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Physical Review Letters

Two theorems on the Hubbard model

Élliott H. Lieb

Published Mar 6, 1989• 2,222 citations
Publisher

Abstract

In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.

DOI
10.1103/physrevlett.62.1201

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Spin and pseudospin towers of the Hubbard model on a bipartite lattice

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In 1989, Lieb proved two theorems about the Hubbard model. One showed that the ground state of the attractive model was a spin singlet state (S = 0) was unique and was positive definite. The other showed that the ground state of the repulsive model on a bipartite lattice at half-filling has a total spin given by [Formula: see text](N[Formula: see text]NB)/2[Formula: see text], corresponding to the difference of the number of lattice sites on the two sublattices divided by two. In the mid to late 1990s, Shen extended these proofs to show that the pseudospin of the attractive model was minimal until the electron number equaled 2NA where it became fixed at J = [Formula: see text](N[Formula: see text]NB)/2[Formula: see text] until the filling became 2NB, where it became minimal again. In addition, Shen showed that a spin tower exists for the spin eigenstates for the half-filled case on a bipartite lattice. The spin tower says the minimal energy state with spin S is higher in energy than the minimal energy state with spin S − 1 until we reach the ground-state spin given above. One long-standing conjecture about this model remains, namely, does the attractive model have such a spin tower for all fillings, which would then imply that the repulsive model has minimal pseudopsin in its ground state. While we do not prove this last conjecture, we provide a quick review of this previous work, provide a constructive proof of the pseudospin of the attractive model ground state, and describe the challenges by proving the remaining open conjecture.

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Ferromagnetism beyond Lieb's theorem

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Ferrimagnetic long-range order of the Hubbard model

Shun-Qing Shen, Zhaoming Qiu, Guang-Shan Tian

In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When ${\mathit{N}}_{\mathit{A}}$>${\mathit{N}}_{\mathit{B}}$(${\mathit{N}}_{\mathit{A}}$ and ${\mathit{N}}_{\mathit{B}}$ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.

Physical Review Letters•Feb 21, 1994•85 citations
In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.
In 1989, Lieb proved two theorems about the Hubbard model. One showed that the ground state of the attractive model was a spin singlet state (S = 0) was unique and was positive definite. The other showed that the ground state of the repulsive model on a bipartite lattice at half-filling has a total spin given by [Formula: see text](N[Formula: see text]NB)/2[Formula: see text], corresponding to the difference of the number of lattice sites on the two sublattices divided by two. In the mid to late 1990s, Shen extended these proofs to show that the pseudospin of the attractive model was minimal until the electron number equaled 2NA where it became fixed at J = [Formula: see text](N[Formula: see text]NB)/2[Formula: see text] until the filling became 2NB, where it became minimal again. In addition, Shen showed that a spin tower exists for the spin eigenstates for the half-filled case on a bipartite lattice. The spin tower says the minimal energy state with spin S is higher in energy than the minimal energy state with spin S − 1 until we reach the ground-state spin given above. One long-standing conjecture about this model remains, namely, does the attractive model have such a spin tower for all fillings, which would then imply that the repulsive model has minimal pseudopsin in its ground state. While we do not prove this last conjecture, we provide a quick review of this previous work, provide a constructive proof of the pseudospin of the attractive model ground state, and describe the challenges by proving the remaining open conjecture.
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction UUU is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``CuO2{\mathrm{CuO}}_{2}CuO2​ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``ddd orbitals'' occupy the vertices of the squares, while ``ppp orbitals'' lie halfway between two ddd orbitals; both ddd and ppp orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of UUU and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up{U}_{d}={U}_{p}Ud​=Up​, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all UUU. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0,Ud=U{U}_{p}=0,{U}_{d}=UUp​=0,Ud​=U and Up=U,Ud=0{U}_{p}=U,{U}_{d}=0Up​=U,Ud​=0. We found that the different environments of ddd and ppp sites lead to a ferromagnetic insulator when Ud=0{U}_{d}=0Ud​=0; by contrast, Up=0{U}_{p}=0Up​=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each ddd site.
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\ensuremath{\Lambda}=\mathcal{A}\ensuremath{\cup}\mathcal{B}$ with $\mathcal{A}\ensuremath{\cap}\mathcal{B}=\ensuremath{\emptyset},$ and bonds connecting any element of A\mathcal{A}A with all the elements of B.\mathcal{B}.B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the A\mathcal{A}A sites and/or of the B\mathcal{B}B sites. This is the Hubbard model defined on the so-called (NA,NB){(N}_{A}{,N}_{B})(NA​,NB​) complete bipartite graph, NA(NB){N}_{A}{(N}_{B})NA​(NB​) being the number of elements in A(B).\mathcal{A}(\mathcal{B}).A(B). In this paper, we analytically find the exact ground state for NA=NB=N{N}_{A}{=N}_{B}=NNA​=NB​=N at half filling for any N; the repulsion has a maximum at a critical N-dependent value of the on-site Hubbard U. The wave function and the energy of the unique, singlet ground state assume a particularly elegant form for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We also calculate the spin-spin correlation function and show that the ground state exhibits an antiferromagnetic order for any nonzero U even in the thermodynamic limit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a Hubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles, and an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature. On the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramagnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this fact can be made explicit by the analytic solutions.
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has NdN_{\rm d}Nd​-fold degenerate ground states. The degeneracy NdN_{\rm d}Nd​ is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.
In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When NA{\mathit{N}}_{\mathit{A}}NA​>NB{\mathit{N}}_{\mathit{B}}NB​(NA{\mathit{N}}_{\mathit{A}}NA​ and NB{\mathit{N}}_{\mathit{B}}NB​ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.