Lattice Fermions Research Articles

Intriguing features of Dirac cones in phagraphene with site specific doping

The non-honeycomb lattice phagraphene is energetically comparable with graphene and has recently been experimentally realized in 1D form. This carbon allotrope exhibits asymmetric Dirac cone feature with tunable Fermi velocity against external strain. This work critically presents an analytical scheme to address the emergence and robustness of Dirac fermions in phagraphene network. In particular, the lattice has judicially been renormalized from twenty atomic sites per unit cell to an equivalent two-level model. The electronic band structure of the renormalized system agrees well with the first principles results near the Fermi level. Besides, the Fermi velocity obtained from the analytical dispersion relation also supports the density functional theory results. Moreover, phagraphene is dynamically stable even after the introduction of boron and nitrogen as foreign atoms. It has been revealed that the degeneracy at the Fermi level is preserved for certain doping sites. Therefore, the criterion of band gap opening in this Dirac system has been explained in terms of appropriate hopping constraints. Furthermore, we have explored that some of the particular lattice sites significantly contribute to the formation of Dirac cones in the low-energy lattice. In addition, proper effective hopping anisotropy between these renormalized sites essentially removes the degeneracy of the Dirac point and opens a bandgap. Our approach is elegant and will serve as an essential reference for realizing the underlying mechanism behind the occurrence of Dirac fermions in the next-generation non-honeycomb lattices.

Naive lattice fermion without doublers

We discuss the naive lattice fermion without the issue of doublers. A local lattice massless fermion action with chiral symmetry and Hermiticity cannot avoid the doubling problem from the Nielsen-Ninomiya theorem. Here we adopt the forward finite-difference deforming the ${\ensuremath{\gamma}}_{5}$-Hermiticity but preserving the continuum chiral symmetry. The lattice momentum is not Hermitian without the continuum limit now. We demonstrate that there is no doubling issue from an exact solution. The propagator only has one pole in the first-order accuracy. Therefore, it is hard to know the avoiding due to the non-Hermiticity. For the second-order, the lattice propagator has two poles as before. This case also does not suffer from the doubling problem. Hence separating the forward derivative from the backward one evades the doublers under the field theory limit. Simultaneously, it is equivalent to breaking the Hermiticity. In the end, we discuss the topological charge and also demonstrate the numerical implementation of the hybrid Monte Carlo.

Restricted Boltzmann machine based on a Fermi sea

In recent years, there has been an intensive research on how to exploit the quantum laws of nature in the machine learning. Models have been put forward which employ spins, photons, and cold atoms. In this work we study the possibility of using the lattice fermions to learn the classical data. We propose an alternative to the quantum Boltzmann machine, the so-called spin-fermion machine (SFM), in which the spins represent the degrees of freedom of the observable data (to be learned), and the fermions represent the correlations between the data. The coupling is linear in spins and quadratic in fermions. The fermions are allowed to tunnel between the lattice sites. The training of SFM can be efficiently implemented since there are closed expressions for the log-likelihood gradient. We find that SFM is more powerful than the classical restricted Boltzmann machine with the same number of physical degrees of freedom. The reason is that SFM has additional freedom due to the rotation of the Fermi sea. We show examples for several data sets.

More about the Grassmann tensor renormalization group

We derive a general formula of the tensor network representation for d-dimensional lattice fermions with ultra-local interactions, including Wilson fermions, staggered fermions, and domain-wall fermions. The Grassmann tensor is concretely defined with auxiliary Grassmann variables that play a role in bond degrees of freedom. Compared to previous works, our formula does not refer to the details of lattice fermions and is derived by using the singular value decomposition for the given Dirac matrix without any ad-hoc treatment for each fermion. We numerically test our formula for free Wilson and staggered fermions and find that it properly works for them. We also find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method.

Stabilizing topological superfluidity of lattice fermions

Attractive interaction between spinless fermions in a two-dimensional lattice drives the formation of a topological superfluid. But the topological phase is dynamically unstable towards phase separation when the system has a high density of states and large interaction strength. This limits the critical temperature to an experimentally challenging regime where, for example, even ultracold atoms and molecules in optical lattices would struggle to realize the topological superfluid. We propose that the introduction of a weaker longer-range repulsion, in addition to the short-range attraction between lattice fermions, will suppress the phase separation instability. Taking the honeycomb lattice as an example, we show that our proposal significantly enlarges the stable portion of the topological superfluid phase and increases the critical temperature by an order of magnitude. Our work opens a route to enhance the stability of topological superfluids by engineering inter-particle interactions.

