Exotic magnetic phases in Mott...
Transcript of Exotic magnetic phases in Mott...
Exotic magnetic phasesin Mott insulators
Sinkovicz PeterWigner Research Centre for Physics
Institute for Solid State Physics and Optics
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Antiferromagnets
Ground-states in antiferromagnetic models:
• Neel phase• Exotic phases (non-classical ground-state)
a) Frustration (small-S spin model)b) Degenerate ground-state configuration (large-S spin model)
⇒ Low coordination number (strong quantum fluctuation)⇒ High coordination number (weak quantum fluctuation)
Order by disordermechanism
Valence Bond Solid(VBS)
Spin Liquid orResonate VBS
(SL)
Exotic magnetic phases in strongly correlated electron systems 2 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
su(N) Heisenberg model
H = J∑〈i,j〉
Sαβ( i )Sβα( j ) J > 0
[Sαβ(i),Sµν (i)
]= δα,νSµβ(i)− δβ,µSαν (i) and
∑α
Sαα(i) = M I
• Simple, still a reliable model
• Gauge theories
• High T superconductivity
• Quantum informations
Exotic magnetic phases in strongly correlated electron systems 3 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
su(N) Heisenberg model
H = J∑〈i,j〉
Sαβ( i )Sβα( j ) J > 0
[Sαβ(i),Sµν (i)
]= δα,νSµβ(i)− δβ,µSαν (i) and
∑α
Sαα(i) = M I
• Simple, still a reliable model
• Gauge theories
• High T superconductivity
• Quantum informations
Simple, but a good model forunderstanding the behavior ofthe quantum magnets.
Exotic magnetic phases in strongly correlated electron systems 3 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
su(N) Heisenberg model
H = J∑〈i,j〉
Sαβ( i )Sβα( j ) J > 0
[Sαβ(i),Sµν (i)
]= δα,νSµβ(i)− δβ,µSαν (i) and
∑α
Sαα(i) = M I
• Simple, still a reliable model
• Gauge theories
• High T superconductivity
• Quantum informations
In spin liquid phase theground-state excitations can bedescribed by gauge theories, so itmakes it possible to study highenergy physics in low energy.
Exotic magnetic phases in strongly correlated electron systems 3 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
su(N) Heisenberg model
H = J∑〈i,j〉
Sαβ( i )Sβα( j ) J > 0
[Sαβ(i),Sµν (i)
]= δα,νSµβ(i)− δβ,µSαν (i) and
∑α
Sαα(i) = M I
• Simple, still a reliable model
• Gauge theories
• High T superconductivity
• Quantum informations
Cuprates (e.g. La2CuO4) has awell above critical temperature(Tc=35K), than the other super-conductors. P. W. Anderson ⇒spin liquid theory
Exotic magnetic phases in strongly correlated electron systems 3 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
su(N) Heisenberg model
H = J∑〈i,j〉
Sαβ( i )Sβα( j ) J > 0
[Sαβ(i),Sµν (i)
]= δα,νSµβ(i)− δβ,µSαν (i) and
∑α
Sαα(i) = M I
• Simple, still a reliable model
• Gauge theories
• High T superconductivity
• Quantum informations
In topological phases robustqubits are constructed. Thesequbits can be controlled byHeisenberg-like models
Exotic magnetic phases in strongly correlated electron systems 3 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Experimental realization of the Heisenberg model
Representations of the Sαβ operator:
• Schwinger fermion
Sαβ(i) = c†i,αci,β {ci,α, c†j,β} = δi,jδα,β
su(N) symmetric phase
ground-state: Fermi liquid → non symmetry breaking
• Schwinger boson
Sαβ(i) = b†i,αbi,β [bi,α, b†j,β] = δi,jδα,β
Order phase (su(N) symmetry breaking)
ground state: Bose condensation (2D+) → symmetry breaking
∑α
Sαα(i) =∑α
f †αfα = ?
Exotic magnetic phases in strongly correlated electron systems 4 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Experimental realization of the Heisenberg model
Representations of the Sαβ operator:
• Schwinger fermion
Sαβ(i) = c†i,αci,β {ci,α, c†j,β} = δi,jδα,β
su(N) symmetric phase
ground-state: Fermi liquid → non symmetry breaking
• Schwinger boson
Sαβ(i) = b†i,αbi,β [bi,α, b†j,β] = δi,jδα,β
Order phase (su(N) symmetry breaking)
ground state: Bose condensation (2D+) → symmetry breaking
∑α
Sαα(i) =∑α
f †αfα = ?
