We present a general derivation of semi-fermionic representation for generators of SU(N) group as a bilinear combination of Fermi operators. The constraints are fulfilled by means of imaginary Lagrange multipliers. The important case of SU(2) group is discussed. We demonstrate how the idea of semi-fermionic representation might be extended to the groups possessing dynamic symmetries. As an example, SO(4) group is considered. We illustrate the application of semi-fermionic representations for various problems of strongly correlated physics.We present a general derivation of semi-fermionic representation for generators of SU(N) group as a bilinear combination of Fermi operators. The constraints are fulfilled by means of imaginary Lagrange multipliers. The important case of SU(2) group is discussed. We demonstrate how the idea of semi-fermionic representation might be extended to the groups possessing dynamic symmetries. As an example, SO(4) group is considered. We illustrate the application of semi-fermionic representations for various problems of strongly correlated physics.

arXiv:cond-mat/0212620Dec 28, 20021 citation

MODEL HAMILTONIANS WITH FERMION INTERACTION

WORLD SCIENTIFIC eBooksAug 19, 20140 citations

Exact diagonalization of Heisenberg SU(N) chains in the fully symmetric and antisymmetric representations

Pierre Nataf, Frédéric Mila

Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg $\mathrm{SU}(N)$ Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of $\mathrm{SU}(N)$ . The method, based on the use of standard Young tableaux, takes advantage of the full $\mathrm{SU}(N)$ symmetry, allowing one to work directly in each irreducible representation of the global $\mathrm{SU}(N)$ group. Since the $\mathrm{SU}(N)$ singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to $N=10$ and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [ $\mathrm{SU}(2)$ with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the $\mathrm{SU}(N)$ level $k$ Wess-Zumino-Witten universality class, where $k$ is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the $\mathrm{SU}(N)$ level 1 WZW universality class.Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg $\mathrm{SU}(N)$ Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of $\mathrm{SU}(N)$. The method, based on the use of standard Young tableaux, takes advantage of the full $\mathrm{SU}(N)$ symmetry, allowing one to work directly in each irreducible representation of the global $\mathrm{SU}(N)$ group. Since the $\mathrm{SU}(N)$ singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to $N=10$ and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [$\mathrm{SU}(2)$ with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the $\mathrm{SU}(N)$ level $k$ Wess-Zumino-Witten universality class, where $k$ is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the $\mathrm{SU}(N)$ level 1 WZW universality class.

arXiv:1601.00922Physical review. B./Physical review. BApr 18, 201630 citations

Semiclassical theory for many-body fermionic systems

Pierre Gaspard, Sudhir R Jain

arXiv:cond-mat/9609187PramanaFeb 1, 199717 citations

Representation of bosons by fermions

Fritz Bopp

Acta Physica Academiae Scientiarum HungaricaeDec 1, 19720 citations

Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg

\mathrm{SU}(N)

Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of

\mathrm{SU}(N)

. The method, based on the use of standard Young tableaux, takes advantage of the full

\mathrm{SU}(N)

symmetry, allowing one to work directly in each irreducible representation of the global

\mathrm{SU}(N)

group. Since the

\mathrm{SU}(N)

singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to

N=10

and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [

\mathrm{SU}(2)

with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the

\mathrm{SU}(N)

level

k

Wess-Zumino-Witten universality class, where

k

is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the

\mathrm{SU}(N)

level 1 WZW universality class.