Form factor approach to the correlation functions of quantum - - PowerPoint PPT Presentation

Form factor approach to the correlation functions of quantum integrable models

• N. Kitanine

IMB, Universit´ e de Bourgogne Quantum Integrable Systems and Geometry Olh˜ ao, September 2012 In collaboration with : K. K. Kozlowski, J.M. Maillet, N. A. Slavnov, V. Terras

– Typeset by FoilT EX – Olh˜ ao, September 2012

• N. Kitanine

Form factor approach

Critical Integrable models

• 1. The XXZ spin-1

2 Heisenberg chain in a magnetic field

Defined on a one-dimensional lattice with M sites, with Hamiltonian, H = H(0) − hSz, H(0) =

M

• m=1
• σx

mσx m+1 + σy mσy m+1 + ∆(σz mσz m+1 − 1)

• ,

Sz = 1 2

M

• m=1

σz

m,

[H(0), Sz] = 0. σx,y,z

m

are the local spin operators (in the spin-1

2 representation) associated with each site

m of the chain and ∆ = cos(ζ), ζ real, is the anisotropy parameter. h - external magnetic field; h > 0. We impose the periodic (or quasi-periodic) boundary conditions

– Typeset by FoilT EX – Olh˜ ao, September 2012 1

• N. Kitanine

Form factor approach

• 2. The non-linear Schr¨
• dinger model
• NLSE ≡ 1D limit of 3D Bose gas.
• Simplest possible interacting massless integrable model.

H =

L

• ∂yΨ†(y)∂yΨ(y) + c Ψ†(y)Ψ†(y)Ψ(y)Ψ(y) − h Ψ†(y)Ψ(y)
• dy,

Ψ(x), Ψ†(x) quantum Bose fields

• Ψ(x), Ψ†(y)
• = δ(x − y),

L: length, c > 0 coupling constant (repulsive regime), h > 0 chemical potential. Both models: solution by algebraic Bethe ansatz, ground state finite Fermi zone

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• N. Kitanine

Form factor approach

Time-dependent correlation functions

• O†(x, t) O(0, 0)
• T =

trH

• O†(x, t) O(0, 0) e−H/kT

trH

• e−H/kT

, T > 0 O†(x, t) O(0, 0) = ψg| O†(x, t) O(0, 0) |ψg, T = 0 where |ψg is the (normalized) ground state. We consider only T = 0 case O(x) - local operators

• For the XXZ spin chain local spin operators σ±

m, σz m ,

• For the Non-linear Schr¨
• dinger Model (NLSM) local fields Ψ(x), Ψ†(x) and local

densities j(x) = Ψ†(x)Ψ(x) O(x, t) ≡ eitHO(x)e−itH

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• N. Kitanine

Form factor approach

Dynamical structure factors

S(k, ω) =

∞

• −∞

dx

∞

• −∞

dt ei(ωt−kx) O†(x, t) O(0, 0) DSF gives the response functions under external perturbations of the system and can be experimentally measured. Numerical computation from the algebraic Bethe ansatz results for the form factors

XXZ - Caux, Hagemans, Maillet (2005) NLSM - Calabrese, Caux (2006) – Typeset by FoilT EX – Olh˜ ao, September 2012 4

• N. Kitanine

Form factor approach

Asymptotic problems

We consider the correlation functions in the thermodynamic limit L → ∞

• Equal time correlation functions t = 0:

Large distance asymptotics x → ∞

• Time-dependent correlation functions:

Long time large distance asymptotics x → ∞, x t = const

• Dynamical structure factors:

Edge exponents (near the excitation dispersion curves ω = εh(k)) S(k, ω) ≃ A(k) H(ω − εh) [ω − εh]θ , θ > 0 Edge exponent near “the hole excitation threshold”.

