Recent Progress in Fermion Quantum Monte Carlo Simulations

The FFLO Phase in 1D Imbalanced Fermion Systems

AntiFerromagnetic Order Parameter of 2D Half-Filled Hubbard Model

Orbitally Selective Mott Transitions

Conclusions

G.G. Batrouni, K. Bouadim (INLN); M.H. Huntley (MIT); V.G. Rousseau (Leiden)Z.J. Bai, S. Chiesa, C.N. Varney, R.T. Scalettar (UC Davis)C.R. Lee (NTHU Taiwan); M. Jarrell (LSU); T. Maier, E. D’Azevedo (ORNL)

Funding:ARO Award W911NF0710576 with funds from the DARPA OLE ProgramDOE SciDAC Program DOE-DE-FC0206ER25793

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I. FFLO Phase- Theoretical Background

Usual Cooper pair: ( k ↑ , −k ↓ )

What happens if N↑ 6= N↓ ?

Mismatched Fermi surfaces: kF, majority 6= kF, minority

Breached Pair

• Superfluid of Cooper pairs ( k ↑ , −k ↓ ) for k < kF, minority.coexists with normal fluid (of excess species).

• Pairs have zero momentum.

• Translationally invariant.

Fulde-Ferrell-Larkin-Ovchinnikov

• Pairs have non-zero momentum kF, majority − kF, minority.

• Spatially inhomogeneous.

• Hard to see in CM systems.

Cold Atom Systems

• Two hyperfine states play role of spin up and down.

• One complication is role of trapping potential.

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FFLO Experimental Motivation - Solid State

Forty years after its theoretical discussion, FFLO phase observed.Heavy fermion system CeCoIn5.Requires very pure and strongly anisotropic single crystals.Apply large field parallel to conducting planes.

H.A. Radovan et al., Nature 425, 51 (2003).

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FFLO Experimental Motivation - Cold Atoms

Fermion 6Li in hyperfine states F = 12, mF = ± 1

2.

Three dimensional, but highly elongated, traps.Tunable interaction strength via Feschbach resonance.Tunable relative mF = ± 1

2populations.

Core of system has uniform pairing (n1 − n2 = 0). Excess atoms sit at edge.G.B. Partridge et al., Science 311, 503 (2006).Also: M.W. Zwierlein et al., Science 311, 492 (2006); Nature 422, 54 (2006).

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The Attractive Fermion Hubbard Hamiltonian

H = −tX

j,σ

(c†jσcj+1σ + c†j+1σcjσ) − |U |X

j

nj↑nj↓ + VT

X

j

j2(nj↑ + nj↓)

Operators c†iσ (ciσ) create (destroy) an electron of spin σ on site i.

Electron kinetic energy t; interaction energy U ; Quadratic confining potential VT .

Ut

Condensed matter: Two spin species σ =↑, ↓.Optically Trapped Atoms: Two hyperfine states “σ” = 1, 2.

Observables

Gσ(l) = 〈c†j+l σcj σ〉 Fourier transform : nσ(k)

Gpair(l) = 〈∆j+l∆†j〉 Fourier transform : npair(k)

∆j = cj2cj1

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Algorithm

Continuous time canonical ‘worm’ algorithm (Rombouts, Van Houcke, Pollet).

No discretization of imaginary time (no ‘Trotter’ errors).

Constant particle number.

Broken world lines (‘worms’) are propagated.

Large moves through configuration space (short correlation times).

Can measure non-local Greens functions.

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FFLO Results for Uniform System

Gσ(l) = 〈c†j+l σcj σ〉 Fourier transform : nσ(k)

Gpair(l) = 〈∆j+l∆†j〉 Fourier transform : npair(k)

∆j = cj2cj1

0 10 20 30|i-j|

0

0.1

Gpa

ir(|i-j|

) =

<∆i∆+ j>

N1=15, N2=15N1=15, N2=19N1=15, N2=23N1=15, N2=27

U = −8Gpair(l) oscillates as cos(qr) with q = kmajority − kminority consistent with LO.

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Begin analysis of nσ(k) and npair(k) by examining unpolarized case

• Weak coupling: n1(k) = n2(k) is sharp.

• Strong coupling: n1(k) = n2(k) rounded.

• npair(k) peaked at k = 0. Peak sharpens with |U |.

