Exclusion processes and quantum phase transitions in XXZ spin - - PowerPoint PPT Presentation
Exclusion processes and quantum phase transitions in XXZ spin chains. 15/07/2014 SSEP and QPT in XXZ spin chains Vivien Lecomte (LPMA Paris VI-VII) Trieste 1 / 25 Marc Cheneau 1 Juan P. Garrahan 2 , Frdric van Wijland 3 Ccile
Marc Cheneau1 Juan P. Garrahan2, Frédéric van Wijland3 Cécile Appert-Rolland4, Bernard Derrida5, Frédéric van Wijland3
1Institut d’Optique, Palaiseau 2Nottingham University 3MSC, Paris 4LPT, Orsay 5LPS, ENS, Paris
Trieste – 15th July 2014
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 1 / 25
Introduction Motivations
Correspondence
· evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain
(Well known at least in the stat. mech. community.) Use: dictionnary between
regimes of large deviations of dynamical observables phases across a Quantum Phase Transition
Perspectives opened / questions raised
fjnite-size efgects large/small scale spectrum
(I will ask questions to you.)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 2 / 25
Introduction Motivations
Correspondence
· evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain
(Well known at least in the stat. mech. community.) Use: dictionnary between
· regimes of large deviations of dynamical observables · phases across a Quantum Phase Transition
Perspectives opened / questions raised
fjnite-size efgects large/small scale spectrum
(I will ask questions to you.)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 2 / 25
Introduction Motivations
Correspondence
· evolution operator for stochastic classical system [particles hopping] · Hamiltonian of quantum XXZ chain
(Well known at least in the stat. mech. community.) Use: dictionnary between
· regimes of large deviations of dynamical observables · phases across a Quantum Phase Transition
Perspectives opened / questions raised
· fjnite-size efgects · large/small scale spectrum
(I will ask questions to you.)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 2 / 25
Exclusion Process System
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Confjgurations: occupation numbers {ni} Exclusion rule: 0 ≤ ni ≤ N Markov evolution for the probability P({ni}, t) ∂tP({ni}, t) = ∑
n′
i
[ W(n′
i → ni)P({n′ i}, t) − W(ni → n′ i)P({ni}, t)
] Large deviation function of time-integrated observables A ⟨e−sA⟩ ∼ et ψ(s) (⇔ determining P(A, t)) A total current Q on time window t #jumps jumps A total activity K on time window t #jumps jumps
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 3 / 25
Exclusion Process System
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Confjgurations: occupation numbers {ni} Exclusion rule: 0 ≤ ni ≤ N Markov evolution for the probability P({ni}, t) ∂tP({ni}, t) = ∑
n′
i
[ W(n′
i → ni)P({n′ i}, t) − W(ni → n′ i)P({ni}, t)
] Large deviation function of time-integrated observables A ⟨e−sA⟩ ∼ et ψ(s) (⇔ determining P(A, t)) A total current Q on time window t #jumps jumps A total activity K on time window t #jumps jumps
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 3 / 25
Exclusion Process System
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Confjgurations: occupation numbers {ni} Exclusion rule: 0 ≤ ni ≤ N Markov evolution for the probability P({ni}, t) ∂tP({ni}, t) = ∑
n′
i
[ W(n′
i → ni)P({n′ i}, t) − W(ni → n′ i)P({ni}, t)
] Large deviation function of time-integrated observables A ⟨e−sA⟩ ∼ et ψ(s) (⇔ determining P(A, t)) A total current Q on time window t #jumps jumps A total activity K on time window t #jumps jumps
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 3 / 25
Exclusion Process System
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Confjgurations: occupation numbers {ni} Exclusion rule: 0 ≤ ni ≤ N Markov evolution for the probability P({ni}, t) ∂tP({ni}, t) = ∑
n′
i
[ W(n′
i → ni)P({n′ i}, t) − W(ni → n′ i)P({ni}, t)
] Large deviation function of time-integrated observables A ⟨e−sA⟩ ∼ et ψ(s) (⇔ determining P(A, t)) A = total current Q on time window [0, t] = #j− − → umps − j← − − umps A = total activity K on time window [0, t] = #j− − → umps + j← − − umps
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 3 / 25
Exclusion Process System
[Schütz & Sandow PRE 49 2726]
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Evolution of probability vector P:
similar to Schrödinger eq. but eq. for the probability instead of the wave function
∂tP = W P W = ∑
1≤k≤L−1
[ σ+
k σ− k+1 + σ− k σ+ k+1 − ˆ
nk(1 − ˆ nk+1) − ˆ nk+1(1 − ˆ nk) ] + α [ σ+
1 − (1 − ˆ
n1) ] + γ [ σ−
1 − ˆ
n1 ] + δ [ σ+
L − (1 − ˆ
nL) ] + β [ σ−
L − ˆ
nL ] σ± = σx ± iσ− and σz = ˆ n − N
2 are spin operators (with j = N 2 )
XXX spin chain Hamiltonian (up to boundary terms and constants).
