dynamic ferromagnetic response of a “high temperature”...

Rajdeep Sensarma

Department of Theoretical Physics,Tata Institute of Fundamental Research, Mumbai

Workshop on Cold Atoms, HRI Allahabad15 February 2014

Dynamic Ferromagnetic Response of a “High Temperature” Quantum Antiferromagnet

Collaborators

Xin (Sunny) Wang

Sankar Das SarmaUniversity of Maryland, College Park

X. Wang, R. Sensarma and S. Das Sarma, arXiv:1308.1091

Outline

1) Cold atoms and quantum simulation of equilibrium many-body physics: an overview.

2) The problem in simulating Fermi Hubbard model 3) Ionic Hubbard model and “high temperature” quantum antiferromagnet.

4) Ferromagnetic Response of an Antiferromagnet

Cold Atoms

Atomic Physics

NuclearPhysics

Chemistry

AstroPhysics

QuantumInformation

CondensedMatter Physics

Quantum Emulator

Approx. Schemes

Slave Boson, Gauge Theory, Variational Monte Carlo, etc.

Experiments

ARPES, transport,neutron scattering, specific heat etc.

Cuprate High Tc SC

Cold atoms provideexperimental implementation

of the model

H = �JX

hiji

c†i�cj� + UX

i

ni"ni#

One Band Hubbard Model onsquare lattice (U >>J)

P. W. Anderson, 1988

➡three band model?➡longer range interaction?➡longer range hopping?➡disorder?Minimal Model

Cold Atom Systems Bosons FermionsRb87, Li7, Na23 K40, Li6Alkali atoms cooled to ~ 10 nK

Laser cooling of atoms

Opposite lasers tuned below atomic transition frequency

Atoms moving toward the light comes into resonance due to Doppler shift

Atoms absorb photon momentum and is slowed down.

Emission in random directions : avg momentum change is 0

Trapped in electric/magnetic fields

Further evaporative cooling by opening the trap

Quantum degenerate gases with 105 - 107 atoms interacting through atomic potentials characterized by s-wave scattering lengths

Bose Einstein Condensation in cold Bosons

M.#H.#Anderson#et#al.#Science,#269,#198#(1995)#

Cornell'Group'JILA'

J.#R.#Abo(Shaeer#et#al.#Science,#292,#476#(2001)#

Ke#erle&Group,&MIT&

Nobel Prize for observing BEC in 1995 to W. Ketterle, C. Wieman and E. Cornell

Tuning Interactions between atoms

๏ Atomic interactions characterized by a (s-wave) scattering length

๏ Multi-channel scattering problem: singlet vs triplet configurations

๏ Energy of Bound state in closed channel tuned by magnetic field

๏ Effective scattering length in open channel diverges when the bound state is at resonance with this channel: Feshbach resonance

Controlled and easily tunable access to strong coupling regime

C.#A.#Regal,#M.#Greiner#and#D.#S.#Jin#PRL,#92,#040403#(2004)#Jin$Group,$JILA$

Fermion Superfluidity and BCS-BEC Crossover

M.#W.#Zwierlein#et#al.#Nature,#435,1047#(2005)#Ke#erle&Group,&MIT&

1kF as

�11 0

Hyperfine states play the role of spins

More Complicated (fun) stuff

Berezinskii-Kosterlitz-Thouless transition in 2D Bose gas

Bose Einstein Condensation of multi-component (spin 1) bosons

BCS-BEC Crossover of Fermions in presence of spin polarization

BCS-BEC Crossover of Fermions with unequal mass ( Na-K mixture)

Bose-Fermi mixtures

Bosons and Fermions with long range dipolar (magnetic or electric) interactions ( 52Cr, Rydberg Atoms, 40K87Rb etc)

176 Yb : Bosons with large spins

Bosons and Fermions in Artificial Gauge Fields

Cold atoms and optical lattices

V0#

λ/2#

Counter-propagating laser beams create periodic potential

Can create 3D, 2D or 1D lattices

Lattice depth can be tuned by changing beam intensity

Lattice geometry (e.g. triangular), spin dependent lattice etc.

