We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo method in the limit when the latter becomes exact.A crucial issue in quantum Monte Carlo methods is the choice of the most convenient stochastic process to be used in the simulation of the dynamics of the system. This aspect is particularly evident in the case of fermion systems [1][2][3][4] due to the anticommutativity of the variables involved which for long time have not lent themselves to direct numerical evaluation. In a recent paper [5] progress has been made in this direction by providing exact probabilistic expressions for quantities involving variables belonging to Grassmann or Clifford algebras. In particular, the real time or the imaginary time evolution operator of a Fermi system or a Berezin integral can be expressed in terms of an associated stochastic dynamics of a collection of Poisson processes. This approach depends on an older general formula to represent probabilistically the solution of a system of ordinary differential equations (ODE) in terms of Poisson processes [8]. In this paper, we present a simple derivation of a similar formula to study fermionic systems, in particular, the Hubbard model. However, the fermionic nature of the Hamiltonian plays no special role and similar representations can be written down for any system described by a finite Hamiltonian matrix. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. For an optimal choice of these parameters we obtain an algorithm which coincides with the well known Green Function Monte Carlo method in the limit when the latter becomes exact [6]. In this way we provide a theoretical characterization of GFMC.Let us consider the Hubbard Hamiltonianwhere Λ ⊂ Z d is a finite d-dimensional lattice with cardinality |Λ|, {1, . . . , |Λ|} some total ordering of the lattice points, and c iσ the usual anticommuting destruction operators at site i and spin index σ. In this paper, we are interested in evaluating the matrix elements n ′ |e −Ht |n where n = (n 1↑ , n 1↓ , . . . , n |Λ|↑ , n |Λ|↓ ) are the occupation numbers taking the values 0 or 1 [7]. The total number of fermions per spin component is a conserved quantity, therefore we consider only configurations n and n ′ such thatIn the following we shall use the mod 2 addition n ⊕ n ′ = (n + n ′ ) mod 2. Let Γ = {(i, j), 1 ≤ i < j ≤ |Λ| : η ij = 0} and |Γ| its cardinality. For simplicity, we will assume that η ij = η if (i, j) ∈ Γ and γ i = γ. By introducingwhere 1 iσ = (0, . . . , 0, 1 iσ , 0, . . . , 0), andthe following representation holds...

We study a model of strongly correlated fermions in one dimension with extended N=2 supersymmetry. The model is related to the spin S = 1/2 XXZ Heisenberg chain at anisotropy ∆ = −1/2 with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.PACS numbers: 05.50.+q,11.30.Pb Supersymmetry is well motivated in high energy physics where it offers a partial solution to fine-tuning problems and improve gauge coupling unification [1]. It also appears in condensed matter models although at a less fundamental level. Examples are disordered systems [2] and models of strongly correlated electrons like extended Hubbard [3] or t-J models [4] where supersymmetry relates fermionic and bosonic composite operators.A typical consequence of unbroken supersymmetry is the prediction of the ground state energy. This is not sufficient to compute the ground state wave function, the relevant quantity for the calculation of vacuum expectation values. It is quite natural to ask whether supersymmetry and the knowledge of the ground state energy are useful to this purpose.In this Letter, we analyze the problem in a recently proposed one dimensional model of itinerant fermions [5] with two supercharges obeying with the Hamiltonian an extended N=2 supersymmetry algebra. The knowledge of the ground state on large finite lattices is important to study its continuum limit where the model is expected to describe a minimal superconformal series.We shall be interested in boundary effects and thus consider mainly free boundary conditions, see [5] for the periodic case. The model can be mapped to the integrable open XXZ Heisenberg spin 1/2 chain with anisotropy ∆ = −1/2 and a suitable real surface magnetic field. In principle, Bethe Ansatz techniques could be applied. The supersymmetry inherited from the fermionic model should allow to compute Baxter function whose zeros give the Bethe quantum numbers [6]. However, the procedure is rather involved with open boundary conditions as discussed in [7].Here, we pursue a different approach starting from the following remarks. The XXZ chain at ∆ = −1/2 is integrable for a large class of boundary conditions. In some specific cases (e.g. twisted or U q (sl(2)) symmetric ones) several remarkable conjectures have been claimed about the combinatorial properties of the ground state wave function [8,9]. They arise from the relation between the XXZ chain and Temperley-Lieb loop models [10].In this Letter, we first show that similar features are present in the fermionic model and in the related XXZ chain with surface magnetic field. We then explain how non standard number theoretical methods ...

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We show that the calculation of Berezin integrals over anticommuting variables can be reduced to the evaluation of expectations of functionals of Poisson processes via an appropriate Feynman-Kac formula. In this way the tools of ordinary analysis can be applied to Berezin integrals and, as an example, we prove a simple upper bound. Possible applications of our results are briefly mentioned.

We study at strong coupling the scaling function describing the large spin anomalous dimension of twist two operators in AE super Yang-Mills theory. In the spirit of AdS/CFT duality, it is possible to extract it from the string Bethe Ansatz equations in the sl´¾µ sector of the Ë ¢ Ë superstring. To this aim, we present a detailed analysis of the Bethe equations by numerical and analytical methods. We recover several short string semiclassical results as a check. In the more difficult case of the long string limit providing the scaling function, we analyze the strong coupling version of the EdenStaudacher equation, including the Arutyunov-Frolov-Staudacher phase. We prove that it admits a unique solution, at least in perturbation theory, leading to the correct prediction consistent with semiclassical string calculations.KEYWORDS: integrable quantum field theory, integrable spin chains (vertex models), quantum integrability (Bethe ansatz).