We investigate the phase diagram of the half-filled $\mathrm{SU}(N)$ Hubbard-Heisenberg model with hopping $t$ , exchange $J$ , and Hubbard $U$ , on a two-dimensional square lattice. In the large- $N$ limit, and as a function of decreasing values of $t∕J$ , the model shows a transition from a $d$ -density wave state to a spin dimerized insulator. A similar behavior is observed at $N=6$ whereas at $N=2$ a spin density wave insulating ground state is stabilized. The $N=4$ model, has a $d$ -density wave ground state at large values of $t∕J$ which as a function of decreasing values of $t∕J$ becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective $(T=0)$ quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.We investigate the phase diagram of the half-filled $\mathrm{SU}(N)$ Hubbard-Heisenberg model with hopping $t$, exchange $J$, and Hubbard $U$, on a two-dimensional square lattice. In the large-$N$ limit, and as a function of decreasing values of $t∕J$, the model shows a transition from a $d$-density wave state to a spin dimerized insulator. A similar behavior is observed at $N=6$ whereas at $N=2$ a spin density wave insulating ground state is stabilized. The $N=4$ model, has a $d$-density wave ground state at large values of $t∕J$ which as a function of decreasing values of $t∕J$ becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective $(T=0)$ quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.

arXiv:cond-mat/0406074Physical Review BFeb 4, 2005139 citations

Large- n limit of the Hubbard-Heisenberg model

John Bradley Marston, Ian K. Affleck

To gain insight into the behavior of the Hubbard model, we define a $\mathrm{SU}(n)$ invariant generalization of the Hubbard-Heisenberg model and, in the large- $n$ limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional $\mathrm{SU}(n)$ Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term $t$ , the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or " $s+id$ " phase is the ground state at half filling for sufficiently large $t$ or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large- $n$ problem to the high- ${T}_{c}$ superconductors.To gain insight into the behavior of the Hubbard model, we define a $\mathrm{SU}(n)$ invariant generalization of the Hubbard-Heisenberg model and, in the large-$n$ limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional $\mathrm{SU}(n)$ Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term $t$, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "$s+id$" phase is the ground state at half filling for sufficiently large $t$ or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-$n$ problem to the high-${T}_{c}$ superconductors.

Physical review. B, Condensed matterJun 1, 1989497 citations

We investigate the phase diagram of the half-filled

\mathrm{SU}(N)

Hubbard-Heisenberg model with hopping

t

, exchange

J

, and Hubbard

U

, on a two-dimensional square lattice. In the large-

N

limit, and as a function of decreasing values of

t∕J

, the model shows a transition from a

d

-density wave state to a spin dimerized insulator. A similar behavior is observed at

N=6

whereas at

N=2

a spin density wave insulating ground state is stabilized. The

N=4

model, has a

d

-density wave ground state at large values of

t∕J

which as a function of decreasing values of

t∕J

becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective

(T=0)

quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.

To gain insight into the behavior of the Hubbard model, we define a

\mathrm{SU}(n)

invariant generalization of the Hubbard-Heisenberg model and, in the large-

n

limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional

\mathrm{SU}(n)

Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term

t

, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "

s+id

" phase is the ground state at half filling for sufficiently large

t

or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-

n

problem to the high-

{T}_{c}

superconductors.