hydrodynamic correlations in low-dimensional interacting ...ffranchi/presentations... ·...

Hydrodynamic Correlations Hydrodynamic Correlations in Lowin Low--Dimensional Dimensional Interacting SystemsInteracting Systems

Doctoral Defense:

1) The Emptiness Formation Probability1) The Emptiness Formation Probability

2) Spin2) Spin--Charge (nonCharge (non--)Separation)Separation

Advisor: Alexander G. Advisor: Alexander G. AbanovAbanov

Fabio FranchiniFabio Franchini

Fabio Franchini

Hydrodynamic Description of Correlators n. 2n. 2 Ph.D. Defense (7-21-2006)

•• Introduction & MotivationsIntroduction & Motivations

•• 1.1. The The EEmptiness mptiness FFormation ormation PProbability in the XY modelrobability in the XY model

•• 2.2. Hydrodynamic Approach to the EFPHydrodynamic Approach to the EFP

•• 3.3. Hydrodynamics for SpinHydrodynamics for Spin--Charge degrees of freedomCharge degrees of freedom

•• Conclusions and direction for future researchConclusions and direction for future research

Fabio Franchini

OutlineOutline


•• 11--D systems, ZeroD systems, Zero--TemperatureTemperature

•• Correlators Correlators →→ Bosonization (linear approximation)Bosonization (linear approximation)

•• Bosonization cannot describe large deviations: Bosonization cannot describe large deviations:

e.g. e.g. EEmptiness mptiness FFormation ormation PProbability or Spinrobability or Spin--Charge InteractionCharge Interaction

•• NonNon--Linear Bosonization Linear Bosonization →→ HydrodynamicsHydrodynamics

•• Integrable model Integrable model →→ Wave function (Wave function (BetheBethe AnsatzAnsatz))

Fabio Franchini

IntroductionIntroduction


•• EFP measures the probability that there are no particles in EFP measures the probability that there are no particles in

a region of length a region of length nn

•• One of the fundamental and simplest correlators in the One of the fundamental and simplest correlators in the

theory of integrable models (Korepin et al.)theory of integrable models (Korepin et al.)

•• Explicit expressions exist, but very complicatedExplicit expressions exist, but very complicated

•• Interesting asymptotic behavior Interesting asymptotic behavior →→ HydrodynamicsHydrodynamics

Fabio Franchini

Part 1:Part 1: Emptiness Formation ProbabilityEmptiness Formation Probability


• 1-d Spin Models: Probability of Formation of a

Ferromagnetic String (PFFS) of length n:

• Mapping to spinless fermion: PFFS becomes the

Emptiness Formation Probability

•• EFP measures the probability that there are no EFP measures the probability that there are no

particles in a region of length particles in a region of length nn

zni

i 1

1P(n)

2=

− σ= ∏

Fabio Franchini

EFP for Spin Systems EFP for Spin Systems

n†

i ii 1

P(n)=

= ψ ψ∏


The Anisotropic XY ModelThe Anisotropic XY Model

• Jordan-Wigner transformation:

spin degrees of freedom

into spinless fermions

• In momentum space, the Hamiltonian becomes:

x x y y zi i 1 i i 1 i

i i

1 1H h2 2+ +

⎡ ⎤+ γ − γ⎛ ⎞ ⎛ ⎞= σ σ + σ σ − σ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∑ ∑

( ) ( )† † †q q q q q q

qH 2 cosq h i sinq − −= − ψ ψ + γ ψ ψ − ψ ψ∑

Fabio Franchini

x yj j j

1 ( i )2

±σ = σ ± σ

†ii

i j

z †j j j

i†

j j

2 1

e <

π ψ ψ+

⎧σ = ψ ψ −⎪∑⎨

σ = ψ⎪⎩


The Anisotropic XY ModelThe Anisotropic XY Model (cont.)(cont.)• A Bogoliubov transformation diagonalizes the Hamiltonian

