymmetric upon the addition of appropriate improvement terms corresponding to the discretization of continuumsurface integrals [41].In our analysis, for the standard choice of the Wilson parameter, we follow the symmetric derivative with a Wilsonmass matrix suggested by Catterall and Gregory in Ref. [42]. The lattice regularized action then takes the form S = a T − (cid:88) i =0 T − (cid:88) j =0 ∇ Sij φ j + Ω (cid:48) i + ψ i T − (cid:88) j =0 (cid:0) ∇ Sij + Ω (cid:48)(cid:48) ij (cid:1) ψ j , (III.6)where the quantity Ω (cid:48) i is defined as Ω (cid:48) i = T − (cid:88) j =0 K ij φ j + W (cid:48) i , (III.7)and its derivative Ω (cid:48)(cid:48) ij is Ω (cid:48)(cid:48) ij = K ij + W (cid:48)(cid:48) ij δ ij . (III.8)The Wilson mass matrix K ij has the form, K ij = mδ ij − ra (cid:3) ij . (III.9)The claim in Eq. (III.5) can be easily verified for this particular Wilson mass prescription, as can be seen later inthe Sec.III B .We can make the variables dimensionless by performing appropriate rescaling. Let us consider the following set ofvariable redefinitions (cid:101) φ = a − / φ, (cid:101) ∇ S = a ∇ S , (cid:101) Ω (cid:48) = √ a Ω (cid:48) , (cid:101) Ω (cid:48)(cid:48) = a Ω (cid:48)(cid:48) . (III.10)Under these rescalings the action becomes (cid:101) S = T − (cid:88) i =0 T − (cid:88) j =0 (cid:101) ∇ Sij (cid:101) φ j + (cid:101) Ω (cid:48) i + ψ i T − (cid:88) j =0 (cid:16) (cid:101) ∇ Sij + (cid:101) Ω (cid:48)(cid:48) ij (cid:17) ψ j . (III.11)It is important to note that in the process of the rescaling of the superpotential, we need to be careful with theparameters of a dimensionless superpotential, and later on, while extracting the physical parameters. B. Theory on a Lattice
Now we will use the lattice regularized action to study certain lattice supersymmetry artifacts, and then write downan expression for the supersymmetric partition function. For convenience, we will not be using the tilde sign on thedimensionless variables; all variables and fields mentioned from now on are understood to be dimensionless. Physicalquantities will be labelled differently.The N = 2 supersymmetry transformations mentioned in Eqs. (II.3) - (II.4) are modified to contain the Wilsonmass terms. For a given lattice site t these transformations are given by Qφ t = ψ t , Qψ t = − N t , Qψ t = 0 , (III.12)and Q φ t = − ψ t , Q ψ t = N t , Q ψ t = 0 , (III.13)where N t = ∇ S φ t + Ω (cid:48) t , and N t = ∇ S φ t − Ω (cid:48) t . (III.14)The supercharges Q and Q satisfy the algebra {Q , Q} = 0 , {Q , Q} = 0 , and {Q , Q} = 2 ∇ S . (III.15)The main obstacle that prevents the preservation of exact lattice SUSY is the failure of Leibniz rule for latticederivatives. Unlike the continuum action given in Eq. (II.9), the lattice regularized action S = T − (cid:88) i =0 T − (cid:88) j =0 ∇ Sij φ j + Ω (cid:48) i + ψ i T − (cid:88) j =0 (cid:0) ∇ Sij + Ω (cid:48)(cid:48) ij (cid:1) ψ j (III.16)preserves only the Q supercharge. The Q supersymmetry is broken for T ≥ . It can also be shown that the actionis only Q invariant. That is QS = 0 (cid:54) = QS . (III.17)We provide a derivation of this in Appendix A.When SUSY is broken, the partition function vanishes, and in that case, the expectation values of the observablesnormalized by the partition function could be ill-defined. In order to overcome this difficulty we will apply periodicboundary conditions for bosons and twisted boundary conditions [43, 44] for fermions. (We will explore the vanishingof the partition function, that is the Witten index and also address whether there is need for twisting the field instudying such one-dimensional supersymmetric models using complex Langevin in Sec. IV. For now we will continueour computations with more general case incorporating the twist.) Imposing twisted boundary conditions (TBC) isanalogues to turning on an external field in the system. We have φ T = φ , (III.18a) ψ T = e iα ψ , (III.18b) ψ T = e − iα ψ , (III.18c)where α denotes the twist parameter.Once the twisted boundary conditions are imposed the partition function given in Eq. (II.1) takes the followingform Z α = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t dψ t dψ t (cid:33) e −S α , (III.19)where S α is the lattice regularized action that respects the twisted boundary conditions.This action can be computed using the following expressions T − (cid:88) j =0 ∇ Sij φ j = 12 ( φ i +1 − φ i − ) , (III.20a) T − (cid:88) j =0 ∇ Sij ψ j = 12 ( ψ i +1 − ψ i − ) , (III.20b) Ω (cid:48) i = T − (cid:88) j =0 K ij φ j + W (cid:48) i = mδ ij φ j − r δ i,j +1 + δ i,j − − δ i,j ) φ j + W (cid:48) i = ( m + r ) φ i − r φ i +1 + φ i − ) + W (cid:48) i , (III.20c) T − (cid:88) j =0 Ω (cid:48)(cid:48) ij ψ j = T − (cid:88) j =0 (cid:2) K ij ψ j + W (cid:48)(cid:48) ij δ ij ψ j (cid:3) = T − (cid:88) j =0 (cid:104) mδ ij ψ j − r δ i,j +1 + δ i,j − − δ ij ) ψ j + W (cid:48)(cid:48) ij δ ij ψ j (cid:105) = ( m + r + W (cid:48)(cid:48) i ) ψ i − r ψ i +1 + ψ i − ) . (III.20d)Upon using the above expressions, the total action with TBC can be written as S α = S B + S Fα , where the bosonicpart of the action takes the form S B = T − (cid:88) i =0 (cid:18)
12 ( φ i +1 − φ i − ) + ( m + r ) φ i − r φ i +1 + φ i − ) + W (cid:48) i (cid:19) , (III.21)and the fermionic part of the action is S Fα = T − (cid:88) i =0 ψ i (cid:18)
12 ( ψ i +1 − ψ i − ) + ( m + r + W (cid:48)(cid:48) i ) ψ i − r ψ i +1 + ψ i − ) (cid:19) . (III.22)Setting r = 1 , the total action becomes S α = T − (cid:88) i =0 (cid:18) φ i − φ i − + mφ i + W (cid:48) i (cid:19) + T − (cid:88) i =0 ψ i (cid:18) ψ i − ψ i − + ( m + W (cid:48)(cid:48) i ) ψ i (cid:19) . (III.23)We note that the r = 1 case matches with that of the backward difference discretization. Let us coalesce the massterm with the potential W , and define a new potential Ξ as Ξ ≡ mφ + W. (III.24)The action with twisted boundary conditions now takes the form S α = T − (cid:88) i =0 (cid:18) T − (cid:88) j =0 ∇ − ij φ j + Ξ (cid:48) i (cid:19) + T − (cid:88) i =0 ψ i (cid:18) T − (cid:88) j =0 ∇ − ij ψ i + Ξ i (cid:19) ψ i . (III.25)Also the expressions for N i and N i become N i = T − (cid:88) j =0 ∇ Sij φ j + Ω (cid:48) i = T − (cid:88) j =0 ∇ − ij φ j + Ξ (cid:48) i , (III.26) N i = T − (cid:88) j =0 ∇ Sij φ j − Ω (cid:48) i = T − (cid:88) j =0 ∇ + ij φ j − Ξ (cid:48) i . (III.27)The partition function given in Eq. (III.19) can be written as the product of a bosonic part and a fermionic part Z α = Z B Z Fα . (III.28)The fermionic part of the partition function, Z Fα , can be computed as follows Z Fα = (cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) exp (cid:32) − T − (cid:88) t =0 ψ t (cid:20) (1 + Ξ (cid:48)(cid:48) t ) ψ t − ψ t − (cid:21)(cid:33) = (cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) T − (cid:89) t =0 (1 + Ξ (cid:48)(cid:48) t ) ψ t ψ t + T − (cid:89) t =0 ψ t − ψ t (cid:124) (cid:123)(cid:122) (cid:125) ψ − = e iα ψ T − = T − (cid:89) t =0 (1 + Ξ (cid:48)(cid:48) t ) − e iα . (III.29)In the last line above we used the integration properties of complex Grassmann variables, that is (cid:90) dψdψ ψψ = 1 , (cid:90) dψdψ = 0 . (III.30)The quantity given in Eq. (III.29) is the determinant of the twisted Wilson fermion matrix W Fα det (cid:2) W Fα (cid:3) = Z Fα = T − (cid:89) t =0 (1 + Ξ (cid:48)(cid:48) t ) − e iα . (III.31)This is in agreement with the expression obtained by Catterall and Gregory in Ref. [42] for periodic boundaryconditions, that is when α = 0 .The full partition function now takes the form Z α = Z B Z Fα = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) exp (cid:34) − (cid:18) φ t − φ t − + Ξ (cid:48) t (cid:19) (cid:35) (cid:32) T − (cid:89) t =0 (1 + Ξ (cid:48)(cid:48) t ) − e iα (cid:33) = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) det (cid:2) W Fα (cid:3) exp (cid:2) −S B (cid:3) = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) exp (cid:2) −S eff α (cid:3) . (III.32)We can write down the expression for the effective action S eff α S α eff = S B − ln (cid:0) det (cid:2) W Fα (cid:3)(cid:1) = T − (cid:88) t =0 (cid:18) φ t − φ t − + Ξ (cid:48) t (cid:19) − ln (cid:32) T − (cid:89) t =0 (1 + Ξ (cid:48)(cid:48) t ) − e iα (cid:33) . (III.33)Given an observable O , we can compute its expectation value as (cid:104)O(cid:105) = lim α → (cid:104)O(cid:105) α = lim α → Z α (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) O exp (cid:2) −S eff α (cid:3) . (III.34)Since we will be interested in updating the fields through complex Langevin dynamics, we also need to computethe gradient of the action. At the k -th lattice site, it takes the form ∂S eff α ∂φ k = ∂S B ∂φ k − W Fα ] ∂ det (cid:2) W Fα (cid:3) ∂φ k = T − (cid:88) t =0 (cid:18) φ t − φ t − + Ξ (cid:48) t (cid:19)(cid:26) δ t,k − δ t − ,k + Ξ (cid:48)(cid:48) t δ t,k (cid:27) − (cid:18) (cid:81) T − t =0 (1 + Ξ (cid:48)(cid:48) t ) (cid:19)(cid:18) (cid:81) T − t =0 (1 + Ξ (cid:48)(cid:48) t ) (cid:19) − e iα (cid:40) T − (cid:88) t =0 Ξ (cid:48)(cid:48)(cid:48) t δ t,k (cid:48)(cid:48) t (cid:41) = (cid:18) φ k − φ k − + Ξ (cid:48) k (cid:19)(cid:26) (cid:48)(cid:48) k (cid:27) − (cid:18) φ k +1 − φ k + Ξ (cid:48) k +1 (cid:19) − (cid:18) (cid:81) T − t =0 (1 + Ξ (cid:48)(cid:48) t ) (cid:19)(cid:18) (cid:81) T − t =0 (1 + Ξ (cid:48)(cid:48) t ) (cid:19) − e iα (cid:40) Ξ (cid:48)(cid:48)(cid:48) k (cid:48)(cid:48) k (cid:41) . (III.35) C. Correlation Functions
