Two-dimensional Fermion Model Research Articles

Effective dimension reduction with mode transformations: Simulating two-dimensional fermionic condensed matter systems with matrix-product states

Tensor network methods have progressed from variational techniques based on matrix-product states able to compute properties of one-dimensional condensed-matter lattice models into methods rooted in more elaborate states such as projected entangled pair states aimed at simulating the physics of two-dimensional models. In this work, we advocate the paradigm that for two-dimensional fermionic models, matrix-product states are still applicable to significantly higher accuracy levels than direct embeddings into one-dimensional systems allow for. To do so, we exploit schemes of fermionic mode transformations and overcome the prejudice that one-dimensional embeddings need to be local. This approach takes the insight seriously that the suitable exploitation of both the manifold of matrix-product states and the unitary manifold of mode transformations can more accurately capture the natural correlation structure. By demonstrating the residual low levels of entanglement in emerging modes, we show that matrix-product states can describe ground states strikingly well. The power of the approach is exemplified by investigating a phase transition of spin-less fermions for lattice sizes up to 10x10.

Numerical continuum tensor networks in two dimensions

We describe the use of tensor networks to numerically determine wave functions of interacting two-dimensional fermionic models in the continuum limit. We use two different tensor network states: one based on the numerical continuum limit of fermionic projected entangled pair states obtained via a tensor network formulation of multi-grid, and another based on the combination of the fermionic projected entangled pair state with layers of isometric coarse-graining transformations. We first benchmark our approach on the two-dimensional free Fermi gas then proceed to study the two-dimensional interacting Fermi gas with an attractive interaction in the unitary limit, using tensor networks on grids with up to 1000 sites.

Machine learning topological invariants of non-Hermitian systems

The study of topological properties by machine learning approaches has attracted considerable interest recently. Here we propose machine learning the topological invariants that are unique in non-Hermitian systems. Specifically, we train neural networks to predict the winding of eigenvalues of four prototypical non-Hermitian Hamiltonians on the complex energy plane with nearly $100\%$ accuracy. Our demonstrations in the non-Hermitian Hatano-Nelson model, Su-Schrieffer-Heeger model and generalized Aubry-Andr\'e-Harper model in one dimension, and two-dimensional Dirac fermion model with non-Hermitian terms show the capability of the neural networks in exploring topological invariants and the associated topological phase transitions and topological phase diagrams in non-Hermitian systems. Moreover, the neural networks trained by a small data set in the phase diagram can successfully predict topological invariants in untouched phase regions. Thus, our work paves the way to revealing non-Hermitian topology with the machine learning toolbox.

Intrinsic spin Hall effect with spin-tensor-momentum coupling

We derive the spin continuity equation by using the Noether’s theorem. A new type of spin-tensor Hall current is found in the continuity equation. The spin-tensor Hall current is originating from the coupling of the spin-tensor and the momentum. The intrinsic spin Hall effect in the two-dimensional fermion model with the spin–orbit coupling and spin-tensor-momentum coupling is studied. The total spin Hall conductivity with the presence of the spin-tensor Hall current is calculated. The numerical results indicate that the total spin Hall conductivity is enhanced by the contribution of the spin-tensor-momentum coupling. The spin-tensor-momentum coupling may increase the spin transport in the intrinsic spin Hall effect.

Logarithmic entanglement growth in two-dimensional disordered fermionic systems

We investigate the growth of the entanglement entropy $S_{\textrm{ent}}$ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case we show that an intermediate weak localization regime exists in which $S_{\textrm{ent}}(t)$ grows logarithmically in time $t$ before Anderson localization sets in. For the case of binary bond disorder near the percolation transition we find additive logarithmic corrections to area and volume laws as well as a scaling at long times which is consistent with an infinite randomness fixed point.

