Abstract
A lattice electron system in which the electron number N is identical to the number of sites \(|\varLambda |\) is said to be half-filled because the maximum possible value of N is \(2|\varLambda |\). Half-filled models represent physically natural situations since a system becomes half-filled if each atom contributes one electron to the system of electrons in consideration. In the present chapter we discuss the theorem of fundamental significance due to Lieb, which sheds light on the origin of antiferromagnetism and ferrimagnetism in interacting itinerant electron systems at half-filling. We start the chapter by a short section (Sect. 10.1) with a formal perturbative argument about the relation between the half-filled Hubbard model and the antiferromagnetic Heisenberg model. Then in Sect. 10.2, we present a thorough discussion of Lieb’s theorems. We shall state main claims, discuss important applications, and give complete and hopefully readable proofs.
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- 1.
Recall that the eigenvalue of \((\hat{\varvec{S}}_\mathrm{tot})^2\) is denoted as \(S_\mathrm{tot}(S_\mathrm{tot}+1)\).
- 2.
It is easy to remove this assumption since the diagonal part \(\sum _{x,\sigma }t_{x,x}\,\hat{c}^\dagger _{x,\sigma }\hat{c}_{x,\sigma }=\sum _xt_{x,x}(\hat{n}_{x,\uparrow }+\hat{n}_{x,\downarrow })\) only shifts the energy of the unperturbed ground states by a constant \(\sum _xt_{x,x}\), and slightly modifies the second order perturbation.
- 3.
The Proof of Theorem 4.1 (p. 75), which establishes the existence of long-range order in the antiferromagnetic Heisenberg model, is also based on the reflection positivity method. But the proof of Lieb’s theorem is much simpler.
- 4.
More precisely this means that, for any \(x,y\in \varLambda \) such that \(x\ne y\), there exists a finite sequence \(z_1, z_2,\ldots , z_n\in \varLambda \) with the properties \(z_1=x\), \(z_n=y\), and \(t_{z_j,z_{j+1}}\ne 0\) for \(j=1,\ldots ,n-1\).
- 5.
It may happen however that the model exhibits phase separation and does not have a unique ground state in the thermodynamic limit.
- 6.
Recall that we used the same strategy in the proof of the Marshall–Lieb–Mattis theorem (Theorem 2.2 in p. 39). See p. 40.
- 7.
Note that we are here using the second convention of transformations of operators as in (A.1.17).
- 8.
Recall that the ground state is a simultaneous eigenstate of the Hamiltonian and \((\hat{\varvec{S}}_\mathrm{tot})^2\). By continuity the eigenvalue of the latter, which is \(S_\mathrm{tot}(S_\mathrm{tot}+1)\), cannot vary.
- 9.
It is essential that the spin operators \(\hat{S}^{(\alpha )}_x\) defined for the Hubbard model, when restricted onto the subspace without any doubly occupied sites, exactly coincide with that of the quantum spin system.
- 10.
See, e.g., [22] for interesting attempts to realize the Lieb lattice by using cold atoms in an optical trap.
- 11.
Ferromagnetism in the Hubbard model due to Nagaoka, which takes place in a singular situation, was already known. See Sect. 11.2.
- 12.
That some tight-binding electron models have flat bands had been known much before. See, e.g., [28].
- 13.
- 14.
It might be interesting to compare the present proof with that of the Perron–Frobenius theorem in Appendix A.4.1. See also an extension of the Perron–Frobenius theorem in [5].
- 15.
A Hermitian matrix \(\mathsf {M}\) is said to be positive definite if all the eigenvalues are positive, or, equivalently, \(\varvec{v}^\dagger \mathsf {M}\varvec{v}>0\) for any nonzero vector \(\varvec{v}\). A Hermitian matrix \(\mathsf {M}\) is said to be negative definite if \(-\mathsf {M}\) is positive definite. A Hermitian matrix \(\mathsf {M}\) is said to be positive semidefinite (or nonnegative) if all the eigenvalues are nonnegative, or, equivalently, \(\varvec{v}^\dagger \mathsf {M}\varvec{v}\ge 0\) for any vector \(\varvec{v}\).
We use the following standard notation here and in what follows: \(\varvec{v}=(v_A)_{A\in {\mathscr {S}}}\) denotes a column vector, and \(\varvec{v}^\dagger \) a row vector. The product \(\varvec{u}^\dagger \varvec{v}=\sum _{A\in {\mathscr {S}}}(u_A)^*v_A\) is a scalar, and \(\varvec{v}\varvec{u}^\dagger \) is a matrix such that \((\varvec{v}\varvec{u}^\dagger )_{A,B}=v_A(u_B)^*\). Recall that \({\text {Tr}}[\varvec{v}\varvec{u}^\dagger ]=\varvec{u}^\dagger \varvec{v}\).
- 16.
Here we used the following easy lemma. Lemma Let \(\mathsf {M}\) and \(\mathsf {M}'\) be \(D\times D\) matrices. If \(\mathsf {M}\) is positive definite and \(\mathsf {M}'\) is positive semidefinite but nonvanishing, then \({\text {Tr}}[\mathsf {M}\mathsf {M}']>0\). Proof Take the orthonormal basis \(\{\varvec{u}_j\}_{j=1,\ldots ,D}\) which consists of the eigenvectors of \(\mathsf {M}'\), i.e., \(\mathsf {M}'\varvec{u}_j=m_j\varvec{u}_j\) with \(m_j\ge 0\) for all j and \(m_j>0\) for some j. Then \({\text {Tr}}[\mathsf {M}\mathsf {M}']=\sum _{j=1}^Dm_j(\varvec{u}_j^\dagger \mathsf {M}\varvec{u}_j)>0\).
- 17.
See footnote 15 for the notation.
- 18.
Since \(\varvec{v}^\dagger \mathsf {I}^{(x)}\mathsf {R}\mathsf {I}^{(x)}\varvec{v}=|\sqrt{\mathsf {R}}\,\mathsf {I}^{(x)}\varvec{v}|^2\), we find \(\sqrt{\mathsf {R}}\,\mathsf {I}^{(x)}\varvec{v}=\varvec{0}\).
- 19.
To be rigorous we need to show that the connectivity of the lattice \(\varLambda \) implies the connectivity of the configuration space \({\mathscr {S}}\). This was indeed done in the more complicated case of spin systems in p. 41, and that proof applies to the present case. (Consider the case with \(S=1/2\), and identify the set of sites with \(\sigma =1/2\) with \(A\subset \varLambda \).) For a direct proof for this simpler situation, see, e.g., (the first half of) the Proof of Lemma in [7].
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Tasaki, H. (2020). Half-Filled Models: Lieb’s Theorems and the Origin of Antiferromagnetism and Ferrimagnetism. In: Physics and Mathematics of Quantum Many-Body Systems. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-41265-4_10
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