one-dimensionalextended Bose-Hubbard modelswith local three-body interactionsTomasz Sowinski

U

CEWQO 2013, Stockholm June 16, 2013

one-dimensional

extended Bose-Hubbard models

with local three-body interactions

Tomasz Sowinski

Institute of Physics of the Polish Academy of Sciences, Warsaw ICFO - The Institute of Photonic Sciences, Barcelona

U

• Hamiltonian of the one-dimensional chain

ˆ

n

= ˆ a

ˆ a

• ultra-cold atoms (bosons) in optical lattice

D. Jaksch et al.: Phys. Rev. Lett. 81, 3108 (1998)

J U

H = J ˆ

X L

i=1

ˆ

a ^† _i (ˆ a _{i 1} + ˆ a _i+1 ) + U

2

X L

i=1

ˆ

n _i (ˆ n _i 1)

- the tunneling amplitude J is determined

by the shape of the lattice potential

- the interaction energy U is determined

by the shape of the lattice and details

of the interaction potential

standard Bose-Hubbard model

U

H = J ˆ

X L

i=1

ˆ

a ^† _i (ˆ a _{i 1} + ˆ a _i+1 ) + U

2

X L

i=1

ˆ

n _i (ˆ n _i 1)

• grand canonical ensemble

µ N ˆ

J / U J_c

J_c

S. Ejima et al.: EPL 93, 3002 (2011)

N ˆ =

X

i=1

ˆ

n

phase diagram

U

J / U J_c

J_c

S. Ejima et al.: EPL 93, 3002 (2011)

H = J ˆ

X L

i=1

ˆ

a ^† _i (ˆ a _{i 1} + ˆ a _i+1 ) + U

2

X L

i=1

ˆ

n _i (ˆ n _i 1) µ N ˆ

• grand canonical ensemble

how the properties of the studied model

will change when local three-body

interactions are taken into account

????

+ W

6

X L

i=1

ˆ

n _i (ˆ n _i 1) (ˆ n _i 2)

phase diagram

U

J. Silva-Valencia, A. Souza: Phys. Rev. A 84, 065601 (2011)

in the presence of three-body interactions

the first insulating lobe remains almost unchanged

DMRG with L up to 512

first insulating lobe

U

T. Sowiński: Phys. Rev. A 85, 065601 (2012)

Exact Diagonalization with L up to 16

phase diagram

shape of the second insulating lobe in

contrast to the first one, depends crucially

on the three-body interactions

U

universality class

local three-body interactions do not change

the characteristic properties of the phase

transition (K-T universality class)

U

attractive two-body forces

U < 0

H = J ˆ

X

i=1

ˆ

a

(ˆ a

+ ˆ a

) + U

2

X

ˆ

n

(ˆ n

1) + W

6

X

ˆ

n

(ˆ n

1) (ˆ n

2) µ ˆ N

U

H = J ˆ

X

ˆ

a

(ˆ a

+ ˆ a

) + U

2

X

ˆ

n

(ˆ n

1) + W

6

X

ˆ

n

(ˆ n

1) (ˆ n

2) µ ˆ N

infinite three-body repulsion limit (W ➜ ∞)

attractive two-body forces

DENSITY

SF PSF

● Mott Insulator (MI)

hˆai 6= 0, hˆa

i 6= 0

hˆai = 0, hˆa

i 6= 0

hˆai = 0, hˆa

i = 0

● pair-superfluid (PSF)

● superfluid (SF)

S. Diehl et al., Phys. Rev. Lett. 104, 165301 (2010) Y. Lee, M. Yang, Phys. Rev. A 81, 061604(R) (2010)

Wha t will ha

ppe n if

W is la rge b ut finite

?

U <

0

U

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

6 108 12 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

SF - PSF phase transition

• indicators of quantum phase transition

F (J, ) = |h G(J) | G(J + ) i|

- ground state fidelity

(J) = @ ² F (J, )

@ ² ₌₀

- fidelity susceptibility ^QPT 1

QPT 0

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

68 1012 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

#  sites

T. Sowiński et al.: ArXiv:1304.4835 (2013)

W ➜ ∞

U

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

68 1012 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

SF - PSF phase transition

T. Sowiński et al.: ArXiv:1304.4835 (2013)

• fidelity susceptibility near QPT

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

68 1012 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

#  sites

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

6 108 12 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

(t, L) ⇠ L ^/⌫ (L ^1/⌫ t)

- finite-size scaling

If all critical parameters were known then by plotting against for different system sizes all curves should collapse to the universal function.

L

L

t

W ➜ ∞

t = J J

J

⇠ ⇠ |t|

⇠ |t|

U

violation of the universality

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0

γ = 2.11

W = ∞

6 108 12 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

6 108 12 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

68 1012 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

#  sites

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0 γ = 2.11

W = ∞

68 1012 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

0 20 40 60 80 100

0 0.1 0.2 0.3

χ

J/|U|

J_c = 0.19 ν = 1.0

γ = 2.11

W = ∞

6 108 12 0

0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

L ^1/γ (J-J _c )/J _c

0 30 60 90 120

0 0.1 0.2 0.3

χ

γ = 2.23

W = 4|U|

0 0.2 0.4 0.6

-8 -6 -4 -2 0 2 4 6

L -γ /ν χ

W ➜ ∞

the critical exponent γ varies with W and is thus

dependent on the MICROSCOPIC details of the model

T. Sowiński et al.: ArXiv:1304.4835 (2013)

U

Conformal Field Theory

S(l) = c

6 log



sin

✓ ⇡l

L

◆

+ s(L) + O

✓ l

L

• entanglement entropy for block of size l ◆

hard constraint

model

DMRG with L up to 256

T. Sowiński et al.: ArXiv:1304.4835 (2013)

for systems with c > 1 the critical exponents

may continuously vary with respect

to the microscopic parameters of the model

U

conclusions

• repulsive two-body interactions (U>0)

• attractive two-body interactions (U<0)

H = J ˆ

X

ˆ

a

(ˆ a

+ ˆ a

) + U

2

X

ˆ

n

(ˆ n

1) + W

6

X

ˆ

n

(ˆ n

1) (ˆ n

2)

- shape of the second insulating lobe in contrast to the first one, depends

crucially on the three-body interactions

- the universality class of MI-SF phase transition does not depend on the

strength of the three-body interactions (at least in 1D)

- critical behavior of the system undergoing a PSF-SF phase transition

depends on the value of the three-body repulsion

- critical exponent and the central charge governing the quantum phase

transitions are shown to have repulsion dependent features

T. Sowiński, R. W. Chhajlany, O. Dutta, L. Tagliacozzo, M. Lewenstein: ArXiv:1304.4835 (2013) T. Sowiński: Phys. Rev. A 85, 065601 (2012)