Exotic magnetic phases in strongly correlated electron systems 4 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Experimental realization of the Heisenberg model
Representations of the Sαβ operator:
• Schwinger fermion
Sαβ(i) = c†i,αci,β {ci,α, c†j,β} = δi,jδα,β
su(N) symmetric phase
ground-state: Fermi liquid → non symmetry breaking
• Schwinger boson
Sαβ(i) = b†i,αbi,β [bi,α, b†j,β] = δi,jδα,β
Order phase (su(N) symmetry breaking)
ground state: Bose condensation (2D+) → symmetry breaking
∑α
Sαα(i) =∑α
f †αfα = ?
Exotic magnetic phases in strongly correlated electron systems 4 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Realization on optical lattices
U � t
〈ni〉 = M
T < Tc
HHub. = −t∑〈i,j〉
(c†i,αcj,α + h.c.
)+
+U
2
∑i
c†i,αc†i,βci,βci,α
Mott phase, spin swapping (second order)
Heisenberg model
Heff. = J∑〈i,j〉
c†i,αcj,αc†j,βci,β J = −4t2
U
Exotic magnetic phases in strongly correlated electron systems 5 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Realization on optical lattices
U � t
〈ni〉 = M
T < Tc
HHub. = −t∑〈i,j〉
(c†i,αcj,α + h.c.
)+
+U
2
∑i
c†i,αc†i,βci,βci,α
Mott phase, spin swapping (second order)
M gives the representation
For M = 1, it recovers the N-dimensi-onal fundamental representation (thecorresponding Young tableau a singlebox)
Exotic magnetic phases in strongly correlated electron systems 5 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground state
e.g. su(2)
S(i)S(j) = S+(i)S−(j) + S−(i)S+(j) + Sz(i)Sz(j)
- Neel ordered
Not eigenstate of the Hamiltonian
Neel like state: classical ordering + small fluctuations
- Valence Bond Solid
Not eigenstate of the Hamiltonian (between two singlet)
VBS like state, e.g. RVB (spin liquid)
For V <∞ ENeel > EVBS
For V =∞ ENeel = EVBS
Exotic magnetic phases in strongly correlated electron systems 6 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
PART I
Exotic magnetic phases in strongly correlated electron systems 7 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 5/2 fermion
• Fundamental representation
• Honeycomb lattice
• Finite temperature
→ su(6)
→ Mott phase: one particle/site
→ Spin liquid, hexagon unit cell
→ Functional integral (τ = −iβ~)
nuclear spin
orbital momentumelectron spin
total electron shell momentum
Experimental realizationAlkaline earth metals
Total spin angular momentum on site i
S(i) = I(i) + J(i) = I(i)
su(N) = su(2 I + 1), 173Yb → su(6)Nat. Phys. 8, 825 (2012), PRL 105, 190401 (2010),
PRL 105, 030402 (2010)
Exotic magnetic phases in strongly correlated electron systems 8 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 5/2 fermion
• Fundamental representation
• Honeycomb lattice
• Finite temperature
→ su(6)
→ Mott phase: one particle/site
→ Spin liquid, hexagon unit cell
→ Functional integral (τ = −iβ~)
One particle per site (〈ni〉 = 1), one every lattice site there is 6internal states
Exotic magnetic phases in strongly correlated electron systems 8 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 5/2 fermion
• Fundamental representation
• Honeycomb lattice
• Finite temperature
→ su(6)
→ Mott phase: one particle/site
→ Spin liquid, hexagon unit cell
→ Functional integral (τ = −iβ~)
M. Hermele et al. made a high N expansion on a square lattice→ for large N they found spin liquid ground-state