– Typeset by FoilT EX – Olh˜ ao, September 2012 5

• N. Kitanine

Form factor approach

Predictions and previous results

• Equal time correlation functions
• 1. Luttinger liquid: 1975 Luther, Peschel, 1981 Haldane
• 2. Conformal field theory: 1984 Cardy
• 3. Algebraic Bethe ansatz and Riemann-Hilbert analisys 2009 N.K., Kozlowski,

Maillet, Slavnov, Terras

• Time-dependent correlation functions
• 1. Conformal field theory: Only for x >> t.
• 2. Algebraic Bethe ansatz and Riemann-Hilbert analisys: 2011 Kozlowski, Terras
• Dynamical structure factors, edge exponents:
• 1. Non-linear Luttinger liquid approach 2008 Imambekov, Glazman

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• N. Kitanine

Form factor approach

Form factor approach

Our goal is to study the behavior of correlation functions in critical models using their form factor expansion O†(x, t)O(0, 0) =

• | ψ′

¯ Fψg ψ′(x, t) Fψ′ ψg(0, 0) Main difficulty : form factors scale to zero in the infinite size limit (L → ∞) for critical models. ¯ Fψg ψ′(x, t) Fψ′ ψg(0, 0) = L−θ eixPex−itEex A(ψ′ , ψg) Analyze the form factor series for large (but finite) system size. Hence we need to describe states that will contribute to the leading behavior of the series in the limits x → ∞ and L → ∞ with x << L, and also to compute the corresponding form factors and their behavior in these limits

– Typeset by FoilT EX – Olh˜ ao, September 2012 7

• N. Kitanine

Form factor approach

Algebraic Bethe ansatz and form factors

• 1. Diagonalise the Hamiltonian using ABA 1979 Faddeev, Sklyanin, Takhtajan

→ key point : Yang-Baxter algebra A(λ), B(λ), C(λ), D(λ) → |ψ = B(λ1) . . . B(λN)|0 with {λ} satisfying Bethe equations

• 2. Describe the ground state and excited states

→ Bethe equations.

• 3. Act with local operators on eigenstates

→ solve the quantum inverse problem 1999 N.K., Maillet, Terras O(x) = f(A, B, C, D) → use Yang-Baxter commutation relations

• 4. Compute the resulting scalar products (determinant representation) 1989 Slavnov

→ determinant representation for the form factors in finite volume 1999 N.K., Maillet, Terras

• 5. Analysis of the form factors in the thermodynamic limit.

2010 N.K., Kozlowski, Maillet, Slavnov, Terras

– Typeset by FoilT EX – Olh˜ ao, September 2012 8

• N. Kitanine

Form factor approach

Algebraic Bethe ansatz

L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan (1979): Algebraic Bethe ansatz

• Yang-Baxter equation:

R12(λ1 − λ2) R13(λ1 − λ3) R23(λ2 − λ3) = =R23(λ2 − λ3) R13(λ1 − λ3) R12(λ1 − λ2).

• Monodromy matrix.

Ta(λ) = A(λ) B(λ) C(λ) D(λ)

• [a]

֒ → Yang-Baxter algebra: ◦ generators A, B, C, D

• commutation relations given by the R-matrix of the model

Rab(λ, µ) Ta(λ)Tb(µ) = Tb(µ)Ta(λ) Rab(λ, µ)

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• N. Kitanine

Form factor approach

• Transfer matrix: T (λ) = tra Ta(λ) = A(λ) + D(λ)

Commuting charges: [T (λ), T (µ)] = 0

• Hamiltonian can be reconstructed from the transfer matrix using the trace identities

Conserved quantities: [H, T (λ)] = 0

• Eigenstates. Construction of the space of states by action of B on a reference state

A(λ)|0 = a(λ)|0, D(λ)|0 = d(λ)|0, C(λ)|0 = 0. States |ψ =

k

B(λk)|0 with {λk} solution of the Bethe equations are eigenstates

• f the transfer matrix: T (µ)|ψ = τ(µ, {λ})|ψ,

And hence of the Hamiltonian: H

k

B(λk)|0 =

• N
• j=1

ε0(λj)

• k

B(λk)|0.

