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Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models

Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models. The extension of Lieb-Mattis theorem [1962] to a frustrated spin system. Tigran Hakobyan Yerevan State University & Yerevan Physics Institute. T. Hakobyan, Phys. Rev. B 75 , 214421 (2007). 2. 4. 1. 5. 7. 3. 6.

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Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models

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  1. Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models The extension of Lieb-Mattis theorem [1962] to a frustrated spin system Tigran Hakobyan Yerevan State University & Yerevan Physics Institute T. Hakobyan, Phys. Rev. B 75, 214421 (2007)

  2. 2 4 1 5 7 3 6 8 9 Heisenberg Spin Models Hamiltonian: interacting sites spin of i-th site spin-spin coupling constants ferromagnetic bond antiferromagnetic bond

  3. antiferromagnetic bonds connect different sites sublattice A sublattice B ferromagnetic bonds connect similar sites Bipartite Lattices The lattice L is called bipartite if it splits into two disjoint sublattices A and Bsuch that: • All interactions between the spins of different sublattices are antiferromagnetic, i. e. • All interactions between the spins within the same sublattice are ferromagnetic, i. e. An example of bipartite system:

  4. Classical Ground State: Néel State Ground state (GS) of the classical Heisenberg model on bipartite lattice is a Néel state, i. e. • The spins within the same sublattice have the same direction. • The spins of different sublattices are in opposite directions. Properties of the Néel state: • Néel state minimizes all local interactions in the classical Hamiltonian. • It is unique up to global rotations. • Its spin is:

  5. Quantum GS: Lieb-Mattis Theorem • The quantum fluctuations destroy Néel state and the ground state (GS) of quantum system has more complicated structure. • However, for bipartite spin systems, the quantum GS inherits some properties of its classical counterpart. • Lieb & Mattis[J. Math. Phys. 3, 749 (1962)] proved that • The quantum GS of a finite-size system is a unique multiplet with total spin • , i. e. . • The lowest-energy in the sector, where the total spin is equal to S, is a monotone increasing function of Sfor any [antiferromagnetic ordering of energy levels]. • All lowest-energy spin-S states form one multiplet for [nondegeneracy of the lowest levels].

  6. Steps of the Proof • Perron-Frobenius theorem: The lowest eigenvalue of any connected matrix having negative or vanishing off-diagonal elements is nondegenerate. Correponding eigenvector is a positive superposition of all basic states. • After the rotation of all spins on one sublattice on , the Hamiltonian reads generate negative off-diagonal elements are diagonal • The matrix of Hamiltonian being restricted to any subspace is connected in the standard Ising basis. • Perron-Frobenius theorem is applied to any subspace: Relative GS

  7. The spin of can be found by constructing a trial state being a positive superposition of (shifted) Ising basic states and having a definite value of the spin. Then it will overlap with . The uniqueness of the relative GS then implies that both states have the same spin. As a result, Outline of the Proof [Lieb & Mattis, 1962] • The multiplet containing has the lowest-energy value among all states with spin . It it nondegenerate. • Antiferromagnetic ordering of energy levels: • The ground state is a unique multiplet with spin

  8. Generalizations The Lieb-Mattis theorem have been generalized to: • Ferromagnetic Heisenberg spin chains B. Nachtergaele and Sh. Starr, Phys. Rev. Lett. 94, 057206 (2005) • SU(n) symmetric quantum chain with defining representation T. Hakobyan, Nucl. Phys. B 699, 575 (2004) • Spin-1/2 ladder model frustrated by diagonal interaction T. Hakobyan, Phys. Rev. B 75, 214421 (2007) The topic of this talk

  9. Frustrates Spin Systems • In frustrated spin models, due to competing interactions, the classical ground state can’t be minimized locally and usually possesses a large degeneracy. • The frustration can be caused by the geometry of the spin lattice or by the presence of both ferromagnetic and antiferromagnetic interactions. ? • Examples of geometrically frustrated systems: • Antiferromagnetic Heisenber spin system on • Triangular lattice, • Kagome lattice, • Square lattice with diagonal interactions.

  10. Frustrated Spin-1/2 Ladder:Symmetries Symmetry axis • The total spin S and reflection parity are good quantum numbers. • So, the Hamiltonian remains invariant on individual sectors with fixed values of both quantum numbers. • Let be the lowest-energy value in corresponding sector.

  11. Frustrated Spin-1/2 Ladder:Generalized Lieb-Mattis Theorem[T. Hakobyan, Phys. Rev. B 75, 214421 (2007)] • The minimum-energy levels are nondegenerate (except perhaps the one with and ) and are ordered according to the rule: N = number of rungs • The ground state in entire sector is a spin singlet while in sector is a spin triplet. In both cases it is unique.

  12. Rung Spin Operators The couplings obey: Reflection-symmetric (antisymmetric) operators Symmetry axis where

  13. Construction of Nonpositive Basis: Rung Spin States We use the following basis for 4 rung states: Rung singlet Rung triplet • We use the basis constructed from rung singlet and rung triplet states: • The reflection operator R is diagonal in this basis. where is the number of rung singlets. Define unitary operator, which rotates the odd-rung spins around z axis on

  14. Construction of Nonpositive Basis: Unitary Shift • Apply unitary shift to the Hamiltnian: generate negative off-diagonal elements All positive off-diagonal elements become negative after applying a sign factor to the basic states are diagonal in our basis

  15. Construction of Nonpositive Basis: Sign Factor • It can be shown that all non-diagonal matrix elements of become nonpositive in the basis = the number of pairs in the sequence where is on the left hand side from . = the number of rung singletsin

  16. Subspaces and Relative Ground States The relative ground state of in subspace is unique and is a positive superposition of all basic states: Due to and reflection R symmetries, the Hamiltonian is invariant on each subspace with the definite values of spin projection and reflection operators, which we call subspace: • The matrix of the Hamiltonian in the basis being restricted on any subspace is connected [easy to verify]. • Perron-Frobenius theorem can be applied to subspace:

  17. Relative ground states • The spin of can be found by constructing a trial state being a positive superposition of defined basic states and having a definite value of the spin. Then it will overlap with . The uniqueness of the relative GS then implies that both states have the same spin. As a result,

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