Lattice Formulation Research Articles

Auxiliary field Quantum Monte Carlo for dilute neutrons on the lattice.

We employ constrained path Auxiliary Field Quantum Monte Carlo (AFQMC) in the pursuit of studying physical nuclear systems using a lattice formalism. Since AFQMC has been widely used in the study of condensed-matter systems such as the Hubbard model, we benchmark our method against published results for both one- and two-dimensional Hubbard model calculations. We then turn our attention to cold atomic and nuclear systems. We use an onsite contact interaction that can be tuned in order to reproduce the known scattering length and effective range of a given interaction. Developing this machinery allows us to extend our calculations to study nuclear systems within a lattice formalism. We perform initial calculations for a range of nuclear systems from two- to few-body neutron systems. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

Action of the Axial U(1) Non-Invertible Symmetry on the ’t Hooft Line Operator: A Lattice Gauge Theory Study

Abstract We study how the symmetry operator of the axial $U(1)$ non-invertible symmetry acts on the ’t Hooft line operator in the $U(1)$ gauge theory by employing the modified Villain-type lattice formulation. We model the axial anomaly by a compact scalar boson, the “QED axion”. For the gauge invariance, the simple ’t Hooft line operator, which is defined by a line integral of the dual $U(1)$ gauge potential, must be “dressed” by the scalar and $U(1)$ gauge fields. A careful consideration on the basis of the anomalous Ward–Takahashi identity containing the ’t Hooft operator with the dressing factor and a precise definition of the symmetry operator on the lattice shows that the symmetry operator leaves no effect when it sweeps out a ’t Hooft loop operator. This result appears inequivalent with the phenomenon concluded in the continuum theory. In an appendix, we demonstrate that the half-space gauging of the magnetic $\mathbb {Z}_N$ 1-form symmetry, when formulated in an appropriate lattice framework, leads to the same conclusion as above. A similar result is obtained for the axion string operator.

Toward QCD on quantum computer: orbifold lattice approach

We propose an orbifold lattice formulation of QCD suitable for quantum simulations. We show explicitly how to encode gauge degrees of freedom into qubits using noncompact variables, and how to write down a simple truncated Hamiltonian in the coordinate basis. We show that SU(3) gauge group variables and quarks in the fundamental representation can be implemented straightforwardly on qubits, for arbitrary truncation of the gauge manifold.

Yet Another Lattice Formulation of 2D U(1) Chiral Gauge Theory via Bosonization

Abstract Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the “magnetic charge” in the bosonization, is represented by a “hole” excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context.

Conserved energy–momentum tensor for real-time lattice simulations

We derive an expression for the energy–momentum tensor in the discrete lattice formulation of pure glue QCD. The resulting expression satisfies the continuity equation for energy conservation up to numerical errors with a symmetric procedure for the time discretization. In the case of the momentum conservation equation, we obtain an expression that is of higher accuracy in lattice spacing (O(a2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(a^2)$$\\end{document}) than the naive discretization where fields in the continuum expressions are replaced by discretized counterparts. The improvements are verified by performing numerical tests on the derived expressions using classical real-time lattice gauge theory simulations. We demonstrate substantial reductions in relative error of one to several orders of magnitude compared to a naive discretization for both energy and momentum conservation equations. We expect our formulation to have applications in the area of pre-equilibrium dynamics in ultrarelativistic heavy ion collisions, in particular for the extraction of transport coefficients such as shear viscosity.

A Lattice Formulation of Weyl Fermions on a Single Curved Surface

Abstract In the standard lattice domain-wall fermion formulation, one needs two flat domain-walls where both of the left- and right-handed massless modes appear. In this work we investigate a single domain-wall system with a nontrivial curved background. Specifically we consider a massive fermion on a 3D square lattice, whose domain-wall is a 2D sphere. In the free theory, we find that a single Weyl fermion is localized at the wall and it feels gravity through the induced spin connection. With a topologically nontrivial U(1) link gauge field, however, we find a zero mode with the opposite chirality localized at the center where the gauge field is singular. In the latter case, the low-energy effective theory is not chiral but vectorlike. We discuss how to circumvent this obstacle in formulating lattice chiral gauge theory in the single domain-wall fermion system.

