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Stability of time-reversal symmetry breaking spinliquid states in high-spin fermionic systems
Edina Szirmai
International School and Workshop on Anyon Physics of Ultracold Atomic GasesKaiserslautern, 12-15. 12. 2014.
Quantum simulations with ultracold atoms
Quantum simulation offundamental models (properties,phenomena).
Novel behavior, completely newphases are expected due to thehigh spin.
High-Tc superconductors
Quantum information
Simulation of gauge theories
Quantum simulations with ultracold atoms
Quantum simulation offundamental models (properties,phenomena).
Novel behavior, completely newphases are expected due to thehigh spin.
A possible explanation for themechanism of high-Tc
superconductivity and their strangebehavior in the non-superconductingphase based on the strong magneticfluctuation in dopped Mott insulators.These fluctuation can be treated withinthe spin liquid concept.
High-Tc superconductors
Quantum information
Simulation of gauge theories
Quantum simulations with ultracold atoms
Quantum simulation offundamental models (properties,phenomena).
Novel behavior, completely newphases are expected due to thehigh spin.
In topological phases of spinliquids the quasiparticles havefractional statistics. They arenonlocal and resist well againstlocal perturbations. Promising qbitcandidates.
High-Tc superconductors
Quantum information
Simulation of gauge theories
Quantum simulations with ultracold atoms
Quantum simulation offundamental models (properties,phenomena).
Novel behavior, completely newphases are expected due to thehigh spin.
Low energy excitations above spinliquids can be described byeffective gauge theories. Aim: tostudy various gauge theories withultracold atoms.
High-Tc superconductors
Quantum information
Simulation of gauge theories
Quantum simulations with ultracold atoms
Atoms loaded into an optical lattice
Periodic potential: standing wave laser light.
Interaction between the neutral atoms:
van der Waals interaction
in case of alkaline-earth atoms:spin independent s-wavecollisions
Easy to control the model parameters:
Interaction strength: Feshbach resonance
Localization: laser intensity
Lattice geometry
Quantum simulations with ultracold atoms
Atoms loaded into an optical lattice
Periodic potential: standing wave laser light.
Interaction between the neutral atoms:
van der Waals interaction
in case of alkaline-earth atoms:spin independent s-wavecollisions
Easy to control the model parameters:
Interaction strength: Feshbach resonance
Localization: laser intensity
Lattice geometry
Quantum simulations with ultracold atoms
Atoms loaded into an optical lattice
Periodic potential: standing wave laser light.
Interaction between the neutral atoms:
van der Waals interaction
in case of alkaline-earth atoms:spin independent s-wavecollisions
Easy to control the model parameters:
Interaction strength: Feshbach resonance
Localization: laser intensity
Lattice geometry
Quantum simulations with ultracold atoms
Atoms loaded into an optical lattice
Periodic potential: standing wave laser light.
Interaction between the neutral atoms:
van der Waals interaction
in case of alkaline-earth atoms:spin independent s-wavecollisions
Easy to control the model parameters:
Interaction strength: Feshbach resonance
Localization: laser intensity
Lattice geometry
Outline
Part IFundamentals of high spin systemsSpin wave descriptionValence bond picture
Part IICompeting spin liquid states ofspin-3/2 fermions in a square latticeCompeting spin liquid states ofspin-5/2 fermions in a honeycomb lattice
Properties of chiral spin liquid stateStability of the spin liquid states beyondthe mean-field approximationFinite temperature behaviorExperimentally measurable quantities
In collaboration with:
M. Lewenstein
P. Sinkovicz
G. Szirmai
A. Zamora
PRA 88(R) 043619 (2013)
PRA 84 011611 (2011)
EPL 93, 66005 (2011)
Fundamental properties of high-spin systems
Fundamental properties of high-spin systems
The Hubbard Hamiltonian with n.n hopping and on-site interaction:
H =−t ∑<i,j>
c†i,αcj,α +∑
iUα,β
γ,δ c†i,αc†
i,β ci,δ ci,γ .
S = |F1−F2|, . . . |F1 + F2|Sz =−S, . . .SF : individual atoms, S: total spin of 2 scattering atoms
4 components: ± 32 , and ± 1
2
F1 = F2 = 32
S = 0: Sz = 0,S = 1: Sz =−1,0,1
S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3
Fundamental properties of high-spin systems
The Hubbard Hamiltonian with n.n hopping and on-site interaction:
H =−t ∑<i,j>
c†i,αcj,α +∑
iUα,β
γ,δ c†i,αc†
i,β ci,δ ci,γ .