Basic Hubbard models with bosons and fermions

H = �JX

hiji

b†i bj +U

2

X

i

ni(ni � 1)� µX

i

ni

H = �JX

hiji

c†i�cj� + UX

i

ni"ni#

bosons

fermions

J ⇠ ER(V0/ER)32 e�p

V0/ER U ⇠ as

�

✓V0

ER

◆3/4

Can tune interactions by tuning either as or V0I Bloch, J Dalibard and W. Zwerger,

RMP 80, 885 (2008)

ER =~2

2m�2

t/U$

Superfluid$Mo/$insulator$

Superfluid Insulator Transition in Bose Hubbard Model

M. Greiner et al., Nature 415 (2002)

J/U

Imaging the Mott Insulator-Superfluid Transition at single atom level

W Bakr et alScience 329,5991 (2010)

Outline


2) The problem with the Fermi Hubbard model 3) Ionic Hubbard model and “high temperature” quantum antiferromagnet.


Fermi Hubbard Model H = �J

X

hiji

c†i�cj� + UX

i

ni"ni# We are interested in this model at strong coupling ( U/J >> 1)

At strong coupling the Fermions are frozen in real space due to the strong onsite repulsion.

At half filling, the Fermion motion is completely suppressed ----> Mott Insulator

UHB$

~U$

LHB$

As U --> ∞, D.O. are projected out of low energy subspace

| i = P| 0i P =Y

i

(1� ni"ni#)

projects out double occupancies

Effects of finite U/J : incorporated through a canonical transformation | i = e�iSP| 0i

A ! A = eiSAe�iS Transform the operators and evaluate them in projected states

S His chosen so that the transformed Hamiltonian does not have any term connecting low and high energy subspaces

At large U/J, expansion of S in powers of J/U --> strong coupling expansion of the effective Hamiltonian

Fermi Hubbard Model at strong coupling H = �J

X

hiji

c†i�cj� + UX

i

ni"ni#

T = T0 + T1 + T�1

H0

[H0, T↵] = ↵UT↵

local part of the Hamiltonian

iS =T1 � T�1

U+ ..

H = eiSHe�iS = H + [iS,H] +1

2[iS, [iS,H]] + ...

H = T0 +2

U[T1, T�1]

At half-filling, there are no vacancies: H =4J2

U

X

hiji

~Si · ~Sj

Spin Flip Process S+i S�

j

Fermion motion is frozen, only spin fluctuations present ---- AF Heisenberg Model

Heisenberg Model

H =4J2

U

X

hiji

~Si · ~Sj

Superexchange Scale

Antiferromagnetic spin order in ground state in 2 and 3 D

Finite Temperature:

•The AF transition temperature (Neel Temperature) TN scales

with the superexchange scale.

•Since 4J2/U << U, the ideas of projection and canonical

transformation still hold. The Hubbard model is reducible to a Heisenberg model at these temperatures.

Evidence of Fermionic Mott Insulator

Suppression*of*double*occupancies**R.#Jordens#et#al.##Nature,#455,#204(2008)###Esslinger#Group,#ETH#Zurich#

Compressibility-measurements-U.#Schneider#et#al.#Science,#322,#1520#(2008)#Bloch#Group,#Mainz/Munich#

AF Spin order is not yet seen in experiments

Problems with emulating the Fermi Hubbard model

In general, fermions are harder to cool than bosons

The superexchange scale 4J2/U goes down with increasing coupling strength

The relevant temperature scale for observing Antiferromagnetism, TN also goes down

Further in simple Bosonic systems like Bose Hubbard model, all interesting quantum many-body physics occur at a scale of J

Possible Solutions:

Search for new ways to cool the system further Magnetic Field Gradient Cooling

Use of microtraps etc.

Search for ways to increase the Neel temperature by modifying the model

Ionic Hubbard model, a close cousin of the standard Hubbard model has higher TN


Outline




The Ionic Hubbard Model

�i = 0

�i = 1 if i is on A sublattice

if i is on B sublattice

V/2

U - V/2

U + V/2

-V/2

-J

Originally introduced to study ionic-neutral transitions in

1D charge transfer compoundslike TTF Chloranil

See: J. Hubbard and J.B. Torrence, PRL 1981 N. Nagaosa and J. Takimoto, JPSJ 1986

A AAAB B B B B B BA A A

We only need a bipartite lattice for this model to work.E.g.: We can think of either square or a cubic lattice

Hopping between A and B sublattice of a bipartite lattice, governed by J

Fermion-Fermion Interaction: the local Hubbard repulsion cost of double occupancies, U

Staggered Ionic Potential on the two sublattices, with a scale V

Explicitly breaks sublattice symmetry in “charge” sector, but not in “spin” sector

H = �JX

hiji�

c†i�cj� + UX

i

ni"ni# +V

2

X

i

(�1)�i(ni" + ni#)

The Ionic Hubbard Model with Cold Atoms

V0#

λ/2#

H = �JX

hiji

c†i�cj� + UX

i

ni"ni#

J ⇠ ER(V0/ER)32 e�p

V0/ER

U ⇠ as

�

✓V0

ER

◆3/4 ER =~2

2m�2

•Use a holographic technique to add a staggered optical potential, V

•This technique is regularly used to create pseudorandom disorder potentials in cold atoms.

