Three-dimensional Lattice Models Research Articles

Flat bands in three-dimensional lattice models with non-trivial Hopf index

We report the presence of exactly and nearly flat bands with non-trivial topology in three-dimensional (3D) lattice models. We first show that an exactly flat band can be realized in a 3D lattice model characterized by a 3D topological invariant, namely Hopf invariant. In contrast, we find another distinct 3D model, exhibiting both 2D Chern and 3D Hopf invariant, namely Hopf-Chern insulator, that can host nearly or perfect flat bands across different 2D planes. Such a Hopf-Chern model can be constructed by introducing specific hopping along the orthogonal direction of a simple two-orbital 2D Chern insulator in the presence of in-plane nearest-neighbor and next-nearest hopping among different orbitals. While the Chern planes host nearly perfect flat bands, the orthogonal planes can host both perfect or nearly perfect flat bands with zero Chern number at some special parameter values. Interestingly, such a 3D lattice construction from 2D allows finite Hopf invariant too. Finally, we show that higher Chern models can also be constructed in the same lattice setup with only nearest and next-nearest hopping, but the appearance of flat bands along high-symmetric path in the Brillouin zone requires longer-range hopping. We close with a discussion on possible experimental platforms to realize the models.

Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models

The Newman-Moore model on the 2D triangular lattice is a classic example of a statistical mechanical model with immobile fractal excitations: fractons. These result in kinetic constraints leading to glassy behavior. Here, the authors generalize this construction to two 3D lattices, trillium and hyperhyperkagome. While these models exhibit a similar phenomenology, the authors show that the underlying fractons have a richer structure and are intimately connected to the matrix cellular automata that describe the manifold of ground states.

Scalar gauge-Higgs models with discrete Abelian symmetry groups.

We investigate the phase diagram and the nature of the phase transitions of three-dimensional lattice gauge-Higgs models obtained by gauging the Z_{N} subgroup of the global Z_{q} invariance group of the Z_{q} clock model (N is a submultiple of q). The phase diagram is generally characterized by the presence of three different phases, separated by three distinct transition lines. We investigate the critical behavior along the two transition lines characterized by the ordering of the scalar field. Along the transition line separating the disordered-confined phase from the ordered-deconfined phase, standard arguments within the Landau-Ginzburg-Wilson framework predict that the behavior is the same as in a generic ferromagnetic model with Z_{p} global symmetry, p being the ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2) universality class for p=2 and p≥4, respectively, while for p=3 only first-order transitions are possible. The results of Monte Carlo simulations confirm these predictions. There is also a second transition line, which separates two phases in which gauge fields are essentially ordered. Along this line we observe the same critical behavior as in the Z_{q} clock model, as it occurs in the absence of gauge fields.

Variational Corner Transfer Matrix Renormalization Group Method for Classical Statistical Models

In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is further verified by investigating several typical phase transitions in classical spin models, where the obtained critical points and critical exponents all agree with the best known results in literature. Its extension to three-dimensional tensor networks or quantum lattice models is straightforward, as also discussed briefly.

Lattice Boltzmann Method for Simulations of Nozzle Flows in Coastal Environments

Sun, P.; Zhou, J.; Liu, Y., and Wang, Y., 2022. Lattice Boltzmann method for simulations of nozzle flows in coastal environments. Journal of Coastal Research, 38(1), 204–217. Coconut Creek (Florida), ISSN 0749-0208. A better understanding of flow in pipes is considered to be critical for marine equipment because it helps to improve pipe performance related to vibration and structure problems. In this study, flow simulations were conducted using a large–eddy-simulation lattice Boltzmann method (LES-LBM) in a nozzle under various Reynolds numbers (Re) and the effect of the three-dimensional cubic lattice models D3Q15, D3Q19, and D3Q27 was examined. The results show that when Re < 200 (i.e. the asymmetry of the results calculated using lattice models D3Q15 and D3Q19 is below 5%) or when the lattice model D3Q27 is adopted, computational results can satisfy the requirement of rotational invariance. When 250 < Re < 300 or the flow suddenly traverses an abrupt expansion, the isotropy of lattice models D3Q15 and D3Q19 experience “collapse”. Defective planes cannot be used to completely solve complex flows and accurately transfer flow information. The resultant problems in the computational results include preference in the circumferential distribution of axial velocity and contraction effects of the phase portrait of axial and radial velocities. More complex flows (e.g., for situations with large Re or fluid traversing abrupt expansion) universally cause more significant lattice model effects, where nonlinear flow phenomena become dominant. Therefore, it is necessary to ensure that solutions in the entire computational domain are independent of lattice orientation during flow simulation using LBM. Overall, the results in this paper have important potential applications in predicting flow characteristics for pipe flows in marine structures such as marine risers in coastal environments and zones farther out to sea.