0 1 2 3k

0

0.5

1

1.5

2

2.5

n1(k), U=-2npair(k), U=-2

n1(k), U=-4npair(k), U=-4

n1(k), U=-8npair(k), U=-8

L=32, N1= N2=15

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Polarized case

npair(k) peaked at kmajority − kminority for all |U |.

Left panel: U = −4

Right panel: U = −10

Symbols: N1 = 7, N2 = 9, L = 32 sites, β = 64.

Lines: N1 = 21, N2 = 27, L = 96 sites, β = 192.

0 1k

0

0.2

0.4

0.6

0.8

1

1.2

0 1k

n1(k)n2(k)npair(k)

U=-4 U=-10

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FFLO pairing no matter how large the polarization is made.

Have not seen “Clogston Limit”.Inset: Peak in npair(k) scales precisely as kF, majority − kF, minority.

0 0.5 1 1.5 2|kF1-kF2|

0

0.5

1

1.5

2

k peak

0 1 2 3k

0

0.5

1

1.5

2

2.5

3

n pair(k

)P=0P=0.0625P=0.1176P=0.1667P=0.21P=0.25P=0.2875P=0.318P=0.3478

L=32, N1=15, U=-9

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Kinetic Energy

As |U | increases:

|Single particle KE| decreases

|Pair KE| increases

Hopping is increasingly “in pairs”.

0 5 10|U|

-0.4

-0.2

0

0.2KE/site N1=9KE/site N2=11Pair KE/site, subtractedPair KE/site<ni1ni2>

Cross-over occurs as double occupancy 〈ni1ni2〉 approaches saturation.

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FFLO Results for Trapped System

Pronounced minimum in density difference at trap center.

npair(k) peaked at nonzero k. (System remains FFLO.)

kpeak consistent with value of local polarization at trap center.

0 10 20 30 40i

0

0.2

0.4

0.6

0.8N

1=13

N2=15

N2-N

1

0 1 2 3

k

0

0.2

0.4

0.6

0.8

npair

(k)

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FFLO Conclusions

Clear evidence for FFLO phase in Imbalanced d = 1 Attractive Hubbard Model

Uniform System

• Spatially oscillating Gpair(r)

• Gpair(k) peaked at k = kF,majority − kF,minority

• No Clogston limit

Trapped System

• Deep minimum in local polarization at trap center.

• System retains signature of FFLO phase: Gpair(k) peaked at k 6= 0.

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II. AntiFerromagnetic Order Parameter of 2D Half-Filled Hubbard Model

Condensed Matter Realization: CuO2 sheets of High-Tc superconductors

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Optical Lattice Realization: In (very) early stages!

Observation and control of superexchange interactions in two site system.J = 2t2/(U + ∆) + 2t2/(U − ∆)

Oscillations in magnetization; Top to bottom: J/U = 1.25, 0.26, 0.048

S. Trotzky et.al., condmat:0712.1853; I. Bloch, Nature 453, 1016 (2008).

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Determinant Quantum Monte Carlo

Compute operator expectation values

〈A〉 = Z−1 Tr [Ae−βH ]

Z = Tr [e−βH ]

Example: single band Hubbard Hamiltonian

H =X

〈ij〉σ

(d†iσdjσ + d†

jσdiσ) + UX

i

(ni↑ −1

2)(ni↓ −

1

2) − µ

X

iσ

(niσ + niσ)

• Inverse Temperature discretized: β = L∆τ

Z = Tr [e−βH ] = Tr [e−∆τH ]L

• Suzuki-Trotter Approximation

e−∆τH ≈ e−∆τKe−∆τP

Extrapolation to ∆τ = 0.

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• (Discrete) Hubbard-Stratonovich Fields decouple interaction:

e−∆τU(ni↑−1

2)(ni↓−

1

2) =

1

2e−

U∆τ

4

X

Siτ

e∆τλSiτ (ni↑−ni↓) = e−∆τPi(τ)

where cosh(∆τλ) = eU∆τ

2 .

• Quadratic Form in fermion operators: Do trace analytically

Z =X

{Siτ}

Tr [e−∆τKe−∆τP(1)e−∆τKe−∆τP(2)e−∆τK . . . e−∆τP(L)]

=X

{Siτ}

detM↑({Siτ})detM↓({Siτ})

dim(Mσ) is the number of spatial sites. For multi-band models

dimension is (number of spatial sites)x(number of orbitals per site).

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• Sample HS field stochastically.

