© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

CRC Press eBooks

Quantum Monte Carlo Studies of the Attractive Hubbard Hamiltonian

Richard T. Scalettar, G. G. Batrouni

Published Nov 2, 2010• 5 citations

Abstract

Richard T. Scalettar Physics Department, University of California, Davis, California 95616, U.S.A. George G. Batrouni INLN, Universite´ de Nice - Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France Originally introduced almost ﬁfty years ago, the Hubbard Hamiltonian, H = −t ∑ 〈ij〉 ∑ i (ni↑− 12)(ni↓− 1 2 )−μ ∑ i (ni↑+ni↓), (11.1) has been widely, and very successfully, used to model many-body eﬀects and quantum phase transitions (QPTs) in condensed matter, despite the simpliﬁcations it embodies [1-4]; see also Chap. 1. The Hamiltonian, given by Eq. (11.1), contains a kinetic energy term t which describes the hopping of two sets of fermionic particles, labeled by σ, on near-neighbor sites 〈ij〉 of the lattice.1 These fermions feel an interaction energy U if they occupy the same site, a term written here in a symmetric form which makes chemical potential μ = 0 correspond to half-ﬁlling ρ = 〈∑σ niσ〉 = 1 at any value of t, U or temperature T , as long as the lattice is bipartite. Many diﬀerent geometries have been studied, including 1D chains, 2D square, triangular, and honeycomb structures, and 3D cubic lattices.

DOI: 10.1201/b10273-18

Related papers

Towards quantum simulation of the Hubbard model with attractively interacting fermions

Debayan Mitra, Peter Brown, Mark Stone +2 more

Bulletin of the American Physical Society•Jun 10, 2015•0 citations

- Quantum Fields on a Lattice

Girish S. Setlur

So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is c†iσc jσ, (9.1) where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as V =U∑ i ni↑ni↓, (9.2) where niσ = c † iσciσ is the number of electrons in site i. These two put together, form the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well. The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as

CRC Press eBooks•Dec 5, 2013•15 citations

Superconductivity and charge density wave order in the two-dimensional Holstein model

Owen Bradley, G. G. Batrouni, Richard T. Scalettar

The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite ${T}_{\mathrm{cdw}}$ in two dimensions. Quantum Monte Carlo calculations have determined ${T}_{\mathrm{cdw}}$ for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in $d=2$ is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine ${T}_{\mathrm{sc}}$ for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where $t$ is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher $T$. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.

arXiv:2011.11703•Physical review. B./Physical review. B•Jun 1, 2021•52 citations

Diagrammatic Monte Carlo Dual Fermions study of the 2D Hubbard model

Sergei Iskakov, Emanuel Gull

Bulletin of the American Physical Society•Mar 15, 2017•0 citations

Semimetal–superfluid quantum phase transitions in 2D and 3D lattices with Dirac points

G Mazzucchi, Luca Lepori, Andrea Trombettoni

We study the superfluid properties of attractively interacting fermions hopping in a family of 2D and 3D lattices in the presence of synthetic gauge fields having π-flux per plaquette. The reason for such a choice is that the π-flux cubic lattice displays Dirac points and that decreasing the hopping coefficient in a spatial direction (say, t z ), these Dirac points are unaltered: it is then possible to study the 3D–2D interpolation towards the π-flux square lattice. We also consider the lattice configuration providing the continuous interpolation between the 2 D π-flux square lattice and the honeycomb geometry. We investigate by a mean-field analysis the effects of interaction and dimensionality on the superfluid gap, chemical potential and critical temperature, showing that these quantities continuously vary along the patterns of interpolation. In the two-dimensional cases at zero temperature and half-filling, there is a quantum phase transition occurring at a critical (negative) interaction U c presenting a linear critical exponent for the gap as a function of | U − U c |. We show that in three dimensions, this quantum phase transition is again retrieved, pointing out that the critical exponent for the gap changes from 1 to 1/2 for each finite value of t z .

arXiv:1306.1441•Journal of Physics B Atomic Molecular and Optical Physics•Jun 24, 2013•13 citations

Quantum Monte Carlo Simulations of the Two-Dimensional Attractive Hubbard Model: Phase Diagram and Spectral Properties

Johannes Maria Singer, T. Schneider, P. F. Meier

Springer eBooks•Jan 1, 1999•0 citations

Richard T. Scalettar Physics Department, University of California, Davis, California 95616, U.S.A. George G. Batrouni INLN, Universite´ de Nice - Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France Originally introduced almost ﬁfty years ago, the Hubbard Hamiltonian, H = −t ∑ 〈ij〉 ∑ i (ni↑− 12)(ni↓− 1 2 )−μ ∑ i (ni↑+ni↓), (11.1) has been widely, and very successfully, used to model many-body eﬀects and quantum phase transitions (QPTs) in condensed matter, despite the simpliﬁcations it embodies [1-4]; see also Chap. 1. The Hamiltonian, given by Eq. (11.1), contains a kinetic energy term t which describes the hopping of two sets of fermionic particles, labeled by σ, on near-neighbor sites 〈ij〉 of the lattice.1 These fermions feel an interaction energy U if they occupy the same site, a term written here in a symmetric form which makes chemical potential μ = 0 correspond to half-ﬁlling ρ = 〈∑σ niσ〉 = 1 at any value of t, U or temperature T , as long as the lattice is bipartite. Many diﬀerent geometries have been studied, including 1D chains, 2D square, triangular, and honeycomb structures, and 3D cubic lattices.

So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is c†iσc jσ, (9.1) where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as V =U∑ i ni↑ni↓, (9.2) where niσ = c † iσciσ is the number of electrons in site i. These two put together, form the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well. The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as

The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite

{T}_{\mathrm{cdw}}

in two dimensions. Quantum Monte Carlo calculations have determined

{T}_{\mathrm{cdw}}

for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in

d=2

is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine

{T}_{\mathrm{sc}}

for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where

t

is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher

T

. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.

We study the superfluid properties of attractively interacting fermions hopping in a family of 2D and 3D lattices in the presence of synthetic gauge fields having π-flux per plaquette. The reason for such a choice is that the π-flux cubic lattice displays Dirac points and that decreasing the hopping coefficient in a spatial direction (say, t z ), these Dirac points are unaltered: it is then possible to study the 3D–2D interpolation towards the π-flux square lattice. We also consider the lattice configuration providing the continuous interpolation between the 2 D π-flux square lattice and the honeycomb geometry. We investigate by a mean-field analysis the effects of interaction and dimensionality on the superfluid gap, chemical potential and critical temperature, showing that these quantities continuously vary along the patterns of interpolation. In the two-dimensional cases at zero temperature and half-filling, there is a quantum phase transition occurring at a critical (negative) interaction U c presenting a linear critical exponent for the gap as a function of | U − U c |. We show that in three dimensions, this quantum phase transition is again retrieved, pointing out that the critical exponent for the gap changes from 1 to 1/2 for each finite value of t z .