The homology of defective crystal lattices and the continuum limit

The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0‐skeleton of a simplicial complex to its higher‐dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid‐state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.     

Interval order and semiorder lattices

When a set of closed intervals of the reals is partially ordered by decreeing that A<B when A lies strictly to the left of B, the resulting structure is called an interval order. Semiorders may be viewed as interval orders that arise from closed intervals having a fixed length. The paper initiates a careful study of interval orders and semiorders that happen also to be lattices. A structure theory is obtained for a class of interval order lattices that includes all such lattices of finite length. Characterizations are given of when these lattices are modular or distributive, as well as when they are semiorders. The theory is of some interest because the completion by cuts of an interval order is necessarily an interval order lattice. Though it is shown that the completion by cuts of a semiorder need not be a semiorder, necessary and sufficient conditions are given for a lattice of finite length to be isomorphic to the completion by cuts of a semiorder.The author wishes to dedicate this paper to the memory of his late colleague Professor Charles H. Randall.     

Local structure theory: Calculation on hexagonal arrays,and interaction of rule and lattice

Local structure theory calculations⁷ are applied to the study of cellular automata on the two-dimensional hexagonal lattice. A particular hexagonal lattice rule denoted (3422) is considered in detail. This rule has many features in common with Conway'sLife. The local structure theory captures many of the statistical properties of this rule; this supports hypotheses raised by a study ofLife itself⁽⁶⁾. As inLife, the state of a cell under (3422) depends only on the state of the cell itself and the sum of states in its neighborhood at the previous time step. This property implies that evolution rules which operate in the same way can be studied on different lattices. The differences between the behavior of these rules on different lattices are dramatic. The mean field theory cannot reflect these differences. However, a generalization of the mean field theory, the local structure theory, does account for the rule-lattice interaction.     

Axioms for quantum theory

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrödinger equations, for phase space amplitudes.     

Molecular mean field theory for liquid water

Attractive bonding interactions between molecules typically have inherent conservation laws which influence the statistical properties of such systems in terms of corresponding sum rules. We have considered lattice water as an example, and we have enunciated the consequences of the sum rule through a general computational procedure called molecular mean field theory. Fluctuations about the mean field are computed and many of the liquid properties have been deduced and compared with Monte Carlo simulation, molecular dynamics, and experimental results. Large correlation lengths are seen to be a consequence of the sum rule in the liquid phase. Long-range Coulomb interactions are shown to have minor effects on our results.     

Factorization symmetry in the lattice Boltzmann method

A non-perturbative algebraic theory of the lattice Boltzmann method is developed based on the symmetry of a product. It involves three steps: (i) Derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher-order Gaussian moments, (ii) A special quasi-equilibrium distribution function found analytically in closed form on the product-lattice in two and three spatial dimensions, and which proves the factorization of quasi-equilibrium moments, and (iii) An algebraic method of pruning based on a one-into-one relation between groups of discrete velocities and moments. Two routes of constructing lattice Boltzmann equilibria are distinguished. The present theory includes previously known limiting and special cases of lattices, and enables automated derivation of lattice Boltzmann models from two-dimensional tables, by finding the roots of one polynomial and solving a few linear systems.     

The Symmetrical Foundation of Measure,Probability, and Quantum Theories

Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalizes addition, and probability theory formalizes inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalized in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.     

Periodic overlayers and moiré patterns: theoretical studies of geometric properties

Single crystal surfaces with periodic overlayers, such as graphene on hexagonal metal substrates, are found to exhibit, apart from their intrinsic periodicity, additional long-range order expressed by approximate surface lattices with large lattice constants. This phenomenon can be described as geometrically analogous to lateral interference effects resulting in periodic moiré patterns which are characterized by two-dimensional moiré lattices. Here we discuss in detail the mathematical formalism determining such moiré patterns based on concepts of two-dimensional Fourier transformation including coincidence lattices. The formalism provides simple relations that allow one to calculate possible moiré lattice vectors in their dependence on rotation angles α and scaling factors p1,p2 for periodic (p1 × p2)Rα overlayers on substrate surfaces described by general Bravais lattices. Specific emphasis will be given to hexagonal lattices where experimental data are available.     