PRL 103, 135301 (2009)
⇒ It predict spin liquid ground state (su(6) singlets) in ourcase, since we have smaller coordination number and large N⇒ minimal model: one hexagon unit cell
Exotic magnetic phases in strongly correlated electron systems 8 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 5/2 fermion
• Fundamental representation
• Honeycomb lattice
• Finite temperature
→ su(6)
→ Mott phase: one particle/site
→ Spin liquid, hexagon unit cell
→ Functional integral (τ = −iβ~)
Grand canonical partition function (F = −kBT lnZβ)
Zβ = Tr[e−βK
]=
∫D[c, c]e−
1~Sβ [c,c]
where K = H− µi(ni − 1) ({µi} Lagrange multipliers) and
Sβ[c, c] =β~∫0
dτ L(τ ; c, c] =
=β~∫0
dτ
{∑i
[ci,α(τ) (~∂τ + µi) ci,α(τ)]− J∑〈i,j〉
ci,α(τ)cj,α(τ)cj,β(τ)ci,β(τ)
}
Exotic magnetic phases in strongly correlated electron systems 8 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Hubbard-Stratonovich transformation
Interaction term
ci,α(τ)cj,α(τ)cj,β(τ)ci,β(τ)
Decoupling:
- spin liquid phase → m.f. variable χ independent of the spin
Hubbard-Stratonovich field: χi,j =∑α
ci,α(τ)cj,α(τ)
Mean field: χi,j = χi,j + δχi,j
Decoupled action
S[c, χ, δχ, . . . ] = S0[c, c] + J−1S1[c, c, δχ, δχ∗] + J−2S2[|δχ|2]
Exotic magnetic phases in strongly correlated electron systems 9 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Saddle point
Mean field equations
δS[... ]δχ
∣∣∣χmf
= 0→ self-consistent equations
→ constraint (one particle per site)
Classification of solutions (local u(1) of the fermions is inherit)
χi,j =∑σ
ci,σcj,σ →∑σ
ci,σcj,σei(ϑj−ϑi) = χi,je
iϕi.j
Wilson loops
Π1 := χ(1)χ(2)χ(3)χ(4)χ(5)χ(6)
Π2 := χ(1)∗χ(8)∗χ(5)∗χ(9)χ(3)∗χ(7)
Π3 := χ(6)∗χ(7)∗χ(4)∗χ(8)χ(2)∗χ(9)∗
6
8
9
7
1
2
34
5
Π 3
Π 1
Π 2
Exotic magnetic phases in strongly correlated electron systems 10 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Saddle point
Mean field equations
δS[... ]δχ
∣∣∣χmf
= 0→ self-consistent equations
→ constraint (one particle per site)
Classification of solutions (local u(1) of the fermions is inherit)
χi,j =∑σ
ci,σcj,σ →∑σ
ci,σcj,σei(ϑj−ϑi) = χi,je
iϕi.j
Wilson loops
Π1 := χ(1)χ(2)χ(3)χ(4)χ(5)χ(6)
Π2 := χ(1)∗χ(8)∗χ(5)∗χ(9)χ(3)∗χ(7)
Π3 := χ(6)∗χ(7)∗χ(4)∗χ(8)χ(2)∗χ(9)∗
6
8
9
7
1
2
34
5
Π 3
Π 1
Π 2
Exotic magnetic phases in strongly correlated electron systems 10 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Saddle point
Mean field equations
δS[... ]δχ
∣∣∣χmf
= 0→ self-consistent equations
→ constraint (one particle per site)
Classification of solutions (local u(1) of the fermions is inherit)
χi,j =∑σ
ci,σcj,σ →∑σ
ci,σcj,σei(ϑj−ϑi) = χi,je
iϕi.j
Wilson loops
Π1 := χ(1)χ(2)χ(3)χ(4)χ(5)χ(6)
Π2 := χ(1)∗χ(8)∗χ(5)∗χ(9)χ(3)∗χ(7)
Π3 := χ(6)∗χ(7)∗χ(4)∗χ(8)χ(2)∗χ(9)∗
6
8
9
7
1
2
34
5
Π 3
Π 1
Π 2
Exotic magnetic phases in strongly correlated electron systems 10 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Mean field solutions
T = 0
E Π1 Π2 Π3
−6.148 riΦ0 riΦ0 riΦ0−6.148 r−iΦ0 r−iΦ0 r−iΦ0
−6.062 r1 r2eiπ r2e
iπ
−6.062 r2eiπ r1 r2e
iπ
−6.062 r2eiπ r2e
iπ r1
−6 1 0 0−6 0 1 0−6 0 0 1
Φ = 2π/3
ΦΦ
Φ
ΦΦ
ΦΦ
ΦΦ
Φ
ΦΦ
Φ
chiral phase
plaquette phase
0
0
0
0
0 0
0
quasi-plaquette phase
Π
Π
Π Π
Π Π
0
0 0
0
0 0
Exotic magnetic phases in strongly correlated electron systems 11 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Mean field solutions
T = 0
E Π1 Π2 Π3
