Fermion Formulation Research Articles

Effective short distance interaction in Calogero-Sutherland quantum fluids

We consider the effective conformal field theory with symmetry W1+∞×W‾1+∞ that describes the thermodynamic limit of the Calogero-Sutherland model. In the repulsive regime of the free fermion formulation, we identify an attractive interaction between opposite moving particle-hole pairs that dominates the short distance behavior and that is proposed as responsible for the destabilization of the ground state, leading to a new one of bosonic nature. The process is described by a Bogoliubov transformation of the free fermion bilinear operators into bosonic ones, preserving the form of the W1+∞ algebra but decoupling the opposite chirality terms in the hamiltonian, as expected in the low energy limit. In coordinate space this interaction has a short range component that arises due to the quantum regularization of the theory. The described dynamical process may be considered as a mechanism of the emergence of the known charge and quantum statistics fractionalization of the low lying excitations of the theory, as predicted in both first and second quantization studies.

Pairing phase favored by magnetic frustration.

The interplay between magnetic frustration and pairing is investigated by adopting a BCS-like pairing mechanism on the frustrated $J_1-J_2$ Ising model on the square lattice. The ground-state and thermal phase transitions of the model are analyzed using a fermionic formulation within a cluster mean-field method. In this approach, the lattice system is divided into identical clusters, where the intracluster dynamic is exactly solved, and the intercluster interactions are replaced by self-consistent mean fields. We introduce a framework with two pairing couplings: an intracluster local coupling, $g$, which controls the electron pairing and its mobility within the clusters, and an intercluster coupling, $g'$, which adjusts the pairing mechanism between clusters. Tuning $g'/g$ allows evaluating how the pairing phase evolves from a weak pairing coupling between clusters (clustered system) to a strong one ($g' \rightarrow g$, homogeneous system). In the range $0 \le g'/g \le 1$, we find that a gradual increase in $g'/g$ favors the pairing phase and induces a change in criticality. In particular, our results reveal the presence of tricriticality for a certain range of $g'/g$. In addition, an increase in competing magnetic interactions weakens the magnetic orders, causing the pairing phase to occur at lower strengths of pairing interactions, especially when $g' = g$. Therefore, our findings support that magnetic frustration favors the pairing phase, contributing to the onset of a superconducting state.

Vacuum energy of nonsupersymmetric S˜ heterotic string models

We use the free fermionic formulation of the heterotic string in four dimensions to study the vacuum structure and energy of nonsupersymmetric tachyon free models that correspond to compactifications of tachyonic vacua of the ten-dimensional heterotic string. We explore the class of heterotic SO(10) nonsupersymmetric models constructed from the S˜ model in the free fermionic formalism and investigate the dependence of the potential on the geometric moduli. This paper will explore a sample of 109 string vacua to find the frequency of viable models, classifying these vacua by the following fertility criteria: tachyon presence, number of spinorial 16/16¯ representations, vectorial 10 states, and top quark mass coupling compatibility. Of these we find those that mimic supersymmetric models with equal number of bosons and fermions at the massless level: a00=0. Tachyon free models occur with a frequency of 5.309×10−3. Furthermore, models that fulfil the rest of the phenomenological fertility conditions and the additional condition on a00 occur with probability 4.0×10−9 We analyze the partition functions and study the moduli dependence of such models, finding that almost all fertile models have finite, positive potential at the free fermionic point, with 2 out of 84 of the fertile cores having negative, finite potential. We demonstrate that the free fermionic point is not necessarily a minimum in the potential. This work provides further evidence that supersymmetry may not be a necessary ingredient of phenomenological models, recreating many of the desirable features of such models without employing supersymmetry. Published by the American Physical Society 2024

Fermionic shift symmetries in (anti) de Sitter space

We study extended shift symmetries that arise for fermionic fields on anti-de Sitter (AdS) space and de Sitter (dS) space for particular values of the mass relative to the curvature scale. We classify these symmetries for general mixed-symmetry fermionic fields in arbitrary dimension and describe how fields with these symmetries arise as the decoupled longitudinal modes of massive fermions as they approach partially massless points. For the particular case of AdS4, we look for non-trivial Lie superalgebras that can underly interacting theories that involve these fields. We study from this perspective the minimal such theory, the Akulov-Volkov theory on AdS4, which is a non-linear theory of a spin-1/2 Goldstino field that describes the spontaneous breaking of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry on AdS4 down to the isometries of AdS4. We show how to write the nonlinear supersymmetry transformation for this theory using the fermionic ambient space formalism. We also study the Lie superalgebras of candidate multi-field examples and rule out the existence of a supersymmetric special galileon on AdS4.

