We describe non-relativistic fermions on the lattice (Hubbard model) in the canonical formulation using transfer matrices in fixed fermion number sectors such that the partition function becomes fully factorized in time. By analytically integrating out the auxiliary Hubbard-Stratanovich field due to the four-fermion interaction, we express the system in terms of discrete, local fermion occupation numbers which are the only remaining degrees of freedom. We show the close relation to the fermion loop and the fermion bag formulation. One can prove that in 1+1 dimension the fermion sign problem is absent. Finally, we construct improved estimators for the ground state energy, 2-point functions, and for the chemical potential.We describe non-relativistic fermions on the lattice (Hubbard model) in the canonical formulation using transfer matrices in fixed fermion number sectors such that the partition function becomes fully factorized in time. By analytically integrating out the auxiliary Hubbard-Stratanovich field due to the four-fermion interaction, we express the system in terms of discrete, local fermion occupation numbers which are the only remaining degrees of freedom. We show the close relation to the fermion loop and the fermion bag formulation. One can prove that in 1+1 dimension the fermion sign problem is absent. Finally, we construct improved estimators for the ground state energy, 2-point functions, and for the chemical potential.

International Symposium on Lattice Field Theory (LATTICE)Jan 4, 20202 citations

Statistical mechanics of the strongly-correlated Hubbard chain. I. formulation

T. Matsubara, Colin J. Thompson

The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.

Journal of Physics A Mathematical and GeneralNov 21, 19910 citations

Lattice theory for nonrelativistic fermions in one spatial dimension

Michael G. Endres

I derive a loop representation for the canonical and grand-canonical partition functions for an interacting four-component Fermi gas in one spatial dimension and an arbitrary external potential. The representation is free of the ``sign problem'' irrespective of population imbalance, mass imbalance, and to a degree, sign of the interaction strength. This property is in sharp contrast with the analogous three-dimensional, two-component interacting Fermi gas, which exhibits a sign problem in the case of unequal masses, chemical potentials, and repulsive interactions. The one-dimensional system is believed to exhibit many phenomena in common with its three-dimensional counterpart, including an analog of the BCS-BEC crossover, and nonperturbative, universal few- and many-body physics at scattering lengths much larger than the range of interaction, making the theory an interesting candidate for numerical study. Positivity of the probability measure for the partition function allows for a mean-field treatment of the model; here, I present such an analysis for the interacting Fermi gas in the SU(4) (unpolarized, mass-symmetric) limit and demonstrate that there exists a phase in which a continuum limit may be defined.I derive a loop representation for the canonical and grand-canonical partition functions for an interacting four-component Fermi gas in one spatial dimension and an arbitrary external potential. The representation is free of the ``sign problem'' irrespective of population imbalance, mass imbalance, and to a degree, sign of the interaction strength. This property is in sharp contrast with the analogous three-dimensional, two-component interacting Fermi gas, which exhibits a sign problem in the case of unequal masses, chemical potentials, and repulsive interactions. The one-dimensional system is believed to exhibit many phenomena in common with its three-dimensional counterpart, including an analog of the BCS-BEC crossover, and nonperturbative, universal few- and many-body physics at scattering lengths much larger than the range of interaction, making the theory an interesting candidate for numerical study. Positivity of the probability measure for the partition function allows for a mean-field treatment of the model; here, I present such an analysis for the interacting Fermi gas in the SU(4) (unpolarized, mass-symmetric) limit and demonstrate that there exists a phase in which a continuum limit may be defined.

arXiv:1204.6182Physical Review AJun 29, 201213 citations

The Hubbard model with fermionic tensor networks

Manuel Schneider, Johann Ostmeyer, Karl Jansen +2 more

Many electromagnetic properties of graphene can be described by the Hubbard model on a honeycomb lattice. However, this system suffers strongly from the sign problem if a chemical potential is included. Tensor network methods are not affected by this problem. We use the imaginary time evolution of a fermionic projected entangled pair state, which allows to simulate both parity sectors independently. Incorporating the fermionic nature on the level of the tensor network allows to fix the particle number to be either even or odd. This way we can access the states at half filling and with one additional electron. We calculate the energy and other observables of both states, which was not possible before with Monte Carlo methods.Many electromagnetic properties of graphene can be described by the Hubbard model on a honeycomb lattice. However, this system suffers strongly from the sign problem if a chemical potential is included. Tensor network methods are not affected by this problem. We use the imaginary time evolution of a fermionic projected entangled pair state, which allows to simulate both parity sectors independently. Incorporating the fermionic nature on the level of the tensor network allows to fix the particle number to be either even or odd. This way we can access the states at half filling and with one additional electron. We calculate the energy and other observables of both states, which was not possible before with Monte Carlo methods.