−6.148 riΦ0 riΦ0 riΦ0−6.148 r−iΦ0 r−iΦ0 r−iΦ0
−6.062 r1 r2eiπ r2e
iπ
−6.062 r2eiπ r1 r2e
iπ
−6.062 r2eiπ r2e
iπ r1
−6 1 0 0−6 0 1 0−6 0 0 1
Φ = 2π/3
ΦΦ
Φ
ΦΦ
ΦΦ
ΦΦ
Φ
ΦΦ
Φ
chiral phase
plaquette phase
0
0
0
0
0 0
0
quasi-plaquette phase
Π
Π
Π Π
Π Π
0
0 0
0
0 0
Ground-state
• su(6) and translation invariant
• Φ = 2π/3 magnetic flux
→ no time reversal symmetry
• topological phase: edge states
→ robust: gap in the spectrum
Exotic magnetic phases in strongly correlated electron systems 11 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Feynman rules in finite temperature
Mean field action
S[c, c, δχ, δχ∗] = S0[c, c] + J−1S1[c, c, δχ, δχ∗] + J−2S2[|δχ|2]
Bare graph elements
• S0[c, c]→ free fermion propagator
G(s→s′)0 (k, n) =
• S2[|δχ|2]→ free boson propagator
D(v→v′)0 (k, n) =
• S1[c, c, δχ, δχ∗]→ bare vertex
Exotic magnetic phases in strongly correlated electron systems 12 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Further results
• Effective action for δχ (one loop)
Seff[δχ, δχ∗] = δχD−11 δχ∗ + δχA
−11 δχ
• Spin-Spin correlation function (neutron scattering)
Σ(r − r′, τ) :=⟨Tτ
(Sz(r, τ)Sz(r
′, 0))⟩
• Elementary excitations
→ Spin-Spin correlation function analytical continuing forcomplex frequencies (iω → ν + iη). Poles = excitations
• Stability analysis (stable if: Imν ≤ 0 and Reν ≥ 0)
→ The three phase around the ground state until T = 0.83J→ Different unit cells (inspirited by Monte Carlo predictions)
a) Hexagon unit cell (6 fermions, 9 HS bosons, 3 Wilson loops)
b) Rectangle unit cell (12 fermions, 18 HS bosons, 6 Wilson loops)
c) Propeller unit cell (18 fermions, 27 HS bosons, 14 Wilson loops)
Exotic magnetic phases in strongly correlated electron systems 13 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
PART II
Exotic magnetic phases in strongly correlated electron systems 14 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 3/2 fermion
• Fundamental representation
• Face centered cubic lattice
• Spin-wave theory
→ su(4)
→ Mott phase: one particle/site
→ Ordered phase, 4 site unit cell
→ 4-flavor boson representation
nuclear spin
orbital momentumelectron spin
total electron shell momentum
Experimental realizationAlkaline earth metals
Total spin angular momentum on site i
S(i) = I(i) + J(i) = I(i)
su(2I + 1),173Yb → su(6)
(4 active and 2 idle components, exchange interaction)
Exotic magnetic phases in strongly correlated electron systems 15 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 3/2 fermion
• Fundamental representation
• Face centered cubic lattice
• Spin-wave theory
→ su(4)
→ Mott phase: one particle/site
→ Ordered phase, 4 site unit cell
→ 4-flavor boson representation
One particle per site (〈ni〉 = 1), one every lattice site there is 4internal states: α = {A,B,C,D}
Exotic magnetic phases in strongly correlated electron systems 15 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 3/2 fermion
• Fundamental representation
• Face centered cubic lattice
• Spin-wave theory
→ su(4)
→ Mott phase: one particle/site
→ Ordered phase, 4 site unit cell
→ 4-flavor boson representation
→ Many experimental and numerical proposal for ordered phasein body centered cubic lattice. A few references andcalculation can be found in T. Yildirim, Turkish Journal ofPhysics, 23, 47-76 (1999).
