Elementary Introduction to the Hubbard Model (original) (raw)
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A fundamental concept in quantum-information processing is the connection of a quantum computational model to a physical system by transformations of closed-operator algebras. The Jordan-Wigner isomorphism gives the correspondence between the algebra of the computational model and the physical model. An electronic system can be represented in terms of the algebra generated by the fermionic creation and annihilation operators or algebra of the Pauli operators. We discuss quantum simulation of the fermionic model by a quantum computer, which is expressed in the language and algebra of quantum spin-1/2 objects Pauli algebra. We apply this method to the paradigmatic cases of 1D and 2D Fermi-Hubbard models, involving couplings with nearest neighbors. We construct the quantum circuit of this model and simulate the system in the IBM quantum computer.
Two-particle states in the Hubbard model
Journal of Physics B: Atomic, Molecular and Optical Physics, 2008
We consider a pair of bosonic particles in a one-dimensional tight-binding periodic potential described by the Hubbard model with attractive or repulsive on-site interaction. We derive explicit analytic expressions for the two-particle states, which can be classified as (i) scattering states of asymptotically free particles, and (ii) interaction-bound dimer states. Our results provide a very transparent framework to understand the properties of interacting pairs of particles in a lattice.
Note on the one-dimensional Hubbard model
Physical Review B, 1974
By use of a Jordan-Signer transformation, the one-dimensional Hubbard model for an itinerant magnet is expressed as an anisotropic Heisenberg model on a double chain with twoand three-spin interactions. The limiting cases={i) free electron, (ii) strong correlation, and (iii) zero-hopping rateare briefly examined in the spin representation. The Hubbard model' is perhaps the simplest model to account for the importance of Coulombic correlations in the theory of itinerant-electron magnetism. The general d-dimensional version of this model is intractableexcept in certain limiting cases. However, progress has been made toward complete characterization of the one-dimensional (1-D) modelnotably by Lieb and Wu.T he 1-D case is of more-than-academic interest, since it is believed to be a good model for various organic charge-transfer salts. s'4 The 1-D Hubba, rd Hamiltonian' may be written as 8= TQQ[C, (f+1)~C, (f) ty /=1 4+1 +C,(l)'C, (l+1)]+U Q n, (l) n, (f), where C, (l)~and C, (f) are the annihilation and creation operators, respectively, for a Fermi
The Hubbard model, spin degeneracy and Ising spins
Philosophical Magazine B, 2002
The Hubbard model is used as a starting point for a study of an electronic system with spin 1. Using a functional integral representation at half-filling it is demonstrated that the spin degeneracy is equivalent to dynamic Ising spins coupled to the fermions. A magnetic phase transition of the model is related to a transition of the Ising spins from a paramagnetic to an antiferromagnetic phase. A metal-insulator transition in the paramagnetic phase can be described within this approach by the Green's function of non-interacting fermions, coupled to Ising spins. This picture is compared with the earlier king spin representation of the Hubbard model by Hirsch and discussed in terms of a small hopping amplitude. Addison-Wesley). Addison-Wesley). 39, 4711.
Thermodynamics and excitations of the one-dimensional Hubbard model
Physics Reports, 2000
We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu equations, paying particular attention to so-called 'string solutions'. Two kinds of string solutions occur: Λ strings, related to spin degrees of freedom and k-Λ strings, describing spinless bound states of electrons. Whereas Λ strings were thoroughly studied in the literature, less is known about k-Λ strings. We carry out a thorough analytical and numerical analysis of k-Λ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach and the quantum transfer matrix approach, respectively. The Yang-Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero temperature limit of the Yang-Yang approach.
On the particle-hole symmetry of the fermionic spinless Hubbard model in D=1
Condensed Matter Physics, 2014
We revisit the particle-hole symmetry of the one-dimensional (D = 1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 X X Z Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the X X Z Heisenberg model. The results presented in this paper are valid for any value of the interaction strength parameter V , which describes the attractive/null/repulsive interaction of neighboring fermions.
