Conditions Leading to Quench Cracking Other Than Severity of Quench
Long-range order and quench dynamics in one-dimensional ...nqs2014.ws/archive/Pres... · Long-range...
Transcript of Long-range order and quench dynamics in one-dimensional ...nqs2014.ws/archive/Pres... · Long-range...
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