Phase Diagram, Symmetry Breaking, and Critical Behavior of Three-Dimensional Lattice Multiflavor Scalar Chromodynamics.

We study the nature of the phase diagram of three-dimensional lattice models in the presence of non-Abelian gauge symmetries. In particular, we consider a paradigmatic model for the Higgs mechanism, lattice scalar chromodynamics with N_{f} flavors, characterized by a non-Abelian SU(N_{c}) gauge symmetry. For N_{f}≥2 (multiflavor case), it presents two phases separated by a transition line where a gauge-invariant order parameter condenses, being associated with the breaking of the residual global symmetry after gauging. The nature of the phase transition line is discussed within two field-theoretical approaches, the continuum scalar chromodynamics, and the Landau-Ginzburg-Wilson (LGW) Φ^{4} approach based on a gauge-invariant order parameter. Their predictions are compared with simulation results for N_{f}=2, 3 and N_{c}=2-4. The LGW approach turns out to provide the correct picture of the critical behavior at the transitions between the two phases.

Topological metal bands with double-triple-point fermions in optical lattices

Novel fermionic quasiparticles with integer pseudospins in some energy bands, such as pseudospin-1 triple-point fermions, recently attract increasing interest since they are beyond the conventional spin-$1/2$ Dirac and Weyl counterparts. In this paper, we propose a class of pseudospin-1 fermioic excitations emerging in topological metal bands, dubbed double-triple-point (DTP) fermions. We first present a general three-band continuum model with $C_4$ symmetry in three dimensions, which has three types of threefold degenerate points in the bands classified by their topological charges $C=\pm4,\pm2,0$, respectively. They are dubbed DTPs as spin-1 generalization of double-Weyl points. We then construct two-dimensional and three-dimensional tight-binding lattice models of topological metal bands with exotic DTP fermions near the DTPs. In two dimensions, the band gaps close at a trivial DTP with zero Berry phase, which occurs at the transition between the normal and topological insulator phases. In three dimensions, the topological properties of three different DTP fermions in lattice systems are further investigated, and the effects of breaking $C_4$ symmetry are also studied, which generally leads to splitting each quadratic DTP into two linear triple points and gives topological phase diagrams. Using ultracold fermionic atoms in optical lattices, the proposed models can be realized and the topological properties of the DTP fermions can be detected.

Simulation of concrete tensile failure under high loading rates using three-dimensional irregular lattice models

The material properties and failure modes of concrete depend on loading rate. For realistic simulation of concrete behavior under dynamic loading, it is essential that this rate dependency be considered. This paper proposes a three-dimensional lattice model—formed from rigid-body-spring elements—that incorporates the rate dependency of concrete strength, elastic modulus, and fracture energy. This is achieved by introducing a visco-elasto-plastic damage unit (formed from a combination of dashpots and Coulomb friction devices) into the rigid-body-spring elements. The results of simulations of elastic wave propagation and comparisons with experimental results reported in the literature validate the model for basic load patterns. Further, changes in the dynamic failure mode with differing loading rates, studied through simulation of the Split Hopkinson Pressure Bar test, are shown to be in agreement with experimental results obtained with respect to strength and fracture locations.