Multiparticle Quantum Walks and Fisher Information in One-Dimensional Lattices.

Recent experiments on quantum walks (QWs) demonstrated a full control over the statistics-dependent walks of single particles and two particles in one-dimensional lattices. However, little is known about the general characterization of QWs at the many-body level. Here, we rigorously study QWs, Bloch oscillations, and the quantum Fisher information for three indistinguishable bosons and fermions in one-dimensional lattices using a time-evolving block decimation algorithm and many-body perturbation theory. We show that such strongly correlated QWs not only give rise to statistics-and-interaction-dependent ballistic transports of scattering states and of two- and three-body bound states but also allow a quantum enhanced precision measurement of the gravitational force. In contrast to the QWs of the fermions, the QWs of three bosons exhibit strongly correlated Bloch oscillations, which present a surprising time scaling t^{3} of the Fisher information below a characteristic time t_{0} and saturate to the fundamental limit of t^{2} for t>t_{0}.

Rotated twisted-mass: a convenient regularization scheme for isospin breaking QCD and QED lattice calculations

We propose a scheme of lattice twisted-mass fermion regularization which is particularly convenient for application to isospin breaking (IB) QCD and QED calculations, based in particular on the so called RM123 approach, in which the IB terms of the action are treated as a perturbation. The main, practical advantage of this scheme is that it allows the calculation of IB effects on some mesonic observables, like e.g. the pi ^+ - pi ^0 mass splitting, using lattice correlation functions in which the quark and antiquark fields in the meson are regularized with opposite values of the Wilson parameter r. These correlation functions are found to be affected by much smaller statistical fluctuations, with respect to the analogous functions in which quark and antiquark fields are regularized with the same value of r. Two numerical application of this scheme, that we call rotated twisted-mass, within pure QCD and QCD + QED respectively, are also provided for illustration.

Nonequilibrium resonant inelastic x-ray scattering study of an electron-phonon model

We use the nonequilibrium dynamical mean field theory formalism to compute the equilibrium and nonequilibrium resonant inelastic X-ray scattering (RIXS) signal of a strongly interacting fermionic lattice model with a coupling of dispersionless phonons to the total charge on a given site. In the atomic limit, this model produces phonon subbands in the spectral function, but not in the RIXS signal. Electron hopping processes however result in phonon-related modifications of the charge excitation peak. We discuss the equilibrium RIXS spectra and the characteristic features of nonequilibrium states induced by photo-doping and by the application of a static electric field. The latter produces features related to Wannier-Stark states, which are dressed with phonon sidebands. Thanks to the effect of field-induced localization, the phonon features can be clearly resolved even in systems with weak electron-phonon coupling.

Numerically exact mimicking of quantum gas microscopy for interacting lattice fermions

A numerical method is presented for reproducing fermionic quantum gas microscope experiments in equilibrium. By employing nested componentwise direct sampling of fermion pseudo-density matrices, as they arise naturally in determinantal quantum Monte Carlo (QMC) simulations, a stream of pseudo-snapshots of occupation numbers on large systems can be produced. There is a sign problem even when the conventional determinantal QMC algorithm can be made sign-problem free, and every pseudo-snapshot comes with a sign and a reweighting factor. Nonetheless, this "sampling sign problem" turns out to be weak and manageable in a large, relevant parameter regime. The method allows to compute distribution functions of arbitrary quantities defined in occupation number space and, from a practical point of view, facilitates the computation of complicated conditional correlation functions. While the projective measurements in quantum gas microscope experiments achieve direct sampling of occupation number states from the density matrix, the presented numerical method requires a Markov chain as an intermediate step and thus achieves only indirect sampling, but the full distribution of pseudo-snapshots after (signed) reweighting is identical to the distribution of snapshots from projective measurements

Reducing autocorrelation time in determinant quantum Monte Carlo using the Wang-Landau algorithm: Application to the Holstein model.

When performing a Monte Carlo calculation, the running time should, in principle, be much longer than the autocorrelation time in order to get reliable results. Among different lattice fermion models, the Holstein model is notorious for its particularly long autocorrelation time. In this paper, we employ the Wang-Landau algorithm in the determinant quantum Monte Carlo to achieve the flat-histogram sampling in the "configuration weight space," which can greatly reduce the autocorrelation time by sacrificing some sampling efficiency. The proposal is checked in the Holstein model on both square and honeycomb lattices. Based on such a Wang-Landau assisted determinant quantum Monte Carlo method, some models with long autocorrelation times can now be simulated possibly.