→ Experimental they have seen ordered phase in Ba2HoSbO6,where Ho3+ form a face centered lattice with spin-3/2, PRB81, 064425 (2010)
⇒ Ordered phase (+ small fluctuations)
⇒ Minimal model: 4 site unit cellExotic magnetic phases in strongly correlated electron systems 15 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Set the problem
• Spin 3/2 fermion
• Fundamental representation
• Face centered cubic lattice
• Spin-wave theory
→ su(4)
→ Mott phase: one particle/site
→ Ordered phase, 4 site unit cell
→ 4-flavor boson representation
Overview of the calculation
− Degenerate mean field ground state
− Choose a one parameter class
− Quantum fluctuations assign the unique ground state⇒ order-by-disorder mechanism
Exotic magnetic phases in strongly correlated electron systems 15 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state I.
In the classical limit, boson operators are characterized by theirmean values: 〈br,α〉 =
√Mξr,α, where ξ ∈ C4
E0 = JM2∑〈r,r′〉
∣∣∣∣∣∑α
ξ∗r,αξr′,α
∣∣∣∣∣2
• Energy of the classical ground state is bounded from below byzero
• Zero energy is realized for mutually orthogonal classicalconfigurations on the neighboring sites
⇒ NOT UNIQUE
Exotic magnetic phases in strongly correlated electron systems 16 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state II.
Fcc lattice contains: octahedra and tetrahedra
Exotic magnetic phases in strongly correlated electron systems 17 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state II.
Fcc lattice contains: octahedra and tetrahedra
i) Consider a single octahedron(a)
(b)
(c) (d)
(e)
Exotic magnetic phases in strongly correlated electron systems 17 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state II.
Fcc lattice contains: octahedra and tetrahedra
i) Single octahedron under-constrainedii) Single tetrahedron
unique configuration
Exotic magnetic phases in strongly correlated electron systems 17 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state II.
Fcc lattice contains: octahedra and tetrahedra
i) Single octahedron under-constrained
ii) Single tetrahedron unique configuration
⇒ Fcc lattice
• Every plane form a bipartite square lattice and can be coloredwith two colors
• Odd and Even plans can be colored with different vectorsEven: ξA = (cosϑz, sinϑz, 0, 0) and ξB = (− sinϑz, cosϑz, 0, 0)
Odd: ξA = (0, 0 cosϑz+1, sinϑz+1) and ξB = (0, 0,− sinϑz+1, cosϑz+1)
Exotic magnetic phases in strongly correlated electron systems 17 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Classical ground-state II.
Fcc lattice contains: octahedra and tetrahedra
i) Single octahedron under-constrained
ii) Single tetrahedron unique configuration
⇒ Fcc lattice
• Every plane form a bipartite square lattice and can be coloredwith two colors
• Odd and Even plans can be colored with different vectorsEven: ξA = (cosϑz, sinϑz, 0, 0) and ξB = (− sinϑz, cosϑz, 0, 0)
Odd: ξA = (0, 0 cosϑz+1, sinϑz+1) and ξB = (0, 0,− sinϑz+1, cosϑz+1)
iii) Helical state: ϑz = zϑ/2 on every plane (ϑ =?)
Exotic magnetic phases in strongly correlated electron systems 17 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground-state
• Technical calculation:- Introduce a canonical transformed boson, which create pure
colors on each site- Introduce Holstein-Primakoff bosons- Semi-classical approximation: If the quantum fluctuations are
small around the supposed classical ground-state- In second order of of the quantum fluctuations the ground-
state energy can be calculated via Bogoliubov transformation
• Results:- numerical result shows ϑ = 0 is the ground state and ϑ = π/2
has the maximum energy- ϑ = 0 and ϑ = π/2 special cases can be calculated analytically
- The analytical calculations point out, that two special casesare effectively 2-dimensional only.Mermin-Wagner-Hohenberg theorem: In 2-dim. there is nolong range order at finite temperature.
Exotic magnetic phases in strongly correlated electron systems 18 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground-state
• Technical calculation:- Introduce a canonical transformed boson, which create pure
colors on each site- Introduce Holstein-Primakoff bosons- Semi-classical approximation: If the quantum fluctuations are
small around the supposed classical ground-state- In second order of of the quantum fluctuations the ground-
state energy can be calculated via Bogoliubov transformation
• Results:- numerical result shows ϑ = 0 is the ground state and ϑ = π/2
has the maximum energy- ϑ = 0 and ϑ = π/2 special cases can be calculated analytically
- The analytical calculations point out, that two special casesare effectively 2-dimensional only.Mermin-Wagner-Hohenberg theorem: In 2-dim. there is nolong range order at finite temperature.