Using the expression given in Eq. (III.34) we can compute the correlation functions. The bosonic and fermioniccorrelation functions are defined as G Bα ( k ) ≡ (cid:104) φ φ k (cid:105) α (III.36)and G Fα ( k ) ≡ (cid:104) ψ ψ k (cid:105) α , (III.37)respectively, at the site k .0The fermionic correlation function has the form (cid:104) ψ ψ k (cid:105) α = 1 Z α (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) × (cid:40)(cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) ψ ψ k exp (cid:34) − T − (cid:88) t =0 ψ t (cid:20) (1 + Ξ (cid:48)(cid:48) t ) ψ t − ψ t − (cid:21)(cid:35)(cid:41)(cid:124) (cid:123)(cid:122) (cid:125) (cid:104) ψ ψ k (cid:105) F × exp (cid:34) − T − (cid:88) t =0 (cid:18) φ t − φ t − + Ξ (cid:48) t (cid:19) (cid:35) . (III.38)Upon following the same procedure as discussed earlier, we can integrate out the fermions and compute the fermionicpart explicitly as (cid:104) ψ ψ k (cid:105) F = (cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) ψ ψ k exp (cid:34) − T − (cid:88) t =0 ψ t (cid:18) (1 + Ξ (cid:48)(cid:48) t ) ψ t − ψ t − (cid:19)(cid:35) = (cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) ψ ψ k T − (cid:89) t =1 (cid:34) − (cid:16) (cid:48)(cid:48) t (cid:17) ψ t ψ t − ψ t − ψ t (cid:35) = (cid:90) (cid:32) T − (cid:89) t =0 dψ t dψ t (cid:33) ψ ψ k (cid:32) k (cid:89) t =1 (cid:2) − ψ t − ψ t (cid:3)(cid:33) (cid:32) T − (cid:89) t = k +1 (cid:104) − (cid:16) (cid:48)(cid:48) t (cid:17) ψ t ψ t (cid:105)(cid:33) = − (cid:32) T − (cid:89) t = k +1 (cid:104) (cid:48)(cid:48) t (cid:105)(cid:33) . (III.39)Putting the above expression back in Eq. (III.38) we get (cid:104) ψ ψ k (cid:105) α = 1 Z α (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) (cid:18) − (cid:104) ψ ψ k (cid:105) F det [ W Fα ] (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) [ ψ ψ k ] Lα exp (cid:104) − S α eff (cid:105) , (III.40)where upon comparison with Eq. (III.34) we define (cid:2) ψ ψ k (cid:3) Lα as our Langevin observable corresponding to the fermioniccorrelator (cid:104) ψ ψ k (cid:105) α . That is, (cid:2) ψ ψ k (cid:3) Lα = − (cid:81) T − t = k +1 (cid:104) (cid:48)(cid:48) t (cid:105)(cid:81) T − t =0 (cid:2) (cid:48)(cid:48) t (cid:3) − e iα . (III.41)Now, for the bosonic correlation function, the computation is trivial. In this case, our Langevin observable is thebosonic correlation function itself. For the k -th lattice site, [ φ φ k ] L = φ φ k , such that (cid:104) φ φ k (cid:105) α = 1 Z α (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t (cid:33) φ φ k exp (cid:104) − S α eff (cid:105) . (III.42) D. Ward Identities
Another set of observables that would help us in the investigations on SUSY breaking is the Ward identities. Inthis section we will derive a non-trivial supersymmetric Ward identity and use it to probe SUSY breaking.For the supersymmetric variation of the fields, Eqs. (III.13) and (III.12), the invariance of the lattice action guidesus to a set of Ward identities that connect the bosonic and fermionic correlators. We can derive these by addingexternal source terms to the action for the corresponding bosonic and fermionic fields. We then use the fact that the1action, measure, and partition function are all invariant under these SUSY transformations (in our case, the theoryhas Q -exact lattice SUSY).The partition function in Eq. (III.19), upon addition of the source terms ( J, θ, θ ), becomes Z α (cid:0) J, θ, θ (cid:1) = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t dψ t dψ t (cid:33) exp (cid:34) −S α + T − (cid:88) t =0 (cid:0) J t φ t + θ t ψ t + θ t ψ t (cid:1)(cid:35) . (III.43)It is easy to see that the variation of the partition function under the Q -transformations vanishes upon turning offthe external sources. That is, Q Z α (cid:0) J, θ, θ (cid:1) = (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t dψ t dψ t (cid:33) exp (cid:34) −S α + T − (cid:88) t =0 (cid:0) J t φ t + θ t ψ t + θ t ψ t (cid:1)(cid:35) × (cid:32) −QS α + T − (cid:88) t =0 (cid:0) J t Q φ t + θ t Q ψ t (cid:1)(cid:33) = 0 . (III.44)In fact, the variation of any derivative of the partition function with respect to these external source terms alsovanishes (upon turning off the sources). This procedure leads us to a non-trivial supersymmetric Ward identity. Wetake the derivative of the partition function with respect to the source terms J j and θ i , that is Q (cid:20) ∂ Z∂J j ∂θ i (cid:21) = Q (cid:20) ∂ Z∂θ i ∂J j (cid:21) = 0= Q (cid:34) (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t dψ t dψ t (cid:33) × exp (cid:32) −S α + T − (cid:88) t =0 (cid:0) J t φ t + θ t ψ t + θ t ψ t (cid:1)(cid:33) φ j ψ i (cid:35) = 0= (cid:18) √ π (cid:19) T (cid:90) (cid:32) T − (cid:89) t =0 dφ t dψ t dψ t (cid:33) × exp (cid:32) −S α + T − (cid:88) t =0 (cid:0) J t φ t + θ t ψ t + θ t ψ t (cid:1)(cid:33) Q (cid:2) φ j ψ i (cid:3) = 0= ⇒ (cid:104) ψ i ψ j (cid:105) + (cid:104) N i φ j (cid:105) = 0 . (III.45)In this work we will consider the following Ward Identity to investigate spontaneous SUSY breaking W : (cid:104) ψ ψ t (cid:105) + (cid:104) N φ t (cid:105) = 0 . (III.46) IV. COMPLEX LANGEVIN SIMULATIONS
In this section we present our simulation results on probing spontaneous SUSY breaking in various supersymmetricquantum mechanics models using complex Langevin dynamics.Let us look at the relevant observables prior to presenting the simulation results. One crucial observable is theexpectation value of the auxiliary field B α = − i (cid:0) ∇ Sij φ j + Ω (cid:48) i (cid:1) = − i (cid:0) ∇ − ij φ j + Ξ (cid:48) i (cid:1) . (IV.1)The non-vanishing (vanishing) nature of the auxiliary field indicates that SUSY is broken (preserved) in the system.That is, (cid:104)B(cid:105) = lim α → (cid:104)B α (cid:105) (cid:40) (cid:54) = 0 SUSY broken = 0
SUSY preserved . (IV.2)2Secondly, we consider the bosonic action S Bα = T − (cid:88) i =0 N i = T − (cid:88) i =0 (cid:18) T − (cid:88) j =0 ∇ Sij φ j + Ω (cid:48) i (cid:19) = T − (cid:88) i =0 (cid:18) T − (cid:88) j =0 ∇ − ij φ j + Ξ (cid:48) i (cid:19) . (IV.3)It has been studied that for exact lattice SUSY, the expectation value of the bosonic action is independent of theinteraction couplings. Thus, the bosonic action expectation value simply counts the number of degrees of freedom[45]. That is, (cid:104)S B (cid:105) = N d.o.f . In supersymmetric quantum mechanics, it is expected that, (cid:104)S(cid:105) = T , and (cid:104)S B (cid:105) = T / where T is the number of sites. Thus we have (cid:104)S B (cid:105) = lim α → (cid:104)S Bα (cid:105) (cid:40) (cid:54) = T / SUSY broken = T / SUSY preserved . (IV.4)The third indicator is the equality of the fermionic and bosonic mass gaps. The mass gaps can be extracted eitherby a cosh (cid:2) ma ( t − T ) (cid:3) fit for the t -th lattice site, or a simple exponential fit over say, the first or last T / time slicesof the respective correlation functions.The last set of observables involve the Ward identity. We expect that W , mentioned in Eq. (III.46), to hold (notto hold) for theories with SUSY preserved (broken). That is, lim α → W (cid:40) −(cid:104) ψ ψ t (cid:105) α (cid:54) = (cid:104) N φ t (cid:105) α SUSY broken −(cid:104) ψ ψ t (cid:105) α = (cid:104) N φ t (cid:105) α SUSY preserved . (IV.5)Only in the limit α → we can comment, using the above set of observables, if the system possesses exact latticeSUSY. Since the partition function is a well-defined quantity for models with SUSY preserved, as expected, wewere able to compute the normalized expectation values of observables, and hence perform numerical investigationsfor α = 0 (PBC) case. The issue in working without the twist field, that is α = 0 (PBC), arises only in modelswhere SUSY is spontaneously broken, since the partition function vanishes, and normalized expectation values of theobservables are ill-defined. A. Supersymmetric Anharmonic Oscillator
Following the discussion in Sec. III B, we consider the the model with supersymmetric anharmonic oscillatorpotential Ξ( φ ) = 12 mφ + 14 gφ . (IV.6)SUSY is preserved in this model according to Ref. [42, 46].First, we simulate SUSY harmonic oscillator for physical parameters m phys = 10 and g phys = 0 , and twist field α = 0 . Simulations were performed for different lattice spacings keeping the (physical) circle size β = 1 . In Tab. Iwe provide the values of the bosonic and fermionic mass gaps. It is clear that m B phys ≈ m F phys indicating that SUSYis preserved in the model. For error estimation we will use the jackknife method for all our models. Fig. 1 showsbosonic (blue triangle) and fermionic (red square) physical mass gaps versus lattice spacing ( a ) and lattice size ( T ).Black dashed line shows the continuum value of SUSY harmonic oscillator mass gaps for the physical parameters m phys = 10 , g phys = 0 that is, m exact = 10 . We see that boson and fermion masses are degenerate within statisticalerrors, and furthermore, as lattice spacing a → , the common mass gap approaches the correct continuum value. Ourresults confirm that the free action has an exact SUSY at finite lattice spacing, which is responsible for the degeneratemass gaps.Now we simulate SUSY anharmonic oscillator for physical parameters m phys = 10 and g phys = 100 , and twist field α = 0 . Simulations were performed for different lattice spacings keeping the (physical) circle size β = 1 . In Tab. IIwe provide the bosonic and fermionic mass gaps. Here also we have m B phys ≈ m F phys indicating that SUSY is preservedin the model. Table III contains the expectation values of the auxiliary field B α and bosonic action S Bα . The meanexpectation value (cid:104)B(cid:105) vanishes in the simulations and thus indicate exact lattice SUSY. This table also contains theexpectation value of the bosonic action S Bα given in Eq. (III.21). For this model, we observe that the expectation3 T a = T − m B m B phys = a − m B m F m F phys = a − m F . . . .