Quantum phase transition of cold atoms trapped in optical lattices

We review our recent theoretical advances in phase transition of cold atoms in optical lattices, such as triangular lattice, honeycomb lattice, and Kagomé lattice. By employing the new developed numerical methods called dynamical cluster approximation and cellular dynamical mean-field theory, the properties in different phases of cold atoms in optical lattices are studied, such as density of states, Fermi surface and double occupancy. On triangular lattice, a reentrant behavior of phase translation line between Fermi liquid state and pseudogap state is found due to the Kondo effect. We find the system undergoes a second order Mott transition from a metallic state into a Mott insulator state on honeycomb lattice and triangular Kagomé lattice. The stability of quantum spin Hall phase towards interaction on honeycomb lattice with spin-orbital coupling is systematically discussed. And we investigate the transition from quantum spin Hall insulator to normal insulator in Kagomé lattice which includes a nearest-neighbor intrinsic spin-orbit coupling and a trimerized Hamiltonian. In addition, we propose the experimental protocols to observe these phase transition of cold atoms in optical lattices.     

Nonequilibrium Green Functions Approach to Strongly Correlated Fermions in Lattice Systems

Quantum dynamics in strongly correlated systems are of high current interest in many fields including dense plasmas, nuclear matter and condensed matter and ultracold atoms. An important model case are fermions in lattice systems that is well suited to analyze, in detail, a variety of electronic and magnetic properties of strongly correlated solids. Such systems have recently been reproduced with fermionic atoms in optical lattices which allow for a very accurate experimental analysis of the dynamics and of transport processes such as diffusion. The theoretical analysis of such systems far from equilibrium is very challenging since quantum and spin effects as well as correlations have to be treated non‐perturbatively. The only accurate method that has been successful so far are density matrix renormalization group (DMRG) simulations. However, these simulations are presently limited to one‐dimensional (1D) systems and short times. Extension of quantum dynamics simulations to two and three dimensions is commonly viewed as one of the major challenges in this field. Recently we have reported a breakthrough in this area [N. Schlünzen et al., Phys. Rev. B (2016)] where we were able to simulate the expansion dynamics of strongly correlated fermions in a Hubbard lattice following a quench of the confinement potential in 1D, 2D and 3D. The results not only exhibited excellent agreement with the experimental data but, in addition, revealed new features of the short‐time dynamics where correlations and entanglement are being build up. The method used in this work are nonequilibrium Green functions (NEGF) which are found to be very powerful in the treatment of fermionic lattice systems filling the gap presently left open by DMRG in 2D and 3D. In this paper we present a detailed introduction in the NEGF approach and its application to inhomogeneous Hubbard clusters. In detail we discuss the proper strong coupling approximation which is given by T ‐matrix selfenergies that sum up two‐particle scattering processes to infinite order. The efficient numerical implemen‐tation of the method is discussed in detail as it has allowed us to achieve dramatic performance gains. This has been the basis for the treatment of more than 100 particles over large time intervals. The numerical results presented in this paper concentrate on the diffusion in 1D to 3D lattices. We find that the expansion dynamics consist of three different phases that are linked with the build‐up of correlations. In the long time limit, a universal scaling with the particle number is revealed. By extrapolating the expansion velocities to the macroscopic limit, the obtained results show excellent agreement with recent experiments on ultracold fermions in optical lattices. Moreover we present results for the site‐resolved behavior of correlations and entanglement that can be directly compared with experiments using the recently developed atomic microscope technique. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A discussion on the origin of quantum probabilities

We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).     

Effects of selective dilution on phase diagram and ground-state magnetizations of an Ising antiferromagnet on triangular and honeycomb lattices

We employ an effective-field theory with correlations in order to study the phase diagram and ground-state magnetizations of a selectively diluted Ising antiferromagnet on triangular and honeycomb lattices. Dilution of different sublattices with generally unequal probabilities results in a rather intricate phase diagram in the sublattice dilution parameters space. In the case of the frustrated triangular lattice antiferromagnet the selective dilution affects the degree of frustration which can lead to some peculiar phenomena, such as reentrant behavior of long-range order or unsaturated sublattice magnetizations at zero temperature. The selectively diluted Ising antiferromagnet on the honeycomb lattice is obtained as a special case when one sublattice of the triangular lattice is completely removed by dilution.     

Thermodynamic Transitions of Antiferromagnetic Ising Model on the Fractional Multi-branched Husimi Recursive Lattice

The multi-branched Husimi recursive lattice is extended to a virtual structure with fractional numbers of branches joined on one site. Although the lattice is undrawable in real space, the concept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interaction on such a set of lattices are calculated to check the critical temperatures (T_c) and ideal glass transition temperatures (T_k) variation with fractional branch numbers. Besides the similar results of two solutions representing the stable state (crystal) and metastable state (supercooled liquid) and indicating the phase transition temperatures, the phase transitions show a well-defined shift with branch number variation. Therefore the fractional branch number as a parameter can be used as an adjusting tool in constructing a recursive lattice model to describe real systems.     