arXiv:2110.15340International Symposium on Lattice Field Theory (LATTICE)May 16, 20223 citations

Grand Canonical Partition Function for Unidimensional Systems: Application to Hubbard Model up to Order beta^3

I. C. Charret, S. M. de Souza, M. T. Thomaz +1 more

We exploit the grassmannian nature of the variables involved in the path integral expression of the grand canonical partition function for self--interacting fermionic models to show, in one-space dimension, a general relation among the terms of it expansion in the high temperature limit and a combination of co-factors of a suitable matrix with commuting entries. As an application, we apply this framework to calculate the exact coefficients, up to order β^3, of the expansion of the grand canonical partition function for the Hubbard model in d=(1+1) in the high temperature limit. The results are valid for any set of parameters that characterize the model.We exploit the grassmannian nature of the variables involved in the path integral expression of the grand canonical partition function for self--interacting fermionic models to show, in one-space dimension, a general relation among the terms of it expansion in the high temperature limit and a combination of co-factors of a suitable matrix with commuting entries. As an application, we apply this framework to calculate the exact coefficients, up to order β^3, of the expansion of the grand canonical partition function for the Hubbard model in d=(1+1) in the high temperature limit. The results are valid for any set of parameters that characterize the model.

arXiv:cond-mat/9607171Jul 24, 19960 citations

Momentum occupation number bounds for interacting fermions

Matthew F. Lapa

We derive rigorous bounds on the average momentum occupation numbers $\langle n_{\mathbf{k}σ}\rangle$ in the Hubbard and Kondo models in the ground state and at non-zero temperature ( $T>0$ ) in the grand canonical ensemble. For the Hubbard model with $T>0$ our bound proves that, when interaction strength $\ll k_B T\ll$ Fermi energy, $\langle n_{\mathbf{k}σ}\rangle$ is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any $T>0$ our bound proves that $\langle n_{\mathbf{k}σ}\rangle$ tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that $\langle n_{\mathbf{k}σ}\rangle$ approaches its non-interacting value as $\mathbf{k}$ moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generalized versions of the Hubbard and Kondo models. Our proofs use the Fermi statistics of the particles in a fundamental way.We derive rigorous bounds on the average momentum occupation numbers $\langle n_{\mathbf{k}σ}\rangle$ in the Hubbard and Kondo models in the ground state and at non-zero temperature ($T>0$) in the grand canonical ensemble. For the Hubbard model with $T>0$ our bound proves that, when interaction strength $\ll k_B T\ll$ Fermi energy, $\langle n_{\mathbf{k}σ}\rangle$ is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any $T>0$ our bound proves that $\langle n_{\mathbf{k}σ}\rangle$ tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that $\langle n_{\mathbf{k}σ}\rangle$ approaches its non-interacting value as $\mathbf{k}$ moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generalized versions of the Hubbard and Kondo models. Our proofs use the Fermi statistics of the particles in a fundamental way.

arXiv:2106.13818Jun 25, 20210 citations

We derive rigorous bounds on the average momentum occupation numbers

\langle n_{\mathbf{k}σ}\rangle

in the Hubbard and Kondo models in the ground state and at non-zero temperature (

T>0

) in the grand canonical ensemble. For the Hubbard model with

T>0

our bound proves that, when interaction strength

\ll k_B T\ll

Fermi energy,

\langle n_{\mathbf{k}σ}\rangle

is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any

T>0

our bound proves that

\langle n_{\mathbf{k}σ}\rangle

tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that

\langle n_{\mathbf{k}σ}\rangle

approaches its non-interacting value as

\mathbf{k}

moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generalized versions of the Hubbard and Kondo models. Our proofs use the Fermi statistics of the particles in a fundamental way.