Lattice Quantum Mechanics Research Articles

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.

Less naive about supersymmetric lattice quantum mechanics

We explain why naive discretization results that have appeared in [hep-lat/0006013] do not appear to yield the desired continuum limit. The fermion propagator on the lattice inevitably yields a diagram with nonvanishing UV degree D=0 contribution in lattice perturbation theory, in contrast to what occurs in the continuum. This diagram gives a finite correction to the boson 2-point function that must be subtracted off in order to obtain the perturbation series of the continuum theory, in the limit where the lattice spacing $a$ vanishes. Using a transfer matrix approach, we provide a nonperturbative proof that this counterterm suffices to yield the desired continuum limit. This analysis also allows us to improve the action to O(a). We demonstrate by Monte Carlo simulation that the spectrum of the continuum theory is well-approximated at finite but small $a$, for both weak and strong coupling. We contrast the above situation for the naive lattice action to what occurs for the supersymmetric lattice action, which preserves a discrete version of half the supersymmetry. There, cancellations between D=0 diagrams occur, obviating the need for counterterms.

On Schrödinger-Hermite Operators in Lattice Quantum Mechanics

Starting from the q-Heisenberg algebra, we derive from a few abstract principles a broad class of Schrodinger operators in lattice quantum mechanics for which one can determine explicit eigenvalues and spectral properties. This happens by algebras of creators and annihilators. Generalized inhomogeneous q-discrete Hermite polynomials occur via their recurrence relations. Within this framework we obtain the special case of an interesting result, proved by Christian Berg in a much larger ge-nerality: The orthogonality measure for q-discrete Hermite polynomials of type II is not uniquely determined on q-exponential lattices.

Lattice Quantum Mechanics: An Application to Bose–Einstein Condensation

A lattice formulation of nonrelativistic quantum mechanics is presented, based on a formal analogy with discrete kinetic theory. The method is applied to the Gross–Pitaevski equation, a specific form of self-interacting nonlinear Schrödinger equation relevant to the study of Bose–Einstein condensation.

On the Integrability of q-Oscillators Based on Invariants of Discrete Fourier Transforms

A discretization of the quantum mechanical phase space is presented in the context of q-noncommutative structures. We give two generalizations of the Heisenberg algebra in the arising lattice phase space. In contrast to ordinary quantum mechanics, there is, a priori, no systematic approach to an integrable oscillator Hamiltonian in lattice quantum mechanics. This is the central obstacle to deal with in this Letter. To do so, we show how in general the integrability of the harmonic oscillator is related to the Fourier transform between momentum and space variables. This will be done in both cases, the continuous and the discrete one. As an application, we finally obtain an integrable lattice Hamiltonian for the harmonic oscillator with generalized Hermite eigenfunctions.

Quantum Groups, Coherent States, Squeezing and Lattice Quantum Mechanics

By resorting to the Fock-Bargmann representation, we incorporate the quantum Weyl-Heisenberg algebra, q-WH, into the theory of entire analytic functions. The q-WH algebra operators are realized in terms of finite difference operators in the z plane. In order to exhibit the relevance of our study, several applications to different cases of physical interest are discussed; squeezed states and the relation between coherent states and theta functions on one side, and lattice quantum mechanics and Bloch functions on the other, are shown to find a deeper mathematical understanding in terms of q-WH. The role played by the finite difference operators and the relevance of the lattice structure in the completeness of the coherent states system suggest that the quantization of the WH algebra is an essential tool in the physics of discretized (periodic) systems.

Variational approach to lattice quantum mechanics.

An alternative approach to lattice calculations is proposed, based on the method of moments. In this approach, instead of working with a lattice of definite (if possible) large size, a sequence of lattices is considered starting from the smallest ones. It is shown, by exploiting the variational aspects of the method of moments, that, through lattices of small size, one can obtain very good approximations for a system with a large correlation length. Furthermore the continuum limit of these approximations is discussed.