Si0τ0→ −Si0τ0

detMσ({Siτ}) → detMσ({Siτ}′)

Computation of new determinant is o(N3). Algorithm is order o(N4L) since NL HSvariables. Local nature of change in HS field: evaluate ratio of determinants in order N2

(“Sherman-Morrison” formula).

Algorithm is order N3L .N ∼ a hundred lattice sites/electronsL = β/∆τ ∼ a hundred imaginary time slices (low temperatures).

• Measurements

〈ciσc†jσ〉 ↔ 〈[M−1σ ]ij〉 = 〈[Gσ]ij〉

• Sign Problem

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We have modernized our legacy DQMC codes

• Optimized linear algebra kernals

• “Delayed Updating” (Jarrell-Maier-D’Azevedo)

• N ∼ five hundred lattice sites/electrons (on a small cluster of workstations)

DOE SciDAC Program

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Updated DQMC codes: Large lattices (N =24x24) allow for good momentum resolution.

U = 2 Fermi function:

ρ = 0.2

-π -π/2 0 π/2 π-π

-π/2

0

π/2

πρ = 0.4

-π/2 0 π/2 π

ρ = 0.6

-π/2 0 π/2 π

ρ = 0.8

-π/2 0 π/2 π 0

0.2

0.4

0.6

0.8

1

n(k)

ρ = 1.0

-π/2 0 π/2 π

U = 2 Gradient of Fermi function:

-π -π/2 0 π/2 π-π

-π/2

0

π/2

π

-π/2 0 π/2 π -π/2 0 π/2 π -π/2 0 π/2 π 0

0.5

1

1.5

2

2.5

∇ n

(k)

-π/2 0 π/2 π

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Fermi surface is smeared further by increasing interaction strength.

U = 4 Fermi function:

ρ = 0.2 β = 8

-π -π/2 0 π/2 π-π

-π/2

0

π/2

πρ = 0.4 β = 8

-π/2 0 π/2 π

ρ = 0.6 β = 6

-π/2 0 π/2 π

ρ = 0.8 β = 4

-π/2 0 π/2 π 0

0.2

0.4

0.6

0.8

1

n(k)

ρ = 1.0 β = 8

-π/2 0 π/2 π

U = 4 Gradient of Fermi function:

-π -π/2 0 π/2 π-π

-π/2

0

π/2

π

-π/2 0 π/2 π -π/2 0 π/2 π -π/2 0 π/2 π 0

0.5

1

1.5

∇ n

(k)

-π/2 0 π/2 π

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Optical Lattice Imaging of Fermi Surface

M. Kohl etal. Phys. Rev. Lett. 94, 080403 (2005).Fermionic 40K atoms

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Interactions reduce the effective hopping/kinetic energy

tefft

=〈c†iσcjσ + c†iσcjσ〉

〈c†iσcjσ + c†iσcjσ〉U=0

.

0

0.2

0.4

0.6

0.8

1

0 4 8 12 16

t eff

/ t

U

10 x 10

β = 24β = 12β = 4

Strong couplingPT

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Local moments form as U increases.

0.5

0.6

0.7

0.8

0.9

1

0 4 8 12 16

C(0

,0)

U

10 x 10

β = 24β = 12β = 4

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Antiferromagnetic spin correlations form at low temperature.

βt = 20 → T = t/20 = W/160 (bandwidth W = 8t)

-0.15

-0.1

-0.05

0

0.05

0.1

(0,0) (10,0) (10,10) (0,0)

C(l

x,ly

)

20 x 20 U = 2.00

(0,0) (10,0)

(10,10)

(a)

β = 32β = 20β = 12

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Antiferromagnetic correlations are well converged for N =20x20 lattices.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

(0,0) (L/2,0) (L/2, L/2) (0,0)

C(l

x,ly

)

U = 2.00 β = 32

(0,0) (L/2,0)

(L/2, L/2)

24 x 2420 x 2016 x 1612 x 12

8 x 8

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Increasing U enhances antiferromagnetic correlations

-0.2

-0.1

0

0.1

0.2

(0,0) (12,0) (12,12) (0,0)

C(l

x,ly

)

24 x 24

(0,0) (12,0)

(12,12)

U = 2.00 β = 32U = 3.00 β = 24U = 4.00 β = 24U = 5.00 β = 20

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Antiferromagnetic structure factor

S(π, π) =1

N

X

ij

〈(ni↑ − ni↓)(nj↑ − nj↓)〉

reaches asymptotic ground state value as β increases. Lower T is required for largerspatial lattices. Correlation length ξ(T ) must exceed linear size.