• The XY Model is essentially Free Fermions

• Correlators for physical quantities involve inverting the

transformation to FF: complications

Fabio Franchini

q q †q q qcos i sin

2 2 −

ϑ ϑχ = ψ + ψ

( )†q q q

qH 1/ 2= ε χ χ −∑ 2 2 2

q (cosq h) sin qε = − + γ


The Phase Diagram of the XY Model

2 2 2q (cosq h) sin qε = − + γ

h

γ

ΣΓ

Ω

Σ

Σ

Ω

Ω

Γ1

1

-1

+

o

-

-

+

o

I

E

Phase Diagram:

• 3 non-critical regions (Σ0, Σ±)

• 3 critical phases:Ω0: Isotropic XYΩ±: Critical magnetic field

Phase Diagram of the XY Model(only γ>0 shown)

Fabio Franchini


n

iq( j k )n 2 2 2

j,k 1

1 cosq h i sinq dqS 1 e2 2(cosq h) sin q

π −

−π

=

⎡ ⎤⎛ ⎞− + γ⎢ ⎥⎜ ⎟= +⎜ ⎟ π⎢ ⎥− + γ⎝ ⎠⎣ ⎦

∫

nP(n) det(S )=

EFP for the XY ModelEFP for the XY Model

• Wick Theorem and Pfaffian properties →EFP as the determinant of a n × n matrix:(Franchini & Abanov (2003))

n†

i ii 1

P(n)=

= ψ ψ∏

Fabio Franchini


Toeplitz MatricesToeplitz Matrices

• Asymptotic behavior of Det Sn: Szegö Theorem, Fisher-Hartwig

conjecture and its generalization, Widom Theorem

• Det Sn depends on the “singularities” of the generating function

iqkk

dqa (q)e2

π

−π= σ

π∫

0 1 2 n 1

1 0 1

2 1 0 2n

1

n 1 2 1 0

a a a aa a aa a a aS

aa a a a

−

−

− −

− + − −

⎛ ⎞⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

…

…

Fabio Franchini

• Matrices like Sn are called

Toeplitz: their elements

depend only upon the

difference of the indices


Toeplitz MatricesToeplitz Matrices TechniquesTechniques

• Szegö Theorem:

• McCoy et Al. (1970) used Toeplitz determinants to calculate

2-point correlators:

• We are the first to use the Generalized Fisher-Hartwig conjecture

Fabio Franchini


The Phase Diagram of the XY Modeland the Asymptotics of the EFP

Fabio Franchini


Interpretation of these resultsInterpretation of these results

• Toeplitz determinant technique is exclusive for XY Model

•• What is the physical meaning of the different behaviors? What is the physical meaning of the different behaviors?

→→ Need for a more physical (general?) approachNeed for a more physical (general?) approach

• Collective description of the system →→ Bosonization Bosonization

• EFP as probability of a collective configuration

Fabio Franchini


EFP as an EFP as an InstantonInstanton SolutionSolution• EFP as a rare fluctuation: P(R) ~ e-S(R)

• Small String: R << 1/T, ξ

S(R) ~ R2

(Gaussian)

• Large String: R >> 1/T or ξ

S(R) ~ R × 1/T or ξ

(exponential)

Fabio Franchini

0

1/T

0

1/T

~ R

~ R0

1/T

~ξ


~R

Limits of BosonizationLimits of Bosonization

• Bosonization as a collective description

• Too large of a fluctuation: Bosonization cannot describe EFP

(New problem: Depletion Formation Probability)

• Correct qualitative behavior:

– Ω0 (XX Model - Critical): Gaussian behavior (Abanov & Korepin, ‘02)

– Crossover for small γ: Gaussian for n << 1/√γ

to Exponential for n >> 1/√γ (Franchini & Abanov, ‘05)

Fabio Franchini


Bosonization for the Isotropic XY Model

α

νDeift et Al. 1997Kitanine et Al. 2002Korepin et Al. 2002Shiroishi et Al. 2002Korepin & Abanov, 2002