S = |F1−F2|, . . . |F1 + F2|Sz =−S, . . .SF : individual atoms, S: total spin of 2 scattering atoms
2 components: ↑, and ↓
F1 = F2 = 12 ,
S = 0: Sz = 0,S = 1: Sz =−1,0,1
S = |F1−F2|, . . . |F1 + F2|Sz =−S, . . .SF : individual atoms, S: total spin of 2 scattering atoms
4 components: ± 32 , and ± 1
2
F1 = F2 = 32
S = 0: Sz = 0,S = 1: Sz =−1,0,1
S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3
Fundamental properties of high-spin systems
The Hubbard Hamiltonian with n.n hopping and on-site interaction:
H =−t ∑<i,j>
c†i,αcj,α +∑
iUα,β
γ,δ c†i,αc†
i,β ci,δ ci,γ .
S = |F1−F2|, . . . |F1 + F2|Sz =−S, . . .SF : individual atoms, S: total spin of 2 scattering atoms
4 components: ± 32 , and ± 1
2
F1 = F2 = 32
S = 0: Sz = 0,S = 1: Sz =−1,0,1
S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3
Fundamental properties of high-spin systems
The Hamiltonian:
H = Hkin + Hint
where Hint contains manytypes of scatteringprocesses.
on-site interaction=⇒
Pauli’s principle
The only nonzero terms arecompletly antisymmetric forthe exchange of the spin ofthe two scattering particles.
Fundamental properties of high-spin systems
The Hamiltonian:
H = Hkin + Hint
where Hint contains manytypes of scatteringprocesses.
on-site interaction=⇒
Pauli’s principle
The only nonzero terms arecompletly antisymmetric forthe exchange of the spin ofthe two scattering particles.
spin-1/2 fermions
Only singlet scatterings areallowed:
Stot = 0 =⇒
H =−t ∑〈i,j〉(c†i,σ cj,σ +H.c.)+∑i U0P
(i)0 , and
P(i)0 = c†
i,↑c†i,↓ci,↓ci,↑ = ni↑ni↓ projects to the singlet
subspace Stot = 0.
Fundamental properties of high-spin systems
The Hamiltonian:
H = Hkin + Hint
where Hint contains manytypes of scatteringprocesses.
on-site interaction=⇒
Pauli’s principle
The only nonzero terms arecompletly antisymmetric forthe exchange of the spin ofthe two scattering particles.
spin-3/2 fermions
Classification of the allowedscattering processes:
Stot = 0
Stot = 2=⇒
H =−t ∑〈i,j〉(c†i,σ cj,σ +H.c.)+∑i
[U0P
(i)0 +U2P
(i)2
],
and P(i)S = c†
i,σ1c†
i,σ2ci,σ3 ci,σ4 P̂S , where P̂S projects to
the subspace Stot = S.T. L. Ho PRL (1998)
T. Ohmi and K. Machida JPSJ (1998)
Fundamental properties of high-spin systems
The Hamiltonian:
H = Hkin + Hint
where Hint contains manytypes of scatteringprocesses.
on-site interaction=⇒
Pauli’s principle
The only nonzero terms arecompletly antisymmetric forthe exchange of the spin ofthe two scattering particles.
spin-5/2 fermions
Classification of the allowedscattering processes:
Stot = 0
Stot = 2
Stot = 4
=⇒
H =−t ∑〈i,j〉(c†i,σ cj,σ +H.c.)
+∑i
[U0P
(i)0 +U2P
(i)2 +U4P
(i)4
],
and P(i)S = c†
i,σ1c†
i,σ2ci,σ3 ci,σ4 P̂S , where P̂S projects to
the subspace Stot = S.T. L. Ho PRL (1998)
T. Ohmi and K. Machida JPSJ (1998)
Effective spin models in the strong coupling limit
The Hubbard Hamiltonian with n.n hopping and on-site interaction:
H =−t ∑<i,j>
c†i,αcj,α + U ∑
ic†
i,αc†i,β ci,αci,β .
spin independent interaction→ SU(N) symmetry
Strongly repulsive limit: U/t → ∞
repulsive interaction, f = 1/(2F +1) filling (# of particles =# of sites)
Perturbation theory up to leading orderwith respect to t .
t preserves S and Sz
nearest-neighbor hopping
Effective spin models in the strong coupling limit
The Hubbard Hamiltonian with n.n hopping and on-site interaction:
H =−t ∑<i,j>
c†i,αcj,α + U ∑
ic†
i,αc†i,β ci,αci,β .
