Reflection Symmetry and Energy-Level Ordering of Frustrated Ladder Models

Tigran HakobyanYerevan State University & Yerevan Physics Institute

The extension of Lieb-Mattis theorem [1962] to a frustrated spin system

T. Hakobyan, Phys. Rev. B 75, 214421 (2007)

Heisenberg Spin Models

Hamiltonian:,

ij i ji j

H J S S

0 :ijJ

0 :ijJ

1

3

2

5

6

7

4

98( ) :x y zi i i iS S S S

interacting sites

spin of i-th site

:ijJ spin-spin coupling constants

, :i j

ferromagnetic bond

antiferromagnetic bond

12J 24J

13J

36J

Bipartite LatticesThe 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. 0 if , or ,ijJ i A j B i B j A

2) All interactions between the spins within the same sublattice are ferromagnetic, i. e. 0 if , or ,ijJ i j A i j B

ferromagnetic bondsconnect similar sites

antiferromagnetic bondsconnect different sites

sublattice A

sublattice B

An example of bipartite system:

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.

Néel state minimizes all local interactions in the classical Hamiltonian.

It is unique up to global rotations.

Its spin is:

( ) max. spin on ( )A BS S A B

Properties of the Néel state:

classicalNeel GS .A BS S S S

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].

SE

GSS S

GS A BS S S

( )SE

quantum classicalGS GS NeelS S S

GSS S

The matrix of Hamiltonian being restricted to any subspace is connected in the standard Ising basis.

Perron-Frobenius theorem is applied to any subspace:

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

Steps of the Proof

Unitaryshif

,

1( ) , 0

2t z z

ij i j i j i j iji j

H J S S S S S S J

generate negative off-diagonal elements are diagonal

zS M

1 11 2 0, , 1, ,i

NN

i

m …m m …m i i i iNMm M

m m m s s… m s

RelativeGS

zS M

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

Outline of the Proof [Lieb & Mattis, 1962]

NeelM

Neel Neel

if

if

A B

M

M M S S SS S

S M S

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,

M

M

MS SM

SE

GS Neel .A BS S S S

1 2 1 2ifS S A BE E S S S S

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)

Generalizations

The topic of this talk

Frustrates Spin Systems

?

Examples of geometrically frustrated systems:

Antiferromagnetic Heisenber spin system on

Triangular lattice,

Kagome lattice,

Square lattice with diagonal interactions.

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.

Frustrated Spin-1/2 Ladder:Symmetries

Symmetry axislJ

lJ

lJ

1,lS

2,lS

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.

1

1 1

1 1 1 2 2 1 1 2 1 1 1 2 1 21 1 1

( ) ( )N N N

l l l l l l l l l l l l ll l l

H J J J

S S S S S S S S S S

SE

Frustrated Spin-1/2 Ladder:Generalized Lieb-Mattis Theorem

[T. Hakobyan, Phys. Rev. B 75, 214421 (2007)]

lJlJ

lJ

1,lS

2,lS

l lJ J

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

1( 1)N 0S

1 2

1 21

1 2

for ( 1) if

for ( 1) if 1

N

S S N

S SE E

S S

The ground state in entire sector is a spin singlet while in

sector is a spin triplet. In both cases it is unique. ( 1)N

1( 1)N

2,NS

1,NS

N = number of rungs

Rung Spin Operators 1 1

1 1 1 2 2 1 1 2 1 1 1 2 1 21 1 1

( ) ( )N N N

l l l l l l l l l l l l ll l l

H J J J

S S S S S S S S S S

Symmetry axis

lJlJ

lJ

1,lS

2,lS

( )1 2

( )1 2

sl l l

al l l

S S S

S S S

Reflection-symmetric (antisymmetric) operators

l lJ J

1( ) ( ) ( ) ( ) ( ) 2

1 11 1

1( ) ( ) ,

2

N Ns s s a a a sl l l l l l l l

l l

H J J J

S S S S S

: 0 : 02 2

s al l l ll l

J J J JJ J

The couplings obey:

where

Construction of Nonpositive Basis: Rung Spin States

1 2 1 0 0N lm m … m m

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.

0( 1 ,)NR 0N

[( 1) 2]( )2 1

1

expN

s zl

l

U i S

11 , 1 , 0

2

1 0

2

Rung singlet

Rung triplet

We use the following basis for 4 rung states:

Define unitary operator, which rotates the odd-rung spins around z axis on

Construction of Nonpositive Basis: Unitary Shift

2) Apply unitary shift to the Hamiltnian:

11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 11

1( ) ( ) ( ) ( ) ( ) 2

1 11 1

1( )

2

1( ) ( )

2

Ns s s s s s a a a a a al l l l l l l l l l l l

l

N Ns s z s z a a z a z sl l l l l l l l

l l

H UHU J S S J S S J S S J S S

J S S J S S J

S

All positive off-diagonal elements become negative after applying a sign factor to the basic states

generate negative off-diagonal elements

are diagonalin our basis

= the number of rung singlets in

Construction of Nonpositive Basis: Sign Factor

3) It can be shown that all non-diagonal matrix elements of become nonpositive in the basis

0 00

sign

[ 2]1 2 1 2

factor

( 1) N NN Nm m … m m m … m

= the number of pairs in the sequence where

is on the left hand side from .

0 0

0

000N 1 2 Nm m … m

0N 1 2 .Nm m … m

H

0

Subspaces and Relative Ground States

,zS M R

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:

zS

, where and, 1 1zS M R M N N … N

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:

( )M

( )M ( )M

1 2 Nm m … m

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

1 1

0

1 2

( 1)

0N N

llN

m …m N m …mMm M

m m … m

H ( )M

Relative ground states

11 if 0 and ( 1)

otherwise

N

M

MS

M

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,

,M

,M