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• N. Kitanine

Form factor approach

The particle-hole spectrum

Ground state solution of the Bethe equations can be described in terms of real rapidities λj densely filling (with a density ρ(λ)) the Fermi zone [−q, q] : Lp0(λj) −

N

• k=1

ϑ(λj − λk) = 2π

• j − N + 1

2

• ,

j = 1, . . . , N. Excited states parametrized by numbers {µℓa}N′

1

with N ′ = N + k, involving other choices of integers ℓ1 < · · · < ℓN′ in the rhs : Lp0(µℓj) −

N′

• k=1

ϑ(µℓj − µℓk) = 2π

• ℓj − N ′ + 1

2

• ,

j = 1, . . . , N ′. ℓa = a , a ∈ {1, . . . , N ′} \ {h1, . . . , hn} ℓha = pa , pa ∈ Z \ {1, . . . , N ′}. To every choice of integers {pa} and {ha} there is an associated configuration of rapidities for the particles {µpa} and for the holes {µha}. We don’t consider complex solutions for XXZ (open problem, can contribute for the dynamical correlation functions).

– Typeset by FoilT EX – Olh˜ ao, September 2012 11

• N. Kitanine

Form factor approach

Thermodynamics of the excited states

• ”holes” in continuous distribution of rapidities at µh1, . . . , µhn
• new ”particle” rapidities at µp1, . . . , µpn
• ×

× × × × ×

• ×
• ×

× ×

• Ground State

Exited

✲ ✛ ✲ ✛

1 Lρ F Lρ

⇒ Excited state’s roots µj shifted infinitesimally in respect to the ground state roots λj. µj − λj = 1 Lρ(λj) · F

• λj
• µp1, . . . , µpn

µh1, . . . , µhn

• + O(L−2)

⇒ Additive excitation spectrum. Pex − PG.S. =

n

• a=1

p(µpa) − p(µha) and Eex − EG.S. =

n

• a=1

ε(µpa) − ε(µha)

– Typeset by FoilT EX – Olh˜ ao, September 2012 12

• N. Kitanine

Form factor approach

Dressed quantities

Lieb equation for the density of Bethe roots ρ(λ) − 1 2π

q

• −q

K(λ − µ) ρ(µ) dµ = 1 2πp′

0(λ),

with K(λ) = ϑ′(λ). Similar integral equations can be written for the dressed charge, dressed momentum, dressed energy and dressed phase:     Z(λ) p(λ) ε(λ) φ(λ, ν)     − 1 2π

q

• −q

K(λ − µ)     Z(µ) p(µ) ε(µ) φ(µ, ν)     dµ =     1 p0(λ) ε0(λ) ϑ(λ − ν)     Boundary value of the dressed charge Z = Z(±q) is related to the Luttinger liquid parameter KLl = Z2

– Typeset by FoilT EX – Olh˜ ao, September 2012 13

• N. Kitanine

Form factor approach

Form factors, Bethe ansatz approach

• Quantum inverse scattering problem (N.K., Maillet, Terras): O(x)←

−T (λ) Model dependent, can be done for the XXZ in a very simple form, more tricky for NLSM.

• Scalar products (Slavnov)

0|

N

• j=1

C(µj)

N

• k=1

B(λk) |0 = detU({µj}, {λk}) detV ({µj}, {λk}) for {λk} a solution of Bethe equations and {µj} an arbitrary set of parameters, : Uab = ∂λaτ(µb, {λk}), Vab = 1 sinh(µb − λa), 1 a, b N, where τ(µb, {λk}) is the eigenvalue of the transfer matrix t(µb)

– Typeset by FoilT EX – Olh˜ ao, September 2012 14

• N. Kitanine

Form factor approach

Computation of form factors

Fψ′ ψg(x) = |ψ′|O(x)|ψg|2 ψ′2 · ψg2

• 1. Determinant representation (quantum inverse problem, scalar product)
• 2. Cauchy determinant extraction: 1999 Izergin, N.K., Maillet, Terras

Fψ′ ψg(x) = det

N

1 λa − µb × Smooth part

• 3. Thermodynamic limit L, N → ∞: 2010 N.K., Kozlowski, Maillet, Slavnov, Terras

¯ Fψg ψ′(x, t) · Fψ′ ψg(0, 0) ∼ L−θ eixPex−itEex S D ,

– Typeset by FoilT EX – Olh˜ ao, September 2012 15

• N. Kitanine

Form factor approach

S - smooth part depends continuously on the rapidities µpj and µhj of the particles and

• holes. Model dependent, explicit expression is rather complicated.