spin independent interaction→ SU(N) symmetry
Strongly repulsive limit: U/t → ∞
repulsive interaction, f = 1/(2F +1) filling (# of particles =# of sites)
Perturbation theory up to leading orderwith respect to t .
t preserves S and Sz
nearest-neighbor hopping
Effective spin models in the strong coupling limit
Effective Hamiltonian (spin-F fermions in the U/t → ∞ limit)
H = J ∑〈i,j〉
c†i,αc†
j,β cj,αci,β
nearest-neighbor interaction
the same spin dependencethat has the original model
no particle transport
Mott state
Without long range spin order/preserved spin rotational invariance:spin liquid state
Effective spin models in the strong coupling limit
Effective Hamiltonian (spin-F fermions in the U/t → ∞ limit)
H = J ∑〈i,j〉
c†i,αc†
j,β cj,αci,β
nearest-neighbor interaction
the same spin dependencethat has the original model
no particle transport
Mott state
Without long range spin order/preserved spin rotational invariance:spin liquid state
Effective spin models in the strong coupling limit
Effective Hamiltonian (spin-F fermions in the U/t → ∞ limit)
H = J ∑〈i,j〉
c†i,αc†
j,β cj,αci,β
nearest-neighbor interaction
the same spin dependencethat has the original model
no particle transport
Mott state
Without long range spin order/preserved spin rotational invariance:spin liquid state
Effective spin models in the strong coupling limit
(ni = c†i,σ ci,σ , and Si = c†
i,σ Fσ ,σ ′ci,σ ′ )
Spin exchange appears explicitly in the Hamiltonian:
F = 12 : AFM Heisenberg model
Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
F = 32 :
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
F = 52 :
HF=5/2eff = ∑〈i,j〉
[a0 ni nj +a1Si Sj +a2(Si Sj)
2 +a3(Si Sj)3 +a4(Si Sj)
4 +a5(Si Sj)5]
Effective spin models in the strong coupling limit
(ni = c†i,σ ci,σ , and Si = c†
i,σ Fσ ,σ ′ci,σ ′ )
Spin exchange appears explicitly in the Hamiltonian:
F = 12 : AFM Heisenberg model
Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
F = 32 :
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
F = 52 :
HF=5/2eff = ∑〈i,j〉
[a0 ni nj +a1Si Sj +a2(Si Sj)
2 +a3(Si Sj)3 +a4(Si Sj)
4 +a5(Si Sj)5]
Effective spin models in the strong coupling limit
(ni = c†i,σ ci,σ , and Si = c†
i,σ Fσ ,σ ′ci,σ ′ )
Spin exchange appears explicitly in the Hamiltonian:
F = 12 : AFM Heisenberg model
Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
F = 32 :
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
F = 52 :
HF=5/2eff = ∑〈i,j〉
[a0 ni nj +a1Si Sj +a2(Si Sj)
2 +a3(Si Sj)3 +a4(Si Sj)
4 +a5(Si Sj)5]
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Let us consider a two-site problem: Hi,j = JSiSj
Site-by-site picture:|Sz
i = F ,Szj =−F
⟩: Ei,j =−JF 2.
BUT this is not an eigenstate of Hi,j = JSzi Sz
j +J2 (S
+i S−j +S−i S+
j ).Let Sz fluctuate to gain energy
⇒ theory of AFM spin waves
Fluctuation above the classical Néel state:
Sublattice magnetization:〈Sz
A〉=−〈SzB〉= F −∆F .
Description: magnon picture.
Large ∆F → "quantum melting"→
Spin liquid
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Let us consider a two-site problem: Hi,j = JSiSj
Site-by-site picture:|Sz
i = F ,Szj =−F
⟩: Ei,j =−JF 2.
BUT this is not an eigenstate of Hi,j = JSzi Sz
j +J2 (S
+i S−j +S−i S+
j ).Let Sz fluctuate to gain energy
⇒ theory of AFM spin waves
Fluctuation above the classical Néel state:
Sublattice magnetization:〈Sz
A〉=−〈SzB〉= F −∆F .
Description: magnon picture.
Large ∆F → "quantum melting"→
Spin liquid
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Let us consider a two-site problem: Hi,j = JSiSj
Site-by-site picture:|Sz
i = F ,Szj =−F
⟩: Ei,j =−JF 2.