Mesoscopic disorder in double-well optical lattices

Double-well optical lattices are considered, each cite of which is formed by a double-well potential. The lattice is assumed to be in an insulating state and order and disorder are defined with respect to the displacement of atoms inside the double-well potential. It is shown that in such lattices, in addition to purely ordered and disordered states, there, can exist an intermediate mixed state, where, inside a generally ordered lattice, there appear disordered regions of mesoscopic size.     

Scaling behaviour of Creutz ratios inSU(2) lattice gauge theory

An analysis of the scaling behaviour of Creutz ratios on large lattices is given forSU(2) gauge theory. The β-interval is 2.5≦β≦2.8. Under a factor 2 scaling test, after multiplicative corrections for lattice artifacts, the Monte Carlo data show deviations from scaling, which are similar for all values of β. The ratios can be fitted successfully by a sum of three perturbative terms and an exponentially decreasing nonperturbative term. For many ratios the latter turns out to be very small, and its size dependence at fixed β is consistent with that of an area term in the Wilson loops. The deviation of the corresponding exponents from the ones expected for an area term gives a coherent cxplanation of the observed departures from scaling. It is well possible that for fixed spatial extension (in lattice units) nonperturbative contributions vanish so fast that they cannot be interpreted as physical effects.     

Height Representation,Critical Exponents,and Ergodicity in the Four-State Triangular Potts Antiferromagnet

We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a (2+2)-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagomé lattice and to bond-coloring models on the triangular and honeycomb lattices. We also calculate critical exponents for the ground-state ensemble of the model. We find that the exponents governing the spin–spin correlation function and spin fluctuations violate the Fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice. We confirm our predictions by extensive Monte Carlo simulations of the model using the Wang–Swendsen–Kotecký cluster algorithm. Although this algorithm is not ergodic on lattices with toroidal boundary conditions, we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order, such as the projective plane. To guard against biases introduced by lack of ergodicity, we perform our simulations on both the torus and the projective plane.     

Sum Rule in a Consistent Relativistic Random-Phase Approximation

A fully consistent relativistic random-phase approximation (RRPA) is studied in the sense that the relativistic mean-field (RMF) wavefunction of nucleus and the particlehole residual interactions in the RRPA are calculated from the same effective Lagrangian. A consistent treatment of RRPA within the RMF approximation, i.e., no sea approximation, has to include also the pairs formed from the Dirac states and occupied Fermi states, which is essential for satisfying the current conservation. The inverse energy-weighted sum rule for the isoscalar giant monopole mode is investigated in the constrained RMF. It is found that the sum rule is fulfilled only in the case where the Dirac state contributions are included.     

Electronic Properties of Strained Double‐Weyl Systems

The effects of strains on the low‐energy electronic properties of double‐Weyl phases are studied in solids and cold‐atom optical lattices. The principal finding is that deformations do not couple, in general, to the low‐energy effective Hamiltonian as a pseudoelectromagnetic gauge potential. The response of an optical lattice to strains is simpler, but still only one of the several strain‐induced terms in the corresponding low‐energy Hamiltonian can be interpreted as a gauge potential. Most interestingly, the strains can induce a nematic order parameter that splits a double‐Weyl node into a pair of Weyl nodes with the unit topological charges. The effects of deformations on the motion of wavepackets in the double‐Weyl optical lattice model are studied. It is found that, even in the undeformed lattices, the wavepackets with opposite topological charges can be spatially split. Strains, however, modify their velocities in a very different way and lead to a spin polarization of the wavepackets.     

Soliton topology versus discrete symmetry in optical lattices

We address the existence of vortex solitons supported by azimuthally modulated lattices and reveal how the global lattice discrete symmetry has fundamental implications on the possible topological charges of solitons. We set a general "charge rule" using group-theory techniques, which holds for all lattices belonging to a given symmetry group. Focusing on the case of Bessel lattices allows us to derive also an overall stability rule for the allowed vortex solitons.     

Anomalous Minimum and Scaling Behavior of Localization Length Near an Isolated Flat Band

Using one‐dimensional tight‐binding lattices and an analytical expression based on the Green's matrix, we show that anomalous minimum of the localization length near an isolated flat band, previously found for evanescent waves in a defect‐free photonic crystal waveguide, is a generic feature and exists in the Anderson regime as well, i.e., in the presence of disorder. Our finding reveals a scaling behavior of the localization length in terms of the disorder strength, as well as a summation rule of the inverse localization length in terms of the density of states in different bands. Most interestingly, the latter indicates the possibility of having two localization minima inside a band gap, if this band gap is formed by two flat bands such as in a double‐sided Lieb lattice.