Stability of time-reversal symmetry breaking spin liquid...
Transcript of Stability of time-reversal symmetry breaking spin liquid...
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.
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.
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)
Part II
Spin liquid phases
Spin liquid phases
Spin liquid phases
Mean-field bonds: χi,j → χ̄i,j
Plaquette operator
Π = χ̄i,j χ̄j,k . . . χ̄l,i = Abs[Π]eiΦ
Φ = n π (n integer): no time reversal symmetry breaking.With nontrivial Φ: time reversal symmetry breaking.
Low lying excitations composite particle(spinon + an elementary flux)→ anyons.Effective gauge theory.
F = 32 system on a square lattice
F = 32 system on square lattice
Let us recall the n.n. Hamiltonian in the limit of strong repulsion:
HF=3/2eff = ∑〈i,j〉
[g0P
(i,j)0 + g2P
(i,j)2
]
where P(i,j)S = c†
i,σ1c†
j,σ2cj,σ3 ci,σ4 P̂S , and P̂S are antisymmetric.
Hamiltonian with 2 parameters.
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
Hamiltonian with 4 parameters
One can exploit the fact that P̂0 and P̂2 are antisymmetric with respectto the exchange of the spin of the colliding particles
leading to a new effective Hamiltonian:
F = 32 system on square lattice
Let us recall the n.n. Hamiltonian in the limit of strong repulsion:
HF=3/2eff = ∑〈i,j〉
[g0P
(i,j)0 + g2P
(i,j)2
]
where P(i,j)S = c†
i,σ1c†
j,σ2cj,σ3 ci,σ4 P̂S , and P̂S are antisymmetric.
Hamiltonian with 2 parameters.
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
Hamiltonian with 4 parameters
One can exploit the fact that P̂0 and P̂2 are antisymmetric with respectto the exchange of the spin of the colliding particles
leading to a new effective Hamiltonian:
F = 32 system on square lattice
Let us recall the n.n. Hamiltonian in the limit of strong repulsion:
HF=3/2eff = ∑〈i,j〉
[g0P
(i,j)0 + g2P
(i,j)2
]
where P(i,j)S = c†
i,σ1c†
j,σ2cj,σ3 ci,σ4 P̂S , and P̂S are antisymmetric.
Hamiltonian with 2 parameters.
HF=3/2eff = ∑〈i,j〉
[a0 ninj + a1SiSj + a2(SiSj)
2 + a3(SiSj)3]
Hamiltonian with 4 parameters
One can exploit the fact that P̂0 and P̂2 are antisymmetric with respectto the exchange of the spin of the colliding particles
leading to a new effective Hamiltonian:
F = 32 system on square lattice Heff = ∑〈i,j〉
[g0P
(i,j)0 +g2P
(i,j)2
]
Heff = ∑<i,j>
[an
(ninj + χ†
i,j χi,j −ni
)+ as
(SiSj + B†
i,jBi,j − 154 ni
)]
E. Sz. and M. Lewenstein EPL 93 66005 (2011)
Site- and bond-centered operators
ni = c†i,α ci,α (particle number at site i)
Si = c†i,α Fα,β ci,β (spin at site i)
χi,j = c†i,α cj,α (scalar valence bonds)
Bi,j = c†i,α Fα,β cj,β (vector valence bonds)
Nonuniform bond centered orders⟨|χi,j |2
⟩∝ 〈Si Sj 〉 → spin-Peierls distorsion B. Marston and J. Affleck PRB (1989)
⟨|Bi,j |2
⟩∝ 〈Qi Qj 〉 → quadrupole-Peierls distorsion
SU(2) bond-ordered states
The nonlocal part of the mean-field Hamiltonian:(
an 〈χj,i〉δα,β + as 〈Bj,i〉Fα,β
)c†
i,αcj,β + H.c.
A more suitable new link parameter: Ui,j = 〈Bi,j〉F, with the usualinner product in the 3 dimensional space of the generators F.