BUT this is not an eigenstate of Hi,j = JSzi Sz
j +J2 (S
+i S−j +S−i S+
j ).Let Sz fluctuate to gain energy
⇒ theory of AFM spin waves
Fluctuation above the classical Néel state:
Sublattice magnetization:〈Sz
A〉=−〈SzB〉= F −∆F .
Description: magnon picture.
Large ∆F → "quantum melting"→
Spin liquid
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Basic concepts of AFM SWT and magnon picture:
Description of the spin wave excitations→ boson operators.
Holstein-Primakoff transformation (and similar for sublattice B):
S+A,j =
(2F −a†
j aj
)1/2aj S−A,j = a†
j
(2F −a†
j aj
)1/2
SzA,j = 2−a†
j aj
Interacting boson system→ (SiSj)p multiboson int.
Linear spin wave theory: no magnon-magnon interaction.
The ground state energy/bond (LSWT): Ei,j =−JF(F + ζ ), with0 < ζ < 1 and geometry-dependent.
If ∆F large from the LSWT:Beyond LSWT.
Change concept.
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Basic concepts of AFM SWT and magnon picture:
Description of the spin wave excitations→ boson operators.
Holstein-Primakoff transformation (and similar for sublattice B):
S+A,j =
(2F −a†
j aj
)1/2aj S−A,j = a†
j
(2F −a†
j aj
)1/2
SzA,j = 2−a†
j aj
Interacting boson system→ (SiSj)p multiboson int.
Linear spin wave theory: no magnon-magnon interaction.
The ground state energy/bond (LSWT): Ei,j =−JF(F + ζ ), with0 < ζ < 1 and geometry-dependent.
If ∆F large from the LSWT:Beyond LSWT.
Change concept.
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Basic concepts of AFM SWT and magnon picture:
Description of the spin wave excitations→ boson operators.
Holstein-Primakoff transformation (and similar for sublattice B):
S+A,j =
(2F −a†
j aj
)1/2aj S−A,j = a†
j
(2F −a†
j aj
)1/2
SzA,j = 2−a†
j aj
Interacting boson system→ (SiSj)p multiboson int.
Linear spin wave theory: no magnon-magnon interaction.
The ground state energy/bond (LSWT): Ei,j =−JF(F + ζ ), with0 < ζ < 1 and geometry-dependent.
If ∆F large from the LSWT:Beyond LSWT.
Change concept.
Antiferromagnetic Heisenberg model: Heff = J ∑〈i,j〉(SiSj − 1
4 ninj)
Basic concepts of AFM SWT and magnon picture:
Description of the spin wave excitations→ boson operators.
Holstein-Primakoff transformation (and similar for sublattice B):
S+A,j =
(2F −a†
j aj
)1/2aj S−A,j = a†
j
(2F −a†
j aj
)1/2
SzA,j = 2−a†
j aj
Interacting boson system→ (SiSj)p multiboson int.
Linear spin wave theory: no magnon-magnon interaction.
The ground state energy/bond (LSWT): Ei,j =−JF(F + ζ ), with0 < ζ < 1 and geometry-dependent.
If ∆F large from the LSWT:Beyond LSWT.
Change concept.
Effective spin models in the strong coupling limit
Even in case of the simplest F = 12 case there exist lattices that
have ground state without breaking of spin rotational invariance(no Néel order) in the thermodynamic limit.Various disordered spin sates occur in e.g.
finite systems,frustrated systems,systems described by Hamiltonian with higher power of SiSj .
To describe these disordered states a powerful possibility:
valence bond picture.
Effective spin models in the strong coupling limit
Even in case of the simplest F = 12 case there exist lattices that
have ground state without breaking of spin rotational invariance(no Néel order) in the thermodynamic limit.Various disordered spin sates occur in e.g.
finite systems,frustrated systems,systems described by Hamiltonian with higher power of SiSj .
To describe these disordered states a powerful possibility:
valence bond picture.
Valence bond pictureF = 1/2 AFM Heisenberg model
Fazekas, Electron Correlation and Magnetism (1999)
Affleck, Kennedy, Lieb, Tasaki, Valence Bond Ground States in Isotropic Quantum Antiferromagnets (1988)
Valence bond picture
Back to the two-site problem: Hi,j = JSiSj
Site-by-site picture:|Sz
i = F ,Szj =−F
⟩: Ei,j =−JF 2.