XXZ Model

Fabio Franchini


• EFP as a rare fluctuation: P(R) ~ e-S0(R)

• From Bosonization:

S0= π2/32 R2 +… ≈ 0.34 R2

• Exact result:

S0 = ½ Ln 2 R2 +… ≈ 0.30 R2

Korepin & Abanov 2002

EFP Crossover from BosonizationEFP Crossover from Bosonization

• We study the crossover for small γ:

• Bosonized Lagrangian:

Fabio Franchini


2 2 2q (cosq h) sin qε = − + γ

− π2 − π

3 − π8π8

π3

π2

q

0.1

0.2

0.3

0.4

0.5

ε

γ=0.05, h=0.5

m2= 2γ: The anisotropy is an effective mass term that opens a gap

EFP Crossover from BosonizationEFP Crossover from Bosonization

• We look for the saddle point solution of

with EFP BC:

• The stationary action shows

the expected crossover

at n ~ 1/√γ :

P(n) ~ e-S(n)

Fabio Franchini


Part 1: Part 1: ConclusionsConclusions

• We express the EFP exactly as a determinant of a matrix

• We calculated the EFP in the whole phase-diagram of the

XY model

• We provided the physical interpretation for the asymptotic

behaviors

• We tested the limits of Bosonization

Fabio Franchini


Part 2: Part 2: Hydrodynamic ApproachHydrodynamic ApproachFabio Franchini

• Collective field description: density ρ and velocity v

• From Galilean Invariance (Landau – 1941):

Continuity Equation

Euler Equation


EFP & HydrodynamicsEFP & Hydrodynamics

• Hydrodynamics keeps non-linearity of the spectrum

• EFP as probability of Instanton configuration (P(R) ~ e-S0(R)) with

BC:

• Bosonization valid for: (ρ0 : equilibrium density)

Fabio Franchini

ε

k


Bosonization

EFP results from HydrodynamicsEFP results from Hydrodynamics• Free Fermions:

• Calogero-Sutherland:

• Exact result known (s ≡ π ρ0 R ) :

Fabio Franchini


(Abanov 2005)

More EFP results from More EFP results from Hydrodynamics?Hydrodynamics?

• Hydrodynamics correctly reproduce the leading behavior of EFP

• Work in progress: calculating EFP for lattice model

(XY and XXZ model)

• Other problems →→ Construct the hydrodynamic description for

more integrable systems

Fabio Franchini


Hydrodynamics from Hydrodynamics from BetheBethe AnsatzAnsatzFabio Franchini


Bethe AnsatzSolution

in current state

Hamiltonian from

BA parameters

ρ and vfrom

BA paramters

Hamiltonianas function of

ρ and v

• From microscopical description:

• Using the Hydrodynamic Hamiltonian:

Hydrodynamics EquationsHydrodynamics EquationsFabio Franchini


Hydrodynamics for Free FermionsHydrodynamics for Free Fermions• Exact Hydrodynamics ( )

• Same Hamiltonian as from non-linear Bosonization:

Fabio Franchini


PhasePhase--Space pictureSpace pictureFabio Franchini


Ground State

BA solution

CurrentState BAsolution

Perturbed(Hydrodynamic)

Current State

solution

Evolutionof thesolutionbeyondhydrodynamicdescription

Hydrodynamics for integrable modelsHydrodynamics for integrable models

• Bethe Ansatz solution: wave function as a superposition of

two-particles scattering

• Essentially Free Fermions with corrected density

• Free Fermions: K(k-p,c)=0 →→

Fabio Franchini


Part 2: Part 2: ConclusionsConclusions• Hydrodynamics correctly reproduce the leading behavior of EFP

• We showed how to construct the hydrodynamic description of

Integrable Models

Fabio Franchini


Future developmentsFuture developments• Construct hydrodynamic description of spin models