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 4 / 25
Exclusion Process System
[Schütz & Sandow PRE 49 2726]
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 Evolution of probability vector P:
similar to Schrödinger eq. but eq. for the probability instead of the wave function
∂tP = W P W = ∑
1≤k≤L−1
[ σ+
k σ− k+1 + σ− k σ+ k+1 − ˆ
nk(1 − ˆ nk+1) − ˆ nk+1(1 − ˆ nk) ] + α [ σ+
1 − (1 − ˆ
n1) ] + γ [ σ−
1 − ˆ
n1 ] + δ [ σ+
L − (1 − ˆ
nL) ] + β [ σ−
L − ˆ
nL ] σ± = σx ± iσ− and σz = ˆ n − N
2 are spin operators (with j = N 2 )
XXX spin chain Hamiltonian (up to boundary terms and constants).
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 4 / 25
Exclusion Process Large deviations
maximal
b b b b b b b b b b b
γ α 1 1 1 1 1 1 β δ 1 L ρ0 ρ1 ⟨ e−sK⟩ ∼ etψ(s) with ψ(s) = max Sp Ws Ws = ∑
1≤k≤L−1
[ e−sσ+
k σ− k+1 + e−sσ− k σ+ k+1 − ˆ
nk(1 − ˆ nk+1) − ˆ nk+1(1 − ˆ nk) ] + α [ e−sσ+
1 − (1 − ˆ
n1) ] + γ [ e−sσ−
1 − ˆ
n1 ] + δ [ e−sσ+
L − (1 − ˆ
nL) ] + β [ e−sσ−
L − ˆ
nL ] XXZ spin chain Hamiltonian
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 5 / 25
Exclusion Process Periodic Boundary Conditions
Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N0 density: ρ0 = N0/L
1 1 1 1 1
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 6 / 25
Exclusion Process Periodic Boundary Conditions
Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N0 density: ρ0 = N0/L
1 1 1 1 1
. . . 2 1 L ≡ 0 . . . Ring geometry
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 6 / 25
Exclusion Process Periodic Boundary Conditions
Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N0 density: ρ0 = N0/L
1 1 1 1 1
Ws =
L−1
∑
k=1
[ e−s( σ+
k σ− k+1 + σ− k σ+ k+1
) − ˆ nk(1 − ˆ nk+1) − (1 − ˆ nk)ˆ nk+1 ] L e
s L k x k x k y k y k z k z k
with es
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 6 / 25
Exclusion Process Periodic Boundary Conditions
Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N0 density: ρ0 = N0/L
1 1 1 1 1
Ws =
L−1
∑
k=1
[ e−s( σ+
k σ− k+1 + σ− k σ+ k+1
) − ˆ nk(1 − ˆ nk+1) − (1 − ˆ nk)ˆ nk+1 ] = L − 1 2 − e−s 2 H∆ H∆ = −
L−1
∑
k=1
[ σx
kσx k+1 + σy kσy k+1 + ∆σz kσz k+1
] with ∆ = es
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 6 / 25
Exclusion Process Periodic Boundary Conditions
SSEP Quantum Spin Chain local occupation number nk (1 ≤ k ≤ L) local spin σz
k (1 ≤ k ≤ L)
nk = 0, 1 ≡ ◦, • σz
k = 1, −1 ≡ ↑, ↓
(fjxed) total occupation N0 ≡ ρ0L (fjxed) total magnetization M ≡ m0L (mesoscopic) density ρ(x) (0 ≤ x ≤ 1) (mesoscopic) magnet. m(x) (0 ≤ x ≤ 1)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 7 / 25
Exclusion Process Periodic Boundary Conditions
SSEP Quantum Spin Chain local occupation number nk (1 ≤ k ≤ L) local spin σz
k (1 ≤ k ≤ L)
nk = 0, 1 ≡ ◦, • σz
k = 1, −1 ≡ ↑, ↓
(fjxed) total occupation N0 ≡ ρ0L (fjxed) total magnetization M ≡ m0L (mesoscopic) density ρ(x) (0 ≤ x ≤ 1) (mesoscopic) magnet. m(x) (0 ≤ x ≤ 1) evolution operator ferromagnetic XXZ Hamiltonian (Jxy = −1) Ws = L − 1 2 − e−s 2 H∆ H∆ =
L−1
∑
k=1
[ Jxy ( σx
kσx k+1 + σy kσy k+1
) + Jzσz
kσz k+1
] = −
L−1
∑
k=1
[ σx
kσx k+1 + σy kσy k+1 + ∆σz kσz k+1
] counting factor ∆ = es of the activity K anisotropy ∆ = −Jz along direction Z cumulant generating function ground state energy ψ(s) = max Sp Ws = L−1
2
− e−s
2 EL(s)