•This technique has also been used by M. Greiner’s group to create the optical lattice (so getting a mask on the scale of lattice spacings is not a problem).

H = �JX

hiji�

c†i�cj� + UX

i

ni"ni# +V

2

X

i

(�1)�i(ni" + ni#)

This model can be implemented with small changes in the setups that simulate the Hubbard model

The Ionic Hubbard ModelThis model has a rich phase diagram. Parameters that can be tuned: U/J, V/U, density, Temp.

We will stick to half-filling (one Fermion per lattice site).

Limiting Cases:

1) Standard Hubbard Model (V=0)

Fermi Liquid at small U/J, with spin-density-wave ordering below exponentially small temperature Tc ~ exp(-J/U)

At large U/J >> 1, Mott insulator with antiferromagnetic ordering below TN ~ J2/U

Nature of transition from metal to insulator is ill-understood in 2 and 3 dimensions.

2) Non-Interacting Model (U=0)

The staggered potential halves the Brillouin zone, system is a band insulator at any finite V.

This can also be thought of as charge-density wave as fermions reside on one sublattice more than the other.

The Ionic Hubbard ModelLimiting Cases:

Stan

dard

Hub

bard

Mod

el

U/J

V/J

Fermi Liquid SDW

Non-Interacting Band Insulator

Mott Insulator

AFM

??

We are interested in this regime

The Ionic Hubbard Model

V/U

We are at U/J >>1, where system is AFM + MI at V=0

MOTTINSULATOR

AF ORDER

•For V/U << 1, physics is dominated by the Hubbard Repulsion

•Double occupancies are not allowed in low energy states, even at the cost of paying potential energy

•Standard Mott Hubbard Physics and AFM

H = �JX

hiji�

c†i�cj� + UX

i

ni"ni# +V

2

X

i

(�1)�i(ni" + ni#)

BANDINSULATOR

•For V/U >> 1, physics is dominated by the staggered ionic potential

•Fermions prefer to live on A sublattice even at the cost of forming double occupancy

• |2,0,2,0,2,0, .......... >

NO SPIN ORDER

Metal?1st vs 2nd order1 or 2 transitions?

T

A. Garg et. al, PRL 2006N Paris et. al, PRL 2007

Use DMFT to study this model

Basics of DMFT

Calculation of optical conductivity of cuprates

Outlines Introduction Formalism Results Conclusion Supplement

Dynamical mean-field theory

Dynamical mean-field theory reduces the lattice problem to an effective impurityproblem plus a self-consistency condition. Output: ”local” approximation to theself-energy, namely the k-dependence is dropped:

⌃(k,!) ! ⌃(!)

The “bath” where the impurity sits is determined self-consistently.

In the impurity model, the local many-body interaction is fully retained; thus itcan capture (at least partially) the physics of strong interactions.

The solution of the impurity problem remains non-trivial: we use quantum MonteCarlo (done by me) and Exact Diagonalization (done by collaborators) techniquesto solve it.

•DMFT reduces an interacting lattice problem to that of a single site (impurity) interacting with a bath (representing all other sites)

•The fact that the impurity site is a randomly chosen site from the lattice leads to a self-consistency between the dynamics of the impurity and that of the bath

•Assume that bath dynamics is governed by a Green’s function

•Solve the impurity problem with on-site interaction to obtain an impurity Green’s fn.

•Use Dyson’s equation to calculate the local impurity self- energy.

•The full lattice problem has k-indep self energy given by

•Obtain the local Green’s function by integrating over momenta.

•Obtain the new bath Green’s fn. from Dyson’s equation.

G0(i!)

Gimp(i!)

⌃imp(i!) = G�10 (i!)�G�1

imp(i!)

⌃l(k, i!) ' ⌃l(i!) ' ⌃imp(i!)