(XY and XXZ Model)

• Calculate EFP for these models

Part 3:Part 3: SpinSpin--Charge SeparationCharge Separation• Luttinger Liquid Spin-Charge separation comes from linear

approximation of Bosonization (Haldane, 1979 & 1981, …):

• Perturbative calculations with spectrum curvature diverge

• Hydrodynamic approach takes into account the whole spectrum

Fabio Franchini


Fermions with contact repulsionFermions with contact repulsion

• Complicated (Nested) Bethe Ansatz:

• The hydrodynamics variables are:

Fabio Franchini


SpinSpin--Charge HydrodynamicsCharge Hydrodynamics• Hydrodynamic Hamiltonian from the Bethe Ansatz:

by inverting the relations

• The commutation relations complete the (implicit) hydrodynamic

description:

Fabio Franchini


Part 3:Part 3: ConclusionsConclusions• We constructed the Hydrodynamic description of

Fermions with contact interaction in implicit form

• We can calculate the spin current carried by a charge

disturbance →→ spin-charge coupling

Fabio Franchini


Future directionsFuture directions• Expand Hamiltonian to quadratic terms (bosonization) and beyond

• Find a close expression for the hydrodynamic Hamiltonian

• Apply this technique to spin Calogero-Sutherland Model

AcknowledgmentsAcknowledgments• A.G. Abanov, J. Verbaarschot, D. Schneble & R. Konik

• Amber Carr

• S. Ratkovic & M. Lerotic

• V.E. Korepin & A.S. Goldhaber

• R. Requist, P. Allen and the rest of the group

• A. Tsvelik, F. Essler & B. McCoy

• The department Staff, Faculty and my fellow Graduate Students

• My parents: Anna & Paolo

Fabio Franchini


PublicationsPublications• A.G. Abanov and F. Franchini; Phys. Lett. A 316 (2003) 342-349,

“Emptiness Formation Probability for the Anisotropic XY Spin Chain in a Magnetic Field”(also available on arXiv:cond-mat/0307001).

• F. Franchini and A.G. Abanov; J. Phys. A 38 (2005) 5069-5096,“Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain”(also available on arXiv:cond-mat/0502015).

• F. Franchini, A. R. Its, B.-Q. Jin and V. E. Korepin; to appear in “Proceedings of the26th International Colloquium on Group Theoretical Methods in Physics”,“Analysis of entropy of XY Spin Chain(also available on arXiv:quant-ph/0606240)

• F. Franchini and A.G. Abanov; In progress,“Coupling of Spin and Charge Degrees of Freedom in a Hydrodynamic Two-Fluid Approach”

• F. Franchini and A.S. Goldhaber; In preparation,“Aharonov-Bohm effect with many vortices'”

Fabio Franchini


Thank You!

In the Thesis, but not in this presentationIn the Thesis, but not in this presentation

• Aharonov-Bohm effect in 2-D medium:

Discrete spectrum;

Exponential decay for a zero-energy particle;

Topological trapping.

• Integrability of gradient-less hydrodynamic theories:

Construction of the conserved currents.

Fabio Franchini


Region Critical γ, h P(n)Zeros of

σ(q)Phase Jumpsof σ(q)

Ω0(known)

Yes γ=0, -1<h<1

E n-1/4 e-αn2 q∉(-kf,kf)

none

Ω− Yes h=-1 E n-1/16 [1+An-1/2 ] e-βn none π

Ω+ Yes h=1 E n-1/16 [1+(-1)n An-1/2 ] e-βn π 0, π

Σ− No h<-1 E e-βn none none

Σ0 No -1<h<1 E e-βn π π

Σ+ No h>1 E [1+(-1)n A] e-βn 0, π 0, πEFP for the Anisotropic XY ModelEFP for the Anisotropic XY Model

Fabio Franchini


Determinant RepresentationDeterminant Representation• From Bethe Ansatz, correlators as complicated