16 0.0625 . . . .
32 0.03125 . . . .
64 0.015625 . . . . TABLE I. Bosonic and fermionic mass gaps for the SUSY harmonic oscillator with physical parameters m phys = 10 and g phys = 0 . Simulations were performed for different lattice spacings, keeping the (physical) circle size β = 1 . We used adaptiveLangevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 , and generation steps N gen = 10 . Measurements weretaken with a gap of steps. m phy s aT m B m F m exact FIG. 1. Bosonic and fermionic physical mass gaps for SUSY harmonic oscillator with physical parameters m phys = 10 and g phys = 0 versus lattice spacing ( a ) and lattice size ( T ). The plot is based on the simulation data provided in Tab. I. value of the bosonic action (cid:104)S B (cid:105) = T , and it was independent of physical parameters g phys and m phys , which againsuggests exact lattice SUSY.Fig. 2 shows bosonic (Left) and fermionic (Right) correlation functions (used to compute the respective mass gaps)versus lattice site ( t ) for various lattice size ( T ) for the SUSY anharmonic oscillator. In Fig. 3 we show the bosonicand fermionic physical mass gaps versus lattice spacing ( a ) and lattice size ( T ). Here we also compare our results withthose obtained by Catterall and Gregory [42] ( m B CG , m F CG ), and find excellent agreement. Black dashed line showsthe continuum value of SUSY anharmonic oscillator mass gaps for the physical parameters m phys = 10 , g phys = 100 that is m exact = 16.865 [41]. We see that boson and fermion masses are degenerate within statistical errors, andfurthermore as lattice spacing a → , the common mass gap approaches the correct continuum value. In Fig. 4 weshow the Langevin time histories of the auxiliary field B α (Left) and the bosonic action S Bα (Right) for lattice size T = 8 . In Fig. 5 we plot the real part of Ward identity W (Left), and its bosonic and fermionic contributions (Right),given in Eq. (III.46), versus the lattice site t for lattices with T values. We observe that the respective bosonic andfermionic contributions cancel out each other within statistical errors, and hence W is satisfied. Our results confirmthat the SUSY anharmonic oscillator has an exact SUSY, which is responsible for the degenerate mass gaps.4 T a = T − m B m B phys = a − m B m F m F phys = a − m F . . . .
16 0.0625 . . . .
32 0.03125 . . . .
64 0.015625 . . . . TABLE II. Bosonic and fermionic mass gaps for SUSY anharmonic oscillator with physical parameters m phys = 10 and g phys = 100 . Simulations were performed for different lattice spacings keeping the (physical) circle size β = 1 . We usedadaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurementswere taken with a gap of steps. < Φ Φ t > t m phys =10.0, g phys =100.0 T=8 T=16 T=32 < ψ – ψ t > t m phys =10.0, g phys =100.0 T=8 T=16 T=32
FIG. 2. Bosonic (Left) and fermionic (Right) correlation functions for SUSY anharmonic oscillator with physical parameters m phys = 10 and g phys = 100 versus lattice site ( t ) for lattice sizes ( T ). These plots are based on the simulation data providedin Tab. II. B. Double-Well Potential
In this section we consider the cases in which the superpotentials have the form of a double-well. We have Ξ (cid:48) ( φ ) = mφ + g (cid:0) φ + µ (cid:1) . (IV.7)According to Ref. [46] SUSY is spontaneously broken in this model.We also consider a complexified double-well potential of the form Ξ (cid:48) ( φ ) = mφ + ig (cid:0) φ + µ (cid:1) . (IV.8)We investigate spontaneous SUSY breaking in these double-well models using complex Langevin method. In Tab.IV we provide the simulation results with physical parameters m phys = 1 , g phys = 3 and µ phys = 2 for lattices with T = 8 , and . For these models, we perform simulations for various twist parameter α to verify the consistencyof our results for the α = 0 case. We noticed that the expectation value of (cid:104)B(cid:105) does not vanish for real double-wellmodel given in Eq. (IV.7), and it vanishes for the complex double well model in Eq. (IV.8). These results indicateSUSY breaking for the real double-well model and unbroken SUSY for the complex double-well model. We alsoobserve that the expectation value of the bosonic action, (cid:104)S B (cid:105) (cid:54) = T , for real double-well and it was also dependenton parameters g phys and µ phys . This again indicates spontaneous SUSY breaking. For the complex double-well modelthe expectation value of the bosonic action (cid:104)S B (cid:105) = T , and it is independent of parameters g phys and µ phys . Thissuggests that SUSY is preserved in this model. In Figs. 6 and 7 we show the Langevin time history of the auxiliaryfield B α (Left) and the bosonic action S Bα (Right) for the real and complex double-well potentials, respectively.5 m phy s aT m B m F m BCG m FCG m exact FIG. 3. Bosonic and fermionic physical mass gaps for SUSY anharmonic oscillator with physical parameters m phys = 10 and g phys = 100 , versus lattice spacing ( a ) and lattice size ( T ). m B CG and m F CG respectively represent bosonic and fermionic massgaps results from Catterall and Gregory [42]. The plot is based on the simulation data provided in Tab. II. Ξ (cid:48) ( φ ) T a = T − α (cid:104)B α (cid:105) (cid:104)S Bα (cid:105) mφ + gφ . .
00 0 . − i . . i . . .
00 0 . i . . i . . .
00 0 . − i . . i . . .
00 0 . − i . . i . TABLE III. Expectation value of the auxiliary field B α and the bosonic action S Bα for SUSY anharmonic oscillator with physicalparameters m phys = 10 and g phys = 100 . Simulations were performed for different lattice spacings keeping the (physical) circlesize β = 1 . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 , and generation steps N gen = 10 . Measurements were taken with a gap of steps. In Fig. 8 we plot the Ward identities for these models. In these plots, on the left column we show completeWard identity W , in the middle and right column we show the real and imaginary parts respectively, of bosonicand fermionic contributions to the Ward identity, as mentioned in Eq. (III.46). For the real double-well potential(Top-Row), the bosonic and fermionic contributions do not cancel out each other and hence Ward identity is notsatisfied. While for the complex double-well potential the bosonic and fermionic contributions cancel out each otherwithin statistical errors and Ward identity is satisfied. Hence, our simulations suggest that SUSY is broken for thereal double-well potential, and it is preserved for the complex double-well potential. C. General Polynomial Potential
In this section, we extend our analysis to the case where the superpotential Ξ (cid:48) ( φ ) takes the form of a degree- k polynomial Ξ (cid:48) ( k ) = g k φ k + g k − φ k − + · · · + g . (IV.9)We begin by considering the polynomial superpotential with real coefficients. For simplicity, we assume the form g k = g , g k − = · · · = g = 0 , g = m and g = 0 (in the same fashion as that of the SUSY anharmonic potential).6 FIG. 4. Langevin time history of the auxiliary field B α (Left) and the bosonic action S Bα (Right) for the SUSY anharmonicoscillator with physical parameters m phys = 10 and g phys = 100 on a lattice with T = 8 . The plots are based on simulationdata provided in Tab. III. -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 [ W ] T t m phys =10.0, g phys =100.0, α =0.0 Re [ W ] Re [ W ] Re [ W ] -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 R e [ < ψ – ψ t > T , < N φ t > T ] t m phys =10.0, g phys =100.0, α =0.0
00 0 . − i . . i . . − . − i . . − i . .
20 0 . − i . . i . .
80 0 . − i . . i . Eq. (IV.7)
12 0 . .
00 0 . − i . . i . .
05 0 . − i . . i . . − . − i . . − i . . − . − i . . − i . . .
00 0 . − i . . − i . . − . − i . . − i . .