EL(s) = min Sp H∆
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 8 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 9 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
Bethe Ansatz:
eigenvector of components ∑
P
A(P)
N0
∏
i=1
[ ζP(i) ]xi eigenvalue ψ(s) = −2N0 + e−s[ ζ1 + . . . + ζN0 ] − e−s [ 1 ζ1 + . . . + 1 ζN0 ] Bethe equations ζL
i = N0
∏
j=1 j̸=i
[ −1 − 2esζi + ζiζj 1 − 2esζj + ζiζj ]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 9 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0
— : infjnite-size limit Repartition of Bethe roots in the complex plane
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 10 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
large deviation function ψ(s) = −2Lρ0(1 − ρ0)s
+ L−2F(u)
+ . . . with u = L2ρ0(1−ρ0)s universal function (singular in u = π2
2 )
F(u) = ∑
k≥2
(−2u)kB2k−2 Γ(k)Γ(k + 1)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 11 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
large deviation function ψ(s) = −2Lρ0(1 − ρ0)s
+ L−2F(u)
+ . . . with u = L2ρ0(1−ρ0)s universal function (singular in u = π2
2 )
u Fu
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 11 / 25
Exclusion Process Microscopic solution
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
large deviation function ψ(s) = −2Lρ0(1 − ρ0)s
+ L−2F(u)
+ . . . with u = L2ρ0(1−ρ0)s universal function (singular in u = π2
2 )
u Fu
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 11 / 25
Macroscopic approach Analogy with quantum mechanics
[Tailleur, Kurchan, VL, JPA 41 505001]
A reminder: propagator in quantum mechanics
⟨fjnal|eitH|initial⟩ dz dzn fjnal ei
t
zn zn ei
t
zn z ei
t
initial p q exp i p q
action
p p x t and q q x t are generically space- & time-dependent fjelds. “semi-classical limit” recovered in the large limit [saddle-point]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 12 / 25
Macroscopic approach Analogy with quantum mechanics
[Tailleur, Kurchan, VL, JPA 41 505001]
A reminder: propagator in quantum mechanics
⟨fjnal|eitH|initial⟩ = ∫ dz1 . . . dzn⟨fjnal|ei∆tH|zn⟩⟨zn−1|ei∆tH|zn−2⟩ . . . . . . ⟨z1|ei∆tH|initial⟩ p q exp i p q
action
p p x t and q q x t are generically space- & time-dependent fjelds. “semi-classical limit” recovered in the large limit [saddle-point]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 12 / 25
Macroscopic approach Analogy with quantum mechanics
[Tailleur, Kurchan, VL, JPA 41 505001]
A reminder: propagator in quantum mechanics
⟨fjnal|eitH|initial⟩ = ∫ dz1 . . . dzn⟨fjnal|ei∆tH|zn⟩⟨zn−1|ei∆tH|zn−2⟩ . . . . . . ⟨z1|ei∆tH|initial⟩ = ∫ DpDq exp{i 1
ℏ S[p, q] action
} p p x t and q q x t are generically space- & time-dependent fjelds. “semi-classical limit” recovered in the large limit [saddle-point]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 12 / 25
Macroscopic approach Analogy with quantum mechanics
[Tailleur, Kurchan, VL, JPA 41 505001]
A reminder: propagator in quantum mechanics
⟨fjnal|eitH|initial⟩ = ∫ dz1 . . . dzn⟨fjnal|ei∆tH|zn⟩⟨zn−1|ei∆tH|zn−2⟩ . . . . . . ⟨z1|ei∆tH|initial⟩ = ∫ DpDq exp{i 1
ℏ S[p, q] action
} p = p(x, t) and q = q(x, t) are generically space- & time-dependent fjelds. “semi-classical limit” recovered in the large 1
ℏ limit
[saddle-point]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 12 / 25