Fredholm integral operators (and minors of them)

• EFP is the simplest correlator, being expressed as the

determinant of such an operator (Korepin et al.):

• With this formula it is possible to find the asymptotic behavior for large n, but only in very special cases

Fabio Franchini


Multiple Integral RepresentationMultiple Integral Representation• For the critical XXZ spin-½ Heisenberg chain (Δ=cosζ)

(Kitanine et al. 2002):

( )

( )1 n

n nnn

... i2 j ka b

a b

Z { },{ }1 iP(n) lim det dn! 2 sinhsinh( )

∞

ζξ ξ → − −∞

<

⎛ ⎞⎜ ⎟λ ξ⎜ ⎟= λ

π⎜ ⎟ζ λ − ξξ − ξ ⎜ ⎟ζ⎝ ⎠

∫∏

( ) ( ) ( )( )

( ) ( )nn n j k j ka b a b

n na 1 b 1 a b

a ba b

isindetsinh sinh isinh sinh i

Z { },{ }sinh i sinh( )= =

<

⎛ ⎞− ζ⎜ ⎟⎜ ⎟λ −ξ λ −ξ − ζλ −ξ λ −ξ − ζ ⎝ ⎠λ ξ = ⋅

λ −λ − ζ ξ −ξ∏∏

∏

Fabio Franchini


Hydrodynamics approachHydrodynamics approach• In general the Bethe Ansatz gives:

• Using Galilean invariance we make a boost:

Fabio Franchini


NonNon--linear Bosonizationlinear Bosonization

• We are interested in bilinears like:

• This is the generator for the currents:

Fabio Franchini


NonNon--linear Bosonization (cont.)linear Bosonization (cont.)• The currents are:

– Density:

– Current Density:

– Hamiltonian:

Fabio Franchini


Linear vs. NonLinear vs. Non--linear Bosonizationlinear Bosonization

• One linearizes the spectrum around the Fermi Points:

• And after Bosonization:

• While from the non-linear procedure we got:

Fabio Franchini


SpinSpin--Charge Separation?Charge Separation?• One defines Spin and Charge degrees of freedom:

• And the linear theory gives

• While from the non-linear one gives:

Fabio Franchini


EFP in the Non Critical RegionsEFP in the Non Critical Regions

( ) dq(h, ) Log (q)2

π

−πβ γ = − σ

π∫

(h, )n,0P(n) E (h, ) e−β γ

− γ∼,0 :−Σ

:+Σ n (h, )nP(n) E (h, ) 1 ( 1) A(h, ) e−β γ+ ⎡ ⎤γ + − γ⎣ ⎦∼

100 200 300 400 500

0.55

0.6

0.65

0.7

0.75

0.8

P(n)n1/16eβn

n

Numerical vs. Analytical results at γ=1, h=1.5(Oscillatory behavior →

Z2 symmetry breakdown at h=1)(defined for γ≠0)

Fabio Franchini


Critical Phase: Critical Phase: ΩΩ00

• Studied by Shiroshi et al. (2001) and in the ‘70s in the

context of Unitary Random Matrices

• For γ=0, σ(q) has only limited support

• Widom’s Theorem →

the behavior is Gaussian with a power law pre-factor:

2

21

c (h) n40

n / 215 13 ( 1) 824 4 1 hP(n) E2 e (1 h) n

2(h) n e

− −′ζ − −α− +⎛ ⎞− =⎜ ⎟⎝ ⎠

∼

Fabio Franchini


EFP in the Critical Phases EFP in the Critical Phases ΩΩ±

• Generalized Fisher-Hartwig conjecture:

• Stretching the conjecture beyond its limits,

we add a subleading term:

