Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models

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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)
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Heisenberg Spin Models

• Hamiltonian

interacting sites
spin of i-th site
spin-spin coupling constants
ferromagnetic bond
antiferromagnetic bond
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Bipartite Lattices

• The lattice L is called bipartite if it splits
  into two disjoint sublattices A and B such that

1. All interactions between the spins of different
   sublattices are antiferromagnetic, i. e.

1. All interactions between the spins within the
   same sublattice are ferromagnetic, i. e.

An example of bipartite system
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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

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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 S for any
  antiferromagnetic ordering of energy levels.
• All lowest-energy spin-S states form one
  multiplet for nondegeneracy of the lowest
  levels.

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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
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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

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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
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Frustrates Spin Systems

• In frustrated spin models, due to competing
  interactions, the classical ground state cant 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.

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Frustrated Spin-1/2 LadderSymmetries
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.

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Frustrated Spin-1/2 LadderGeneralized
Lieb-Mattis TheoremT. Hakobyan, Phys. Rev. B
75, 214421 (2007)
N number of rungs

• The minimum-energy levels are nondegenerate
  (except perhaps the one with
  and ) and are ordered according to the
  rule

• The ground state in entire sector is a
  spin singlet while in sector
  is a spin triplet. In both cases it is unique.

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Rung Spin Operators
The couplings obey
Reflection-symmetric (antisymmetric) operators
Symmetry axis
where
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Construction of Nonpositive Basis Rung Spin
States
We use the following basis for 4 rung states
Rung singlet
Rung triplet

1. 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
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Construction of Nonpositive Basis Unitary Shift

1. 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
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Construction of Nonpositive Basis Sign Factor

1. 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 singlets
in
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Subspaces and Relative
Ground 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

• The relative ground state of in
  subspace is unique and is a positive
  superposition of all basic states

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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,