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Statistical Physics Journal Club:
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 1 / 19
1 The Richardson Model and integrability
2 Application of algebraic Bethe Ansatz
3 Some Results
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 2 / 19
Anderson’s question“paradigm“ for superconductivity:the Bardeen, Cooper, Schrieffer model
Order parameter: ∆
Anderson: number of levels involved information of Cooper pairs ≈ ∆/d
[HBCS ,N] 6= 0
Question:
How do pairing correlations change when the size of the superconductor isdecreased from the bulk to only a few electrons?
Nuclear Superconductivity
Superconducting nanograins (Al)(Ralph, Black, Tinkham - Phys. Rev. Lett. 74, 3241 / 76 688 / 78, 4087)
Al d ∼ 0.45meV , Ec ∼ 46meV , ∆ ∼ 0.38meV
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 3 / 19
Anderson’s question“paradigm“ for superconductivity:the Bardeen, Cooper, Schrieffer model
Order parameter: ∆
Anderson: number of levels involved information of Cooper pairs ≈ ∆/d
[HBCS ,N] 6= 0
Question:
How do pairing correlations change when the size of the superconductor isdecreased from the bulk to only a few electrons?
Nuclear Superconductivity
Superconducting nanograins (Al)(Ralph, Black, Tinkham - Phys. Rev. Lett. 74, 3241 / 76 688 / 78, 4087)
Al d ∼ 0.45meV , Ec ∼ 46meV , ∆ ∼ 0.38meV
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 3 / 19
the Richardson hamiltonianspin 1/2 fermions c†ασN single-particle energy levels εαNf fermions
H =∑α
∑σ=↑↓
εαc†ασcασ − g∑α,β
c†α↑c†α↓cβ↓cβ↑
integrable (Cambiaggio, Rivas, Saraceno 1997)
S−α = cα,↓cα,↑ S+α = c†α,↑c
†α,↓ Sz
α = 1− 2Nα
HBCS(εα, g) =N∑α=1
εαSzα − g
N∑α,β=1
S+α S−β .
Explicit construction of the integrals of motion
τα = −g
2Szα +
∑β 6=α
1
εα − εβ
(S+α ⊗ S−β + S−α ⊗ S+
β + 2Szα ⊗ Sz
β
)
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 4 / 19
the Richardson hamiltonianspin 1/2 fermions c†ασN single-particle energy levels εαNf fermions
H =∑α
∑σ=↑↓
εαc†ασcασ − g∑α,β
c†α↑c†α↓cβ↓cβ↑
integrable (Cambiaggio, Rivas, Saraceno 1997)
S−α = cα,↓cα,↑ S+α = c†α,↑c
†α,↓ Sz
α = 1− 2Nα
HBCS(εα, g) =N∑α=1
εαSzα − g
N∑α,β=1
S+α S−β .
Explicit construction of the integrals of motion
τα = −g
2Szα +
∑β 6=α
1
εα − εβ
(S+α ⊗ S−β + S−α ⊗ S+
β + 2Szα ⊗ Sz
β
)Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 4 / 19
Algebraic Bethe Ansatz approach
Compute directly
matrix elements of the physical observablesquench matrix
Eigenstates of the form
|wj〉 =M∏
k=1
C(wk)| ↓↓↓ ... ↓〉 ≡M∏
k=1
(N∑α=1
S+α
wk − εα
)| ↓↓↓ ... ↓〉
with energy: E (wj) = 2M∑
j=1
wj −N∑α=1
εα
the set of “rapidities” wj satisfies the Bethe-Richardson equations
− 1
g=
N∑α=1
1
wj − εα−
M∑k 6=j
2
wj − wkj = 1, . . . ,M
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 5 / 19
The electrostatic analogyQuantum Pairing Model Classical 2D Electrostatic Picture
Effective single particle energy εα Orbiton position zα = εαPair energy wj Pairon position zj = wj
Pairing strength g Electric field strength e = ± 1g
U =
eM∑
j=1
qjRe(z j) + eN∑α=1
qαRe(zα)−N∑α=1
M∑j=1
qαqj ln |zj − zα|
−1
2
∑α6=β
qαqβ ln |zα − zβ | −1
2
∑i 6=j
qiqj ln |zi − zj | .
equilibrium position of the free pairons in the presenceof the fixed orbitons:
e =∑α
qαzα − zj
+∑k 6=j
qk
zj − zk
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 6 / 19
The large-N limit and BCS
∫Ω
ρ(ε)
ε− wdε−
∫Γ
r(w ′)
w ′ − wdw − 1
g= 0
−→∫
Ω
ρ(ε) dε√(ε− λ)2 + ∆2
=1
g
density of energy states:∫Ω
ρ(ε)dε = N
density of roots:∫Γ
r(w)dw = M
∫Γ
wr(w)dw = E
→ M =
∫Ω
(1− ε− λ√
(ε− λ)2 + ∆2
)ρ(ε)dε
→ E = −∆2
g
+
∫Ω
ε(1− ε− λ√(ε− λ)2 + ∆2
)ρ(ε)dε
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 7 / 19
Quantum quenches
Hg0 → Hg
Evolution of states:
|ψ(t)〉 =∑ν
e−iωνt〈ψνg |ψµg0〉|ψνg 〉.