20 0 . − i . . i . . − . − i . . − i . . . − . − i . . − i . . − . − i . . − i . . − . − i . . − i . . − . − i . . − i . Eq. (IV.8)
12 0 . . − . i . . i . . − . i . . i . . − . i . . i . . − . i . . i . . . − . i . . i . . − . i . . i . . − . i . . i . . − . i . . i . TABLE IV. Expectation value of the auxiliary field B α and the bosonic action S Bα for the real and complex double-wellpotentials given in Eq. (IV.7) and (IV.8), respectively, with physical parameters m phys = 1 , g phys = 3 and µ phys = 2 .Simulations were performed for different lattice spacings keeping the (physical) circle size β = 1 . We used adaptive Langevinstep size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken witha gap of steps. we provide the Langevin time history of the auxiliary field B α (Left-Column) and the bosonic action S Bα (Right-Column) for the even- (Top-Row) and odd-(Bottom-Row) degree real polynomial potentials on a lattice with T = 8 ,respectively. In Tab. VI we provide the bosonic and fermionic mass gaps for odd-degree real polynomial potentials.Here we have m B phys ≈ m F phys indicating that SUSY is preserved in the model with odd-degree real polynomial potential.In Fig. 10 we plot the Ward identities for these models. In these plots, on the left column we show real part ofcomplete Ward identity and on the right column we show the real part of bosonic and fermionic contributions to theWard identity, as mentioned in Eq. (III.46). For the even-degree potential (Top-Row), the bosonic and fermioniccontributions do not cancel out each other and hence Ward identity is not satisfied. While for the odd-degree poten-tial (Bottom-Row) the bosonic and fermionic contributions cancel out each other within statistical errors and Wardidentity is satisfied. Hence, our simulations suggest that SUSY is broken for the even-degree, and preserved for theodd-degree real-polynomial superpotential.Let us now consider models with complex potentials. We begin with the double-well inspired real and complex8 FIG. 6. Langevin time history of the mean auxiliary field B α (Left) and the bosonic action S Bα (Right) for the real double-wellpotential given in Eq. (IV.7) with physical parameters m phys = 1 , g phys = 3 and µ phys = 2 for a lattice with T = 8 . The plotsare based on simulation data provided in Tab. IV.FIG. 7. Langevin time history of the mean auxiliary field B α (Left) and the bosonic action S Bα (Right) for the complex double-well potential given in Eq. (IV.8), with physical parameters m phys = 1 , g phys = 3 and µ phys = 2 for T = 8 . The plots are basedon the simulation data provided in Tab. IV. cubic superpotentials of the form Ξ (cid:48) (3)Real = gφ (cid:0) φ + µ (cid:1) , (IV.12) Ξ (cid:48) (3)Complex = igφ (cid:0) φ + µ (cid:1) (IV.13)and investigate SUSY breaking in these models.In Tab. VII, we provide the simulation results for these double-well inspired real and complex cubic superpotentialswith physical parameters m phys = 1 , g phys = 3 and µ phys = 2 for lattices with T = 8 , and . We performedsimulations for various α values to verify the consistency of our results for the α = 0 case. We noticed that theauxiliary field expectation value (cid:104)B(cid:105) vanishes for real double-well inspired cubic model given in Eq. (IV.12), and itdoes not vanish for the complex double-well inspired cubic model given in Eq. (IV.13). These results indicate thatSUSY is preserved for the real double-well inspired cubic model and it is broken for complex double-well inspiredcubic model. We also observe that for the real double-well inspired cubic model, the expectation value of bosonicaction (cid:104)S B (cid:105) = T , and it is independent of parameters g and µ , thus verifying that SUSY is preserved in this model.The expectation value of the bosonic action, (cid:104)S B (cid:105) (cid:54) = T for the complex double-well inspired cubic model and it9 -3-2-1 0 1 2 0 2 4 6 8 10 12 14 16 [ W ] T t g phys =3.0, m phys =1.0, µ phys =2.0, α =0.0 Re [ W ] Im [ W ] Re [ W ] Im [ W ] Re [ W ] Im [ W ] -3-2-1 0 1 2 0 2 4 6 8 10 12 14 16 R e [ < ψ – ψ t > T , < N φ t > T ] t g phys =3.0, m phys =1.0, µ phys =2.0, α =0.0
It has been studied in the literature that it is possible to generalize the raising and lowering operator method withthe help of SUSY and shape invariance to handle many more potentials of physical interest [47, 48]. In this sec-tion, we will simulate a particular type of Shape Invariant Potential (SIP) and investigate spontaneous SUSY breaking.We consider the
Scarf-I superpotential, which has the form Ξ (cid:48) ( φ ) = A tan ( µφ ) − B sec ( µφ ) , − π ≤ µφ ≤ π , (IV.14)where A > B ≥ and µ > . We will focus on the case B = 0 . That is, Ξ (cid:48) ( φ ) = gµ tan ( µφ ) , − π ≤ µφ ≤ π . (IV.15)The parameter µ has the dimension of square root of energy and g is the dimensionless coupling.0 Ξ (cid:48) ( φ ) T a = T − α (cid:104)B α (cid:105) (cid:104)S Bα (cid:105) . .
00 0 . − i . . i . .
05 0 . − i . . . − i . . .
20 0 . − i . − . − . − i . −
12 0 . .
00 0 . − i . . i . . − . − i . . − i . . − . − i . . . − i . . .
80 0 . − i . − Ξ (cid:48) t (4)
16 0 . .
00 0 . − i . . i . .
05 0 . − i . . − i . . − . − i . . . i . . . . − . − i . . i . . . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . (cid:48) t (5)
16 0 . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . TABLE V. Expectation value of the auxiliary field B α and the bosonic action S Bα for the even- and odd-degree superpotentialsgiven in Eqs. (IV.10) and (IV.11), respectively, for physical parameters m phys = 10 and g phys = 100 . Simulations were performedfor different lattice spacings keeping the (physical) circle size β = 1 . We used adaptive Langevin step size ∆ τ ≤ × − ,thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken with a gap of steps. In Tab. VIII we provide the simulation results with physical parameters g phys = 10 and µ phys = 5 for lattices with T = 8 , and . We perform simulations for various twist parameter α to verify the consistency of our results forthe α = 0 case. Our simulations show that the auxiliary field expectation value (cid:104)B(cid:105) vanishes for Scarf-I model givenin Eq. (IV.15). We also observe that the expectation value of the bosonic action (cid:104)S B (cid:105) = T , and it is independentof g and µ . These results indicate that SUSY is preserved in this model. In Fig. 13 we provide the Langevin timehistory of the auxiliary field B α (Left) and bosonic action S Bα (Right).In Fig. 14 we plot the Ward identity for this model. In these plots, on the left column we show the complete Wardidentity W , on the middle and right columns we show the real and imaginary parts respectively, of the bosonic andfermionic contributions to the Ward identity, as mentioned in Eq. (III.46). Our simulations show that the bosonicand fermionic contributions cancel out each other within statistical errors and hence the Ward identity is satisfied,suggesting that SUSY is preserved for the model with Scarf-I potential.1
FIG. 9. Langevin time history of the auxiliary field B α (Left-Column) and the bosonic action S Bα (Right-Column) for the even-(Top-Row) and odd-(Bottom-Row) degree real polynomial potentials given in Eqs. (IV.10) and (IV.11) for physical parameters m phys = 10 and g phys = 100 on a lattice with T = 8 . The plots are based on the simulation data provided in Tab. V. T a = T − α m B m B phys = a m B m F m F phys = a m F .
125 0 . . . . . . . . . . . . . . . . . TABLE VI. Bosonic and fermionic mass gaps, both physical and dimensionless, for the odd-degree superpotential Ξ (cid:48) (5) given inEq. (IV.11) for physical parameters m phys = 10 and g = 100 . Simulations were performed for different lattice spacings keepingthe (physical) circle size β = 1 . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 andgeneration steps N gen = 10 . Measurements were taken with a gap of steps. E. PT -Symmetric Models In this section we will look at one-dimensional actions that exhibits the so-called PT -symmetry, where P and T denote the parity symmetry and time reversal invariance, respectively. The motivation for considering PT -symmetrictheories is the following. Imposing PT -symmetric boundary conditions on the functional-integral representation of thefour-dimensional − λφ theory can give a spectrum that is bounded below [49]. Such an interaction leads to a quantumfield theory that is perturbatively renormalizable and asymptotically free, with a real and bounded spectrum. These2 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0 2 4 6 8 10 12 14 16 [ W ] T t k=4, m phys = 10.0, g phys =100.0, α =0.0, T=8 Re [ W ] Re [ W ] Re [ W ] -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0 2 4 6 8 10 12 14 16 R e [ < ψ – ψ t > T , < N φ t > T ] t k=4, m phys = 10.0, g phys =100.0, α =0.0, T=8
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . Eq. (IV.12)
16 0 . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . . .
00 1 . i . . − i . .
05 1 . i . . − i . . − . − i . . − i . .
80 1 . i . . − i . . . − . − i . . − i . . − . − i . . − i . . − . − i . . − i . . − . − i . . − i . Eq. (IV.13)
16 0 . .
