Odd Number Of Fermions Research Articles

Simulating both parity sectors of the Hubbard model with tensor networks

Tensor networks are a powerful tool to simulate a variety of different physical models, including those that suffer from the sign problem in Monte Carlo simulations. The Hubbard model on the honeycomb lattice with non-zero chemical potential is one such problem. Our method is based on projected entangled pair states (PEPS) using imaginary time evolution. We demonstrate that it provides accurate estimators for the ground state of the model, including cases where Monte Carlo simulations fail miserably. In particular it shows near to optimal, that is linear, scaling in lattice size. We also present a novel approach to directly simulate the subspace with an odd number of fermions. It allows to independently determine the ground state in both sectors. Without a chemical potential this corresponds to half filling and the lowest energy state with one additional electron or hole. We identify several stability issues, such as degenerate ground states and large single particle gaps, and provide possible fixes.

Improved treatment of blocking effect at finite temperature

The blocking effect caused by the odd particle on the pairing properties of systems with odd number of fermions at finite temperature interacting via the monopole pairing force is studied within several approximations. The results are compared with the predictions obtained by using the exact solutions of the pairing Hamiltonian. The comparison favors the approximation with the odd particle occupying the top level, which is the closest to the Fermi surface and whose occupation number decreases with increasing temperature.

Entanglement in fermionic Fock space

We propose a generalization of the usual stochastic local operations and classical communications (SLOCC) and local unitary (LU) classifications of entangled pure state fermionic systems based on the spin group. Our generalization relies on the fact that there is a representation of this group acting on the fermionic Fock space, which, when restricted to fixed particle number subspaces, naturally recovers the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space, which change the particle number. The classification scheme built on the spin group naturally prohibits entanglement between states containing even and odd numbers of fermions. In our scheme the problem of the classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so-called pure spinors. We construct the basic invariants of the spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states; an intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to the configurations of wrapped membranes reinterpreted as qubits.

Quantum dots for demonstration of additional dimensions of spacetime

By the example of electron mesoscopic systems, we show the impossibility of constraints of the quantum principle of superposition imposed by the superselection rule. This rule was introduced by Wick, Wightman, and Wigner in order to avoid the violation of Lorentz invariance due to the absence of physical invariance under rotations by an angle of 2π in states which are a coherent superposition of states with an even and odd number of fermions. We describe a mesoscopic system (a semiconductor double quantum dot at low temperatures) where such superpositions are realized; this is confirmed by experiments. We suggest a new experiment which explicitly demonstrates the absence of physical invariance under rotations by an angle of 2π. We note that an alternative to the superselection rule is the existence (along with x, y, z, and t) of additional spinor (Grassmann) dimensions of spacetime introduced in quantum field theory for realization of supersymmetry. It is proved that additional dimensions are real; their physical meaning is clarified for nonrelativistic systems of fermions.

Can an odd number of fermions be created due to the chiral anomaly?

We describe a possible creation of an odd number of fractionally charged fermions in the 1+1 dimensional Abelian Higgs model. We point out that for 1+1 dimensions this process does not violate any symmetries of the theory, nor makes it mathematically inconsistent. We construct the proper definition of the fermionic determinant in this model and outline how to generalize it for calculation of preexponent in realistic mathematically consistent 3+1 dimensional models with the creation of an even number of fermions. © 2006 The American Physical Society.

Existence of Baryons, Baryon Spectrum and Mass Splitting in Strong Coupling Lattice QCD

We consider a functional integral formulation for one-flavor lattice Quantum Chromodynamics in d=2,3 space dimensions and imaginary time, and work in the regime of the small hopping parameter , and zero plaquette coupling. Following the standard construction, this model exhibits positivity which is used to obtain the underlying physical Hilbert space . Then, using a Feynman-Kac formalism, we write the correlation functions for the model as functional integrals over the space of Grassmannian (fermionic) fields for one quark specie and the SU(3) gauge fields. We determine the energy-momentum spectrum associated with gauge invariant local baryon (anti-baryon) fields which are composites of three quark (anti-quark) fields. With the associated correlation functions, we establish a Feynman-Kac formula, and a spectral representation for the Fourier transform of the two-point functions. This representation allows us to show that baryons and anti-baryons arise as tightly bound, bound states of three (anti-)quarks. Labelling the components of the baryon fields by s=3/2,1/2,-1/2,-3/2, we show that the baryon and anti-baryon mass spectrum only depends on |s|, and the associated masses are given by M s = −3lnκ+r s (κ), where r s (κ) is real analytic in κ, for each d=2,3. The mass splitting is M 3/2 −M 1/2 =18κ6, for d=2 and, if any, is at least of (κ7), for d=3. In the subspace o ⊂ generated by an odd number of fermions, the baryon and anti-baryon energy-momentum dispersion curves are isolated up to near the baryon-meson threshold −5lnκ (upper gap property), identical and are determined up to (κ5). The symmetries of coordinate reflections, spatial lattice rotations, parity and charge conjugation are established for the correlation functions, and are shown to be implemented on by unitary (anti-unitary, for time reversal) operators.

