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arXiv:1003.2147 · Symmetry Integrability and Geometry Methods and Applications

Ordering of Energy Levels for Extended SU(N) Hubbard Chain

Tigran Hakobyan

Published Mar 20, 2010• 6 citations
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Abstract

The Lieb-Mattis theorem on the antiferromagnetic ordering of energy levels is generalized to SU (N ) extended Hubbard model with Heisenberg exchange and pair-hopping terms.It is proved that the minimum energy levels among the states from equivalent representations are nondegenerate and ordered according to the dominance order of corresponding Young diagrams.In particular, the ground states form a unique antisymmetric multiplet.The relation with the similar ordering among the spatial wavefunctions with different symmetry classes of ordinary quantum mechanics is discussed also.

DOI
10.3842/sigma.2010.024

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The Lieb-Mattis theorem on the antiferromagnetic ordering of energy levels is generalized to SU (N ) extended Hubbard model with Heisenberg exchange and pair-hopping terms.It is proved that the minimum energy levels among the states from equivalent representations are nondegenerate and ordered according to the dominance order of corresponding Young diagrams.In particular, the ground states form a unique antisymmetric multiplet.The relation with the similar ordering among the spatial wavefunctions with different symmetry classes of ordinary quantum mechanics is discussed also.
The Lieb-Mattis theorem is generalized to an antiferromagnetic spin-ladder model with four-spin cyclic exchange interaction. We prove that for J>2KJ>2\phantom{\rule{0.3em}{0ex}}\text{K}J>2K, the antiferromagnetic ordering of energy levels takes place separately in two sectors, which remain symmetric and antisymmetric under the reflection with respect to the longitudinal axis of the ladder. We prove also that at the self-dual point J=2KJ=2\phantom{\rule{0.3em}{0ex}}\text{K}J=2K, the Lieb-Mattis rule holds in the sectors with fixed number of rung singlets. In both cases, it agrees with the similar rule for Haldane chain with appropriate spin number.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N = 4n + 2 (n an integer) with periodic boundary conditions or N = 4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains with spins of arbitrary magnitude, thus extending our previous result for the spin-1/2 chain. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.
The Lieb-Mattis theorem orders the lowest-energy states of total spin s of a system of P interacting fermions. We generalize these predictions to fermionic mixtures of P particles with more than N=2 spinor components or species in the thermodynamic limit P→∞. The lowest-energy state inside each permutation symmetry sector h, arising in the P-fold tensor product decomposition, is well approximated by a U(N) coherent (quasiclassical variational) state, especially in the limit P→∞. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau h_{0}) configuration. We exemplify our construction with the N=3 level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and U(N)-invariant quantum Hall ferromagnets. In the limit P→∞, each lowest-energy state within each permutation symmetry sector h undergoes a quantum phase transition for a critical value λ_{c}(h) of the exchange coupling constant λ, depending on h. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector h_{0}.
Based on the exact solution of one-dimensional Fermi gas systems with SUSUSU(N) symmetry in a hard wall, we demonstrate that we are able to sort the ordering of the lowest-energy eigenvalues of states with all allowed permutation symmetries, which can be solely marked by certain quantum numbers in the Bethe ansatz equations. Our results give examples beyond the scope of the generalized Lieb-Mattis theorem, which can only compare the ordering of energy levels of states belonging to different symmetry classes if they are comparable according to the pouring principle. In the strongly interacting regime, we show that the ordering of energy levels can be determined by an effective spin-exchange model, and we extend our results to a nonuniform system trapped in a harmonic potential.