RF dressing with two-component fermions in optical lattices

RF dressing with two-component fermions in optical lattices

Chiral lattice fermions from staggered fields

We describe a proposal for constructing a lattice theory that we argue may be capable of yielding free Weyl fermions in the continuum limit. The model employs reduced staggered fermions and uses site parity dependent Yukawa interactions of Fidkowski-Kitaev type to gap a subset of the lattice fermions without breaking symmetries. The possibility for such symmetric mass generation is tied to the cancellation of certain discrete anomalies arising in the continuum limit. The latter place strong constraints on the number of lattice fermions -- constraints that are satisfied by this model. We present numerical results for the model in two dimensions which support this sc

SU(3) vs. SU(2) fermions in optical lattices: Color-Hall vs. spin-Hall topological insulators

We contrast topological insulating phases of SU(3) and SU(2) fermions in optical lattices under artificial magnetic fields. For this purpose, we compare the phase diagram of chemical potential vs. color-flip fields for SU(3) colored fermions (D 10 representation) for fixed color-orbit coupling to the phase diagram of chemical potential vs. spin-flip fields for SU(2) spin- fermions (D 1 representation) for fixed spin-orbit coupling. To classify the emergent topological phases, we obtain the Chern matrix and identify three distinct topological invariants for SU(3) colored fermions: the charge-charge, the color-charge and the color-color Chern numbers. This is in sharp contrast to the SU(2) spin- case, where only two topological invariants are necessary: the charge-charge and the spin-charge Chern numbers. Although we focus on SU(3) colored and SU(2) spin- fermions, our results indicate the existence of three independent Chern numbers associated with the quantum Hall response for higher fermionic irreducible representations D p and D p0 (p > 1) of SU(2) and SU(3), respectively. We suggest to investigate this comparison using the internal states of ultracold Fermi atoms ytterbium-173 and strontium-87 .

Lattice-fermionic Casimir effect and topological insulators

The Casimir effect arises from the zero-point energy of particles in momentum space deformed by the existence of two parallel plates. For degrees of freedom on the lattice, its energy-momentum dispersion is determined so as to keep a periodicity within the Brillouin zone, so that its Casimir effect is modified. We study the properties of Casimir effect for lattice fermions, such as the naive fermion, Wilson fermion, and overlap fermion based on the M\"obius domain-wall fermion formulation, in the $1+1$-, $2+1$-, and $3+1$-dimensional space-time with the periodic or antiperiodic boundary condition. An oscillatory behavior of Casimir energy between odd and even lattice size is induced by the contribution of ultraviolet-momentum (doubler) modes, which realizes in the naive fermion, Wilson fermion in a negative mass, and overlap fermions with a large domain-wall height. Our findings can be experimentally observed in condensed matter systems such as topological insulators and also numerically measured in lattice simulations.

Metal and insulator states of SU(6) × SU(2) clusters of fermions in one-dimensional optical lattices

We studied the behavior of mixtures of 173Yb (with symmetry up to SU(6)) and 171Yb (up to SU(2)) fermionic isotopes loaded in one-dimensional (1D) optical lattices. To do so, we solved the Schrödinger equation describing different systems using a diffusion Monte Carlo technique. We considered continuous Hamiltonians in which the interactions between atoms of different species (isotopes and/or spins) were modeled by contact potentials with parameters derived from their experimental scattering lengths. This implies that we can find both attractive and repulsive interactions between fermion pairs in the same cluster. The strength of those interactions can be changed by varying the transverse confinement, leading to different cluster behaviors. Only balanced clusters, i.e. with the same number of 173Yb and 171Yb atoms were considered. We found that the standard state for these clusters is a metallic-like one with different populations of 173Yb–171Yb molecule-like pairs in each optical lattice potential well. However, for big enough clusters, insulator-like states are also possible.

Strong boundary and trap potential effects on emergent physics in ultra-cold fermionic gases

The field of quantum simulations in ultra-cold atomic gases has been remarkably successful. In principle it allows for an exact treatment of a variety of highly relevant lattice models and their emergent phases of matter. But so far there is a lack in the theoretical literature concerning the systematic study of the effects of the trap potential as well as the finite size of the systems, as numerical studies of such non periodic, correlated fermionic lattices models are numerically demanding beyond one dimension. We use the recently introduced real-space truncated unity functional renormalization group to study these boundary and trap effects with a focus on their impact on the superconducting phase of the 2D Hubbard model. We find that in the experiments not only lower temperatures need to be reached compared to current capabilities, but also system size and trap potential shape play a crucial role to simulate emergent phases of matter.

