Odd Lattices Research Articles

Topological edge states in Su-Schrieffer-Heeger chain via non-Hermitian coupling

We study the effect of non-Hermition coupling on topological and Hermitian characteristics in a non-Hermitian Su-Schrieffer-Heeger (SSH) chain. Different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases are presented. The odevity of the number of lattices is also considered. We find that in the even number lattices case the system undergoes an abrupt transition from the Hermitian states to non-Hermitian states at a certain critical point when increasing the strength of the imaginary potential in non-Hermitian coupling. But in the odd number lattices case the “critical points” are actually pseudo critical points, which is proved by the amplitudes distributions calculated by coupled mode theory (CMT).

Chern-Simons invariants from ensemble averages

We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.

Quantum walk on a toral phase space

A quantum walk on a toral phase space involving translations in position and its conjugate momentum is studied in the simple context of a coined walker in discrete time. The resultant walk, with a family of coins parametrized by an angle is such that its spectrum is exactly solvable with eigenangles for odd parity lattices being equally spaced, a feature that is remarkably independent of the coin. The eigenvectors are naturally specified in terms using the q-Pochhammer symbol, but can also be written in terms of elementary functions, and their entanglement can be analytically found. While the phase-space walker shares many features in common with the well-studied case of a coined walker in discrete time and space, such as ballistic growth of the walker position, it also presents novel features such as exact periodicity, and formation of cat-states in phase space. The participation ratio (PR), a measure of delocalization in walker space, is studied in the context of both kinds of quantum walks; while the classical PR increases as there is a time interval during which the quantum walks display a power-law growth ∼t0.825. Studying the evolution of coherent states in phase space under the walk enables us to identify an Ehrenfest time after which the coin-walker entanglement saturates.

2- and 3-Modular lattice wiretap codes in small dimensions

A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are considered, too, and three lattices are found to outperform the best unimodular lattices.

On the Parity Problem in One-Dimensional Cellular Automata

We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem.

Packing dimers on(2p+1)×(2q+1)lattices

We use computational method to investigate the number of ways to pack dimers on m x n lattices, where both m and n are odd. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on (2k+1) x (2k+1) odd square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on 2k x 2k even square lattices, for which an exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd-by-odd lattice strips with any width n > or =1. This logarithmic term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer-dimer problem might be solvable.

First-passage method for the study of the efficiency of a two-channel reaction on a lattice

We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are fully confirmed, including the existence of a parity effect. For even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events, while for odd lattices the reaction time is minimized by a purely synchronous process. We provide an intuitive explanation for this behavior. In addition, we give explicit expressions for the variance of the reaction time. The latter displays a similar even-odd behavior, suggesting that the parity effect extends to higher order moments.

Jacobi identities, modular lattices, and modular towers

We give first a simple proof of a generalized Jacobi identity for n-dimensional odd diagonal lattices which specializes to the classical Jacobi identity for the lattice Z 2. For Z + ℓ Z , it recovers a one-parameter family of Jacobi identities discovered recently by Chan, Chua and Solé, used to deduce two quadratically converging algorithms for computing π corresponding to elliptic functions for the cubic and septic bases. Next, motivated by strongly modular lattices for the ten special levels ℓ, where σ 1(ℓ)∣24, we derive quadratic iterations in these ten special levels generalizing the cubic and septic cases. This also gives a uniform proof of the equations used by N.D. Elkies for 13 of his explicit modular towers. They correspond exactly to the case where all eta terms occur to the same power in his list. This provides a link between strongly modular lattices and modular towers.

The Shadow Theory of Modular and Unimodular Lattices

It is shown that ann-dimensional unimodular lattice has minimal norm at most 2[n/24]+2, unlessn=23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to stronglyN-modular even lattices forNin{1, 2, 3, 5, 6, 7, 11, 14, 15, 23}, (*)and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial forN=1 and 2). ForN>1 in (*), lattices meeting the new bound are constructed that are analogous to the “shorter” and “odd” Leech lattices. These include an odd associate of the 16-dimensional Barnes–Wall lattice and shorter and odd associates of the Coxeter–Todd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu groupM23.

A Note on Optimal Unimodular Lattices

The highest possible minimal norm of a unimodular lattice is determined in dimensionsn⩽33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8.1020in dimension 33). Unimodular lattices with no roots exist if and only ifn⩾23,n≠25.

SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD

We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

Hecke eigenforms and representation numbers of arbitrary rank lattices

Introduction. Given a totally positive quadratic form Q over a totally real number field K, one can obtain a Hilbert modular form by restricting Q to a lattice L and forming the theta series attached to L; the Fourier coefficients of the theta series are the representation numbers of Q on L. The space of Hilbert modular forms generated by all theta series attached to lattices of the same weight, level and character is invariant under a subalgebra of the Hecke algebra, hence one can (in theory) diagonalize this space of modular forms with respect to an appropriate Hecke subalgebra and infer relations on the representation numbers of the lattices. In a previous paper the author found such relations by constructing eigenforms from theta series attached to lattices of even rank which are “nice” at dyadic primes; the purpose of this paper is to extend the previous results to all lattices. We begin by proving a lemma (Lemma 1.1) which allows us to remove the restriction regarding dyadic primes. Then using our previous work we find that associated to any even rank lattice L is a family of lattices famL which is partitioned into nuclear families (which are genera when the ground field is Q), and the averaged representation numbers of these nuclear families satisfy arithmetic relations (Theorem 1.2). In §2 we define “Fourier coefficients” attached to integral ideals for a half-integral weight Hilbert modular form. Then in analogy to the case K = Q, we describe the effect of the Hecke operators on these Fourier coefficients (Theorem 2.5). In §3 we use theta series attached to odd rank lattices to construct eigenforms for the Hecke operators; the results of §2 then give us arithmetic relations on the representation numbers of the odd rank lattices. When the ground field is Q, we may assume Q(L) ⊆ Z and then these relations may be stated as

Covariant heterotic strings and odd self-dual lattices

We investigate the implications of modular invariance for covariantly formulated heterotic strings. It is shown that modular invariant heterotic strings are characterized by odd self-dual lorentzian lattices which include charges of the bosonized superconformal ghosts. The proof of modular invariance involves the anomaly in the ghost number current in a crucial way.

Odd unimodular lattices

One investigates the relation between even and odd unimodular lattices in ℝ n . One constructs the operation of taking even associates for odd lattices. One computes the action of this operation upon the theta-series. One investigates the even associates of lattices of the form Λ⊕ℤ s .

Random walks with sign reversals and the intrinsic angular momentum behaviour on lattices

Classical uniaxial rotators on lattices are considered. The assumption is made that the rotators exchange angular momentum only in a certain sector of relative orientations (collision sector) because of the finite range of the anisotropic interaction potential; the latter may in particular be a hard core potential (instantaneous collisions). Since the axes of the rotators are supposed to be parallel but not collinear, during each collision the exchange of angular momentum occurs with sign reversal, i.e. is of the cog-wheel type. In a first approximation all correlations between successive collisions are neglected. This amounts to considering a random walk on a lattice in which the walker carries with him a variable whose sign is changed at each step. Then, one is led to distinguish two types of lattices, depending upon the possibility for returning to the initial site after an odd number of steps. For two-dimensional lattices, the asymptotic solution of the autocorrelation function of the angular momentum is analytically given, and its behaviour at shorter times is obtained by a numerical method. For the odd lattices, the asymptotic behaviour is generally exponential, whereas there exist algebraic long tails for the even lattices. Finally, coming back to the problem where the successive collisions are not independent, some simulation results for two-dimensional lattices of soft core diatomics are described.