D - discreet part depends on the set of integers appearing in the logarithmic Bethe equations, when the rapidities of the particles or holes approach the Fermi surface, a microscopic (of order 1/M) deviation of a particle (or hole) rapidity leads to a macroscopic change in Da. Universal and rather simple. Exponent θℓ can be written in terms of the shift function It is solely the kinematical factor D (together with the values of θ for the various form factors) that drives the asymptotic behavior, while the S part enters only the corresponding amplitude. For the particle-hole contribution we get: O†(x, t) O2(0, 0)ph = lim

L→∞

• {µp},{µh}

L−θ eixPex−itEex S( {µp},{µh} ) D( {µp},{µh}|{p},{h} )

– Typeset by FoilT EX – Olh˜ ao, September 2012 16

• N. Kitanine

Form factor approach

Critical form factors

• Equal time correlation functions: In the large distance limit x → ∞, the oscillatory

character of the form factor sums localizes the particle and hole rapidities, in the absence

• f any other saddle point of the oscillating exponent, around the Fermi boundaries ±q.
• Time-dependent correlation functions,: asymptotic regime x → +∞ and x/t

fixed. Excitations (holes, particles): dressed momentum p, dressed energy ε

• scillating phase xp(λ) − tε(λ) has a unique simple saddle-point λ0:

xp′(λ0) − tε′(λ0) = 0. the particle and hole rapidiies localize, around the saddle point and around the Fermi boundaries ±q

• Dynamical structure factors, edge exponents:

One particle (hole) fixed other excitations close to the Fermi boundaries

– Typeset by FoilT EX – Olh˜ ao, September 2012 17

• N. Kitanine

Form factor approach

Computation of the 2-point functions

We consider excited states containing n±

p particles, resp. n± h holes, with rapidities equal

to ±q such that n+

p − n+ h = n− h − n− p = ℓ,

ℓ ∈ Z. Sum of the critical form factors O†(x) O(0)cr= lim

L→∞ ∞

• ℓ=−∞

L−θℓ exp

• 2ixℓkF

Fℓ

• 2 fℓ(Fℓ,+, w)fℓ(Fℓ,−, w)
• w=exp(2πix

L )

• Fℓ
• 2 - special renormalized form factor of class ℓ (correlation amplitude).

Corresponds to the excited state, with ℓ particles and ℓ holes located on the opposite ends

• f the Fermi zone ±q: Bethe roots {µ1, . . . , µN} defined by the integers ˜

nj = nj ± ℓ. The sum of the discreet parts is hidden in the hypergeometric series fℓ(ν, w).

– Typeset by FoilT EX – Olh˜ ao, September 2012 18

• N. Kitanine

Form factor approach

Discreet parts: fℓ(ν, w) =

∞

• n,k=0

k−n=ℓ

• p1<···<pn

pa∈N∗

• h1<···<hk

ha∈N∗

w

n

• j=1

(pj−1)+ k

• j=1

hj sin πν

π 2n ×

n

• a>b

(pa − pb)2

k

• a>b

(ha − hb)2

n

• a=1

k

• b=1

(pa + hb − 1)2 Γ2 {pa + ν}, {hb − ν} {pa}, {hb}

• .

We use standard hypergeometric notations for ratios of Γ functions: Γ a1 , . . . , ar b1 , . . . , bs

• =

r

• k=1

Γ(ak) ·

s

• k=1

Γ(bk)−1.

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• N. Kitanine

Form factor approach

Hypergeometric identity

Main identity (beauty of integrability) fℓ(ν, w) = wℓ(ℓ−1)/2G2(1 + ℓ + ν) G2(1 + ν) (1 − w)−(ν+ℓ)2 . Here G(x) is the Barnes function G(x + 1) = Γ(x)G(x). In particular f0(ν, w) = (1 − w)−ν2 . Proof: Toeplitz and Hankel determinant approach. First appeared in a completely different context: 2001 Borodin, Olshansky The thermodynamic limit becomes easy to handle leading to the asymptotic results

– Typeset by FoilT EX – Olh˜ ao, September 2012 20

• N. Kitanine

Form factor approach

Equal-time correlation functions. XXZ

• Correlation function σz

1σz m+1

Asymptotic expansion: σz

1σz m+1cr = (2D − 1)2 − 2Z2

π2m2 + 2

∞

• ℓ=1

cos(2mℓkF )

• Fz

ℓ

• 2

1

• 2πm

2ℓ2Z2, where

• Fz

ℓ

• 2

= lim

L→∞ L2ℓ2Z2 |ψg|σz 1|ψℓ|2

ψg|ψgψℓ|ψℓ, and the Bethe roots corresponding to the ℓ-shifted ground state |ψℓ are defined as usual Lp0(µj) −

N

• k=1

ϑ(µj − µk) = 2π

• j + ℓ − N + 1

2

• ,

j = 1, . . . , N.