BUT this is not an eigenstate of Hi,j = JSzi Sz
j +J2 (S
+i S−j +S−i S+
j ).Let Sz fluctuate to gain energy
⇒ theory of AFM spin waves
Bond-by-bond picture:Hi,j =−JF(F +1)+ J
2 (Si +Sj)2 ⇒ |Si + Sj |= 0 (singlet)
Ei,j =−JF(F + 1)
⇒ theory of valence bonds
Valence bond picture
Back to the two-site problem: Hi,j = JSiSj
Site-by-site picture:|Sz
i = F ,Szj =−F
⟩: Ei,j =−JF 2.
BUT this is not an eigenstate of Hi,j = JSzi Sz
j +J2 (S
+i S−j +S−i S+
j ).Let Sz fluctuate to gain energy
⇒ theory of AFM spin waves
Bond-by-bond picture:Hi,j =−JF(F +1)+ J
2 (Si +Sj)2 ⇒ |Si + Sj |= 0 (singlet)
Ei,j =−JF(F + 1)
⇒ theory of valence bonds
Valence bond picture
Let us consider now larger lattice: H = ∑〈i,j〉Hi,j = J ∑〈i,j〉SiSj
[ij] = 1√2[α(i)β (j)−β (i)α(j)] = ji singlet bond
Usually longer bonds appear.
VB state: |VB⟩
= ∏pairs|[i, j]
⟩RVB state: |RVB
⟩= ∑
conf∏pairs
A (|i− j|)|[i, j]⟩
Approximation: nearest-neighbor RVB
P. W. Anderson (1973)
Valence bond picture
Let us consider now larger lattice: H = ∑〈i,j〉Hi,j = J ∑〈i,j〉SiSj
[ij] = 1√2[α(i)β (j)−β (i)α(j)] = ji singlet bond
Singlet state on a lattice: letting all spins participate in pair bonds.
The singlet basis for L = 4:
[12][34] =
1 2
4 3
and [23][41] =
1 2
4 3
Usually longer bonds appear.
VB state: |VB⟩
= ∏pairs|[i, j]
⟩RVB state: |RVB
⟩= ∑
conf∏pairs
A (|i− j|)|[i, j]⟩
Approximation: nearest-neighbor RVB
P. W. Anderson (1973)
Valence bond picture
Let us consider now larger lattice: H = ∑〈i,j〉Hi,j = J ∑〈i,j〉SiSj
[ij] = 1√2[α(i)β (j)−β (i)α(j)] = ji singlet bond
Singlet state on a lattice: letting all spins participate in pair bonds.
The singlet basis for L = 4:
[12][34] =
1 2
4 3
and [23][41] =
1 2
4 3
One spin – one bond.
Only non-crossing bonds:
1 2
4 3
−=
1 2
4 3
1 2
4 3
Usually longer bonds appear.
VB state: |VB⟩
= ∏pairs|[i, j]
⟩RVB state: |RVB
⟩= ∑
conf∏pairs
A (|i− j|)|[i, j]⟩
Approximation: nearest-neighbor RVB
P. W. Anderson (1973)
Valence bond picture
Let us consider now larger lattice: H = ∑〈i,j〉Hi,j = J ∑〈i,j〉SiSj
[ij] = 1√2[α(i)β (j)−β (i)α(j)] = ji singlet bond
F = 12
SiSj =−12P
(i,j)0 + 1
4 , and P(i,j)0 projects to the singlet subspace
P(2,3)0
=
1 2 1 2
4 3 4 3
or P(4,6)0
1 2
4 3
6
5
=
1 2
4 3
6
5
Resonate between different configurations of the valence (singlet) bonds.
L. Hulthén (1938)
Usually longer bonds appear.
VB state: |VB⟩
= ∏pairs|[i, j]
⟩RVB state: |RVB
⟩= ∑
conf∏pairs
A (|i− j|)|[i, j]⟩
Approximation: nearest-neighbor RVB
P. W. Anderson (1973)
Valence bond picture
Let us consider now larger lattice: H = ∑〈i,j〉Hi,j = J ∑〈i,j〉SiSj
[ij] = 1√2[α(i)β (j)−β (i)α(j)] = ji singlet bond
Usually longer bonds appear.
VB state: |VB⟩
= ∏pairs|[i, j]
⟩RVB state: |RVB
⟩= ∑
conf∏pairs
A (|i− j|)|[i, j]⟩
Approximation: nearest-neighbor RVB
P. W. Anderson (1973)
Valence bond picture
1) Simplest RVB system:
L = 4, isotropic HM, F = 1/2
Ψ0 = 1√3
([12][34] + [23][41]) = 1√3
1 2
4 3
1 2
4 3
+
E0 =−2J.⟨Ψ0|Sz
j |Ψ0⟩
= 0, for ∀j .〈Φ0|Sz
1Sz2 |Ψ0〉=−1/6, and 〈Φ0|Sz
1Sz3 |Ψ0〉= 1/12
⇒ No magnetic order but there is an AF correlation.
