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

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.

Quantum information

In topological phases of spinliquids the quasiparticles havefractional statistics. They arenonlocal and resist well againstlocal perturbations. Promising qbitcandidates.

Quantum information

Low energy excitations above spinliquids can be described byeffective gauge theories. Aim: tostudy various gauge theories withultracold atoms.

Quantum information

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

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

F1 = F2 = 32

S = 0: Sz = 0,S = 1: Sz =−1,0,1

S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3

H =−t ∑<i,j>

c†i,αcj,α +∑

iUα,β

γ,δ c†i,αc†

i,β ci,δ ci,γ .

2 components: ↑, and ↓

F1 = F2 = 12 ,

S = 0: Sz = 0,S = 1: Sz =−1,0,1

F1 = F2 = 32

S = 0: Sz = 0,S = 1: Sz =−1,0,1

S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3

H =−t ∑<i,j>

c†i,αcj,α +∑

iUα,β

γ,δ c†i,αc†

i,β ci,δ ci,γ .

F1 = F2 = 32

S = 0: Sz = 0,S = 1: Sz =−1,0,1

S = 2: Sz =−2, . . . ,2S = 3: Sz =−3, . . . ,3

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.

The Hamiltonian:

H = Hkin + Hint

Pauli’s principle

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.

The Hamiltonian:

H = Hkin + Hint

Pauli’s principle

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

(i)0 +U2P

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)

The Hamiltonian:

H = Hkin + Hint

Pauli’s principle

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)0 +U2P

(i)2 +U4P

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

H =−t ∑<i,j>

c†i,αcj,α + U ∑

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

H =−t ∑<i,j>

c†i,αcj,α + U ∑

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 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

H = J ∑〈i,j〉

c†i,αc†

j,β cj,αci,β

Mott state

H = J ∑〈i,j〉

c†i,αc†

j,β cj,αci,β

Mott state

(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]

4 ninj)

F = 32 :

HF=3/2eff = ∑〈i,j〉

2 + a3(SiSj)3]

F = 52 :

HF=5/2eff = ∑〈i,j〉

4 +a5(Si Sj)5]

4 ninj)

F = 32 :

HF=3/2eff = ∑〈i,j〉

2 + a3(SiSj)3]

F = 52 :

HF=5/2eff = ∑〈i,j〉

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

4 ninj)

i = F ,Szj =−F

⟩: Ei,j =−JF 2.

j +J2 (S

+i S−j +S−i S+

A〉=−〈SzB〉= F −∆F .

4 ninj)

i = F ,Szj =−F

⟩: Ei,j =−JF 2.

j +J2 (S

+i S−j +S−i S+

A〉=−〈SzB〉= F −∆F .

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†

)1/2aj S−A,j = a†

(2F −a†

SzA,j = 2−a†

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.

4 ninj)

S+A,j =

(2F −a†

)1/2aj S−A,j = a†

(2F −a†

SzA,j = 2−a†

Change concept.

4 ninj)

S+A,j =

(2F −a†

)1/2aj S−A,j = a†

(2F −a†

SzA,j = 2−a†

Change concept.

4 ninj)

S+A,j =

(2F −a†

)1/2aj S−A,j = a†

(2F −a†

SzA,j = 2−a†

Change concept.

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.

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

i = F ,Szj =−F

⟩: Ei,j =−JF 2.

j +J2 (S

+i S−j +S−i S+

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

Back to the two-site problem: Hi,j = JSiSj

i = F ,Szj =−F

⟩: Ei,j =−JF 2.

j +J2 (S

+i S−j +S−i S+

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

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)

Singlet state on a lattice: letting all spins participate in pair bonds.

The singlet basis for L = 4:

[12][34] =

and [23][41] =

VB state: |VB⟩

= ∏pairs|[i, j]

RVB state: |RVB⟩

= ∑conf

∏pairs

A (|i− j|)|[i, j]⟩

Singlet state on a lattice: letting all spins participate in pair bonds.

The singlet basis for L = 4:

[12][34] =

and [23][41] =

One spin – one bond.

Only non-crossing bonds:

VB state: |VB⟩

= ∏pairs|[i, j]

RVB state: |RVB⟩

= ∑conf

∏pairs

A (|i− j|)|[i, j]⟩

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

Resonate between different configurations of the valence (singlet) bonds.