00 1 . i . . − i . . − . − i . . − i . . − . − i . . − i . . − . − i . . − i . TABLE VII. Expectation of the auxiliary field B α and bosonic action S Bα for double-well inspired real and complex cubicpotentials given in Eqs. (IV.12) - (IV.13) for physical parameters m phys = 1 , g phys = 3 and µ phys = 2 . Simulations wereperformed for different lattice spacings (keeping the (physical) circle size β = 1 ) and various values of α . We used adaptiveLangevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements weretaken with a gap of steps. the physical parameters g phys = 0 . for lattices with T = 4 , , and α = 0 . We noticed that the auxiliary fieldexpectation value (cid:104)B(cid:105) vanishes, thereby suggesting that SUSY is preserved for these models. We also observe thatthe expectation value of bosonic action (cid:104)S B (cid:105) = T , and it is independent of g . These results indicate that SUSY ispreserved for PT -symmetry inspired δ -potentials. In Figs. 15, 16, 17, and 18 (Top-Row) we provide the Langevintime history of the auxiliary field B α (Top-Left) which fluctuates around zero and the bosonic action S Bα (Top-Right)fluctuating around T for δ = 1 , , , , respectively on a lattice with T = 8 . In the simulations for δ = 1 , in order toavoid huge excursions in the values of the fields and observables, we introduced a rejection criterion on the complexLangevin evolution; we rejected the field configurations until the magnitude of the drift ( u ) is less than a drift cut( u cut ), that is u ≤ u cut , we chose u cut = 5 . T = 4) , . T = 8 , . In Figs. 15, 16, 17, and 18, we also plot the Ward identity (Bottom-Row) for these models. In these plots, weshow complete Ward identity W (Bottom-Left), and the real (Bottom-Middle) and imaginary (Bottom-Right) partsrespectively, of bosonic and fermionic contributions to the Ward identity, as mentioned in Eq. (III.46). Our simulationsshow that the bosonic and fermionic contributions cancel out each other within statistical errors for δ = 1 , , , andhence the Ward identities are satisfied, again suggesting that SUSY is preserved for PT -symmetry inspired δ -potentialmodels.4 FIG. 11. Langevin time history of the auxiliary field B α (Left-Column) and the bosonic action S Bα (Right-Column) for thedouble-well inspired real (Top-Row) and complex (Bottom-Row) cubic potentials given in Eqs. (IV.12) - (IV.13) for physicalparameters m phys = 1 , g phys = 3 , µ phys = 2 on a lattice with T = 8 . The plots are based on simulation data provided in Tab.VII. V. CONCLUSIONS
In this paper we have investigated the absence or presence of SUSY breaking various quantum mechanical models,with real and complex superpotentials, with the help of complex Langevin method. We arrived at the following con-clusions after performing complex Langevin simulations on these models. We find that SUSY is preserved in quantummechanics models with (i.) harmonic and anharmonic oscillator potentials, (ii.) complex double-well potential, (iii.)odd-powered polynomial potential, (iv.) double-well inspired cubic potential, (v.)
Scarf-I (shape invariant) potential,and (vi.) PT -symmetry inspired potentials. We find that SUSY is broken in (i.) double-well potential, (ii.) even-powered polynomial potential, and (ii.) double-well inspired complex cubic potential. We also checked the reliabilityof our simulations by studying the Fokker-Planck equation as the correctness criterion and the exponential fall off ofthe drift terms. They are provided in Appendix C. Our conclusion is that complex Langevin method can be reliablyused to probe non-perturbative SUSY breaking in various quantum mechanical models.It would be interesting to extend our investigations to models in higher dimensions, and especially quantum fieldtheoretic systems in four dimensions, such as QCD with finite temperature and baryon/quark chemical potentials.Work is in progress to examine dynamical SUSY breaking in two-dimensional quantum field theories with complexactions [56].5 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 16 [ W ] T t g phys =3.0, m phys =1.0, µ phys =2.0, α =0.0 Re [ W ] Im [ W ] Re [ W ] Im [ W ] Re [ W ] Im [ W ] -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 16 R e [ < ψ – ψ t > T , < N φ t > T ] t g phys =3.0, m phys =1.0, µ phys =2.0, α =0.0
We thank discussions with Takehiro Azuma, Pallab Basu, and Navdeep Singh Dhindsa. The work of AJ wassupported in part by the Start-up Research Grant (No. SRG/2019/002035) from the Science and Engineering ResearchBoard (SERB), Government of India, and in part by a Seed Grant from the Indian Institute of Science Education andResearch (IISER) Mohali. AK was partially supported by IISER Mohali and a CSIR Research Fellowship (FellowshipNo. 517019).6 Ξ (cid:48) ( φ ) T a = T − α (cid:104)B α (cid:105) (cid:104)S Bα (cid:105) . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . gµ tan ( µφ ) 12 0 . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . . .
00 0 . i . . i . .
05 0 . i . . i . .
20 0 . i . . i . .
80 0 . i . . i . TABLE VIII. Mean expectation of the auxiliary field B α and the bosonic action S Bα for the model with Scarf-I potential givenin Eq. (IV.15) for physical parameters g phys = 10 and µ phys = 5 . Simulations were performed for different lattice spacings(keeping the (physical) circle size β = 1 ) and various twist parameters. We observe that the expectation values are independentof the twist field. In the simulations we used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken with a gap of steps. -0.06-0.04-0.02 0 0.02 0.04 0.06 0 2 4 6 8 10 12 14 16 [ W ] T t g phys =10.0, µ phys =5.0, α =0.0 Re [ W ] Im [ W ] Re [ W ] Im [ W ] Re [ W ] Im [ W ] -0.06-0.04-0.02 0 0.02 0.04 0.06 0 2 4 6 8 10 12 14 16 R e [ < ψ – ψ t > T , < N φ t > T ] t g phys =10.0, µ phys =5.0, α =0.0
Scarf-I potential given in Eq. (IV.15) for physical parameters g phys = 10 . and µ phys = 5 . . Simulations were performed on a lattice with T = 8 , and , and twist parameter α = 0 . Theplots are based on the simulation data provided in Tab. VIII. Appendix A: SUSY Invariance of the Action
In this section we study the invariance of continuum and lattice-regularized actions under their respective SUSYtransformations.Let us begin by verifying the Q - and QQ -exact forms of the continuum action given in Eq. (II.6). We use the formof the action in which the auxiliary field B has been integrated out using its equation of motion, i B = ∂ τ φ + W (cid:48) ( φ ) . S = −Q (cid:90) β dτ ψ (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) = QQ (cid:90) β dτ (cid:18) ψψ + W ( φ ) (cid:19) . (A.1)7 Ξ (cid:48) ( φ ) = − ig ( iφ ) (1+ δ ) T a = T − α (cid:104)B α (cid:105) (cid:104)S Bα (cid:105) .
25 0 . − . − i . . i . δ = 1 8 0 .
125 0 . − . − i . . − i . . . − . − i . . − i . .
25 0 . . i . . − i . δ = 3 8 0 .
125 0 . . i . . − i . . . . i . . i . TABLE IX. Expectation value of the auxiliary field B α and bosonic action S Bα for the PT -symmetric models given in Eq. (IV.16)with δ = 1 , and g phys = 0 . . Simulations were performed on lattices with T = 4 , and , keeping β = 1 and α = 0 . For δ = 1 we used adaptive Langevin step size ∆ τ ≤ × − ( T = 4) , − ( T = 8) , × − ( T = 12) , thermalization steps N therm = 10 ,generation steps N gen = 10 ( T = 4 , , ( T = 8) , and measurements were taken every T = 4) , T = 8 , steps.For δ = 3 we used adaptive Langevin step size ∆ τ ≤ − ( T = 4) , − ( T = 8) , × − ( T = 12) , thermalization steps N therm = 10 , generation steps N gen = 10 ( T = 4 , , ( T = 12) and measurements were taken every steps. Ξ (cid:48) ( φ ) = − ig ( iφ ) (1+ δ ) T a = T − α (cid:104)B α (cid:105) (cid:104)S Bα (cid:105) .
25 0 . . i . . i . δ = 2 8 0 .
125 0 . . i . . i . . . . − i . . i . .
25 0 . . i . . i . δ = 4 8 0 .
125 0 . . i . . i . . . . − i . . i . TABLE X. Expectation value of the auxiliary field B α and bosonic action S Bα for the PT -symmetric models given in Eq.(IV.16) with δ = 2 , and g phys = 0 . . Simulations were performed on lattices with T = 4 , and , keeping β = 1 and α = 0 .For δ = 2 we used adaptive Langevin step size ∆ τ ≤ × − ( T = 4 , , − ( T = 12) , thermalization steps N therm = 10 ,generation steps N gen = 10 ( T = 4 , , ( T = 12) , and measurements were taken every steps. For δ = 4 we usedadaptive Langevin step size ∆ τ ≤ × − ( T = 4) , − ( T = 8 , , thermalization steps N therm = 10 , generation steps N gen = 10 ( T = 4) , ( T = 8 , and measurements were taken every steps. The Q - and QQ -exact forms can be verified as −Q (cid:90) β dτ ψ (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) = − (cid:90) β dτ (cid:20)(cid:0) Q ψ (cid:1) (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) − ψ (cid:18) Q ∂φ∂τ + Q W (cid:48) ( φ ) (cid:19)(cid:21) = − (cid:90) β dτ (cid:34) − (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) − ψ (cid:18) ∂ψ∂τ + W (cid:48)(cid:48) ( φ ) ψ (cid:19)(cid:35) = ⇒ S (A.2)8 -2-1 0 1 2 0 2 4 6 8 10 12 [ W ] T t δ =1.0, g phys =0.5, α =0.0 Re [ W ] Im [ W ] Re [ W ] Im [ W ] Re [ W ] Im [ W ] -2-1 0 1 2 0 2 4 6 8 10 12 R e [ < ψ – ψ t > T , < N φ t > T ] t δ =1.0, g phys =0.5, α =0.0
12 ¯ ψ (cid:0) Q ψ (cid:1) + Q W ( φ ) (cid:19) = Q (cid:90) β dt (cid:18) − ψ (cid:18) ∂φ∂τ − W (cid:48) ( φ ) (cid:19) − W (cid:48) ( φ ) ψ (cid:19) = −Q (cid:90) β dτ ψ (cid:18) ∂φ∂τ + W (cid:48) ( φ ) (cid:19) = ⇒ S. (A.3)Now we would like to verify Eq. (III.17), that is, we need to check whether the supercharges Q and Q acting onthe lattice regularized action vanishes or not under the SUSY transformations given in Eq. (III.12) and (III.13). Wehave the lattice regularized action S = T − (cid:88) i =0 T − (cid:88) j =0 ∇ Sij φ j + Ω (cid:48) i + ψ i T − (cid:88) j =0 (cid:0) ∇ Sij + Ω (cid:48)(cid:48) ij (cid:1) ψ j . (A.4)9 -3-2-1 0 1 2 3 0 2 4 6 8 10 12 [ W ] T t δ =2.0, g phys =0.5, α =0.0 Re [ W ] Im [ W ] Re [ W ] Im [ W ] Re [ W ] Im [ W ] -3-2-1 0 1 2 3 0 2 4 6 8 10 12 R e [ < ψ – ψ t > T , < N φ t > T ] t δ =2.0, g phys =0.5, α =0.0
N=3, Re [ Φ ]N=3, Im [ Φ ] N=4, Re [ Φ ]N=4, Im [ Φ ] -1-0.5 0 0.5 0 200 400 600 800 1000 Φ Langevin Time
N=4, Im [ Φ ]N=4, Re [ Φ ] N=3, Im [ Φ ]N=3, Re [ Φ ] -1-0.5 0 0.5 0 200 400 600 800 1000 Φ Langevin Time
N=4, Re [ Φ ]N=4, Im [ Φ ] N=3, Re [ Φ ]N=3, Im [ Φ ] -1.5-1-0.5 0 0.5 1 0 200 400 600 800 1000 Φ Langevin Time
N=4, Re [ Φ ]N=3, Re [ Φ ] N=3, Im [ Φ ]N=4, Im [ Φ ] FIG. 20. Langevin time history of the k -point equal-time correlation functions for bosonic PT -symmetric potential given inEq. (B.1) for δ = 1 , ( N = 2 + δ = 3 , ) and coupling g = 1 . . These history plots are based on the simulation data providedin Tab. XI. Appendix C: Reliability of Simulations
In this section we investigate the reliability of complex Langevin simulations carried out in this work. We make useof two of the methods proposed in the recent literature: one is based on the Fokker-Planck equation as a correctnesscriterion [34, 35, 58] and the other is based on the probability distribution of the magnitude of the drift term [59, 60].