Evolution of observables:
〈O(t)〉 ≡ 〈Ψ(t)|O|Ψ(t)〉 =∑ν,ν′
e−i(ων−ων′)t〈ψνg |ψµg0
〉〈ψµg0|ψν′
g 〉〈ψν′
g |O|ψνg 〉
Probability distribution of the work:
P(W ) =∑µ
|〈ψ0g0|ψµg 〉|2δ(W − ωµg + ω0
g0)
Not easy:
Compute eigenstates and eigenvalues of both Hamltonians
Compute overlaps between them
Sum over the full (huge!) Hilbert spaceFrancesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 8 / 19
Static and dynamical correlation functions
GOαβ=〈GS |O†αOβ |GS〉〈GS |GS〉
GOαβ(t) =
〈GS |O†α(t)Oβ(0)|GS〉〈GS |GS〉
=∑w
〈w|Oα|GS〉∗〈w|Oβ |GS〉e i(ωw−ωGS )t
〈GS |GS〉〈w|w〉
focus on
G dzz(t) =
N∑α=1
〈GS |Szα(t)Sz
α(0)|GS〉〈GS |GS〉
G d+−(t) =
N∑α=1
〈GS |S+α (t)S−α (0)|GS〉〈GS |GS〉
G od+−(t) =
N∑α,β=1
〈GS |S+α (t)S−β (0)|GS〉〈GS |GS〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 9 / 19
Solving the Richardson equations
Dimension of the Hilbert space
D(N,M) =
(NM
)at half-filling :D(16, 8) = 12870D(32, 16) = 601 080 390
Scanning procedure: start fromthe (known) g = 0 solutions andincrease slowly the coupling
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 10 / 19
The quench matrixThe Yang-Baxter algebra A(w), B(w), C(w), D(w) admits aconjugation operation 〈0|
∏b B(wb)
∏a C(va)|0〉
↓
Slavnov’s formula:
〈wj|vj〉 =1√
DetG [wj]DetG [vj]
M∏a 6=b
(vb − wa)
M∏a<b
(wa − wb)M∏
a>b
(va − vb)
DetJ
Jab =vb − wb
vb − wa
N∑k=1
1
(wa − εk)(vb − εk)− 2
∑c 6=a
1
(wa − wc)(va − wc)
the C(w) operators∑Nα=1
S+α
wk−εα have no explicit dependence on g
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 11 / 19
Form Factors and observables
this is what we would like: 〈w|Szα|v〉 = 〈0|
M∏b=1
B(wb)Szα
M∏a=1
C(va)|0〉
this is what we have:
A(wk) =−1
g+
N∑α=1
Szα
wk − εαB(wk) =
N∑α=1
S−αwk − εα
,
C(wk) =N∑α=1
S+α
wk − εαD(wk) =
1
g−
N∑α=1
Szα
wk − εα+ Yang-Baxter algebra + | ↓↓↓ ... ↓〉
〈~w |Szm|~v〉 = − lim
u→εα(u − εα)〈w|D(u)|v〉
〈~w |S+α |~v〉 = lim
u→εα(u − εα)〈w|C(u)|v〉
〈~w |S−α |~v〉 = limu→εα
(u − εα)〈w|B(u)|v〉
why not? 〈~w |S+α S−β |~v〉 〈~w |Sz
α Szβ |~v〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 12 / 19
Form Factors and observables
this is what we would like: 〈w|Szα|v〉 = 〈0|
M∏b=1
B(wb)Szα
M∏a=1
C(va)|0〉
this is what we have:
A(wk) =−1
g+
N∑α=1
Szα
wk − εαB(wk) =
N∑α=1
S−αwk − εα
,
C(wk) =N∑α=1
S+α
wk − εαD(wk) =
1
g−
N∑α=1
Szα
wk − εα+ Yang-Baxter algebra + | ↓↓↓ ... ↓〉
〈~w |Szm|~v〉 = − lim
u→εα(u − εα)〈w|D(u)|v〉
〈~w |S+α |~v〉 = lim
u→εα(u − εα)〈w|C(u)|v〉
〈~w |S−α |~v〉 = limu→εα
(u − εα)〈w|B(u)|v〉
why not? 〈~w |S+α S−β |~v〉 〈~w |Sz
α Szβ |~v〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 12 / 19
Form Factors and observables
this is what we would like: 〈w|Szα|v〉 = 〈0|
M∏b=1
B(wb)Szα
M∏a=1
C(va)|0〉
this is what we have:
A(wk) =−1
g+
N∑α=1
Szα
wk − εαB(wk) =
N∑α=1
S−αwk − εα
,
C(wk) =N∑α=1
S+α
wk − εαD(wk) =
1
g−
N∑α=1
Szα
wk − εα+ Yang-Baxter algebra + | ↓↓↓ ... ↓〉
〈~w |Szm|~v〉 = − lim
u→εα(u − εα)〈w|D(u)|v〉
〈~w |S+α |~v〉 = lim
u→εα(u − εα)〈w|C(u)|v〉
〈~w |S−α |~v〉 = limu→εα
(u − εα)〈w|B(u)|v〉
why not? 〈~w |S+α S−β |~v〉 〈~w |Sz
α Szβ |~v〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 12 / 19
Form Factors and observables
this is what we would like: 〈w|Szα|v〉 = 〈0|
M∏b=1
B(wb)Szα
M∏a=1
C(va)|0〉
this is what we have:
A(wk) =−1
g+
N∑α=1
Szα
wk − εαB(wk) =
N∑α=1
S−αwk − εα
,
C(wk) =N∑α=1
S+α
wk − εαD(wk) =
1
g−
N∑α=1
Szα
wk − εα+ Yang-Baxter algebra + | ↓↓↓ ... ↓〉