L. Hulthén (1938)

VB state: |VB⟩

= ∏pairs|[i, j]

RVB state: |RVB⟩

= ∑conf

∏pairs

A (|i− j|)|[i, j]⟩

VB state: |VB⟩

= ∏pairs|[i, j]

RVB state: |RVB⟩

= ∑conf

∏pairs

A (|i− j|)|[i, j]⟩

1) Simplest RVB system:

L = 4, isotropic HM, F = 1/2

Ψ0 = 1√3

([12][34] + [23][41]) = 1√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.

1) Simplest RVB system:

L = 4, isotropic HM, F = 1/2

Ψ0 = 1√3

([12][34] + [23][41]) = 1√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.

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(J1,J2) = J1 ∑j

SiSj+1 + J2 ∑j

SiSj+2

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.

H(J1,J2) = J1 ∑j

SiSj+1 + J2 ∑j

SiSj+2

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.

H(J1,J2) = J1 ∑j

SiSj+1 + J2 ∑j

SiSj+2

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

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

⟩decay algebraically→ quasi-long range AF correlations.

3 4 5 6 7 81 2

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.

3 4 5 6 7 81 2

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.

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

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〉

(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〉

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:

HF=3/2eff = ∑〈i,j〉

(i,j)0 + g2P

(i,j)2

i,σ1c†

HF=3/2eff = ∑〈i,j〉

2 + a3(SiSj)3]

HF=3/2eff = ∑〈i,j〉

(i,j)0 + g2P

(i,j)2

i,σ1c†

HF=3/2eff = ∑〈i,j〉

2 + a3(SiSj)3]

F = 32 system on square lattice Heff = ∑〈i,j〉

(i,j)0 +g2P

(i,j)2

Heff = ∑<i,j>

(ninj + χ†

i,j χi,j −ni

(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

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

as =(g2−g0)

gS =− 4tUS

as > 0

an = 0

an < 0

as = 0−an ≈ 2.6as

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

quadrupole dimer/box

box state

0.2 0.4 0.6 0.8

energy

g0 = −0.01

g2 = −0.4

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)

Linear Zeeman energy: ωL = gF µBB

Quadratic Zeeman energy: ωq =ω2

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

µB: Bohr magneton

ωL� 1

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

µB: Bohr magneton

ωL� 1

ξ = (V0/ωR)1/4

V0 potential depth,

ωhf/ωL ∼ 104−107

µB: Bohr magneton

ωL� 1

ξ = (V0/ωR)1/4

V0 potential depth,

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...

j,β ci,β

Z =∫

Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉

j,β ci,β

Z =∫

Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉

j,β ci,β

Z =∫

Z [ϕ,χ] = exp(−∫ β0 dτ ∑〈i,j〉

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

tr log(βG−1) = tr log[β (G−1

(0) −Σ)]

= tr log(

βG−1(0)

∑n=1

tr(G(0)Σ)n

δSeff[{δψ}]δψ

∣∣∣∣{δψ}=0

(0) −Σ)]

= tr log(

βG−1(0)

∑n=1

tr(G(0)Σ)n

δSeff[{δψ}]δψ

∣∣∣∣{δψ}=0

(0) −Σ)]

= tr log(

βG−1(0)

∑n=1

tr(G(0)Σ)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

Finite temperature behavior

chiral state quasiplaquette state plaquette state

00 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 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

-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

00 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 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

-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)

-0.500

0.000 0.000

0.200 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)

-0.500

0.000 0.000

0.200 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

0.170 0.170

0.170 0.1700.170

0.180 0.180 0.180

0.1800.1

0.1900.190

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

0.170 0.170

0.170 0.1700.170

0.180 0.180 0.180

0.1800.1

0.1900.190

Experimental measurable quantities

Spectral density: ρtot(ω) = ∑k ImS(k,ω)

0 1 2 3 4 50.0

0 1 2 3 4 50

Experimental measurable quantities

Spectral density: ρtot(ω) = ∑k ImS(k,ω)

0 1 2 3 4 50.0

0 1 2 3 4 50

Thank you for your attention

Stability of time-reversal symmetry breaking spin liquid...

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