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.

Dynamical chirality production in one dimension

We discuss the quantum computation of dynamical chirality production in lattice gauge theory. Although the chirality of a lattice fermion is complicated in general dimensions, it can be simply formulated on a one-dimensional lattice. The chiral fermion formalism enables us to extract the physical part of the chirality production that would be interpreted as the chiral anomaly in the continuous theory. We demonstrate the computation of the Z2 lattice gauge theory on a classical emulator. Published by the American Physical Society 2024

From free-fermionic constructions to orbifolds and back

We systematically develop the explicit map between string vacua constructed in the Free Fermionic Formulation and their ℤ2N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_2^N $$\\end{document} toroidal orbifold counterparts. We illustrate the map in various example classes of models, including cases relevant for string phenomenology, as well as in theories where space-time supersymmetry is broken by the stringy Scherk-Schwarz mechanism.

Frustrated fermionic [formula omitted] model with pairing interaction

We study the effects of a BCS-like pairing mechanism on the frustrated J1−J2 model on the square lattice. The model is expressed in a fermionic formulation, in which the fermions that represent the magnetic moments can also participate in the pair formation. A cluster mean-field approach allows to get an effective self-consistent problem in the grand canonical potential at half-filling. The competing antiferromagnetic (AF) interactions lead to a phase diagram with different AF orders separated by a discontinuous phase transition at the highly frustrated regime. As the strength of pairing increases, the criticality of the highly frustrated region is affected, with the arising of a triple point. Beyond a pairing strength threshold, the system exhibits absence of magnetic orders with a high density of double occupied sites near the highly frustrated point. Therefore, the frustration can assist the pairing by weakening the magnetic orders, giving possibility to a nonmagnetic ground-state.

Spectroscopy in the 2+1D Thirring model with N=1 domain wall fermions

We employ the domain wall fermion (DWF) formulation of the Thirring model on a lattice in $2+1+1$ dimensions and perform $N=1$ flavor Monte Carlo simulations. At a critical interaction strength the model features a spontaneous $\mathrm{U}(2)\ensuremath{\rightarrow}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$ symmetry breaking; we analyze the induced spin-0 mesons, both Goldstone and non-Goldstone, as well as the correlator of the fermion quasiparticles, in both resulting phases. Crucially, we determine the anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}\ensuremath{\approx}3$ at the critical point, in stark contrast with the Gross-Neveu model in 3D and with results obtained with staggered fermions. Our numerical simulations are complemented by an analytical treatment of the free fermion correlator, which exhibits large early-time artifacts due to branch cuts in the propagator stemming from unbound interactions of the fermion with its heavy doublers. These artifacts are generalizable beyond the Thirring model, being an intrinsic property of DWF, or more generally Ginsparg-Wilson fermions.

Recent advances in fermionic hierarchical equations of motion method for strongly correlated quantum impurity systems

Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.

Mass spectroscopy of excited light mesons using truncated overlap fermions

We study excited light mesons by quenched lattice quantum chromodynamics (QCD) simulations with a truncated overlap fermion formalism based on domain wall fermions. Truncated overlap fermions satisfy lattice chiral symmetry instead of chiral symmetry in continuum field theory, as for domain wall fermions, but offer lower simulation costs. Our results show good agreement with the experimental values for the excited state of a 1, ρ, and π mesons, and demonstrate that a 1(1260) and a 1(1640) are simple two-quark states, whereas a 1(1420) may have a more complicated structure. The results are similar to those of previous dynamical studies using clover-Wilson fermions or chirally improved fermions, even though our lattice QCD calculations are performed with the quenched approximation. The study shows that lattice QCD simulations using truncated overlap fermions are essential in lattice studies of excited states.

Non-Hermitian skin effect in quadratic Lindbladian systems: An adjoint fermion approach

The skin effect has been discovered in non-Hermitian Hamiltonian systems where all the eigenstates have their amplitudes concentrating to the open boundaries of the systems and decaying exponentially into the bulk. Later, certain open systems obeying the quadratic Lindblad equation has also been found to exhibit the skin effect, which is manifested in the ``chiral damping" phenomenon as the particle populations, decaying from their initial uniform unity values, show asymmetry with respect to the open boundaries. However, in those open systems, each cell couples to the environment in an identical way. It is natural to expect that the long time steady state of those open systems shall have spatially uniform particle populations. Furthermore, particle population variations due to the excitation of normal modes on top of the steady state shall also not show asymmetry with respect to the open boundaries. To reconcile the natural expectations with the skin effect, we employ an adjoint fermion formalism to study the quadratic Lindbladian systems. We work out the long time steady state and the normal modes on top of it, which exhibit no asymmetry as expected. We show that it is the interference between the normal modes that gives rise to the skin effect.