Ui,j is a member of SU(2)4×4 matrix for F = 3/2 fermionic atoms
The SU(2) plaquette: ΠSU(2) = Ui,jUj,k Uk ,lUl,i .
The SU(2) plaquette ΠSU(2) is also invariant under the U(1)gauge transformation defined above: ci,σ → ci,σ eiφi ,〈χi,j〉 → 〈χi,j〉ei(φj−φi ), and Ui,j → Ui,jei(φj−φi ).
Mean-field phase diagram: F = 3/2, 1/4 filling, square lattice
Parameters
an =5g2−g0
4
as =(g2−g0)
3
gS =− 4tUS
as > 0
an = 0
AFM
(a)
(b)
an < 0
as = 0−an ≈ 2.6as
−g0
−g2
box phase
Nonzero order parameters
AFM: 〈Si〉box phase: 〈χi,j〉
valence bond solid stateplaquettes with flux Φ = 0, or ±π
For the SU(4) line: C. Wu MPL (2006)E. Sz. and M. Lewenstein EPL 93 66005 (2011)
In presence of magnetic field
Hamiltonian with magnetic field:
Hh = HMF + h∑i
Si .
quadrupole dimer/box
box state
FM
0.2 0.4 0.6 0.8
−24
−20
−16
−12
−8
energy
h
g0 = −0.01
g2 = −0.4
(b)
(a)
Order parametersSU(2) dimer/Quadrupole dimer:〈Si〉, 〈χi,j〉, 〈Bi,j〉SU(2) plaquette/Quadrupole plaquette:〈Si〉, 〈χi,j〉, 〈Bi,j〉
E. Sz. and M. Lewenstein EPL 93 66005 (2011)
In presence of magnetic field
Linear Zeeman energy: ωL = gF µBB
Quadratic Zeeman energy: ωq =ω2
Lωhf
.
ωhf : hyperfine splitting (1-10 GHz)
gF : gyromagnetic factor
µB: Bohr magneton
The quadraticZeeman term canbe neglected ifωLωq� 1⇒ ωhf
ωL� 1
Magnetic field was considered in units of t .
t has a maximum atV0 ≈ ωR ,where t ∼ ωR ∼ 1−100 kHz.(ωR/2π ∼ 400.98 kHz for 9Be)
In optical lattice: t = ωR2π ξ 3e−2ξ 2
ξ = (V0/ωR)1/4
V0 potential depth,
ωR recoil energyResult: the SU(2) plaquette state has the lowest enegy at ωL ∼ 0.1−1t (0.1-100 kHz)
ωhf/ωL ∼ 104−107
In presence of magnetic field
Linear Zeeman energy: ωL = gF µBB
Quadratic Zeeman energy: ωq =ω2
Lωhf
.
ωhf : hyperfine splitting (1-10 GHz)
gF : gyromagnetic factor
µB: Bohr magneton
The quadraticZeeman term canbe neglected ifωLωq� 1⇒ ωhf
ωL� 1
Magnetic field was considered in units of t .
t has a maximum atV0 ≈ ωR ,where t ∼ ωR ∼ 1−100 kHz.(ωR/2π ∼ 400.98 kHz for 9Be)
In optical lattice: t = ωR2π ξ 3e−2ξ 2
ξ = (V0/ωR)1/4
V0 potential depth,
ωR recoil energy
Result: the SU(2) plaquette state has the lowest enegy at ωL ∼ 0.1−1t (0.1-100 kHz)
ωhf/ωL ∼ 104−107
In presence of magnetic field
Linear Zeeman energy: ωL = gF µBB
Quadratic Zeeman energy: ωq =ω2
Lωhf
.
