Long-range order and quench

dynamics in one-dimensional quantum

systems with power-law interactions

Masaki TEZUKA (Dept. of Physics, Kyoto Univ.)

Alejandro M. Lobos, MT and Antonio M. Garcia-Garcia, Phys. Rev. B 88, 134506 (2013). MT, Antonio M. Garcia-Garcia, and Miguel A. Cazalilla, Phys. Rev. A 90, 053618 (2014).

• Motivation: dynamics of condensate decay

• Power-law interaction in trapped ions

• Bosonization & DMRG analysis of

1D attractively interacting fermions: Long-range order

1D hard-core bosons: Quench dynamics from condensate

Contents:

Introduction: Dynamics of condensate

formation / decay

• Light-induced superconductivity in stripe-ordered cuprate – [D. Fausti et al.: Science 331, 189 (2011)]

• Time evolution of the superconducting gap after excitation – [C. L. Smallwood et al.: Science 336, 1137 (2012)]

• Dynamically split quasi-1D Bose gases – [J. Schmiedmayer and coworkers: Nature 449, 324 (2007); Science 337, 1318 (2012); …]

Theory: several works using Gross-Pitaevskii eqn., Bogoliubov approx., etc. as well as

• Generalized Gibbs ensemble approaches – [e.g. M. Rigol et al.: Nature 452, 854 (2008), M. A. Cazalilla: PRL 97, 156403 (2006)]

• Boundary conformal field theory [e.g. P. Calabrese and J. Cardy: PRL 96, 136801 (2006)]

• Nonequilibrium DMFT [Tsuji et al.: PRL 110, 136404 (2013); PRB 88, 165115 (2013)]

• AdS/CFT approach, etc.

Quench from a condensate?

Condensate (Ground state (T=0)

or low-T state)

Hamiltonian H<

Time τ

Hamiltonian H>: interaction or external parameter changed

Initial reaction

Late stages of time evolution?

Thermalization? (with generally

higher T)

Thermalization? (with generally

higher T)

τ = 0

if interactions are short-ranged

1D: numerically exact approach possible, but no long-range order?

[Mermin-Wagner-Hohenberg theorem forbids spontaneous symmetry breaking]

“Dimensional quench” from ordered state to short-range 1D?

Spin-1/2 system and hard-core bosons

Up spin

Down spin

Boson

No boson

Interaction (z) Siz Sj

z

Siz = ni

↑ - ni↓ ni = bi† bi

Local magnetization Local population

ni nj Density interaction

x Component Six = (Si

+ - Si-)/2

Raising operator Si+ = Si

x + iSiy bi

† Creation operator

Interaction (xy) Si+ Sj

- bi†bj Intersite hopping

𝑏𝑖 †𝑏𝑗 Single-particle correlation 𝑆𝑖

+𝑆𝑖 −

Spin correlation (xy)

Manipulation of power-law interaction in trapped ion systems

• Ion crystal (for a recent review, see R. C. Thompson: 1411.4945)

J. W. Britton (NIST) et al.: Nature 484, 489 (2012)

S=1/2 9Be+ ions in 2D trap: Spin interaction ∝ d-a

R. Islam, C. Monroe (NIST) et al.: Science 340, 583 (2013) 171Yb+ ions in 1D trap: Long-range order

Long-range order in 1D spin by power-law interaction (forbidden for short-range interaction) Genuine long-range off-diagonal order in hard-core boson language

[Mermin & Wagner; Hohenberg]

Exponent determined by detuning from ion crystal phonon modes

Our models

• Effective dimension:

Fermions: Attractive Hubbard model + power-law hopping

Bosons: Nearest neighbor interaction + power-law hopping

f > 0 for τ < 0, f = 0 for τ > 0

• Large |U|/t limit: Maps to a spin model with

• Long-range superconducting order expected for

κ corresponds to 2α; Long-range order expected for

Related models in the literature Power-law hopping for repulsive U Hubbard [Gebhard and Ruckenstein: PRL 68, 244 (1992)] Non-interacting systems [Mirlin et al.: PRE 54, 3221 (1996); Mirlin: Phys. Rep. 326, 259 (2000)] Spinless fermions, random hopping [Khatami, Rigol, Relaño, and García-García: PRE 85, 050102 (R) (2012)]

1D fermions + power-law hopping case

Bosonization + self-consistent harmonic approximation approach: Long-range order expected for |U|>>t and exponent α < 3/2

Pair

co

rrel

atio

n

[A. M. Lobos, M. Tezuka, and A. M. García-García: PRB 88, 134506 (2013)]

Real-space distance

(hopping ∝ r-α) ~

Constant beyond length scale:

1D fermions + power-law hopping case Bosonization + self-consistent harmonic approximation approach: Long-range order expected for |U|>>t and exponent α < 3/2

Attractively interacting S=1/2 fermions: long-range superconducting order realized by power-law hopping

Pair

co

rrel

atio

n

DMRG result for U = -20 t (N=34 per spin, L=234 sites) ground state:

[A. M. Lobos, M. Tezuka, and A. M. García-García: PRB 88, 134506 (2013)]

Real-space distance

Correlation function almost constant for α = 0.5 and 0.8; much slower decay than the short-range model

~

𝑡𝑙𝑚 = 𝑡′

𝑙 − 𝑚 𝜅

Our model for condensed initial state:

1D bosons + power-law hopping Soft-core bosons

Hard-core bosons with nearest neighbor interaction

𝐻 = − 𝑡 𝑙−𝑚𝑙 − 𝑚 𝜅

𝑏𝑙 𝑏𝑚 +H. c.