Valence bond picture
1) Simplest RVB system:
L = 4, isotropic HM, F = 1/2
Ψ0 = 1√3
([12][34] + [23][41]) = 1√3
1 2
4 3
1 2
4 3
+
E0 =−2J.⟨Ψ0|Sz
j |Ψ0⟩
= 0, for ∀j .〈Φ0|Sz
1Sz2 |Ψ0〉=−1/6, and 〈Φ0|Sz
1Sz3 |Ψ0〉= 1/12
⇒ No magnetic order but there is an AF correlation.
Valence bond picture
2) One-dimensional J1-J2 isotropic AFM HM, F = 1/2
H(J1,J2) = J1 ∑j
SiSj+1 + J2 ∑j
SiSj+2
Majumdar-Ghosh model: J1 = 2J2 ≡ J.
Valence bond picture
2) One-dimensional J1-J2 isotropic AFM HM, F = 1/2
H(J1,J2) = J1 ∑j
SiSj+1 + J2 ∑j
SiSj+2
Majumdar-Ghosh model: J1 = 2J2 ≡ J.
H(J,J/2) =J4 ∑
j(Sj−1 + Sj + Sj+1)2 + const.
The total spin of a "trion of site j" is either Strion = 1/2 or 3/2
E = J4 LStrion (Strion + 1) + const. ⇒ S(j)
trion(GS) = 1/2.
Valence bond picture
2) One-dimensional J1-J2 isotropic AFM HM, F = 1/2
H(J1,J2) = J1 ∑j
SiSj+1 + J2 ∑j
SiSj+2
Majumdar-Ghosh model: J1 = 2J2 ≡ J.
H(J,J/2) =J4 ∑
j(Sj−1 + Sj + Sj+1)2 + const.
The total spin of a "trion of site j" is either Strion = 1/2 or 3/2
E = J4 LStrion (Strion + 1) + const. ⇒ S(j)
trion(GS) = 1/2.
Construct such a state.
Valence bond picture
2) One-dimensional J1-J2 isotropic AFM HM, F = 1/2
H(J1,J2) = J1 ∑j
SiSj+1 + J2 ∑j
SiSj+2
Majumdar-Ghosh model: J1 = 2J2 ≡ J.
S(j)trion(GS) = 1/2
3 4 5 6 7 81 2
2-fold degeneracy.
No magnetic order.
(Discrete) translational inv. is broken!
spin liquid↔
valence bond solid
Valence bond picture
3 4 5 6 7 81 2
Dimer order parameter: Dj = A〈Sj−1Sj −SjSj+1〉 (with A =−4/3J).
Why there is no resonance?
Cancellations only in the MG case (J2/J1 = 1/2).
Dimer ground state above J2/J1 ≈ 0.24.Inhomogeneous: Dj 6= 0.Only short range AF correlations.
RVB ground state below J2/J1 ≈ 0.24.Homogeneous: Dj = 0.⟨Sz
j Szl
⟩decay algebraically→ quasi-long range AF correlations.
Valence bond picture
3 4 5 6 7 81 2
Dimer order parameter: Dj = A〈Sj−1Sj −SjSj+1〉 (with A =−4/3J).
Why there is no resonance?
Cancellations only in the MG case (J2/J1 = 1/2).Dimer ground state above J2/J1 ≈ 0.24.
Inhomogeneous: Dj 6= 0.Only short range AF correlations.
RVB ground state below J2/J1 ≈ 0.24.Homogeneous: Dj = 0.⟨Sz
j Szl
⟩decay algebraically→ quasi-long range AF correlations.
Valence bond picture
3 4 5 6 7 81 2
Dimer order parameter: Dj = A〈Sj−1Sj −SjSj+1〉 (with A =−4/3J).
Why there is no resonance?
Cancellations only in the MG case (J2/J1 = 1/2).Dimer ground state above J2/J1 ≈ 0.24.
Inhomogeneous: Dj 6= 0.Only short range AF correlations.
RVB ground state below J2/J1 ≈ 0.24.Homogeneous: Dj = 0.⟨Sz
j Szl
⟩decay algebraically→ quasi-long range AF correlations.