ωhf : hyperfine splitting (1-10 GHz)
gF : gyromagnetic factor
µB: Bohr magneton
The quadraticZeeman term canbe neglected ifωLωq� 1⇒ ωhf
ωL� 1
Magnetic field was considered in units of t .
t has a maximum atV0 ≈ ωR ,where t ∼ ωR ∼ 1−100 kHz.(ωR/2π ∼ 400.98 kHz for 9Be)
In optical lattice: t = ωR2π ξ 3e−2ξ 2
ξ = (V0/ωR)1/4
V0 potential depth,
ωR recoil energyResult: the SU(2) plaquette state has the lowest enegy at ωL ∼ 0.1−1t (0.1-100 kHz)
ωhf/ωL ∼ 104−107
In presence of magnetic field
Linear Zeeman energy: ωL = gF µBB
Quadratic Zeeman energy: ωq =ω2
Lωhf
.
ωhf : hyperfine splitting (1-10 GHz)
gF : gyromagnetic factor
µB: Bohr magneton
The quadraticZeeman term canbe neglected ifωLωq� 1⇒ ωhf
ωL� 1
Magnetic field was considered in units of t .
t has a maximum atV0 ≈ ωR ,where t ∼ ωR ∼ 1−100 kHz.(ωR/2π ∼ 400.98 kHz for 9Be)
In optical lattice: t = ωR2π ξ 3e−2ξ 2
ξ = (V0/ωR)1/4
V0 potential depth,
ωR recoil energyResult: the SU(2) plaquette state has the lowest enegy at ωL ∼ 0.1−1t (0.1-100 kHz)
ωhf/ωL ∼ 104−107
Competing spin liquid states of SU(6) symmetricspin-5/2 fermions on a honeycomb lattice
Néel state vs. spin liquid state
Néel state
Néel state is not energeticallyfavorable.
M. Hermele, V. Gurarie, A. M. Rey, PRL (2009).
Spin liquid - VB state
The ground state expectedlyconsists SU(6) singlets
(6 atoms with F = 52 can form
an SU(6) singlet)
Ground state spin liquid states — Technical details
Finite temperature field theory
H[c,c†] =−J ∑〈i,j〉 c†i,αcj,αc†
j,β ci,β
Partition function: Z =∫
[dc][dc]exp(−∫ β0 dτ L[c,c])
L[c,c] = ∑i c i,α∂τci,α + H and #atoms= #sites
Decoupling procedure:
Hubbard-Stratonovich transformation: L[c,c]→ L[c,c;ϕ,χ]auxiliary fields: ϕi (on-site, real), and χi,j (link, complex)
Z =∫
[dϕ][dχ][dc][dc]exp(−∫ β0 dτ L[c,c;ϕ,χ]) =
∫[dϕ][dχ]Z [ϕ,χ]
Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉
[1J |χi,j |2 + lndetGi,j(τ)
])
Saddle-point→ and beyond...
Ground state spin liquid states — Technical details
Finite temperature field theory
H[c,c†] =−J ∑〈i,j〉 c†i,αcj,αc†
j,β ci,β
Partition function: Z =∫
[dc][dc]exp(−∫ β0 dτ L[c,c])
L[c,c] = ∑i c i,α∂τci,α + H and #atoms= #sites
Decoupling procedure:
Hubbard-Stratonovich transformation: L[c,c]→ L[c,c;ϕ,χ]auxiliary fields: ϕi (on-site, real), and χi,j (link, complex)
Z =∫
[dϕ][dχ][dc][dc]exp(−∫ β0 dτ L[c,c;ϕ,χ]) =
∫[dϕ][dχ]Z [ϕ,χ]
Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉
[1J |χi,j |2 + lndetGi,j(τ)
])
Saddle-point→ and beyond...
Ground state spin liquid states — Technical details
Finite temperature field theory
H[c,c†] =−J ∑〈i,j〉 c†i,αcj,αc†
j,β ci,β
Partition function: Z =∫
[dc][dc]exp(−∫ β0 dτ L[c,c])
L[c,c] = ∑i c i,α∂τci,α + H and #atoms= #sites
Decoupling procedure:
Hubbard-Stratonovich transformation: L[c,c]→ L[c,c;ϕ,χ]auxiliary fields: ϕi (on-site, real), and χi,j (link, complex)
Z =∫
[dϕ][dχ][dc][dc]exp(−∫ β0 dτ L[c,c;ϕ,χ]) =
∫[dϕ][dχ]Z [ϕ,χ]
Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉
[1J |χi,j |2 + lndetGi,j(τ)
])
Saddle-point→ and beyond...