Gl(i!) =

Zdk[i! � ✏k � ⌃imp(i!)]

�1

G�10 (i!) = G�1

l (i!) + ⌃imp(i!)

Keep Iterating till self-consistency is achieved

DMFT for Ionic Hubbard Model

We clearly need to distinguish spin and sublattice degrees of freedom.

This leads to 4 X 4 matrices for all the Green’s function.

We only keep diagonal elements of the self energy (this is not a cluster DMFT)

We use continuous-time Monte Carlo techniques to solve the impurity problem.

We are always working in imaginary time (frequency) and use MAXENT based methods to get frequency dependent quantities.

(A", A#, B", B#)

The final numerical output of the DMFT comes in the form of spin and sub-lattice dependent local Green’s functions in imaginary time.

G�,A(B)(⌧)

The sub-lattice and spin dependent number density can be obtained from the Green’s functions as

n�↵ = 1 +G�↵(⌧ ! 0+)

Loss of Antiferromagnetism

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

(a)

(b)

V/t = 0

V/t = 5

V/t = 7

V/t = 7.6

∆n

m

T/t

T/J

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14

U/t = 8

U/t = 10

U/t = 12

U/t = 14

U/t = 16

TN/t

V/t

TN/J

V/J

For U/J >>1, for V/U <<1, the Neel temperature rises with V/U

It reaches an optimum value at ~ V/U=0.6, before crashing down.

For U/J = 16, TN rises by 40% compared to usual Hubbard model

V/J = 0, 5, 7, 7.6U/J = 10

U/J = 8, 10, 12, 14, 16

for V/U ~1, the staggered magnetization comes down to zero very fast.

m = (nA" � nA# � nB" + nB#)/4

A Perturbative Argument for high TN

T = T0 + T1 + T�1

T↵ = TAB↵ + TBA

↵

iS(1) =TAB1 � TBA

�1

U + V+

TBA1 � TAB

�1

U � V

At small V/U << 1, the Hubbard repulsion is the largest energy scale in the problem

So, the basic ideas of projection and canonical transformation to get effects of finiteU/J can be justified here.

H = �JX

hiji�

c†i�cj� + UX

i

ni"ni# +V

2

X

i

(�1)�i(ni" + ni#)

H0

A B B A

[H0, TAB(BA)↵ ] = [↵U + (�)V ]TAB(BA)

↵

H =4J2

U(1� V 2/U2)

X

hiji

~Si · ~Sj

Heisenberg Model with increases superexchange scale

A Perturbative Argument for high TN

A B

J

U-V

J

A B

J

U+VJ

2 different spin-flip processes corresponding to

2J2/(U-V) 2J2/(U+V)+= 4J2 U/(U2-V2) = (4J2 /U) [1/(1-V2/U2)]

S�AS+

B

In Heisenberg Model, TN ~ superexchange scale

For small V/U, TN~ (J2/U)[1-V2/U2]-1

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14

U/t = 8

U/t = 10

U/t = 12

U/t = 14

U/t = 16

TN/t

V/t

TN/J

V/J

U/J = 8, 10, 12, 14, 16

Perturbative argument holds at small V/UDMFT is required to obtain the full curve

and optimal TN

Outline




An Antiferromagnet gives Ferromagnetic Response

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

(a) (b)

(c) (d)

!AA↑

!AA↓

!AB↑

!AB↓

V/t = 0 V/t = 5

V/t = 7 V/t = 7.6

! A(ω

=0)t

T/t

A�↵(! = 0) =1

⇡TG�↵[⌧ = 1/(2T )]

At V/U ~ 1, the low energy single-particle density of statesshow striking asymmetry betweenthe 2 spin species

The spin polarization of the zero frequency DOS increases with increasing temperature beforegoing to zero at the AF transition temperature

The system has static AF spin order, but as far as low energy dynamics is concerned, it behaves like a Ferromagnet.

If a current is set up, it will be carried by spin-polarized carriers ---- finite spin conductivity

T/J

JA(!

=0)

JA(!