Lattice Realization of the Axial U(1) Noninvertible Symmetry

Abstract In U(1) lattice gauge theory with compact U(1) variables, we construct the symmetry operator, i.e. the topological defect, for the axial U(1) noninvertible symmetry. This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg–Wilson relation. The invariance of the symmetry operator under the gauge transformation of the gauge field on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern–Simons term on the defect over smooth lattice gauge transformations. The projection operator for allowed magnetic fluxes on the defect then emerges with lattice regularization. The resulting symmetry operator is manifestly invariant under lattice gauge transformations. In an appendix, we give another way of constructing the symmetry operator on the basis of a 3D $\mathbb {Z}_N$ topological quantum field theory, the level-N BF theory on the lattice.

Crystallographic map: A general lattice and basis formalism enabling efficient and discretized exploration of crystallographic phase space

Crystallographic map: A general lattice and basis formalism enabling efficient and discretized exploration of crystallographic phase space

Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain

Abstract The Schwinger model, 1D quantum electrodynamics, has CP symmetry at θ = π due to the topological nature of the θ term. At zero temperature, it is known that as the fermion mass increases, the system undergoes a second-order phase transition to the CP broken phase, which belongs to the same universality class as the quantum Ising chain. In this paper, we obtain the phase diagram near the quantum critical point (QCP) in the temperature and fermion mass plane using first-principle Monte Carlo simulations, while avoiding the sign problem by using the lattice formulation of the bosonized Schwinger model. Specifically, we perform a detailed investigation of the correlation function of the electric field near the QCP and find that its asymptotic behavior can be described by the universal scaling function of the quantum Ising chain. This finding indicates the existence of three regions near the QCP, each characterized by a specific asymptotic form of the correlation length, and demonstrates that the CP symmetry is restored at any nonzero temperature, entirely analogous to the quantum Ising chain. The range of the scaling behavior is also examined and found to be particularly wide.

Monte Carlo study of Schwinger model without the sign problem

Monte Carlo study of the Schwinger model (quantum electrodynamics in one spatial dimension) with a topological θ term is very difficult due to the sign problem in the conventional lattice formulation. In this paper, we point out that this problem can be circumvented by utilizing the lattice formulation of the bosonized Schwinger model, initially invented by Bender et al. in 1985. After conducting a detailed review of their lattice formulation, we explicitly validate its correctness through detailed comparisons with analytical and previous numerical results at θ = 0. We also obtain the θ dependence of the chiral condensate and successfully reproduce the mass perturbation result for small fermion masses m/g ≲ 0.125. As an application, we perform a precise calculation of the string tension and quantitatively reveal the confining properties in the Schwigner model at finite temperature and θ region for the first time. In particular, we find that the string tension is negative for noninteger probe charges around θ = π at low temperatures.

Lattice scalar field theory at complex coupling

Lattice scalar field theories encounter a sign problem when the coupling constant is complex. This is a close cousin of the real-time sign problems that afflict the lattice Schwinger-Keldysh formalism, and a more distant relative of the fermion sign problem that plagues calculations of QCD at finite density. We demonstrate the methods of complex normalizing flows and contour deformations on scalar fields in $0+1$ and $1+1$ dimensions, respectively. In both cases, intractable sign problems are readily bypassed. These methods extend to negative couplings, where the partition function can be defined only by analytic continuation. Finally, we examine the location of partition function zeros, and discuss their relation to the performance of these algorithms.

Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model

We revisit the lattice formulation of the Schwinger model using the Kogut-Susskind Hamiltonian approach with staggered fermions. This model, introduced by Banks et al. Phys. Rev. D 13, 1043 (1976), contains the mass term ${m}_{\mathrm{lat}}{\ensuremath{\sum}}_{n}{(\ensuremath{-}1)}^{n}{\ensuremath{\chi}}_{n}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\chi}}_{n}$, and setting it to zero is often assumed to provide the lattice regularization of the massless Schwinger model. We instead argue that the relation between the lattice and continuum mass parameters should be taken as ${m}_{\mathrm{lat}}=m\ensuremath{-}\frac{1}{8}{e}^{2}a$. The model with $m=0$ is shown to possess a discrete chiral symmetry that is generated by the unit lattice translation accompanied by the shift of the $\ensuremath{\theta}$ angle by $\ensuremath{\pi}$. While the mass shift vanishes as the lattice spacing $a$ approaches zero, we find that including this shift greatly improves the rate of convergence to the continuum limit. We demonstrate the faster convergence using both numerical diagonalizations of finite lattice systems, as well as extrapolations of the lattice strong coupling expansions.

Non-Hermitian skin clusters from strong interactions

Strong, non-perturbative interactions often lead to new exciting physics, as epitomized by emergent anyons from the Fractional Quantum hall effect. Within the actively investigated domain of non-Hermitian physics, we provide a family of states known as non-Hermitian skin clusters. Taking distinct forms as Vertex, Topological, Interface, Extended and Localized skin clusters, they generically originate from asymmetric correlated hoppings on a lattice, in the strongly interacting limit with quenched single-body energetics. Distinct from non-Hermitian skin modes which accumulate at boundaries, our skin clusters are predominantly translation invariant particle clusters. As purely interacting phenomena, they fall outside the purview of generalized Brillouin zone analysis, although our effective lattice formulation provides alternative analytic and topological characterization. Non-Hermitian skin clusters originate from the fragmentation structure of the Hilbert space and may thus be of significant interest in modern many-body contexts such as the Eigenstate thermalization hypothesis (ETH) and quantum scars.

From Asymptotic Freedom to θ Vacua: Qubit Embeddings of the O(3) Nonlinear σ Model.

Conventional lattice formulations of θ vacua in the 1+1-dimensional O(3) nonlinear sigma model suffer from a sign problem. Here, we construct the first sign-problem-free regularization for arbitrary θ. Using efficient lattice MonteCarlo algorithms, we demonstrate how a Hamiltonian model of spin-1/2 degrees of freedom on a two-dimensional spatial lattice reproduces both the infrared sector for arbitrary θ, as well as the ultraviolet physics of asymptotic freedom. Furthermore, as a model of qubits on a two-dimensional square lattice with only nearest-neighbor interactions, it is naturally suited for studying the physics of θ vacua and asymptotic freedom on near-term quantum devices. Our construction generalizes to θ vacua in all CP(N-1) models, solving a long-standing sign problem.

Wave propagation in viscoelastic metamaterials via added-state formulation

Mechanical metamaterials composed of a non dissipative periodic microstructure of flexible ligaments and stiff rings, viscoelastically coupled with local resonators, are considered. By following a variational approach, the linear damped dynamics are described according to a beam lattice formulation, valid for a generic coordination number. A consistent theoretical description of the viscoelastic ring-resonator coupling is introduced, leading to an integral–differential form of the governing equations. The enlargement of the model dimension by addition of viscoelastic states allows to conveniently describe the metamaterial dynamics through a system of ordinary differential equations. Therefore, the application of the Z-transform and a suitable mapping in the wavevector space allows to obtain the system of ordinary differential equations governing the free and forced propagation of damped waves. Subsequently, the polynomial eigenproblem ruling the harmonic free wave propagation in the complex frequency domain is stated by applying the bilateral Laplace transform, whereas an analytical procedure is outlined to solve the forced wave propagation problem in the time domain. The triangular beam lattice metamaterial is considered as significant benchmark for application of the theoretical and methodological framework. Considering first the free wave propagation, the properties of the dynamic stiffness matrix are discussed and the dispersion frequency spectrum is parametrically investigated, with focus on the spectral effects of the viscoelastic coupling. The dependence of the stop bandwidth amplitude between the low and high-frequency spectral branches on the relaxation time characterizing the viscous function is also highlighted and discussed. Moreover, the forced response to a harmonic external point source is numerically investigated in the time domain. The results are qualitatively compared and quantitatively discussed by distinguishing the fundamental cases of non-resonant, resonant and quasi-resonant excitation frequency.

