Minkowski Lattice Research Articles

Lattice renormalization of quantum simulations

With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with Qiskit noiseless simulators for a $2+1$D discrete non-Abelian $D_4$ gauge theory with two spatial plaquettes.

Self-assembly of a space-tessellating structure in the binary system of hard tetrahedra and octahedra.

We report the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation, it had not previously been observed in self-assembly simulations. Both known one-component phases - the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice - are found to coexist with the binary phase. Apart from an alternative, monoclinic packing of octahedra, no additional crystalline phases were observed.

Integer-Forcing MIMO Linear Receivers Based on Lattice Reduction

A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multiple-output (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the 2 × 2 and 4× 4 MIMO channels, we compare the coded-block error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors.

Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

Understanding how polyhedra pack into extended arrangements is integral to the design and discovery of crystalline materials at all length scales. Much progress has been made in enumerating and characterizing the packing of polyhedral shapes, and the self-assembly of polyhedral nanocrystals into ordered superstructures. However, directing the self-assembly of polyhedral nanocrystals into densest packings requires precise control of particle shape, polydispersity, interactions and driving forces. Here we show with experiment and computer simulation that a range of nanoscale Ag polyhedra can self-assemble into their conjectured densest packings. When passivated with adsorbing polymer, the polyhedra behave as quasi-hard particles and assemble into millimetre-sized three-dimensional supercrystals by sedimentation. We also show, by inducing depletion attraction through excess polymer in solution, that octahedra form an exotic superstructure with complex helical motifs rather than the densest Minkowski lattice. Such large-scale Ag supercrystals may facilitate the design of scalable three-dimensional plasmonic metamaterials for sensing, nanophotonics and photocatalysis.

Interleaving schemes for multidimensional cluster errors

We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t/spl equiv/0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t/spl ne/0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice.

New formulation of the Dirac equation on a Minkowski lattice

The Dirac equation is formulated on (1+{ital D})-dimensional discrete Minkowski space-time. We find that the minimum number of components of the fermion field is 2{sup D{minus}1} in the massless case, which is smaller than 2{sup D+1} of the Kogut-Susskind fermion. There exist no extra poles in the fermion propagator. The action is not Hermitian but the quantization can be performed consistently. In the (1+3)-dimensional massless case the equation describes a single species of Dirac particle in the continuous space-time limit. In the (1+1)-dimensional massless case the equation is the same as the automaton equation by {close_quote}t Hooft and describes a chiral fermion. The time evolution operator is unitary and the norm is conserved. For interacting fermions with gauge fields the evolution operator is not unitary. But in the continuum limit the unitarity will be recovered. The consequences of loosening the unitarity condition on the time evolution operator are discussed. {copyright} {ital 1997} {ital The American Physical Society}

Commutator anomalies in finite-element quantum electrodynamics

The equations of motion for four-dimensional quantum electrodynamics have been formulated on a hypercubic Minkowski lattice. This model is used to investigate the lattice current algebra. Regularization of the current is shown to arise in a natural and nonarbitrary manner.

Two-dimensional quantum field theory

By using an explicit finite-difference scheme we get approximate expressions of the unequal-time commutators for short time intervals and some spectrum estimations for the two-dimensional quantum field theory \ensuremath{\square}\ensuremath{\varphi}+\ensuremath{\sigma}\ensuremath{\varphi}+\ensuremath{\lambda}${\ensuremath{\varphi}}^{3}$=0.

Dirac field on a Minkowski lattice.

The properties of several explicit finite-difference schemes for the free massive two-dimensional Dirac equation are considered.Received 13 May 1985DOI:https://doi.org/10.1103/PhysRevD.32.2066©1985 American Physical Society