Macroscopic approach Hydrodynamic limit
[Tailleur, Kurchan, VL, JPA 41 505001]
For exclusion processes Using SU(2) coherent states:
⟨ρf|etW|ρi⟩ = ∫ ρ(t)=ρf
ρ(0)=ρi
DρDˆ ρ exp{L S[ˆ ρ, ρ]
action
} e
sK f et
s
i t
f i
exp L
s action
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 13 / 25
Macroscopic approach Hydrodynamic limit
[Tailleur, Kurchan, VL, JPA 41 505001]
For exclusion processes Using SU(2) coherent states:
⟨ρf|etW|ρi⟩ = ∫ ρ(t)=ρf
ρ(0)=ρi
DρDˆ ρ exp{L S[ˆ ρ, ρ]
action
} ⟨e−sK⟩ ∼ ⟨ρf|etWs|ρi⟩ = ∫ ρ(t)=ρf
ρ(0)=ρi
DρDˆ ρ exp{L Ss[ˆ ρ, ρ]
action
} Again: use saddle-point to handle the large L limit.
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 13 / 25
Macroscopic approach Hydrodynamic limit
[Tailleur, Kurchan, VL, JPA 41 505001]
For exclusion processes Same Ss[ˆ ρ, ρ] as the MSR action of the Langevin evolution:
∂tρ(x, t) = −∂x [ − ∂xρ(x, t) + ξ(x, t) ] ⟨ξ(x, t)ξ(x′, t′)⟩ = 1 Lρ(x, t) ( 1 − ρ(x, t) ) δ(x′ − x)δ(t′ − t)
One recovers the action of fmuctuating hydrodynamics
[Spohn, Bertini De Sole Gabrielli Jona-Lasinio Landim]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 13 / 25
Macroscopic approach Hydrodynamic limit
[Tailleur, Kurchan, VL, JPA 41 505001]
For exclusion processes Same Ss[ˆ ρ, ρ] as the MSR action of the Langevin evolution:
∂tρ(x, t) = −∂x [ − ∂xρ(x, t) + ξ(x, t) ] ⟨ξ(x, t)ξ(x′, t′)⟩ = 1 Lρ(x, t) ( 1 − ρ(x, t) ) δ(x′ − x)δ(t′ − t)
One recovers the action of fmuctuating hydrodynamics
[Spohn, Bertini De Sole Gabrielli Jona-Lasinio Landim]
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 13 / 25
Macroscopic approach Large deviation function
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
Periodic boundary conditions More general fmuctuating hydrodynamics 1 Lt⟨Q⟩ ∝ D(ρ) (Fourier’s law) 1 Lt⟨Q2⟩c = σ(ρ) (For the SSEP, σ(ρ) = ρ(1 − ρ)) Saddle point evaluation e
sK
exp L
s
Large deviation function [assuming uniform profjle x ] s s K c t
at saddle-point
L D u
fmuctuations
with u L s D
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 14 / 25
Macroscopic approach Large deviation function
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
Periodic boundary conditions More general fmuctuating hydrodynamics 1 Lt⟨Q⟩ ∝ D(ρ) (Fourier’s law) 1 Lt⟨Q2⟩c = σ(ρ) (For the SSEP, σ(ρ) = ρ(1 − ρ)) Saddle point evaluation ⟨e−sK⟩ ∼ ∫ DρDˆ ρ exp{L Ss[ˆ ρ, ρ]} Large deviation function [assuming uniform profjle x ] s s K c t
at saddle-point
L D u
fmuctuations
with u L s D
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 14 / 25
Macroscopic approach Large deviation function
[Appert, Derrida, VL, van Wijland, PRE 78 021122]
Periodic boundary conditions More general fmuctuating hydrodynamics 1 Lt⟨Q⟩ ∝ D(ρ) (Fourier’s law) 1 Lt⟨Q2⟩c = σ(ρ) (For the SSEP, σ(ρ) = ρ(1 − ρ)) Saddle point evaluation ⟨e−sK⟩ ∼ ∫ DρDˆ ρ exp{L Ss[ˆ ρ, ρ]} Large deviation function [assuming uniform profjle ρ(x) = ρ] ψ(s) = −s⟨K⟩c t
at saddle-point
+ L−2DF(u)
fmuctuations
with u = L2sσ(ρ0)σ′′(ρ0) 8D2
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 14 / 25
Macroscopic approach Large deviation function
Correspondence between the (Gaussian) integration of small fmuctuations AND discreteness of Bethe root repartition. More general? Repartition of Bethe roots for s sc? Fluctuating hydrodynamics for quantum chains?