Topological Gauge Actions on the Lattice as Overlap Fermion Determinants

Overlap fermion on the lattice has been shown to properly reproduce topological aspects of gauge fields. In this paper, we review the derivation of Overlap fermion formalism in a torus of three space-time dimensions. Using the formalism, we show how to use the Overlap fermion determinants in the massless and infinite mass limits to construct different continuum topological gauge actions, such as the level-k Chern–Simons action, “half-CS” term and the mixed Chern–Simons (BF) coupling, in a gauge-invariant lattice UV regulated manner. Taking special Abelian and non-Abelian background fields, we demonstrate numerically how the lattice formalism beautifully reproduces the continuum expectations, such as the flow of action under large gauge transformations.

Topology induced first-order phase transitions in lattice quantum gravity

Causal Dynamical Triangulations (CDT) is a lattice formulation of quantum gravity, suitable for Monte-Carlo simulations which have been used to study the phase diagram of the model. It has four phases characterized by different dominant geometries, denoted phase A, B, C and Cb. In this article we analyse the A − B and the B − C phase transitions in the case where the topology of space is that of the three-torus. This completes the phase diagram of CDT for such a spatial topology. We observe that the order of a phase transition of spacetime geometries can depend on the topology of spacetime.

Lattice fermions as spectral graphs

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four­dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.

Large-[formula omitted] Chern insulators: Lattice field theory and quantum simulation approaches to correlation effects in the quantum anomalous Hall effect

Four-Fermi quantum field theories in (2+1) dimensions lie among the simplest models in high-energy physics, the understanding of which requires a non-perturbative lattice formulation addressing their strongly-coupled fixed points. These lattice models are also relevant in condensed matter, as they offer a neat playground to explore strong correlations in the quantum anomalous Hall (QAH) effect. We give a detailed description of our multidisciplinary approach to understand the fate of the QAH phases as the four-Fermi interactions are increased, which combines strong-coupling and effective-potential techniques, unveiling a rich phase diagram with large-N Chern insulators and Lorentz-breaking fermion condensates. Moreover, this toolbox can be enlarged with recent advances in quantum information science, as we show that tensor-network algorithms based on projected entangled pairs can be used to improve our understanding of the strong-coupling limit. We also present a detailed scheme that uses ultra-cold atoms in optical lattices with synthetic spin–orbit coupling to build quantum simulators of these four-Fermi models. This yields a promising alternative to characterize the strongly-coupled fixed points and, moreover, could also explore real-time dynamics and finite-fermion densities.

Diffusion-induced stress concentrations in diffusional creep

The lattice continuum formulation for diffusional creep is implemented into the computational framework based on the finite difference method for space and the predictor–corrector algorithm for time discretization, to solve the coupled elasticity-diffusion problem with moving boundaries. The numerical scheme is implemented and tested considering 3D periodic structures without grain boundary sliding. Of primary interest in this paper are the stress non-uniformities resulting from nonuniform composition eigenstrains.We consider two regimes: the one where the rate limiting process is bulk diffusion of vacancies, and the one where the rate is controlled by vacancy nucleation/annihilation at grain boundaries (nucleation-controlled creep). We found that the stress concentration factor for diffusion-controlled creep is independent of the applied stress and grain size. No stress concentrations are present for the nucleation-controlled creep case.Stress and grain size dependence of minimum strain creep rates are determined by the present model for a variety of applied stresses, grain sizes, different driving processes (diffusion and nucleation-controlled creep) and compared with the classical theory for diffusional creep. We found that steady-state creep rates varied linearly with applied stresses for both diffusion and nucleation-controlled creep. The numerical results show good correspondence to analytical predictions for idealized diffusional Nabarro-Herring creep. Significantly lower steady-state strain rates were computed for nucleation-controlled creep. These results demonstrate the ability of the present model to reproduce the stress and grain size dependence of the steady-state strain creep rates.