In this work we extend the notion of what is meant by a meanfield. Meanfields are approximately maps - through some self consistency relation - of a complex, usually manybody, problem to a simpler more readily solvable problem. This mapping can then be solved to represent properties of the complex many body problem using some self consistency relations. Prototypical examples of simpler meanfield problems (meanfield systems) are the single site and free particle problems. Here we propose a new class of simple meanfield systems where the simple problem to be solved is a 1D spin chain. These meanfields are particularly useful for studying quasi-1D models, where there is a 3D system composed of weakly coupled 1D spin chains with the coupling in the transverse direction weaker than in the 1D direction. We illustrate this idea by considering meanfields for the Ising (of any coupling sign) and ferromagnetic Heisenberg models with one direction coupled much more strongly then the other directions (quasi-1D systems) which map at meanfield level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also consider more exotic models to illustrate other methods of solving 1D systems, namely the $N$ -state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling $Γ$ and the transverse directions have coupling $J$ with $λ\sim\fracΓ{J}\gg1$ then regular meanfield would predict the transition temperature to be $k_{B}T_{c}\simΓ$ for all three models while 1D meanfield predicts temperatures of $k_{B}T_{c}\sim\fracΓ{\log\left(λ\right)}$ for the Ising and Potts models and $k_{B}T_{c}\sim\fracΓ{\sqrtλ}$ for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.In this work we extend the notion of what is meant by a meanfield. Meanfields are approximately maps - through some self consistency relation - of a complex, usually manybody, problem to a simpler more readily solvable problem. This mapping can then be solved to represent properties of the complex many body problem using some self consistency relations. Prototypical examples of simpler meanfield problems (meanfield systems) are the single site and free particle problems. Here we propose a new class of simple meanfield systems where the simple problem to be solved is a 1D spin chain. These meanfields are particularly useful for studying quasi-1D models, where there is a 3D system composed of weakly coupled 1D spin chains with the coupling in the transverse direction weaker than in the 1D direction. We illustrate this idea by considering meanfields for the Ising (of any coupling sign) and ferromagnetic Heisenberg models with one direction coupled much more strongly then the other directions (quasi-1D systems) which map at meanfield level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also consider more exotic models to illustrate other methods of solving 1D systems, namely the $N$-state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling $Γ$ and the transverse directions have coupling $J$ with $λ\sim\fracΓ{J}\gg1$ then regular meanfield would predict the transition temperature to be $k_{B}T_{c}\simΓ$ for all three models while 1D meanfield predicts temperatures of $k_{B}T_{c}\sim\fracΓ{\log\left(λ\right)}$ for the Ising and Potts models and $k_{B}T_{c}\sim\fracΓ{\sqrtλ}$ for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.

arXiv:2403.08389Mar 13, 20240 citations

Study of the triangular lattice tV model near x=13

Olexei I. Motrunich, Patrick A. Lee

We study the extended Hubbard model on a triangular lattice near doping $x=\frac{1}{3}$ , which may be relevant for the recently discovered superconductor ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}y{\mathrm{H}}_{2}\mathrm{O}$. By generalizing this model to $N$ fermionic species, we formulate a meanfield description in the limit of large $N$ . In meanfield, we find two possible phases: a renormalized Fermi liquid and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ charge density wave state. The transition between the two phases is driven by increasing the nearest-neighbor repulsion and is found to be first order for doping $x=\frac{1}{3}$ , but occurs close to the point of the local instability of the uniform liquid. We also study fluctuations about the uniform meanfield state in a systematic $1∕N$ expansion, focusing on the residual interaction of quasiparticles and possible superconducting instabilities due to this interaction. Upon moving towards the charge density wave instability, the increasing charge fluctuations favor a particular $f$ -wave triplet state. (This state was recently discussed by Tanaka et al., cond-mat/0311266.) We also report a direct Gutzwiller wave function study of the spin- $\frac{1}{2}$ model.We study the extended Hubbard model on a triangular lattice near doping $x=\frac{1}{3}$, which may be relevant for the recently discovered superconductor ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}y{\mathrm{H}}_{2}\mathrm{O}$. By generalizing this model to $N$ fermionic species, we formulate a meanfield description in the limit of large $N$. In meanfield, we find two possible phases: a renormalized Fermi liquid and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ charge density wave state. The transition between the two phases is driven by increasing the nearest-neighbor repulsion and is found to be first order for doping $x=\frac{1}{3}$, but occurs close to the point of the local instability of the uniform liquid. We also study fluctuations about the uniform meanfield state in a systematic $1∕N$ expansion, focusing on the residual interaction of quasiparticles and possible superconducting instabilities due to this interaction. Upon moving towards the charge density wave instability, the increasing charge fluctuations favor a particular $f$-wave triplet state. (This state was recently discussed by Tanaka et al., cond-mat/0311266.) We also report a direct Gutzwiller wave function study of the spin-$\frac{1}{2}$ model.