Loop braiding statistics in exactly soluble three-dimensional lattice models

We construct two exactly soluble lattice spin models that demonstrate the importance of three-loop braiding statistics for the classification of 3D gapped quantum phases. The two models are superficially similar: both are gapped and both support particle-like and loop-like excitations similar to that of charges and vortex lines in a $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory. Furthermore, in both models the particle excitations are bosons, and in both models the particle and loop excitations have the same mutual braiding statistics. The difference between the two models is only apparent when one considers the recently proposed three-loop braiding process in which one loop is braided around another while both are linked to a third loop. We find that the statistical phase associated with this process is different in the two models, thus proving that they belong to two distinct phases. An important feature of this work is that we derive our results using a concrete approach: we construct string and membrane operators that create and move the particle and loop excitations and then we extract the braiding statistics from the commutation algebra of these operators.

Emergent Spacetime Supersymmetry in 3D Weyl Semimetals and 2D Dirac Semimetals.

Supersymmetry (SUSY) interchanges bosons and fermions but no direct evidence of it has been revealed in nature yet. In this Letter, we observe that fluctuating pair density waves (PDW) consist of two complex order parameters which can be superpartners of the unavoidably doubled Weyl fermions in three-dimensional lattice models. We construct explicit fermionic lattice models featuring 3D Weyl fermions and show that PDW is the leading instability via a continuous phase transition as short-range interactions exceed a critical value. Using a renormalization group, we theoretically show that N=2 space-time SUSY emerges at the continuous PDW transitions in 3D Weyl semimetals, which we believe is the first realization of emergent (3+1)D space-time SUSY in microscopic lattice models. We further discuss possible routes to realize such lattice models and experimental signatures of emergent SUSY at the PDW criticality.

Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity

Models of three-dimensional lattices with long-range interactions of Grunwald-Letnikov type for fractional gradient elasticity of non-local con- tinuum are suggested. The lattice long-range interac- tions are described by fractional-order difference operators. Continuous limit of suggested three-dimen- sional lattice equations gives continuum differential equations with the Grunwald-Letnikov derivatives of non-integer orders. The proposed lattice models give a new microstructural basis for elasticity of materials with power-law type of non-locality. Moreover these lattice models allow us to have a unified microscopic description for fractional and usual (non-fractional) gradient elasticity continuum.

Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

Entanglement spectrum of topological Weyl semimetal

The concept of entanglement spectrum, which was previously associated with gapped topological phases, is extended to the gapless Weyl semimetal phase. By dimensional reduction, we numerically investigate entanglement spectra of Weyl semimetal phases in two three-dimensional (3D) lattice models consisting of coupled layers of 2D quantum anomalous Hall insulators. It is found that for the gapless phase, there still exists a correspondence between the entanglement surface state and the physical surface state; and the “trace index”, defined as the discontinuity in the trace of the single-particle entanglement spectrum, remains equivalent to the Chern number of the 2D insulator layers.

Truncation errors and the rotational invariance of three-dimensional lattice models in the lattice Boltzmann method

The application of the lattice Boltzmann method (LBM) in three-dimensional isothermal hydrodynamic problems often adopts one of the following models: D3Q15, D3Q19, or D3Q27. Although all of them retrieve consistent Navier–Stokes dynamics in the continuum limit, they are expected to behave differently at discrete level. The present work addresses this issue by performing a LBM truncation error analysis. As a conclusion, it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non-linear truncation errors. While reduced lattice schemes, such as D3Q15 and D3Q19, introduce spurious angular dependencies through non-linear truncation errors, the complete three-dimensional cubic lattice D3Q27 is absent from such features. This result justifies the superiority of the D3Q27 lattice scheme to cope with the rotational invariance principle in three-dimensional isothermal hydrodynamic problems, particularly when convection is not negligible. Such a theoretical conclusion also finds support in numerical tests presented in this work: a Poiseuille duct flow and a weakly-rotating duct flow.