1. Reliability using Fokker-Planck Operator
The holomorphic observables of the theory O k [ φ, τ ] at k -th lattice site evolve in the following way [34, 35, 58] ∂ O k [ φ, τ ] ∂τ = (cid:101) L k O k [ φ, τ ] , (C.1)where (cid:101) L k is the Langevin operator for the k -th site. It is defined as (cid:101) L k = (cid:18) ∂∂φ k − ∂ S eff [ φ ] ∂φ k (cid:19) ∂∂φ k . (C.2)Once the equilibrium distribution is reached, assuming that it exists, we can remove the τ dependence from the4observables. Then we have C O k ≡ (cid:104) (cid:101) L k O k [ φ ] (cid:105) = 0 , (C.3)and this can be used as a criterion for correctness of the complex Langevin method. This criterion has been investigatedin various models in Refs. [34, 35, 58]. The criterion for correctness, in principle, needs to be satisfied for a completeset of observables O [ φ ] , in a suitably chosen basis [35]. It leads to an infinite tower of identities, which as a collection,resembles to the Schwinger-Dyson equations.If we take the auxiliary field B k at the k -th site as the observable O k , we then have (cid:101) L k O k = (cid:101) L k B k = − i Ξ (cid:48)(cid:48)(cid:48) k + i Ξ (cid:48)(cid:48) k ∂ S eff ∂φ k . (C.4)The observable (cid:101) L B respects translational symmetry on the lattice and hence we have averaged the values over alllattice sites. In Tab. XII and Fig. 21, we provide the expectation values and Langevin time history of (cid:101) L B , respectivelyfor the simulations of supersymmetric anharmonic oscillator model mentioned in Sec. IV A. In Tab. XV and Fig.22, we provide the expectation values and Langevin time history of (cid:101) L B , respectively for the simulations of real andcomplex double-well models mentioned in Sec. IV B. In Tab. XIV and Fig. 23, we provide the expectation values andLangevin time history of (cid:101) L B , respectively for the simulations of real even- and odd-degree polynomial superpotentialsmentioned in Sec. IV C. In Tab. XV and Fig. 24, we provide the expectation values and Langevin time history of (cid:101) L B , respectively for the simulations of real and complex double-well inspired cubic potentials mentioned in Sec. IV C.In Tab. XVI and Fig. 25, we show the expectation values and the Langevin time history of (cid:101) L B , respectively for thesimulations of Scarf-I model mentioned in Sec. IV D. In Tab. XVII, we tabulate the expectation values of (cid:101) L B , andin Fig. 26 and 27 we show the Langevin time history for δ = 1 , , , for the simulations of PT -symmetry inspired δ -potential mentioned in Sec. IV E. Ξ (cid:48) ( φ ) T a = T − (cid:104) (cid:101) L B α (cid:105) mφ + gφ . i .
16 0.0625 . − i .
32 0.03125 . i .
64 0.015625 . i . TABLE XII. Expectation value of (cid:101) L B α for a SUSY anharmonic oscillator with parameters m phys = 10 . and g phys = 100 . .Simulations were performed for different lattice spacings with β = 1 and α = 0 . . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken with a gap of steps.
2. Decay of the Drift Terms
Another test to check the correctness of the complex Langevin dynamics, as proposed in Refs. [59, 60], is to lookat the probability distribution P ( u ) of the magnitude of the drift term u at large values of the drift.We have the magnitude of the mean drift u = (cid:118)(cid:117)(cid:117)(cid:116) T T − (cid:88) i =0 (cid:12)(cid:12)(cid:12)(cid:12) ∂ S eff ∂φ i (cid:12)(cid:12)(cid:12)(cid:12) . (C.5)In Refs. [59, 60] the authors demonstrated, in a few simple models, that the probability of the drift term shouldbe suppressed exponentially at larger magnitudes in order to guarantee the correctness of complex Langevin method.In Fig. 28, we show the probability distribution P ( u ) of the magnitude of the mean drift term u , for the simulationsof supersymmetric anharmonic (Left) and harmonic (Right) superpotential mentioned in Sec. IV A. In Fig. 29,5 FIG. 21. (cid:101) L B α against Langevin time for the SUSY anharmonic oscillator with potential given in Eq. (IV.6) for physicalparameter m phys = 10 , g phys = 100 . Simulations were performed for T = 8 and α = 0 . . Plot is based on simulation parametermentioned in Tab. XII. Ξ (cid:48) ( φ ) T a = T − (cid:104) (cid:101) L B α (cid:105) . i . Eq. (IV.7) 12 0.0833 . i .
16 0.0625 . i . − . i . Eq. (IV.8) 12 0.0833 − . − i .
16 0.0625 − . − i . TABLE XIII. Expectation value of (cid:101) L B α for the real (Left) and complex (Right) double-well potential given in Eq. (IV.7) and(IV.8), respectively, with m phys = 1 . , g phys = 3 . and µ phys = 2 . . Simulations were performed for different lattice spacingswith β = 1 and twist parameter α = 0 . . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken with a gap of steps. we show the probability distribution P ( u ) of the magnitude of the mean drift term u , for the simulations of real(Left) and complex (Right) double-well superpotential mentioned in Sec. IV B. In Fig. 30, we show the probabilitydistribution P ( u ) of the magnitude of the mean drift term u , for the simulations of even-(Left) and odd-(Right) degreereal polynomial superpotential mentioned in Sec. IV C. In Fig. 31, we show the probability distribution P ( u ) of themagnitude of the mean drift term u , for the simulations of real (Left) and complex (Right) cubic double-well potentialmentioned in Sec. IV C. In Fig. 32, we show the probability distribution P ( u ) of the magnitude of the mean driftterm u , for the simulations of Scarf-I potential mentioned in Sec. IV D. In Fig. 33 and 34 we show the probabilitydistribution P ( u ) of the magnitude of mean drift term u , for the simulations of PT -symmetry inspired δ -potentialmentioned in Sec. IV E. [1] P. de Forcrand and O. Philipsen, “The QCD phase diagram for small densities from imaginary chemical potential,” Nucl.Phys. B (2002) 290–306, arXiv:hep-lat/0205016 .[2] M. Cristoforetti, F. Di Renzo, and L. Scorzato, “New approach to the sign problem in quantum field theories: Highdensity qcd on a lefschetz thimble,”
Phys. Rev. D (Oct, 2012) 074506. FIG. 22. Langevin time history of (cid:101) L B α for the real (Left) and complex (Right) double-well potential given in Eq. (IV.7) and(IV.8), respectively, with physical parameter m phys = 1 . , g phys = 3 . , µ phys = 2 . on a lattice with T = 8 . The plots are basedon simulation parameters mentioned in Tab. XIII. Ξ (cid:48) ( φ ) T a = T − (cid:104) (cid:101) L B α (cid:105) . i . . (cid:48) t (4)
12 0.0833 . i .
16 0.0625 . i . . − i . (cid:48) t (5)
12 0.0833 . i .
16 0.0625 . − i . TABLE XIV. Expectation value of (cid:101) L B α for the even- and odd-degree superpotentials given in Eqs. (IV.10) and (IV.11),respectively, for the physical parameters m phys = 10 . and g phys = 100 . . Simulations were performed for different latticespacings with β = 1 and α = 0 . . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 andgeneration steps N gen = 10 . Measurements were taken with a gap of k = 4) , k = 5) steps.[3] J. R. Klauder, “STOCHASTIC QUANTIZATION,” Acta Phys. Austriaca Suppl. (1983) 251–281.[4] J. R. Klauder, “A Langevin Approach to Fermion and Quantum Spin Correlation Functions,” J. Phys.
A16 (1983) L317.[5] J. R. Klauder, “Coherent State Langevin Equations for Canonical Quantum Systems With Applications to the QuantizedHall Effect,”
Phys. Rev.
A29 (1984) 2036–2047.[6] G. Parisi, “ON COMPLEX PROBABILITIES,”
Phys. Lett. (1983) 393–395.[7] P. H. Damgaard and H. Huffel, “Stochastic Quantization,”
Phys. Rept. (1987) 227.[8] C. E. Berger, L. RammelmŸller, A. C. Loheac, F. Ehmann, J. Braun, and J. E. Drut, “Complex Langevin and otherapproaches to the sign problem in quantum many-body physics,” arXiv:1907.10183 [cond-mat.quant-gas] .[9]
AuroraScience
Collaboration, M. Cristoforetti, F. Di Renzo, and L. Scorzato, “New approach to the sign problem inquantum field theories: High density QCD on a Lefschetz thimble,”
Phys. Rev.