Exotic magnetic phases in strongly correlated electron systems 18 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Ground-state
• Technical calculation:- Introduce a canonical transformed boson, which create pure
colors on each site- Introduce Holstein-Primakoff bosons- Semi-classical approximation: If the quantum fluctuations are
small around the supposed classical ground-state- In second order of of the quantum fluctuations the ground-
state energy can be calculated via Bogoliubov transformation
• Results:- numerical result shows ϑ = 0 is the ground state and ϑ = π/2
has the maximum energy- ϑ = 0 and ϑ = π/2 special cases can be calculated analytically- The analytical calculations point out, that two special cases
are effectively 2-dimensional only.Mermin-Wagner-Hohenberg theorem: In 2-dim. there is nolong range order at finite temperature.
Exotic magnetic phases in strongly correlated electron systems 18 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Solution for the finite T instability
• Extend the model by a next-nearest-neighbor term
Hext. = J∑〈i,j〉
Sαβ( i )Sβα( j ) + J2
∑〈〈i,j〉〉
Sαβ( i )Sβα( j )
where J > 0 AF coupling and J2 = −0.023J F coupling
• The helical state is supposed in the classical configuration
⇒ ϑ = 0 is still the ground-state
⇒ Now, it has a truly 3-d structure
⇒ Fluctuations are small enough, stable
Exotic magnetic phases in strongly correlated electron systems 19 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Solution for the finite T instability
• Extend the model by a next-nearest-neighbor term
Hext. = J∑〈i,j〉
Sαβ( i )Sβα( j ) + J2
∑〈〈i,j〉〉
Sαβ( i )Sβα( j )
where J > 0 AF coupling and J2 = −0.023J F coupling
• The helical state is supposed in the classical configuration
⇒ ϑ = 0 is still the ground-state
⇒ Now, it has a truly 3-d structure
⇒ Fluctuations are small enough, stable
Exotic magnetic phases in strongly correlated electron systems 19 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Solution for the finite T instability
• Extend the model by a next-nearest-neighbor term
Hext. = J∑〈i,j〉
Sαβ( i )Sβα( j ) + J2
∑〈〈i,j〉〉
Sαβ( i )Sβα( j )
where J > 0 AF coupling and J2 = −0.023J F coupling
• The helical state is supposed in the classical configuration
⇒ ϑ = 0 is still the ground-state
⇒ Now, it has a truly 3-d structure
⇒ Fluctuations are small enough, stable
Exotic magnetic phases in strongly correlated electron systems 19 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Further results
• Spin reduction at finite temperature (for both model)⟨SAA(r)
⟩= M −
∑α 6=A
b†r,αbr,α r ∈ ΛA
⇒ The reduction of the magnetization is a measure of howgood our approximation is.
• Classical spin-spin correlation function
Σ(r) =
⟨∑α,β
[Sβα(0)− M
4δα,β
] [Sαβ (r)− M
4δα,β
]⟩cl
⇒ Measurable with scattering experiments
⇒ The order state can be characterized with the it’s peeks inFourier-space (ordering wave vectors)
Exotic magnetic phases in strongly correlated electron systems 20 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Summary
Spin liquid phases of alkaline-earth-metal atoms at finite T
• We studied SU(6) Heisenberg antiferromagnet on honeycomblattice
• We determined some spin liquid phases with lowest freeenergy and study their finite temperature behaviours
Supervisor: Szirmai Gergely
Order-by-disorder of four-flavor antiferromagnetism on a fcc lattice
• We studied SU(4) Heisenberg antiferromagnet on fcc lattice
• We found highly degenerate Neel state, but the flavor wavesselect one symmetry-breaking ground-state
Supervisor: Penc Karlo
Exotic magnetic phases in strongly correlated electron systems 21 / 22
Motivation Heisenberg model Spin liquid on a honeycomb lattice Ordered phase on a fcc lattice Summary
Outlook
Quantized recurrence time in iterated open quantum dynamics
• We analyzed the expected recurrence time in iterated openquantum systems
• Grunbaumet al. have showed that the expected first returntime in unitary random walk is an integer number
• We have generalized this statement for unital dynamics
• We have shown that the expected return time is equal to thedimension of the Hilbert space, which is explored by thesystem over the time of the whole dynamics
Supervisor: Asboth Janos
Exotic magnetic phases in strongly correlated electron systems 22 / 22