Lattice modelling of fracture in composite concrete pavements and overlays

Recent research in the USA has focused on the design, construction and performance of composite concrete pavements – that is, two heterogeneous concrete layers placed sequentially using ‘wet-on-wet’ techniques. In addition, other research into rehabilitation practices has focused on improving the design and analysis of bonded concrete overlays of concrete pavements and bridge decks. While these techniques can offer many benefits to pavements and bridges, they also introduce some uncertainties, including the possibility of thermally, hygrally, or mechanically induced fracture and separation at the interface of the concrete layers. In this work, the potential for mixed-mode fracture at the interface between concrete layers is evaluated using three-dimensional lattice models. This discrete approach is coupled with a finite element model for plate behaviour away from the potential cracking zone. The model, using two damage criteria, was verified and validated against experimental data for failure in notched concrete beams in three-point testing. In addition, simulations were conducted using each criterion for interface failure in a composite concrete pavement. The simulation results suggest that interface fracture is unlikely, even when considering unfavourable conditions such as substandard material properties and large temperature differences between the pavement layers.

Phase transitions in three-dimensional topological lattice models with surface anyons

We study the phase diagrams of a family of 3D "Walker-Wang" type lattice models, which are not topologically ordered but have deconfined anyonic excitations confined to their surfaces. We add a perturbation (analogous to that which drives the confining transition in Z_p lattice gauge theories) to the Walker-Wang Hamiltonians, driving a transition in which all or some of the variables associated with the loop gas or string-net ground states of these models become confined. We show that in many cases the location and nature of the phase transitions involved is exactly that of a generalized Z_p lattice gauge theory, and use this to deduce the basic structure of the phase diagram. We further show that the relationship between the phases on opposite sides of the transition is fundamentally different than in conventional gauge theories: in the Walker-Wang case, the number of species of excitations that are deconfined in the bulk can increase across a transition that confines only some of the species of loops or string-nets. The analogue of the confining transition in the Walker-Wang models can therefore lead to bulk deconfinement and topological order.

Three-dimensional topological lattice models with surface anyons

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang [arXiv:1104.2632v2]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $\nu=1/2$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of $bF$ theories. Our special example of the 3D semion model captures much of the behavior of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum hall effect. We find that in these models a fermion is always deconfined in the three dimensional bulk.

Lattice model for the surface states of a topological insulator with applications to magnetic and exciton instabilities

A surface of a strong topological insulator (STI) is characterized by an odd number of linearly dispersing gapless electronic surface states. It is well known that such a surface cannot be described by an effective two-dimensional lattice model (without breaking the time-reversal symmetry), which often hampers theoretical efforts to quantitatively understand some of the properties of such surfaces, including the effect of strong disorder, interactions and various symmetry-breaking instabilities. Here we formulate a lattice model that can be used to describe a {\em pair} of STI surfaces and has an odd number of Dirac fermion states with wavefunctions localized on each surface. The Hamiltonian consists of two planar tight-binding models with spin-orbit coupling, representing the two surfaces, weakly coupled by terms that remove the extra Dirac points from the low-energy spectrum. We illustrate the utility of this model by studying the magnetic and exciton instabilities of the STI surface state driven by short-range repulsive interactions and show that this leads to results that are consistent with calculations based on the continuum model as well as three-dimensional lattice models. We expect the model introduced in this work to be widely applicable to studies of surface phenomena in STIs.

Topological insulators from complex orbital order in transition-metal oxides heterostructures

Topological band insulators which are dynamically generated by electron-electron interactions have been theoretically proposed in two- and three-dimensional lattice models. We present evidence that the two-dimensional version can be stabilized in digital (111) heterostructures of transition-metal oxides as a result of purely local interactions. The topological phases are accompanied by spontaneous ordering of complex orbitals and we discuss their stability with respect to the Hund's rule coupling, Jahn-Teller interaction, and inversion-symmetry breaking terms. As main competitors we identify spin-nematic and magnetic phases.

Discrete breathers on symmetry-determined invariant manifolds for scalar models on the plane square lattice

A group-theoretical approach for studying localized periodic and quasiperiodic vibrations in two- and three-dimensional lattice dynamical models is developed. This approach is demonstrated for the scalar models on the plane square lattice. The symmetry-determined invariant manifolds admitting existence of localized vibrations are found, and some types of discrete breathers are constructed on these manifolds. A general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed. This method allows one to decompose the corresponding system of linear differential equations with time-dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system.