D86 (2012) 074506, arXiv:1205.3996[hep-lat] .[10] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu, and T. Sano, “Hybrid Monte Carlo on Lefschetz thimbles - Astudy of the residual sign problem,”
JHEP (2013) 147, arXiv:1309.4371 [hep-lat] .[11] F. Di Renzo and G. Eruzzi, “Thimble regularization at work: from toy models to chiral random matrix theories,” Phys.Rev.
D92 no. 8, (2015) 085030, arXiv:1507.03858 [hep-lat] .[12] Y. Tanizaki, Y. Hidaka, and T. Hayata, “Lefschetz-thimble analysis of the sign problem in one-site fermion model,”
NewJ. Phys. no. 3, (2016) 033002, arXiv:1509.07146 [hep-th] .[13] H. Fujii, S. Kamata, and Y. Kikukawa, “Monte Carlo study of Lefschetz thimble structure in one-dimensional Thirringmodel at finite density,” JHEP (2015) 125, arXiv:1509.09141 [hep-lat] . [Erratum: JHEP09,172(2016)].[14] A. Alexandru, G. Basar, and P. Bedaque, “Monte Carlo algorithm for simulating fermions on Lefschetz thimbles,” Phys. -4000-2000 0 2000 4000 0 10000 20000 30000 40000 50000 L ˜ B α Langevin Time k=4, m phys =10.0, g phys =100.0, α =0.0, T=8 Im [L˜ B α ] Re [L˜ B α ] Exact Re [L˜ B α ]Exact Im [L˜ B α ] -400-200 0 200 400 0 10000 20000 30000 40000 50000 L ˜ B α Langevin Time k=5, m phys =10.0, g phys =100.0, α =0.0, T=8 Im [L˜ B α ] Re [L˜ B α ] Exact Re [L˜ B α ]Exact Im [L˜ B α ] FIG. 23. Langevin time history of (cid:101) L B α for models with even- (Left) and odd- (Right) degree polynomial potentials given inEqs. (IV.10) and (IV.11) with physical parameter m phys = 10 . , g phys = 100 . on a lattice with T = 8 and α = 0 . . The plotsare based on simulation data provided in Tab. XIV. Ξ (cid:48) ( φ ) T a = T − (cid:104) (cid:101) L B α (cid:105) . − i . Eq. (IV.12) 12 0.0833 . − i .
16 0.0625 . − i . − . − i . Eq. (IV.13) 12 0.0833 − . i .
16 0.0625 − . − i . TABLE XV. Expectation value of (cid:101) L B α for the real (Left) and complex (Right) cubic double-well potential given in Eq. (IV.12)and (IV.13), respectively, with m phys = 1 . , g phys = 3 . , and µ phys = 2 . . Simulations were performed for various lattice sizeand twist parameter α = 0 . . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 andgeneration steps N gen = 10 . Measurements were taken with a gap of steps. Rev.
D93 no. 1, (2016) 014504, arXiv:1510.03258 [hep-lat] .[15] J. Berges and I. O. Stamatescu, “Simulating nonequilibrium quantum fields with stochastic quantization techniques,”
Phys. Rev. Lett. (2005) 202003, arXiv:hep-lat/0508030 [hep-lat] .[16] J. Berges, S. Borsanyi, D. Sexty, and I. O. Stamatescu, “Lattice simulations of real-time quantum fields,” Phys. Rev.
D75 (2007) 045007, arXiv:hep-lat/0609058 [hep-lat] .[17] J. Berges and D. Sexty, “Real-time gauge theory simulations from stochastic quantization with optimized updating,”
Nucl. Phys.
B799 (2008) 306–329, arXiv:0708.0779 [hep-lat] .[18] J. Bloch, J. Glesaaen, J. J. M. Verbaarschot, and S. Zafeiropoulos, “Complex Langevin Simulation of a Random MatrixModel at Nonzero Chemical Potential,”
JHEP (2018) 015, arXiv:1712.07514 [hep-lat] .[19] G. Aarts and I.-O. Stamatescu, “Stochastic quantization at finite chemical potential,” JHEP (2008) 018, arXiv:0807.1597 [hep-lat] .[20] C. Pehlevan and G. Guralnik, “Complex Langevin Equations and Schwinger-Dyson Equations,” Nucl. Phys. B (2009) 519–536, arXiv:0710.3756 [hep-th] .[21] G. Aarts, “Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential,”
Phys. Rev. Lett. (2009) 131601, arXiv:0810.2089 [hep-lat] .[22] G. Aarts, “Complex Langevin dynamics at finite chemical potential: Mean field analysis in the relativistic Bose gas,”
JHEP (2009) 052, arXiv:0902.4686 [hep-lat] .[23] G. Aarts and K. Splittorff, “Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finitechemical potential,” JHEP (2010) 017, arXiv:1006.0332 [hep-lat] .[24] G. Aarts and F. A. James, “Complex Langevin dynamics in the SU(3) spin model at nonzero chemical potentialrevisited,” JHEP (2012) 118, arXiv:1112.4655 [hep-lat] . FIG. 24. Langevin time history of (cid:101) L B α for the real (Left) and complex (Right) cubic double-well potential given in Eq. (IV.12)and (IV.13), respectively, with physical parameters m phys = 1 . , g phys = 3 . , and µ phys = 2 . on a lattice with T = 8 and α = 0 . . The plots are based on the simulation data provided in Tab. XV. Ξ (cid:48) ( φ ) T a = T − (cid:104) (cid:101) L B α (cid:105) . − i . . gµ tan ( µφ )
12 0.0833 . − i . .
16 0.0625 . − i . . TABLE XVI. Expectation value of (cid:101) L B α for Scarf-I potential (IV.15) with g phys = 10 . and µ phys = 5 . . Simulations wereperformed for different lattice spacings with β = 1 and twist parameter α = 0 . . We used adaptive Langevin step size ∆ τ ≤ × − , thermalization steps N therm = 10 and generation steps N gen = 10 . Measurements were taken with a gap of steps.[25] Y. Ito and J. Nishimura, “The complex Langevin analysis of spontaneous symmetry breaking induced by complexfermion determinant,” JHEP (2016) 009, arXiv:1609.04501 [hep-lat] .[26] Y. Ito and J. Nishimura, “Spontaneous symmetry breaking induced by complex fermion determinant — yet anothersuccess of the complex Langevin method,” PoS
LATTICE2016 (2016) 065, arXiv:1612.00598 [hep-lat] .[27] K. N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura, and S. K. Papadoudis, “Complex Langevin analysis of thespontaneous symmetry breaking in dimensionally reduced super Yang-Mills models,”
JHEP (2018) 151, arXiv:1712.07562 [hep-lat] .[28] K. N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura, T. Okubo, and S. Kovalkov Papadoudis, “Complex Langevinanalysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model,” JHEP (2020) 069, arXiv:2002.07410 [hep-th] .[29] P. Basu, K. Jaswin, and A. Joseph, “Complex Langevin Dynamics in Large N Unitary Matrix Models,”
Phys. Rev.
D98 no. 3, (2018) 034501, arXiv:1802.10381 [hep-th] .[30] D. J. Gross and E. Witten, “Possible Third Order Phase Transition in the Large N Lattice Gauge Theory,”
Phys. Rev.
D21 (1980) 446–453.[31] S. R. Wadia, “A Study of U(N) Lattice Gauge Theory in 2-dimensions,” arXiv:1212.2906 [hep-th] .[32] S. R. Wadia, “ N = Infinity Phase Transition in a Class of Exactly Soluble Model Lattice Gauge Theories,” Phys. Lett. (1980) 403–410.[33] A. Joseph and A. Kumar, “Complex Langevin Simulations of Zero-dimensional Supersymmetric Quantum FieldTheories,”
Phys. Rev. D (2019) 074507, arXiv:1908.04153 [hep-th] .[34] G. Aarts, E. Seiler, and I.-O. Stamatescu, “The Complex Langevin method: When can it be trusted?,”
Phys. Rev.
D81 (2010) 054508, arXiv:0912.3360 [hep-lat] .[35] G. Aarts, F. A. James, E. Seiler, and I.-O. Stamatescu, “Complex Langevin: Etiology and Diagnostics of its MainProblem,”
Eur. Phys. J.
C71 (2011) 1756, arXiv:1101.3270 [hep-lat] .[36] K. Nagata, J. Nishimura, and S. Shimasaki, “Justification of the complex Langevin method with the gauge coolingprocedure,”
PTEP no. 1, (2016) 013B01, arXiv:1508.02377 [hep-lat] . -15000-10000-5000 0 5000 10000 15000 0 100 200 300 400 500 L ˜ B α Langevin Time g phys =10.0, µ phys =5.0, α =0.0, T=8 Im [L˜ B α ] Re [L˜ B α ] Exact Re [L˜ B α ]Exact Im [L˜ B α ] FIG. 25. Langevin time history of (cid:101) L B α for the Scarf-I potential given in Eq. (IV.15) for physical parameter g phys = 10 . and µ phys = 5 . on a lattice with T = 8 and α = 0 . . The plots are based on the simulation data provided in Tab. XVI. Ξ (cid:48) ( φ ) = − ig ( iφ ) (1+ δ ) T a = T − (cid:104) (cid:101) L B α (cid:105) .
25 0 . − i . δ = 1 8 0 .
125 0 . i . . . i . . − . − i . δ = 2 8 0 . − . i . . − . − i . . − . − i . δ = 3 8 0 .
125 0 . − i . . . . − i . . .
25 0 . i . δ = 4 8 0 .
125 0 . i . . . − i . TABLE XVII. Expectation value of (cid:101) L B α for the PT -symmetric potentials given in Eq. (IV.16) with δ = 1 , , , and g phys = 0 . .Simulations were performed for different lattice spacings with β = 1 and twist parameter α = 0 . . Data is based on thesimulations parameters mentioned in Tab. IX, X.[37] J. R. Klauder and W. P. Petersen, “NUMERICAL INTEGRATION OF MULTIPLICATIVE NOISE STOCHASTICDIFFERENTIAL EQUATIONS,” SIAM J. Num. Anal. (1985) 1153–1166.[38] J. R. Klauder and W. P. Petersen, “SPECTRUM OF CERTAIN NONSELFADJOINT OPERATORS AND SOLUTIONSOF LANGEVIN EQUATIONS WITH COMPLEX DRIFT,” J. Stat. Phys. (1985) 53–72.[39] H. Gausterer and J. R. Klauder, “Complex Langevin Equations and Their Applications to Quantum Statistical andLattice Field Models,” Phys. Rev.
