Statistical Physics Journal Club · Statistical Physics Journal Club: Francesco Buccheri (SISSA)...

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


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


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


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

)


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


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


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


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〉


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


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


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〉




M∏b=1

B(wb)Szα

M∏a=1

C(va)|0〉


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




〈~w |S+α |~v〉 = lim



(u − εα)〈w|B(u)|v〉


α Szβ |~v〉




M∏b=1

B(wb)Szα

M∏a=1

C(va)|0〉


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




〈~w |S+α |~v〉 = lim



(u − εα)〈w|B(u)|v〉


α Szβ |~v〉




M∏b=1

B(wb)Szα

M∏a=1

C(va)|0〉


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




〈~w |S+α |~v〉 = lim



(u − εα)〈w|B(u)|v〉


α Szβ |~v〉


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


Truncation mechanismquench matrix:

dynamical correlation functions:


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 〉


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)〉


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


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


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


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


Date post:	26-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Statistical Physics Journal Club · Statistical Physics Journal Club: Francesco Buccheri (SISSA)...

Documents