Thermal transitions, pseudogap behavior, and BCS-BEC crossover in Fermi-Fermi mixtures

We study the mass imbalanced Fermi-Fermi mixture within the framework of a two-dimensional lattice fermion model. Based on the thermodynamic and species dependent quasiparticle behavior we map out the finite temperature phase diagram of this system and show that unlike the balanced Fermi superfluid there are now two different pseudogap regimes as PG-I and PG-II. While within the PG-I regime both the fermionic species are pseudogapped, PG-II corresponds to the regime where pseudogap feature survives only in the light species. We believe that the single particle spectral features that we discuss in this paper are observable through the species resolved radio frequency spectroscopy and momentum resolved photo emission spectroscopy measurements on systems such as, 6$_{Li}$-40$_{K}$ mixture. We further investigate the interplay between the population and mass imbalances and report that at a fixed population imbalance the BCS-BEC crossover in a Fermi-Fermi mixture would require a critical interaction (U$_{c}$), for the realization of the uniform superfluid state. The effect of imbalance in mass on the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase has been probed in detail in terms of the thermodynamic and quasiparticle behavior of this phase. It has been observed that in spite of the s-wave symmetry of the pairing field a nodal superfluid gap is realized in the LO regime. Our results on the various thermal scales and regimes are expected to serve as benchmarks for the experimental observations on 6$_{Li}$-40$_{K}$ mixture.

Critical Theory of Two-Dimensional Mott Transition: Integrability and Hilbert Space Mapping

We reconsider the Mott transition in the context of a two-dimensional fermion model with density-density coupling. We exhibit a Hilbert space mapping between the original model and the Double Lattice Chern-Simons theory at the critical point by use of the representation theory of the q-oscillator and Weyl algebras. The transition is further characterized by the ground state modification. The explicit mapping provides a new tool to further probe and test the detailed physical properties of the fermionic lattice model considered here and to enhance our understanding of the Mott transition(s).

Production of black holes and string balls in a two-dimensional split-fermion model

We present cross-sections for the black hole and string ball production in proton-proton collisions in a TeV-scale gravity model with split fermions in two dimensions. The cross-section for black hole and string ball production in the split-fermion model is smaller than in the non-split-fermion model. The drop of the cross-section for the string ball production can be one to two orders of magnitude with the increase of the width of the brane from $L=0$ to 15 $\text{TeV}^{-1}$. The cross-section for string ball production in two-dimensional split fermion model reduces more in comparison to black holes. Black holes are quite hard to be observed at the LHC. In fact, taking into account the current experimental limits on the fundamental Planck scale, black holes cannot be produced at the LHC. Cross-section for string ball production depends significantly on string coupling constant, making it very model dependent. We investigate the range of values of string coupling constant from 0.02 to 0.4. There has been no evidence for production of string balls at $\sqrt{s}=8$ TeV. A two-dimensional split fermion model with a extremely thick brane can account for the absence of signature of string balls for a world with the value of fundamental Planck scale even as low as 1 TeV.

Monte Carlo simulations of two-dimensional fermion systems with string-bond states

We describe an application of variational Monte Carlo to two-dimensional fermionic systems within the recently developed tensor-network string-bond state (SBS) ansatz. We use a combination of variational Monte Carlo and stochastic optimization to optimize the matrix-product state matrices representing the ground state. We present results for a two-dimensional spinless fermion model including nearest-neighbor Coulomb interactions and determine using finite-size scaling the phase boundary between charge-ordered insulating and metallic phases. This approach can treat frustrated systems and be easily extended to for fermions with spin.

Integrable theories and generalized graded Maillet algebras

We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the subsector of the string theory on AdS5 × S5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday–Arutyunov–Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet’s r- and s-matrices formalism, and derive a well-defined algebra of transition matrices, which allows for the lattice formulation of the theory. We illustrate our formalism on the examples of the bosonic Wadati–Konno–Ichikawa–Shimizu (WKIS) model and the two-dimensional free massive Dirac fermion model, which can be obtained by a consistent reduction of the full AAF model, and give the explicit forms of their corresponding r-matrices.

