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


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




H = J∑〈i,j〉


[Sαβ(i),Sµν (i)


∑α

Sαα(i) = M I


• Gauge theories



Simple, but a good model forunderstanding the behavior ofthe quantum magnets.




H = J∑〈i,j〉


[Sαβ(i),Sµν (i)


∑α

Sαα(i) = M I


• Gauge theories



In spin liquid phase theground-state excitations can bedescribed by gauge theories, so itmakes it possible to study highenergy physics in low energy.




H = J∑〈i,j〉


[Sαβ(i),Sµν (i)


∑α

Sαα(i) = M I


• Gauge theories



Cuprates (e.g. La2CuO4) has awell above critical temperature(Tc=35K), than the other super-conductors. P. W. Anderson ⇒spin liquid theory




H = J∑〈i,j〉


[Sαβ(i),Sµν (i)


∑α

Sαα(i) = M I


• Gauge theories



In topological phases robustqubits are constructed. Thesequbits can be controlled byHeisenberg-like models



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 = ?



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



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)



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



PART I



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)



Set the problem





→ su(6)




One particle per site (〈ni〉 = 1), one every lattice site there is 6internal states



Set the problem





→ su(6)




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



Set the problem





→ su(6)




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)

}



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]



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



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



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



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



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)



PART II



Set the problem



• Face centered cubic lattice

• Spin-wave theory

→ su(4)


→ 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)



Set the problem





→ su(4)




One particle per site (〈ni〉 = 1), one every lattice site there is 4internal states: α = {A,B,C,D}



Set the problem





→ su(4)




→ 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


Set the problem





→ su(4)




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



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



Classical ground-state II.

Fcc lattice contains: octahedra and tetrahedra





i) Consider a single octahedron(a)

(b)

(c) (d)

(e)





i) Single octahedron under-constrainedii) Single tetrahedron

unique configuration





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)





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 =?)



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.



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.



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



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)



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



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


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