Symmetry Classes of Open Fermionic Quantum Matter

We present a full symmetry classification of fermion matter in and out of thermal equilibrium. Our approach starts from first principles, the ten different classes of linear and anti-linear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification are then the generators of reversible dynamics, dissipation and fluctuations featuring in the generally irreversible and interacting dynamical equations. A sharp distinction between the symmetries of equilibrium and out of equilibrium dynamics, respectively, arises from the different role played by `time' in these two cases: In unitary quantum mechanics as well as in `micro-reversible' thermal equilibrium, anti-linear transformations combined with an inversion of time define time reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while anti--linear transformations in Fock space remain physically significant, and hence must be considered in autonomy. The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in non-equilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium. At large times, the dynamical generators thus symmetry classified determine the steady state non-equilibrium distributions for arbitrary interacting systems. To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions. More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states. This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.

Eigenspectrum, Chern numbers and phase diagrams of ultracold color-orbit-coupled SU(3) fermions in optical lattices

We study ultracold color fermions with three internal states Red, Green and Blue with ${\rm SU(3)}$ symmetry in optical lattices, when color-orbit coupling and color-flip fields are present. This system corresponds to a generalization of two-internal state fermions with ${\rm SU(2)}$ symmetry in the presence of spin-orbit coupling and spin-flipping Zeeman fields. We investigate the eigenspectrum and Chern numbers to describe different topological phases that emerge in the phase diagrams of color-orbit coupled fermions in optical lattices. We obtain the phases as a function of artificial magnetic, color-orbit and color-flip fields that can be independently controlled. For fixed artificial magnetic flux ratio, we identify topological quantum phases and phase transitions in the phase diagrams of chemical potential versus color-flip fields or color-orbit coupling, where the chirality and number of midgap edge states changes. The topologically non-trivial phases are classified in three groups: the first group has total non-zero chirality and exhibit only the quantum charge Hall effect; the second group has total non-zero chirality and exhibit both quantum charge and quantum color Hall effects; and the third group has total zero chirality, but exhibit the quantum color Hall effect. These phases are generalizations of the quantum Hall and quantum spin Hall phases for charged spin-$1/2$ fermions. Lastly, we also describe the color density of states and a staircase structure in the total and color filling factors versus chemical potential for fixed color-orbit, color-flip and magnetic flux ratio. We show the existence of incompressible states at rational filling factors precisely given by a gap-labelling theorem that relates the filling factors to the magnetic flux ratio and topological quantum numbers.

Hamiltonian models of lattice fermions solvable by the meron-cluster algorithm

We introduce a half-filled Hamiltonian of spin-half lattice fermions that can be studied with the efficient meron-cluster algorithm in any dimension. As with the usual bipartite half-filled Hubbard models, the na\"{\i}ve $U(2)$ symmetry is enhanced to $SO(4)$. On the other hand, our model has a novel spin-charge flip ${\mathbb{Z}}_{2}^{C}$ symmetry which is an important ingredient of free massless fermions. In this work we focus on one spatial dimension and show that our model can be viewed as a lattice-regularized two-flavor chiral-mass Gross-Neveu model. Our model remains solvable in the presence of the Hubbard coupling $U$, which maps to a combination of Gross-Neveu and Thirring couplings in one dimension. Using the meron-cluster algorithm we find that the ground state of our model is a valence bond solid when $U=0$. From our field theory analysis, we argue that the valence bond solid forms inevitably because of an interesting frustration between spin and charge sectors in the renormalization group flow enforced by the ${\mathbb{Z}}_{2}^{C}$ symmetry. This state spontaneously breaks translation symmetry by one lattice unit, which can be identified with a ${\mathbb{Z}}_{2}^{\ensuremath{\chi}}$ chiral symmetry in the continuum. We show that increasing $U$ induces a quantum phase transition to a critical phase described by the $SU(2{)}_{1}$ Wess-Zumino-Witten theory. The quantum critical point between these two phases is known to exhibit a novel symmetry enhancement between spin and dimer. Here we verify the scaling relations of these correlation functions near the critical point numerically. Our study opens up the exciting possibility of numerical access to similar novel phase transitions in higher dimensions in fermionic lattice models using the meron-cluster algorithm.

Effective free-fermionic form factors and the XY spin chain

We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.