Manifestation of additional dimensions of space-time in mesoscopic systems

We note that the existence of physical states which are coherent superpositions of states with even and odd numbers of fermions means the existence, together with x,y,z,t, of additional spinor dimensions of space-time. A system with variable number of electrons is described in which such superpositions are realized. Experiments with mesoscopic condensed matter systems are suggested which generalize the experiment of Nakamura et al. and may provide direct observation of such superpositions and, thereby, justify the reality of a superspace with additional spinor dimensions introduced in quantum field theory to account for supersymmetry. The nature of additional dimensions of space-time is elucidated for nonrelativistic systems.

Many-body systems interacting via a two-body random ensemble. I. Angular momentum distribution in the ground states

In this paper, we discuss the angular momentum distribution in the ground states of many-body systems interacting via a two-body random ensemble. Beginning with a few simple examples, a simple approach to predict P(I)'s, angular momenta I ground state (g.s.) probabilities, of a few solvable cases, such as fermions in a small single-j shell and d boson systems, is given. This method is generalized to predict P(I)'s of more complicated cases, such as even or odd number of fermions in a large single-j shell or a many-j shell, d-boson, sd-boson or sdg-boson systems, etc. By this method we are able to tell which interactions are essential to produce a sizable P(I) in a many-body system. The g.s. probability of maximum angular momentum $I_{max}$ is discussed. An argument on the microscopic foundation of our approach, and certain matrix elements which are useful to understand the observed regularities, are also given or addressed in detail. The low seniority chain of 0 g.s. by using the same set of two-body interactions is confirmed but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even fermions in a single-j shell. Preliminary results by taking a displaced two-body random ensemble are presented for the I g.s. probabilities.

Extreme location of F drip line and disappearance of theN=20magic structure

The drip line of F isotopes is discussed from the viewpoint of a unified description of exotic nuclei with $N\ensuremath{\sim}20.$ The structure of those nuclei is described by means of large-scale shell model calculations, most of which are Monte Carlo shell model. It is shown that, due to the narrowing of the $N=20$ shell gap towards smaller Z, the mixing among normal, intruder, and higher intruder configurations can be enhanced significantly, and can produce crucial effects, including the F drip line located extremely far from the O drip line. The feasibility of Monte Carlo shell model is demonstrated for systems with an odd number of fermions.

Quon Statistics for Composite Systems and a Limit on the Violation of the Pauli Principle for Nucleons and Quarks

The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons. The parameter $q$ attached to a quon labels a smooth interpolation between bosons ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+1$) and fermions ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$). Wigner and Ehrenfest and Oppenheimer showed that a composite system of identical bosons and fermions is a fermion if it contains an odd number of fermions and is a boson otherwise. We generalize this and show that ${q}_{\mathrm{composite}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q}_{\mathrm{constituent}}^{{n}^{2}}$ for a system of $n$ identical quons. Using this generalization, we find bounds on possible violations of the Pauli exclusion principle for nucleons and quarks based on such bounds for nuclei.

Random matrix theory and three-dimensional QCD.

We suggest that the special properties near zero virtuality of three-dimensional QCD follow from a Hermitian random matrix model. The exact spectral density is derived for this family of random matrix models for both an even and odd number of fermions. New sum rules for the inverse powers of the eigenvalues of the Dirac operator are obtained. The issue of anomalies in random matrix theories is discussed.

Boson-fermion mappings for odd systems from supercoherent states.

We extend the formalism whereby boson mappings can be derived from generalized coherent states to boson-fermion mappings for systems with an odd number of fermions. This is accomplished by constructing supercoherent states in terms of both complex and Grassmann variables. In addition to a known mapping for the full so(2N+1) algebra, we also uncover some other formal mappings, together with mappings relevant to collective subspaces.

Microscopic derivation of boson-fermion Hamiltonians.

A recent analysis of the correspondence between fermion and boson spaces is extended to systems with an odd number of fermions. A fermion space is now mapped onto a boson-fermion space. The case is discussed of the SD-j and sd-j spaces, i.e., spaces in which an odd particle moving in a j orbit is coupled to systems of S,D pairs and s,d bosons, respectively. The sd-j image of the quadrupole-quadrupole interaction is constructed both for systems of identical particles and for proton-neutron systems. Numerical comparisons are shown between spectra in the SD-j and sd-j spaces. The role of the exchange term in the sd-j Hamiltonian is analyzed. The comparison with other mapping techniques as well as with phenomenological Hamiltonians is discussed.

Dynamical breakdown of symmetries in three-dimensional QED.

We analyze the possibility of dynamical breakdown of parity in three-dimensional QED. We find that for an even number of fermions only chiral symmetry is broken dynamically while parity is conserved. For an odd number of fermions, where a parity-conserving mass generation is not possible, we find no consistent nontrivial solution for the Schwinger-Dyson equations, leading us to conjecture that at least one fermion remains massless and no mass is induced for the photon in this case.