D33 (1986) 3678.[40] D. Baumgartner and U. Wenger, “Supersymmetric quantum mechanics on the lattice: I. Loop formulation,”
Nucl. Phys.B (2015) 223–253, arXiv:1412.5393 [hep-lat] .[41] G. Bergner, T. Kaestner, S. Uhlmann, and A. Wipf, “Low-dimensional Supersymmetric Lattice Models,”
Annals Phys. (2008) 946–988, arXiv:0705.2212 [hep-lat] .[42] S. Catterall and E. Gregory, “A Lattice path integral for supersymmetric quantum mechanics,”
Phys. Lett. B (2000)349–356, arXiv:hep-lat/0006013 .[43] T. Kuroki and F. Sugino, “Spontaneous supersymmetry breaking in large-N matrix models with slowly varyingpotential,”
Nucl. Phys. B (2010) 434–473, arXiv:0909.3952 [hep-th] . FIG. 26. Langevin time history of (cid:101) L B α for the PT -symmetric potentials given in Eq. (IV.16) with δ = 1 (Left) and δ = 3 (Right) and physical coupling g phys = 0 . on a lattice with T = 8 and α = 0 . . The plots are based on simulation parametersmentioned in Tab. IX.FIG. 27. Langevin time history of (cid:101) L B α for the PT -symmetric potentials given in Eq. (IV.16) with δ = 2 (Left) and δ = 4 (Right) and physical coupling g phys = 0 . on a lattice with T = 8 and α = 0 . . The plots are based on simulation parametersmentioned in Tab. X.[44] T. Kuroki and F. Sugino, “Spontaneous supersymmetry breaking in matrix models from the viewpoints of localizationand Nicolai mapping,” Nucl. Phys. B (2011) 409–449, arXiv:1009.6097 [hep-th] .[45] S. Catterall, D. B. Kaplan, and M. Unsal, “Exact lattice supersymmetry,”
Phys. Rept. (2009) 71–130, arXiv:0903.4881 [hep-lat] .[46] E. Witten, “Dynamical Breaking of Supersymmetry,”
Nucl. Phys.
B188 (1981) 513.[47] R. Dutt, A. Khare, and U. P. Sukhatme, “Supersymmetry, Shape Invariance and Exactly Solvable Potentials,”
Am. J.Phys. (1988) 163–168.[48] F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rept. (1995) 267–385, arXiv:hep-th/9405029 .[49] C. M. Bender, K. A. Milton, and V. M. Savage, “Solution of Schwinger-Dyson equations for PT symmetric quantum fieldtheory,”
Phys. Rev. D (2000) 085001, arXiv:hep-th/9907045 .[50] N. S. Dhindsa and A. Joseph, “Probing Non-perturbative Supersymmetry Breaking through Lattice Path Integrals,” arXiv:2011.08109 [hep-lat] .[51] D. Kadoh and N. Ukita, “General solution of the cyclic Leibniz rule,” PTEP no. 10, (2015) 103B04, arXiv:1503.06922 [hep-lat] .[52] D. Kadoh and K. Nakayama, “Lattice study of supersymmetry breaking in N = 2 supersymmetric quantum mechanics,” Nucl. Phys. B (2019) 114783, arXiv:1812.10642 [hep-lat] .[53] D. Kadoh and K. Nakayama, “Direct computational approach to lattice supersymmetric quantum mechanics,”
Nucl. -5 -4 -3 -2 -1
0 5 10 15 20 25 P [ u ] u g phys =100.0, m phys = 10.0, α =0.0 T = 8 (cid:9) T = 16 T = 32 T = 64 10 -5 -4 -3 -2 -1
0 2 4 6 8 10 P [ u ] u g phys =0.0, m phys = 10.0, α =0.0 T = 8 (cid:9) T = 16 T = 32 T = 64
FIG. 28. The probability distribution P ( u ) of the magnitude of the drift term u for the SUSY anharmonic oscillator (Left) withthe physical parameters m phys = 10 . , and g phys = 100 . and SUSY harmonic oscillator (Right) with the physical parameters m phys = 10 . , and g phys = 0 . , given in Eq.(IV.6). The plots are based on simulation data provided in Tab. I, II. -5 -4 -3 -2 -1
0 20 40 60 80 100 P [ u ] u g phys =3.0, µ phys =2.0, m phys =1.0 α =0.0 T = 8 T = 12 T = 16 10 -5 -4 -3 -2 -1
0 2 4 6 8 10 P [ u ] u g phys =3.0, µ phys =2.0, m phys =1.0 α =0.0 T = 8 T = 12 T = 16
FIG. 29. The probability distribution P ( u ) of the magnitude of the drift term u for the real (Left) and complex (Right)double-well potential given in Eq. (IV.7) and (IV.8), respectively with physical parameters m phys = 1 . , g phys = 3 . , and µ phys = 2 . . The plots are based on simulation data provided in Tab. IV. Phys. B (2018) 278–297, arXiv:1803.07960 [hep-lat] .[54] D. Kadoh, T. Kamei, and H. So, “Numerical analyses of N = 2 supersymmetric quantum mechanics with a cyclic Leibnizrule on a lattice,” PTEP no. 6, (2019) 063B03, arXiv:1904.09275 [hep-lat] .[55] C. M. Bender and K. A. Milton, “Model of supersymmetric quantum field theory with broken parity symmetry,”
Phys.Rev.
D57 (1998) 3595–3608, arXiv:hep-th/9710076 [hep-th] .[56] A. Joseph and A. Kumar, “In progress,”.[57] C. W. Bernard and V. M. Savage, “Numerical simulations of PT symmetric quantum field theories,”
Phys. Rev. D (2001) 085010, arXiv:hep-lat/0106009 .[58] G. Aarts, P. Giudice, and E. Seiler, “Localised distributions and criteria for correctness in complex Langevin dynamics,” Annals Phys. (2013) 238–260, arXiv:1306.3075 [hep-lat] .[59] K. Nagata, J. Nishimura, and S. Shimasaki, “Argument for justification of the complex Langevin method and thecondition for correct convergence,”
Phys. Rev.
D94 no. 11, (2016) 114515, arXiv:1606.07627 [hep-lat] .[60] K. Nagata, J. Nishimura, and S. Shimasaki, “Testing the criterion for correct convergence in the complex Langevinmethod,”
JHEP (2018) 004, arXiv:1802.01876 [hep-lat] . -5 -4 -3 -2 -1
0 20 40 60 80 100 P [ u ] u k=4, g phys =100.0, m phys =10.0, α =0.0 T = 8 T = 12 T = 16 10 -5 -4 -3 -2 -1
0 5 10 15 20 P [ u ] u k=5, g phys =100.0, m phys = 10.0, α =0.0 T = 8 T = 12 T = 16
FIG. 30. The probability distribution P ( u ) of the magnitude of the drift term u for the even (Left) and odd (Right) realpolynomial potential given in Eq. (IV.12) and (IV.8) respectively, with physical parameters m phys = 10 . , g phys = 100 . . Theplots are based on simulation data provided in Tab. V. -5 -4 -3 -2 -1
0 2 4 6 8 10 P [ u ] u g phys =3.0, µ phys =2.0, m phys =1.0, α =0.0 (cid:9)(cid:9)(cid:9)T = 8 (cid:9)(cid:9)(cid:9)T = 12 (cid:9)(cid:9)(cid:9)T = 16 -5 -4 -3 -2 -1
0 10 20 30 40 50 60 70 80 P [ u ] u g phys =3.0, µ phys =2.0, m phys =1.0, α =0.0 (cid:9)(cid:9)(cid:9)T = 8 (cid:9)(cid:9)(cid:9)T = 12 (cid:9)(cid:9)(cid:9)T = 16 FIG. 31. The probability distribution P ( u ) of the magnitude of the drift term u for the real (Left) and complex (Right) cubicdouble-well potential given in Eq. (IV.12) and (IV.8) respectively, with physical parameters m phys = 1 . , g phys = 3 . , and µ phys = 2 . . The plots are based on simulation data provided in Tab. VII. -5 -4 -3 -2 -1
0 20 40 60 80 100 P [ u ] u g phys =10.0, µ phys =5.0, α =0.0 T = 8 T = 12 T = 16
FIG. 32. The probability distribution P ( u ) of the magnitude of the drift term u for the Scarf-I potential given in Eq.(IV.15)for physical parameters g phys = 10 . , and µ phys = 5 . . The plots are based on simulation data provided in Tab. VIII. -7 -6 -5 -4 -3 -2 -1
0 5 10 15 20 P [ u ] u δ =1.0, g phys =0.5, α =0.0 T = 4 (cid:9) T = 8 T = 12 10 -7 -6 -5 -4 -3 -2 -1
0 50 100 150 200 P [ u ] u δ =3.0, g phys =0.5, α =0.0 T = 4 (cid:9) T = 8 T = 12
FIG. 33. The probability distribution P ( u ) of the magnitude of the drift term u for the PT -symmetric superpotentials givenin Eq. (IV.16) with δ = 1 (Left), and δ = 3 (Right) and physical coupling g phys = 0 . . The plots are based on simulation dataprovided in Tabs. IX. -8 -7 -6 -5 -4 -3 -2 -1
0 5 10 15 20 P [ u ] u δ =2.0, g phys =0.5, α =0.0 T = 4 (cid:9) T = 8 T = 12 10 -8 -7 -6 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 P [ u ] u δ =4.0, g phys =0.5, α =0.0 T = 4 (cid:9) T = 8 T = 12
FIG. 34. The probability distribution P ( u ) of the magnitude of the drift term u for the PT -symmetric superpotentials givenin Eq. (IV.16) with δ = 2 (Left), and δ = 4 (Right) and physical coupling g phys = 0 .5