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 15 / 25
Macroscopic approach Large deviation function
Correspondence between the (Gaussian) integration of small fmuctuations AND discreteness of Bethe root repartition. More general? Repartition of Bethe roots for s > sc? Fluctuating hydrodynamics for quantum chains?
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 15 / 25
Macroscopic approach Large deviation function
Correspondence between the (Gaussian) integration of small fmuctuations AND discreteness of Bethe root repartition. More general? Repartition of Bethe roots for s > sc? Fluctuating hydrodynamics for quantum chains?
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 15 / 25
Beyond the critical point Scaling
Rescaling of the large deviation function
[singularity at λc > 0 as L → ∞ ]
φ(λ) = lim
L→∞ Lψ(λ/L2)
Using the correct non-uniform saddle-point profjle for λ > λc
10 20 30 40 50
5
non-uniform profile uniform profile
λc =
π2 σ(ρ0)
(can be large!)
[Bodineau, Derrida, PRE 78 021122] phase transition in WASEP for large dev.
(non-stationary profjle)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 16 / 25
Beyond the critical point Scaling
Rescaling of the large deviation function
[singularity at λc > 0 as L → ∞ ]
φ(λ) = lim
L→∞ Lψ(λ/L2)
Using the correct non-uniform saddle-point profjle for λ > λc
10 20 30 40 50
5
non-uniform profile uniform profile
λc =
π2 σ(ρ0)
(can be large!)
[Bodineau, Derrida, PRE 78 021122] phase transition in WASEP for large dev.
(non-stationary profjle)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 16 / 25
Beyond the critical point Scaling
Optimal (i.e. saddle-point) profjle
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 17 / 25
Beyond the critical point Scaling
SSEP Quantum Spin Chain local occupation number nk (1 ≤ k ≤ L) local spin σz
k (1 ≤ k ≤ L)
nk = 0, 1 ≡ ◦, • σz
k = 1, −1 ≡ ↑, ↓
(fjxed) total occupation N0 ≡ ρ0L (fjxed) total magnetization M ≡ m0L (mesoscopic) density ρ(x) (0 ≤ x ≤ 1) (mesoscopic) magnet. m(x) (0 ≤ x ≤ 1) evolution operator Ws ferromagnetic XXZ Hamiltonian H∆ cumulant generating function ψ(s) ground state energy EL(s)
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 18 / 25
Beyond the critical point Scaling
SSEP Quantum Spin Chain local occupation number nk (1 ≤ k ≤ L) local spin σz
k (1 ≤ k ≤ L)
nk = 0, 1 ≡ ◦, • σz
k = 1, −1 ≡ ↑, ↓
(fjxed) total occupation N0 ≡ ρ0L (fjxed) total magnetization M ≡ m0L (mesoscopic) density ρ(x) (0 ≤ x ≤ 1) (mesoscopic) magnet. m(x) (0 ≤ x ≤ 1) evolution operator Ws ferromagnetic XXZ Hamiltonian H∆ cumulant generating function ψ(s) ground state energy EL(s) minimal activity phase (s → +∞) Ising ferromagnetic order (∆ → +∞)
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↑ ↑ ↑ ↑ ↑ maximal activity phase (s → −∞) XY degenerate groundstate (∆ = 0)
↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ & ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ • (in fact, superp. of eiθ ↓ ↑ ↓ ↑ ↓ ↑ ↓ . . .) time t (steady state: t → +∞) inverse temp. β (zero-temp. limit: β → +∞) dynamical partition function partition function ⟨e−sK⟩ ≃ Tr etWs ZXXZ
β
(∆) = Tr e−β H∆
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 19 / 25
Beyond the critical point Scaling
[VL, Garrahan, van Wijland, JPA 45 175001]
Saddle-point equations for the profjle ρ(x) take the form ( ∂xρ(x) )2 + EP ( ρ(x) ) = 0 Motion in “time” x of a particle of “position” in a “oscillations” depict the non-uniform profjle x
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 20 / 25
Beyond the critical point Scaling
[VL, Garrahan, van Wijland, JPA 45 175001]