〈~w |Szm|~v〉 = − lim
u→εα(u − εα)〈w|D(u)|v〉
〈~w |S+α |~v〉 = lim
u→εα(u − εα)〈w|C(u)|v〉
〈~w |S−α |~v〉 = limu→εα
(u − εα)〈w|B(u)|v〉
why not? 〈~w |S+α S−β |~v〉 〈~w |Sz
α Szβ |~v〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 12 / 19
Quench: Weak to strong, strong to weak coupling
Figure: First column of the quench matrix (ground-state overlaps) for several quenches. In all plots N = 16,M = 8 and theground state energies (represented by vertical lines) have been shifted for clarity. Top: Decomposition of the g = 0 ground-statewith states at g = 0.05, 0.5, 0.95. Center: Decomposition of several initial ground-state g0 = 0, 0.15, 0.3, 0.5 in terms of thestates at g = 1. Bottom: Decomposition of the g0 = 1 ground-state in terms of g = 0.95, 0.55, 0.15, 0 states
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 13 / 19
Truncation mechanismquench matrix:
dynamical correlation functions:
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 14 / 19
The BCS order parameterChoice of the parameters:
Equidistant levels εα = α α ∈ NHalf filling Np = N/2
Ψ(t) =∑α
√14 + 〈Sz
α〉2 =∑α
√〈S−α S+
α 〉〈S+α S−α 〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 15 / 19
Statistics of the work done, Order parameter evolution
P(W ) =∑µ
|〈ψ0g0|ψµg 〉|2δ(W−Ωµ)
Ωµ = ωµg − ω0g0
ΨOD(t) = 〈ψ(t)| 1Nr
∑Nαβ=1 S+
α S−β |ψ(t)〉
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 16 / 19
Double quench: occupation and work
Aβ =∑γ∈Hg
e−iωγg tq〈ψβg0|ψγg 〉〈ψγg |ψ0
g0〉
Iq,r =
∑α>0 |Aα|2q
(∑
α>0 |Aα|2)q
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 17 / 19
The paper in a few words
The integrable Richardson model
Application of algebraic Bethe Ansatz to dynamics
Slavnov formula for the quench matrix: YBg ↔ YBg0
Evolution of observables → solve the quantum inverse problem!
Thanks Mauro
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 18 / 19
The paper in a few words
The integrable Richardson model
Application of algebraic Bethe Ansatz to dynamics
Slavnov formula for the quench matrix: YBg ↔ YBg0
Evolution of observables → solve the quantum inverse problem!
Thanks Mauro
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 18 / 19
References
A. Faribault, P. Calabrese, J. Caux - Bethe Ansatz approach to quenchdynamics in the Richardson model - J. Math. Phys. 50, 095212 (2009)
A. Faribault, P. Calabrese, J. Caux - Quantum quenches from integrability:the fermionic pairing model - J. Stat. Mech. (2009) P03018
A. Faribault, P. Calabrese, J. Caux - Exact Mesoscopic correlation functionsof the pairing model - Phys. Rev. B 77, 064503 (2008)
A. Faribault, P. Calabrese, J. Caux - Dynamical correlation functions of themesoscopic pairing model - Arxiv:1003.0582v1
Links, Zhou, McKenzie, Gould - Algebraic Bethe ansatz method for theexact calculation of energy spectra and form factors: applications to modelsof Bose-Einstein condensates and metallic nanograins - J. Phys A 36 (2003)R63-R104
Roman, Sierra, Dukelski - Large N limit of the exactly solvable BCS model:analytics versus numerics - Nucl.Phys. B634 (2002) 483-510
J. Dukelsky, S. Pittel, G. Sierra - Rev.Mod.Phys.76:643-662,2004
Von Delft, Ralph - Spectroscopy of discrete energ levels in ultrasmallmetallic grains - Phys. Rep. 345 (2001) 61-173
Francesco Buccheri (SISSA) JC: Bethe Ansatz and quantum quenches March 12, 2010 19 / 19