Topological qubits in graphenelike systems

The fermion-doubling problem can be an obstacle to getting half a qubit in two-dimensional fermionic tight-binding models in the form of Majorana zero modes bound to the core of superconducting vortices. We argue that the number of such Majorana zero modes is determined by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological charge for a family of two-dimensional fermionic tight-binding models ranging from noncentrosymmetric materials to graphene. This charge depends on the dimension of the representation (i.e., the number of species of Dirac fermions---where the doubling problem enters) and the parity of the Chern number induced by breaking time-reversal symmetry. We show that in graphene there are as many as ten order parameters that can be used in groups of four to change the topological number from even to odd.

Stable topological superconductivity in a family of two-dimensional fermion models

We show that a large class of two-dimensional spinless fermion models exhibit topological superconducting phases characterized by a non-zero Chern number. More specifically, we consider a generic one-band Hamiltonian of spinless fermions that is invariant under both time-reversal, $\mathbb{T}$, and a group of rotations and reflections, $\mathbb{G}$, which is either the dihedral point-symmetry group of an underlying lattice, $\mathbb{G}=D_n$, or the orthogonal group of rotations in continuum, $\mathbb{G}={\rm O}(2)$. Pairing symmetries are classified according to the irreducible representations of $ \mathbb{T} \otimes \mathbb{G}$. We prove a theorem that for any two-dimensional representation of this group, a time-reversal symmetry breaking paired state is energetically favorable. This implies that the ground state of any spinless fermion Hamiltonian in continuum or on a square lattice with a singly-connected Fermi surface is always a topological superconductor in the presence of attraction in at least one channel. Motivated by this discovery, we examine phase diagrams of two specific lattice models with nearest-neighbor hopping and attraction on a square lattice and a triangular lattice. In accordance with the general theorem, the former model exhibits only a topological $(p + ip)$-wave state, while the latter shows a doping-tuned quantum phase transition from such state to a non-topological, but still exotic $f$-wave superconductor.

Functional integral approach of the two-dimensional massive fermion model with Thirring interaction among different species

We use the Abelian reduction of the Wess–Zumino–Witten theory to perform the functional integral bosonization of the two-dimensional fermion model with Thirring interaction among N different massive Fermi field species. The operator solution for the quantum equations of motion is reconstructed from the functional integral formulation. The fermion–boson correspondences, obtained earlier by Halpern, are generalized to the case of quartic interaction between different Fermi field species. For the massless model, the general Wightman functions of the Fermi field are displayed. The partition function and the equation of state of the statistical mechanical system associated with the effective bosonized theory are obtained. The charge screening mechanism for the Thirring field is discussed by considering the model with local gauge symmetry. The present approach provides a general guideline in order to obtain the operator solution of two-dimensional Abelian models, since we only need to know the free-field fermion–boson correspondences in order to reconstruct from the functional integral formulation the operator solution for the quantum equations of motion.

Absence of vortex condensation in a two dimensional fermionicXYmodel

Motivated by a puzzle in the study of two-dimensional lattice quantum electrodynamics with staggered fermions, we construct a two-dimensional fermionic model with a global U(1) symmetry. Our model can be mapped into a model of closed packed dimers and plaquettes. Although the model has the same symmetries as the XY model, we show numerically that the model lacks the well-known Kosterlitz-Thouless phase transition. The model is always in the gapless phase showing the absence of a phase with vortex condensation. In other words the low energy physics is described by a noncompact U(1) field theory. We show that by introducing an even number of layers one can introduce vortex condensation within the model and thus also induce a Kosterlitz-Thouless transition.