Saddle-point equations for the profjle ρ(x) take the form ( ∂xρ(x) )2 + EP ( ρ(x) ) = 0 Motion in “time” x of a particle of “position” ρ in a
0.2 0.4 0.6 0.8 1.0
0.5 1.0
“oscillations” depict the non-uniform profjle x
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 20 / 25
Beyond the critical point Scaling
[VL, Garrahan, van Wijland, JPA 45 175001]
Saddle-point equations for the profjle ρ(x) take the form ( ∂xρ(x) )2 + EP ( ρ(x) ) = 0 Motion in “time” x of a particle of “position” ρ in a
0.2 0.4 0.6 0.8 1.0
0.5 1.0
“oscillations” depict the non-uniform profjle ρ(x)
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 20 / 25
Beyond the critical point Scaling
[Cheneau, VL, work in progress]
What about solutions with more than one kink+anti-kink? φ(λ) corresponding profjles ρ(x)
Λc 4 Λc
50 100 150 200 80 60 40 20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
λ x
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 21 / 25
Beyond the critical point Smaller sizes
Aim: experimental realizations with cold atoms → non-periodic (but isolated, 1D) system → smaller sizes & fjnite-temperature & excited state
10 20 30
2 4
increasing L
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 22 / 25
Beyond the critical point Smaller sizes
L = 9 sites infjnite-size ground state N0 = 3 particles infjnite-size excited states gathering(?) of microscopic eigenvalues macroscopic (L ) states
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 23 / 25
Beyond the critical point Smaller sizes
L = 9 sites infjnite-size ground state N0 = 3 particles infjnite-size excited states
50 100 150
gathering(?) of microscopic eigenvalues macroscopic (L ) states
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 23 / 25
Beyond the critical point Smaller sizes
L = 9 sites infjnite-size ground state N0 = 3 particles infjnite-size excited states
50 100 150
gathering(?) of microscopic eigenvalues macroscopic (L ) states
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 23 / 25
Beyond the critical point Smaller sizes
L = 9 sites infjnite-size ground state N0 = 3 particles infjnite-size excited states
50 100 150
gathering(?) of microscopic eigenvalues − → macroscopic (L = ∞) states
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 23 / 25
Conclusion
Microscopic approach:
⋆ operator formalism ⋆ XXZ spin chain ⋆ Bethe Ansatz
Macroscopic approach:
⋆ action of fmuctuating hydrodynamics ⋆ saddle-point method, dynamical phase transition
Questions:
Finite-size crossover around a quantum phase transition? Between:
Luttinger Liquid (s ) Phase-separated ferromagnet (s )
Across the transition: continuum spectrum gaped spectrum? XXZ transition not at but at L Are scaling exponents/functions known? Are they interesting?
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 24 / 25
Conclusion
Microscopic approach:
⋆ operator formalism ⋆ XXZ spin chain ⋆ Bethe Ansatz
Macroscopic approach:
⋆ action of fmuctuating hydrodynamics ⋆ saddle-point method, dynamical phase transition
Questions:
⋆ Finite-size crossover around a quantum phase transition? Between:
· Luttinger Liquid (s → −∞) · Phase-separated ferromagnet (s → +∞)
⋆ Across the transition: continuum spectrum → gaped spectrum? ⋆ XXZ transition not at ∆ = 1 but at ∆ = 1 + O(L−2) ⋆ Are scaling exponents/functions known? Are they interesting?
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 24 / 25
Conclusion
References: ⋆ Marc Cheneau, Vivien Lecomte et al. work in progress (2014) ⋆ Vivien Lecomte, Juan P. Garrahan, Frédéric van Wijland
⋆ Cécile Appert-Rolland, Bernard Derrida, Vivien Lecomte and Frédéric van Wijland
Vivien Lecomte (LPMA – Paris VI-VII) SSEP and QPT in XXZ spin chains 15/07/2014 25 / 25