=0)

0

0.2

0.4

0.6

-15 -10 -5 0 5 10 15

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

(a) (b)

(c) (d)

AA↑

AA↓

AB↑

AB↓

(wide)

T/t = 0.2 T/t = 0.2

T/t = 0.18 T/t = 0.1

A(ω

)t

ω/t

Frequency Dependent Density of States

An Antiferromagnet gives Ferromagnetic Response

No spin polarization above TN

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

(a) (b)

(c) (d)

!AA↑

!AA↓

!AB↑

!AB↓

V/t = 0 V/t = 5

V/t = 7 V/t = 7.6

! A(ω

=0)t

T/t

Gapless up-spin DOS and soft gap in down-spin DOS

just below TN0

0.2

0.4

0.6

-15 -10 -5 0 5 10 15

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

(a) (b)

(c) (d)

AA↑

AA↓

AB↑

AB↓

(wide)

T/t = 0.2 T/t = 0.2

T/t = 0.18 T/t = 0.1

A(ω

)t

ω/t

0

0.2

0.4

0.6

-15 -10 -5 0 5 10 15

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

0

0.2

0.4

0.6

-4 -2 0 2 4

(a) (b)

(c) (d)

AA↑

AA↓

AB↑

AB↓

(wide)

T/t = 0.2 T/t = 0.2

T/t = 0.18 T/t = 0.1

A(ω

)t

ω/tSoft gap in up-spin DOS and hard gap in down-spin DOS

well below TN

T/J =0.18

T/J =0.2

T/J =0.1

T/J =0.2

V/J =7.6

T/J !/J

!/J

!/J

JA(!)

JA(!)

JA(!)

JA(!

=0)

See A. Garg et. al arXiv: 1307.2693for ferrimagnetic states at finite doping

Role of Double Occupancies

•At V/U << 1, the Hubbard repulsion dominates the physics and double occupancies are projected out of the low energy subspace.

•At half-filling this leads to a Heisenberg model with superexchange scale J=4t2/U

•AF is lost due to spin fluctuations, the system remains in projected subspace even above TN

•As V increases, the Fermions gain potential energy by preferentially staying on A sublattice.

•So, for V~U, the potential energy gain can compensate for the Hubbard repulsion.

•For U-V ~J, double occupancies on A sublattice are part of the low energy subspace.

•Double Occupancies on B sublattice have energy ~U+V and are projected out.

•The increasing weight of states with double occupancies play a crucial role in destroying AF and in providing the dynamic ferromagnetic response.

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

(a)

(b)

V/t = 0

V/t = 5

V/t = 7

V/t = 7.6

∆n

m

T/t

V/J = 0, 5, 7, 7.6 U/J = 10

T/J

Density Asymmetry and Double Occupancies

�n = nA � nB

At half-filling:

States in the subspace with double occupancies projected out have Δn = 0

The perfect density ordered state |2,0,2,0,2,0,.....> has Δn =2

The density asymmetry can stand as a proxy for no of double occupancies.

At low V/U << 1, Δn, and hence no. of D.O. does not change as T is increases through TND.O. does not play a role in destroying AF order

For V/U ~1, Δn increases from a low temp asymptotic value to an asymptotic high temp value as T is increased upto TN and then remains constant.

Increasing D.O. shows that they play important role in degrading AF order.

A Sublattice

The Ferromagnetic Response

B Sublattice A Sublattice B Sublattice

V ~ UV << U

0

-V/2

U-V

V/2

U+V

U+V

U-V

V/2

-V/2

The Ferromagnetic Response V ~ U

B

E =0

E =U-V

Low Energy Process

A B

E =0

E =U+V

High Energy Process

A

As long as there is AF order, only majority spins on the B sublattice can move in low energy dynamics

Dynamic Ferromagnetic response is due to resonance between projected AF ordered state and state with DO

The staggered spin order + staggered potential ----> Spin polarization of low energy DOS. Crucial role of DO, not

present in standard Heisenberg model.

This can be verified in Spin Resolved RF Spectroscopy or Spin Resolved Transport Measurements

0

0.1

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0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0 0.1 0.2 0.3 0.4

(a) (b)

(c) (d)

!AA↑

!AA↓

!AB↑

!AB↓

V/t = 0 V/t = 5

V/t = 7 V/t = 7.6

! A(ω

=0)t

T/t

ConclusionsCold Atoms provide a novel playground to study strongly correlated systems

Ionic Hubbard model, which can be implemented with cold atoms, show a larger Neel temperature than standard Hubbard model.

The staggered potential (for V << U) increases the scale of spin fluctuations.

The Neel temperature can be 40% higher than that of standard Hubbard model.

For V ~ U, the system shows spin polarization in the low energy DOS.

This is understood as resonance between AF state and state with double occupancy


dynamic ferromagnetic response of a “high temperature”...

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