An introduction to the Hubbard model
The Hubbard model is very important for the study of of magnetic phenomena and strongly correlated electron systems. This work serves as an introduction to the Hubbard model and a presentation of the elements necessary to reach it. Here it is applied to a simple case to see how you work with it.
States and energies of the periodic Hubbard model at half filling
Nuclear Physics B, 1996
The representation of the periodic Hubbard model in the Clifford algebra leads to explicit expressions for several families of non-trivial half-filled states in any number of dimensions. A generalization of these expressions explains the structure of the spectrum of the general Hubbard hamiltonian.
The Hubbard model: bosonic excitations and zero-frequency constants
Physica C: Superconductivity, 2004
A fully self-consistent calculation of the bosonic dynamics of the Hubbard model is developed within the Composite Operator Method. From one side we consider a basic set of fermionic composite operators (Hubbard fields) and calculate the retarded propagators. On the other side we consider a basic set of bosonic composite operators (charge, spin and pair) and calculate the causal propagators. The equations for the Green's functions (GF) (retarded and causal), studied in the polar approximation, are coupled and depend on a set of parameters not determined by the dynamics. First, the pair sector is self-consistently solved together with the fermionic one and the zero-frequency constants (ZFC) are calculated not assuming the ergodic value, but fixing the representation of the GF in such a way to maintain the constrains required by the algebra of the composite fields. Then, the scheme to compute the charge and spin sectors, ZFCs included, is given in terms of the fermionic and pair correlators.
In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral t, on-site repulsion U , and N 2 a ≫ 1 sites consistent with its exact global SO(3) × SO(3) × U (1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron-rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t > 0 the occupancy configurations of these objects generate the energy eigenstates that span the one-and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spindensity ground state of one-and two-electron operators. Our description involves three basic objects: charge c fermions, spin-1/2 spinons, and η-spin-1/2 η-spinons. Independent spinons and independent η-spinons are invariant under the above unitary transformation. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the η-spinon (and spinons) that are not invariant under that transformation have η spin 1/2 (and spin 1/2) but are part of η-spinneutral (and spin-neutral) 2ν-η-spinon (and 2ν-spinon) composite ην fermions (and sν fermions) where ν = 1, 2, ... is the number of η-spinon (and spinon) pairs. The occupancy configurations of the c fermions, independent spinons and 2ν-spinon composite sν fermions, and independent η-spinons and 2ν-η-spinon composite ην fermions correspond to the state representations of the U (1), spin SU (2), and η-spin SU (2) symmetries, respectively, associated with the model SO(3) × SO(3) × U (1) = [SU (2)×SU (2)×U (1)]/Z 2 2 global symmetry. The components of the αν fermion discrete momentum values qj = [qj x1 , qj x2 ] are eigenvalues of the corresponding set of αν translation generators in the presence of fictitious magnetic fields Bαν. Our operator description has been constructed to inherently the αν translation generatorsˆ q αν in the presence of the fictitious magnetic field Bαν commuting with the momentum operator, consistently with their component operatorsqαν x 1 and qαν x 1 commuting with each other. In turn, unlike for the 1D model such generators not commute in general with the Hamiltonian, except for the Hubbard model on the square lattice in the oneand two-electron subspace. Concerning one-and two-electron excitations, the picture that emerges is that of a two-component quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions. The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x = 0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x > 0. For 0 < x ≪ 1 the latter short-range spin order has an incomensurate-spiral character. Our results are of interest for studies of ultra-cold fermionic atoms on optical lattices and elsewhere evidence is provided that upon addition of a small three-dimensional anisotropy plane-coupling perturbation to the square-lattice quantum liquid considered here its short-range spin order coexists for low temperatures and a well-defined range of hole concentrations with a long-range superconducting order so that the use of the general description introduced in this paper contributes to the further understanding of the role of electronic correlations in the unusual properties of the hole-doped cuprates.