Form factors of B→πℓν and a determination of |Vub| with Möbius domain-wall fermions

Using a fully relativistic lattice fermion action, we compute the form factors of the semileptonic decay $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\ell}\ensuremath{\nu}$, which is required for the determination of the Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{ub}|$. We employ the M\"obius domain-wall fermion formalism for the generation of lattice ensembles with $2+1$ sea quark flavors as well as for the valence heavy and light quarks. We compute the form factors at various values of the lattice spacing and multiple light and heavy quark masses, and extrapolate the results to the physical point. We combine our lattice results with the available experimental data to obtain $|{V}_{ub}|=(3.93\ifmmode\pm\else\textpm\fi{}0.41)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$.

Limit shape phase transitions: a merger of arctic circles

We consider a free fermion formulation of a statistical model exhibiting a limit shape phenomenon. The model is shown to have a phase transition that can be visualized as the merger of two liquid regions – arctic circles. We show that the merging arctic circles provide a space-time resolved picture of the phase transition in lattice QCD known as Gross–Witten–Wadia transition. The latter is a continuous phase transition of the third order. We argue that this transition is universal and is not spoiled by interactions if parity and time-reversal symmetries are preserved. We refer to this universal transition as the merger transition.

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.

The plain and simple parquet approximation: single-and multi-boson exchange in the two-dimensional Hubbard model

The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16times 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation.Graphical abstract

Particle physics and cosmology of the string derived no-scale flipped SU(5)

In a recent paper, we identified a cosmological sector of a flipped SU(5) model derived in the free fermionic formulation of the heterotic superstring, containing the inflaton and the goldstino superfields with a superpotential leading to Starobinsky type inflation, while SU(5){times }U(1) is still unbroken. Here, we study the properties and phenomenology of the vacuum after the end of inflation, where the gauge group is broken to the Standard Model. We identify a set of vacuum expectation values, triggered by the breaking of an anomalous U(1)_A gauge symmetry at roughly an order of magnitude below the string scale, that solve the F and D-flatness supersymmetric conditions up to 6th order in the superpotential which is explicitly computed, leading to a successful particle phenomenology. In particular, all extra colour triplets become superheavy guaranteeing observable proton stability, while the Higgs doublet mass matrix has a massless pair eigenstate with realistic hierarchical Yukawa couplings to quarks and leptons. The supersymmetry breaking scale is constrained to be high, consistent with the non observation of supersymmetry signals at the LHC.

Channel order in the two-channel Kondo lattice

We discuss the two-channel Kondo lattice model where a single localized spin per unit cell is coupled to two different conduction electron orbitals per unit cell. The calculation is done using the bond fermion formalism. For a Hamiltonian which is symmetric under exchange of the conduction channels we find a spontaneous breaking of this symmetry, i.e., the Kondo singlets are formed almost exclusively between the localized spin and only one of the two conduction orbitals. In addition to this channel-ferromagnetic phase we also find a channel-antiferromagnetic phase where the preferred conduction electron orbital alternates between sublattices. The parameter which determines the transition between these phases is the interchannel hybridization.

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.

Ground states of Heisenberg spin clusters from projected Hartree-Fock theory

We apply the projected Hartree-Fock theory (PHF) for approximating ground states of Heisenberg spin clusters. Spin-rotational, point-group, and complex-conjugation symmetry are variationally restored from a broken-symmetry mean-field reference, where the latter corresponds to a product of local spin states. A fermionic formulation of the Heisenberg model furnishes a conceptual connection to PHF applications in quantum chemistry and detailed equations for a self-consistent field optimization of the reference state are provided. Different PHF variants are benchmarked for ground-state energies and spin-pair correlation functions of antiferromagnetic spin rings and three different polyhedra, with various values of the local spin-quantum number $s$. Although PHF is not suitable to study the thermodynamic limit (where it reduces to the conventional HF results), the low computational cost and the compact wave-function representation make PHF a promising complement to existing approaches for ground states of finite spin clusters, particularly for large local spin $s$ and a moderately large number of sites $N$. The present work may also motivate future explorations of more accurate post-PHF methods for Heisenberg spin clusters.