arXiv:cond-mat/0401160Physical Review BJul 22, 200450 citations

Ground state fluctuations of quantum mean field systems

A.K. Callebaut, M. Fannes

QP-PQ, quantum probability and white noise analysisJul 4, 19941 citation

Gauge theory of the Hubbard model on honeycomb lattice and its instanton effect

Abolhassan Vaezi, Xiao-Gang Wen

In this paper we investigate possible spin disordered phase in the Hubbard model on the honeycomb lattice. Using a slave-particle theory that include the charge fluctuations, we find a meanfield spin disordered phase in a range of on-site repulsion $U$ . The spin disordered state is described by gapped fermions coupled to compact U(1) gauge field. We study the confinement/deconfinement problem of the U(1) gauge theory due to the instantons proliferation. We calculate all allowed instanton terms and compute their quantum numbers. It is shown that the meanfield spin disordered phase is unstable. The instantons proliferation induce a translation symmetry breaking.In this paper we investigate possible spin disordered phase in the Hubbard model on the honeycomb lattice. Using a slave-particle theory that include the charge fluctuations, we find a meanfield spin disordered phase in a range of on-site repulsion $U$. The spin disordered state is described by gapped fermions coupled to compact U(1) gauge field. We study the confinement/deconfinement problem of the U(1) gauge theory due to the instantons proliferation. We calculate all allowed instanton terms and compute their quantum numbers. It is shown that the meanfield spin disordered phase is unstable. The instantons proliferation induce a translation symmetry breaking.

arXiv:1101.1662Jan 9, 20112 citations

N

-state Potts model. Magnetic phase transition temperatures and are obtained for all three models, we see that they significantly differ from the usual meanfield estimates. Indeed if the 1D direction has coupling

Γ

and the transverse directions have coupling

J

with

λ\sim\fracΓ{J}\gg1

then regular meanfield would predict the transition temperature to be

k_{B}T_{c}\simΓ

for all three models while 1D meanfield predicts temperatures of

k_{B}T_{c}\sim\fracΓ{\log\left(λ\right)}

for the Ising and Potts models and

k_{B}T_{c}\sim\fracΓ{\sqrtλ}

for the ferromagnetic Heisenberg model. Cluster 1D meanfield extensions are also proposed.

We study the extended Hubbard model on a triangular lattice near doping

x=\frac{1}{3}

, which may be relevant for the recently discovered superconductor ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}y{\mathrm{H}}_{2}\mathrm{O}$. By generalizing this model to

N

fermionic species, we formulate a meanfield description in the limit of large

N

. In meanfield, we find two possible phases: a renormalized Fermi liquid and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ charge density wave state. The transition between the two phases is driven by increasing the nearest-neighbor repulsion and is found to be first order for doping

x=\frac{1}{3}

, but occurs close to the point of the local instability of the uniform liquid. We also study fluctuations about the uniform meanfield state in a systematic

1∕N

expansion, focusing on the residual interaction of quasiparticles and possible superconducting instabilities due to this interaction. Upon moving towards the charge density wave instability, the increasing charge fluctuations favor a particular

f

-wave triplet state. (This state was recently discussed by Tanaka et al., cond-mat/0311266.) We also report a direct Gutzwiller wave function study of the spin-

\frac{1}{2}

model.

U

. The spin disordered state is described by gapped fermions coupled to compact U(1) gauge field. We study the confinement/deconfinement problem of the U(1) gauge theory due to the instantons proliferation. We calculate all allowed instanton terms and compute their quantum numbers. It is shown that the meanfield spin disordered phase is unstable. The instantons proliferation induce a translation symmetry breaking.