• In ordered phase S(π, π) grows with N .

0

2

4

6

8

10

12

14

4 8 12 16 20 24 28 32

S

b

U = 2.00

4 x 4 6 x 6 8 x 8

10 x 1012 x 1214 x 1416 x 1618 x 1820 x 20

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Finite size scaling of AF structure factor

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.05 0.1

1 / L

U = 4.00 β = 24

(b)

(c)

S(π,π) / L2

C(L/2,L/2)

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U dependence of AF order parameter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 ∞

maf

U / t

This workHirsch/Tang

HFxSWRPASWT

Heisenberg

Possible use for benchmarking optical lattice emulations

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III. Orbitally selective Mott Transitions

Multiple bands with different degree of electronic correlation, e.g.

• One band with U > UcMott

• Another with U > UcMott

What happens with interband hybridization or interactions?

H = −tX

〈ij〉σ

( c†iσcjσ + c†jσciσ ) − µX

i

(ni↑ + ni↓)

+X

i

h

JzSzi (ni↑ − ni↓) + J⊥( S+

i c†i↓ci↑ + S−i c†i↑ci↓ )

i

+ UX

i

(ni↑ −1

2)(ni↓ −

1

2)

Itinerant band (adjustable U) and a fully localized orbital (spin- 12

degrees of freedom)

OSMT: Is band metallic or localized?

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Local Moment

DMFT (Costi) shows first order OSMT at Uc

Signaled by discontinuous jump in 〈S2z 〉

DQMC Results:

0 0.5 1 1.5 U

0.5

0.6

0.7

Sz2

β=10β=12β=14

0 0.5 1 1.5 U

J/U=0.1J/U=0.2J/U=0.4

J/U=0.2 β=14

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Spectral Function

In same range of U values

• Temperature dependence of A(ω = 0) changes sign

• Gap opens in A(ω)

0 0.5 1 U

0

0.2

0.4

0.6

0.8

1

A(0

)

β=10β=12β=14

-4 -2 0 2 4 ω

0

0.2

0.4

U=0.50U=0.75U=1.00

J/U=0.2

J/U=0.2 β=14

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Conductivity

Change in sign of dσ(T )/dT provides additional signal of MIT.

0 0.5 1 1.5 U

0

5

10

15

20

σ

β=10β=12β=14

0 0.5 1 1.5U

β=10β=12β=14

0 0.5 1 1.5U

β=10β=12β=14

J/U= 0.1 J/U= 0.2 J/U= 0.4

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Scaling of AF Order Parameter

Change in transport at OSMT accompanied by a paramagnet-antiferromagnet transition.

0 0.05 0.1 0.15 0.2

N-1/2

0

0.05

0.1

0.15

0.2

Szz

(π,π

) / N

U=0.55U=0.5U=0.45U=0.4

0 0.05 0.1 0.15 0.2

N-1/2

0.05

0.1

0.15

0.2U=0.2U=0.25U=0.3U=0.35

J/U=0.2

J/U=0.4

β=20

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Phase Diagram

Different measurements provide numerically consistent picture of OSMT

0 0.2 0.4 0.6 0.8J/U

0

0.5

1

1.5

2U

σdcA(ω=0)Szz(π,π)

Paramagnetic Metal

AFzz Insulator

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CONCLUSIONS

• Observe FFLO Phase in Spin Polarized Attractive Hubbard Hamiltonian

• Large lattice (N ∼ 24x24) of half-filled Repulsive Hubbard Hamiltonian

→ Antiferromagnetic Order Parameter 〈m2af(U)〉

• Observation of OSMT in Two Orbital Repulsive Hubbard Hamiltonian

G.G. Batrouni, K. Bouadim (INLN)M.H. Huntley (MIT)V.G. Rousseau (Leiden)Z.J. Bai, S. Chiesa, C.N. Varney, R.T. Scalettar (UC Davis)C.R. Lee (NTHU Taiwan), M. Jarrell (LSU)T. Maier, E. D’Azevedo (ORNL)

Funding:ARO Award W911NF0710576 with funds from the DARPA OLE ProgramDOE SciDAC Program DOE-DE-FC0206ER25793

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