Ground state spin liquid states — Technical details
Finite temperature field theory
H[c,c†] =−J ∑〈i,j〉 c†i,αcj,αc†
j,β ci,β
Partition function: Z =∫
[dc][dc]exp(−∫ β0 dτ L[c,c])
L[c,c] = ∑i c i,α∂τci,α + H and #atoms= #sites
Decoupling procedure:
Hubbard-Stratonovich transformation: L[c,c]→ L[c,c;ϕ,χ]auxiliary fields: ϕi (on-site, real), and χi,j (link, complex)
Z =∫
[dϕ][dχ][dc][dc]exp(−∫ β0 dτ L[c,c;ϕ,χ]) =
∫[dϕ][dχ]Z [ϕ,χ]
Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉
[1J |χi,j |2 + lndetGi,j(τ)
])
Saddle-point→ and beyond...
Ground state spin liquid states — Technical details
Ground state spin liquid states — Technical details
Ground state spin liquid states — Technical details
Integrating out the fermion fields:
Saddle-point approximation:
χi,j(q̂) = βV χ̄i,jδq̂,0 + δ χi,j(q̂),
ϕi(q̂) = βV ϕ̄iδq̂,0 + δϕi(q̂).
δSeff[{δψ}]δψ
∣∣∣∣{δψ}=0
= 0
tr log(βG−1) = tr log[β (G−1
(0) −Σ)]
= tr log(
βG−1(0)
)+
∞
∑n=1
tr(G(0)Σ)n
n.
Ground state spin liquid states — Technical details
Integrating out the fermion fields:
Saddle-point approximation:
χi,j(q̂) = βV χ̄i,jδq̂,0 + δ χi,j(q̂),
ϕi(q̂) = βV ϕ̄iδq̂,0 + δϕi(q̂).
δSeff[{δψ}]δψ
∣∣∣∣{δψ}=0
= 0
tr log(βG−1) = tr log[β (G−1
(0) −Σ)]
= tr log(
βG−1(0)
)+
∞
∑n=1
tr(G(0)Σ)n
n.
Ground state spin liquid states — Technical details
Integrating out the fermion fields:
Saddle-point approximation:
χi,j(q̂) = βV χ̄i,jδq̂,0 + δ χi,j(q̂),
ϕi(q̂) = βV ϕ̄iδq̂,0 + δϕi(q̂).
δSeff[{δψ}]δψ
∣∣∣∣{δψ}=0
= 0
tr log(βG−1) = tr log[β (G−1
(0) −Σ)]
= tr log(
βG−1(0)
)+
∞
∑n=1
tr(G(0)Σ)n
n.
Ground state spin liquid states
χi,j = J tr(G0
∂Σ∂ χ∗i,j
)
Π1 = χ1χ2χ3χ4χ5χ6 = |Π1|eiφ1
b) c)a)
Ground state spin liquid states
χi,j = J tr(G0
∂Σ∂ χ∗i,j
)
Π1 = χ1χ2χ3χ4χ5χ6 = |Π1|eiφ1
b) c)a)
Staggered flux state
Box state
Chiral spin liquid state
Chiral spin liquid state
Chiral spin liquid state
Finite temperature behavior
chiral state quasiplaquette state plaquette state
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
00 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
0
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
0
1
0 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3 0 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3
a) b) c)
All the spin liquid phases "melt" around the same critical temperature.
No new state occurs as lowest freeenergy SP solution.
The SP free energies approach eachother without crossing.
The chiral state remains the lowestfree energy solution even at T > 0. 0
-6chiral spin liquid phasequasi plaquette phaseplaquette phase
-7
-8
-9
-10
-11
-12
-13
-14
-150.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Finite temperature behavior
chiral state quasiplaquette state plaquette state
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
00 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
0
0.7
0.8
0.9
0.4
0.5
0.6
0.1
0.2
0.3
0
1
0 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3 0 0.7 0.8 0.90.4 0.5 0.60.1 0.2 0.3
a) b) c)
All the spin liquid phases "melt" around the same critical temperature.