Ring exchange, the exciton Bose liquid, and bosonization in two dimensions

Motivated by the high-${T}_{c}$ cuprates, we consider a model of bosonic Cooper pairs moving on a square lattice via ring exchange. We show that this model offers a natural middle ground between a conventional antiferromagnetic Mott insulator and the fully deconfined fractionalized phase that underlies the spin-charge separation scenario for high-${T}_{c}$ superconductivity. We show that such ring models sustain a stable critical phase in two dimensions, the exciton Bose liquid (EBL). The EBL is a compressible state, with gapless but uncondensed boson and ``vortex'' excitations, power-law superconducting and charge-ordering correlations, and broad spectral functions. We characterize the EBL with the aid of an exact plaquette duality transformation, which motivates a universal low-energy description of the EBL. This description is in terms of a pair of dual bosonic phase fields, and is a direct analog of the well known one-dimensional bosonization approach. We verify the validity of the low-energy description by numerical simulations of the ring model in its exact dual form. The relevance to the high-${T}_{c}$ superconductors and a variety of extensions to other systems are discussed, including the bosonization of a two-dimensional fermionic ring model.

Relativistic Bound State Solutions of a Two-Dimensional Fermion Model with Color SU(3) Symmetry

We implement color SU(3) symmetry on a relativistic two-dimensional fermion model proposed by Glockle, Nogami and Fukui (GNF). We find that it is possible to construct solutions for several multiplets of two and three particles by using the solutions for the GNF model. Though our model is based on the single-particle theory (i.e., the model does not contain any anti-particles), we regard a particle belonging to 3 as an anti-particle while the particles belong to 3 representation of the SU(3) group. There are conditions for the coupling constant that must be satisfied in order that a solution is acceptable as a physical bound state. We construct bound state solutions of 3 for a particle-particle pair, 1 for a particle-anti-particle pair and 1 for a three-particle system. We find that the singlet states form the tightest bound states among these solutions. We believe that our result provides some useful information for understanding quark confinement.

A New Bosonization Procedure of Two-Dimensional Fermion Models in the Path Integral Formulation

We present a new bosonization procedure for two-dimensional fermion models in the path integral formulation. We introduce external sources for the fermion currents instead of the auxiliary boson fields used in the conventional procedure. The calculations become much simpler because the external sources do not interact with themselves, while the auxiliary boson fields do, and we can find the equivalent boson models directy. All the formulations are done in Minkowski space-time. The extended Bukhvostov-Lipatov model and the massive Thirring model are discussed here as typical two-dimensional fermion models with currentcurrent-type interactions. The Rothe and Stamatescu model is also discussed, though this is slightly out of the context of bosonization, and the relation of the fermion current in the Heisenberg picture with that in the interaction picture is derived almost straightforwardly. Usually in the operator formalism, this relation is derived using the splitting procedure, which requires detailed calculations. It is expected that we can derive the boson model equivalent to the fermion model with SU(N) symmetry more easily and without any guess work.

Plateau transitions in the pairing model: Topology and selection rule

Based on the two-dimensional lattice fermion model, we discuss transitions between different pairing states. Each phase is labeled by an integer which is a topological invariant and characterized by vortices of the Bloch wavefunction. The transitions between phases with different integers obey a selection rule. Basic properties of the edge states are revealed. They reflect the topological character of the bulk. Transitions driven by randomness are also discussed numerically.

The Wess-Zumino-Witten term in non-commutative two-dimensional fermion models

We study the effective action associated to the Dirac operator in two dimensional non-commutative Field Theory. Starting from the axial anomaly, we compute the determinant of the Dirac operator and we find that even in the U(1) theory, a Wess-Zumino-Witten like term arises.

Bogoliubov-quasiparticle spectra of the effectived-wave model for cuprate superconductivity

An exact-diagonalization technique on finite-size clusters is used to study the ground state and excitation spectra of the two-dimensional effective fermion model, a fictious model of hole quasiparticles derived from numerical studies of the two-dimensional t-J model at low doping. We show that there is actually a reasonable range of parameter values where the $d_{x^2-y^2}$-wave pairing of holes occurs and the low-lying excitation can be described by the picture of Bogoliubov quasiparticles in the BCS pairing theory. The gap parameter of a size $\Delta_d\simeq 0.13|V|$ (where $V$ is the attractive interaction between holes) is estimated at low doping levels. The paired state gives way to the state of clustering of holes for some stronger attractions.