1 1c n C (1, )n16 2P(n) E ( )n 1 ( 1) A ( )n e

− − −β γ+ +

⎡ ⎤γ + − γ⎢ ⎥

⎣ ⎦∼ Numerical vs. Analytical

results at γ=1, h=1

100 200 300 400 500

0.76

0.78

0.8

0.82

0.84

0.86

P(n)n1/16eβn

n

1(h, )n16P(n) E( )n e

− −β γγ∼

1 1c c ( 1, )n16 2P(n) E ( )n 1 A ( )n e

− − −β − γ− −

⎡ ⎤γ + γ⎢ ⎥

⎣ ⎦∼

:−Ω

:+Ω

Fabio Franchini


The FisherThe Fisher--HartwigHartwig ConjectureConjecture• We parametrize the generating function as:

where τ(q) is a smooth, non-zero function with winding number 0

• The asymptotic behavior of the determinant is:

( ) ( ) rr r

R(q )mod2

rr 1

(q) (q) e 2 2cos(q ) λ− κ π− −θ π

=

σ = τ − − θ∏

2 2r r( ) [ ]n

n ndet(S ) E[ , , ] n eΣ λ −κ −β τ

→∞τ κ λ∼

( ) dq[ ] Log (q)2

π

−πβ τ = − τ

π∫

Fabio Franchini


The generalized FH ConjectureThe generalized FH Conjecture• When more than one parametrization exists:

the asymptotic behavior of the determinant is expressedas a sum of terms:

( ) ( )aarrr

Ri (q )mod2a

rr 1

(q) (q) e 2 2cos(q ) λ− κ π− −θ π

=

σ = τ − − θ∏

aa a a [ ]nn n a T

det(S ) E[ , , ] n e ;Ω −β τ

→∞∈

τ κ λ∑∼ ( )a a dq[ ] Log (q)2

π

−πβ τ = − τ

π∫

( ) ( ){ }a 2 a 2 j 2 j 2r r r rj

T a; ( ) ( ) max ( ) ( )= Σ λ − κ = Σ λ − κ = Ω

Fabio Franchini


Non Critical Regions: Non Critical Regions: ΣΣ−−, Σ, Σ00 andand ΣΣ++• The generating function has different structures:

−3 −2 −1 1 2 3q

0.2

0.4

0.6

0.8

1AbsHσL

−3 −2 −1 1 2 3q

−1.5

−1

−0.5

0.5

1

1.5

ArgHσL

Abs(σ) and Arg(σ) for γ=1, h=-1.5; γ=1, h=0.5; γ=1, h=1.5

−3 −2 −1 1 2 3q

0.050.10.150.20.250.30.35

AbsHσL

−3 −2 −1 1 2 3q

−1.5

−1

−0.5

0.5

1

1.5

ArgHσL

�3 �2 �1 1 2 3q

0.94

0.95

0.96

0.97

0.98

0.99

Abs�Σ�

�3 �2 �1 1 2 3q

�0.3

�0.2

�0.1

0.1

0.2

0.3

Arg�Σ�

−Σ 0Σ +Σ

Fabio Franchini


Critical Phases: Critical Phases: ΩΩ00 , , ΩΩ−− , , ΩΩ++

Abs(σ) and Arg(σ) for γ=1, h=1

• The generating function presents the following behavior:

−3 −2 −1 1 2 3q

0.1

0.2

0.3

0.4

0.5

0.6

0.7

AbsHσL

−3 −2 −1 1 2 3q

−1.5

−1

−0.5

0.5

1

1.5

ArgHσL

Abs(σ) and Arg(σ) for γ=1, h=-1

−3 −2 −1 1 2 3q

0.7

0.75

0.8

0.85

0.9

0.95

AbsHσL

−3 −2 −1 1 2 3q

−0.75

−0.5

−0.25

0.25

0.5

0.75

ArgHσL

−Ω +Ω

Fabio Franchini

�3 �2 �1 1 2 3q

0.2

0.4

0.6

0.8

1Abs�Σ�

�3 �2 �1 1 2q

�1

�0.5

0.5

1Arg�Σ�

Abs(σ) and Arg(σ) for γ=0, h=0.5

0Ω


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