𝐿

𝑙≠𝑚

− 𝜇 𝑛𝑙

𝐿

𝑙=1

+ 𝑉 𝑛𝑙 𝑛𝑙+1

𝐿−1

𝑙=1

𝐻 = − 𝑡 𝑙−𝑚𝑙 − 𝑚 𝜅

𝑏𝑙 𝑏𝑚 +H. c.

𝐿

𝑙≠𝑚

− 𝜇 𝑛𝑙

𝐿

𝑙=1

+ 𝑈 𝑛𝑙 𝑛𝑙 − 1

𝐿

𝑙=1

𝑡𝑟≥2 = 𝑡′

κ of bosonic model ~ 2α for fermions

Bosons: Large condensate fraction for

the ground state (DMRG results)

Almost constant correlation function up to κ ~ 1.5

Distance r=|i-j|

No density-wave order

f = 0.1

f = t’/t = 1

N=40, L=80, f = 0.1 Hard-core bosons; V = -1.9t

Condensate fraction:

Largest eigenvalue of 𝑏𝑖 †𝑏𝑗

divided by N (even larger for soft-core bosons)

0.27

0.23

𝑏𝑖 †𝑏𝑗

𝑛𝑖 𝑛𝑗 − 𝑛𝑖 𝑛𝑗

κ = 1 κ = 2

0.5

Condensate

fra

ction

Quench dynamics (hardcore bosons)

(2) Time-dependent DMRG – Works best for Hamiltonian only with short-range terms

– Long-range terms in τ<0 Hamiltonian has been quenched

Remove all hoppings other than nearest neighbor at τ=0 Time evolution of the 1D system?

Short time τ: ρ(τ) ~ exp(-γ τ2) (Gaussian curve; also obtained from Bogoliubov appr.)

Longer time: ρ(τ) ~ exp(-γ’ τ(3-κ)/2) (slower than exponential) Slower decay for larger κ expected

(1) Bosonization approach for condensate fraction ρ(τ)

Phase field θ(x)

Density field ∂xφ(x)

Take continuum limit, expand to leading quadratic order, neglect density fluctuation of the initial state, and introduce self-consistent harmonic approx.

T(x): self-consistently determined

Diagonalize the obtained effective quadratic Hamiltonian Time evolution of the state under post-quench Hamiltonian simulated

(SCTA)

Time-dependent DMRG

)()2/ˆexp()2/ˆexp(

)ˆexp()2/ˆexp()2/ˆexp(

)ˆexp()2/ˆexp()2/ˆexp(

)ˆexp(

ˆˆˆˆˆ

3

12/,2/1,2

,11,23,2

2,13,22/,12/

,11,23,22,1

OHiHi

HiHiHi

HiHiHi

Hi

HHHHH

LLLL

LLLL

LL

LLLL

)2/ˆexp( 12/,2/ LLHi

)2/ˆexp( 22/,12/ LLHi

)ˆexp( ,1 LLHi

)2/ˆexp( 1,2 LLHi

)ˆexp( 2,1Hi

Suzuki-Trotter decompotision

)2/ˆexp( 3,2Hi

)2/ˆexp( 12/,2/ LLHi

T/τ finite system iterations to reach time T

Application of exp(-iτHi,j) is almost exact if Hi,j only affects neighboring sites i, j

White and Feiguin: PRL 93, 076401 (2004)

Quench: time-dependent DMRG

Smaller exponent for larger κ

Gaussian 𝜌 𝜏 ∝ 𝑒−𝛾𝜏

2

Slower than exponential decay

N=40, L=80, f = 0.1, V=-1.2

[M. Tezuka, A. M. Garcia-Garcia, and M. A. Cazalilla, Phys. Rev. A 90, 053618 (2014)]

Conclusion • Effective dimension quench of Bose gas

Long-range order in the initial state: Ground state for Hamiltonian with

power-law hopping

1D short-range hopping model:

ground state does not have long-range order

Alejandro M. Lobos, Masaki Tezuka, and Antonio M. García-García: PRB 88, 134506 (2013) Masaki Tezuka, Antonio M. García-García, and Miguel A. Cazalilla: PRA 90, 053618 (2014)

motivated by ion-trap experiments

• After initial Gaussian-like decay, bosonization approach predicts stretched exponential decay of condensate (slower for larger κ)

• Time-dependent DMRG study possible; qualitatively consistent behavior of the condensate fraction obtained