No new state occurs as lowest freeenergy SP solution.
The SP free energies approach eachother without crossing.
The chiral state remains the lowestfree energy solution even at T > 0. 0
-6chiral spin liquid phasequasi plaquette phaseplaquette phase
-7
-8
-9
-10
-11
-12
-13
-14
-150.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Stability analysis
Stability matrix : Cµ,ν ∼∂ 2(G0Σ)2
∂ χµ∂ χν
chiral spin liquid quasiplaquettea) b)
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0
-0.500
-0.500
-0.500
-0.500
-0.500
0.000
0.000 0.000
0.000
0.000 0.000
0.000
0.200
0.200
0.200
0.200
0.200
0.200
0.200 0
.200
0.200
0.200
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0
The quasiplaquette state collapses into the lower free energy chiralspin liquid state.
Stability analysis
Stability matrix : Cµ,ν ∼∂ 2(G0Σ)2
∂ χµ∂ χν
chiral spin liquid quasiplaquettea) b)
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.400
0.400
0.400
0.400
0.400
0.400
0.400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0
-0.500
-0.500
-0.500
-0.500
-0.500
0.000
0.000 0.000
0.000
0.000 0.000
0.000
0.200
0.200
0.200
0.200
0.200
0.200
0.200 0
.200
0.200
0.200
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0
The quasiplaquette state collapses into the lower free energy chiralspin liquid state.
Experimentally measurable quantities
Structure factor: S(r,τ; r′,0) = 〈Sz(r,τ)Sz(r′,0)〉chiral spin liquid quasiplaquette
a) b)0.160
0.160
0.160
0.160
0.160
0.160
0.170
0.170 0.170
0.170
0.170 0.1700.170
0.180 0.180 0.180
0.180
0.1800.1
80
0.180
0.150
0.156
0.162
0.168
0.174
0.180
0.186
0.192
0.198
0.204
0.210
0
0.1900.190
0.190
0.190
0.190
0.190
0.190
0.200
0.200
0.200
0.200
0.200
0.150
0.156
0.162
0.168
0.174
0.180
0.186
0.192
0.198
0.204
0.210
0
Unambiguous features→ Suitable tool to distinguish the phases.
Experimentally measurable quantities
Structure factor: S(r,τ; r′,0) = 〈Sz(r,τ)Sz(r′,0)〉chiral spin liquid quasiplaquette
a) b)0.160
0.160
0.160
0.160
0.160
0.160
0.170
0.170 0.170
0.170
0.170 0.1700.170
0.180 0.180 0.180
0.180
0.1800.1
80
0.180
0.150
0.156
0.162
0.168
0.174
0.180
0.186
0.192
0.198
0.204
0.210
0
0.1900.190
0.190
0.190
0.190
0.190
0.190
0.200
0.200
0.200
0.200
0.200
0.150
0.156
0.162
0.168
0.174
0.180
0.186
0.192
0.198
0.204
0.210
0
Unambiguous features→ Suitable tool to distinguish the phases.
Experimental measurable quantities
Spectral density: ρtot(ω) = ∑k ImS(k,ω)
0 1 2 3 4 50.0
0.4
0.8
1.2
0 1 2 3 4 50.0
0.1
0.2
0.3
0.4
0 1 2 3 4 50
2
4
6
8
Unambiguous features→ Suitable tool to distinguish the phases.
Experimental measurable quantities
Spectral density: ρtot(ω) = ∑k ImS(k,ω)
0 1 2 3 4 50.0
0.4
0.8
1.2
0 1 2 3 4 50.0
0.1
0.2
0.3
0.4
0 1 2 3 4 50
2
4
6
8
Unambiguous features→ Suitable tool to distinguish the phases.
Thank you for your attention