– Typeset by FoilT EX – Olh˜ ao, September 2012 21

• N. Kitanine

Form factor approach

• Correlation function σ−

1 σ+ m+1

The boundary values of the shift function are F−,ℓ = ℓ(Z − 1) − 1 2Z, F+,ℓ = ℓ(Z − 1) + 1 2Z. Asymptotic expansion σ−

1 σ+ m+1cr =

(−1)m

• 2πm
• 1

2Z2 ∞

• ℓ=−∞

(−1)ℓe2imℓ kF

• F+

ℓ

• 2

1

• 2πm

2ℓ2Z2. The special form factor of the ℓ class and the ℓ-shifted ground state |ψ′

ℓ

• F+

ℓ

• 2

= lim

L→∞ L

• 2ℓ2Z2+

1 2Z2

ψg|σ+

1 |ψ′ ℓ

• 2

ψg|ψgψ′

ℓ|ψ′ ℓ,

Lp0(µj) −

N+1

• k=1

ϑ(µj − µk) = 2π

• j + ℓ + 1 − N

2

• ,

j = 1, . . . , N + 1.

– Typeset by FoilT EX – Olh˜ ao, September 2012 22

• N. Kitanine

Form factor approach

Time-dependent correlation functions. NLSM

Two regimes, to distinguish we introduce a parameter τ:

• Space-like regime |x/t| > vF , the saddle point is outside the Fermi zone |λ0| > q,

τ = 1

• Time-like regime |x/t| < vF , , the saddle point is inside the Fermi zone |λ0| < q,

τ = −1 Correlation function of fields: Ψ†(x, t) Ψ(0, 0) =

∞

• n0=0

∞

• ℓ=−∞

( √ 2πρ(λ0))n0 G(1 + n0) |t ε′′(λ0) − x p′′(λ0)|n2

0/2

× eiπ

2 ϕτ (n0,ℓ) |FΨ τ,ℓ|2 exp

• ix(2ℓ + τn0)kF + iτn0(x p(λ0) − t ε(λ0))
• |2π(x − vF t)|(ℓZ− 1

2Z +τn0 Φ+)2|2π(x + vF t)|(ℓZ+ 1 2Z +τn0 Φ−)2, – Typeset by FoilT EX – Olh˜ ao, September 2012 23

• N. Kitanine

Form factor approach

Correlation function of densities: j(x, t) j(0, 0) = kF π 2 − Z2 2π2 x2 + v2

F t2

(x2 − v2

F t2)2

+

∞

• n0=0

∞

• ℓ=−∞

|ℓ|+n0>0

( √ 2πρ(λ0))n0 G(1 + n0) |t ε′′(λ0) − x p′′(λ0)|n2

0/2

×

• Fj

τn0,ℓ

• 2eix(2ℓ+n0τ)kF +iτn0(x p(λ0)−t ε(λ0))+iπ

2 ϕτ (n0,ℓ)

• 2π|x − vF t|

(ℓZ+τn0Φ+)2 2π|x + vF t| (ℓZ+τn0Φ−)2 . Here Z - Fermi boundary value of the dressed charge, φ(λ, µ) - dressed phase and Φ± = −φ(±q, λ0) + Z ∓ Z−1 2 ,

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• N. Kitanine

Form factor approach

Edge exponents. NLSM

Two structure factors: density structure factor S(k, ω) and spectral function A(k, ω) (Fourier transforms of the density and field correlation functions) Two thresholds: particle (upper) and hole (lower). Edge exponents and amplitudes can be computed for both functions and both thresholds. Example: density structure factor on the hole threshold S(k, ω)hole = H(δω)

• Fj

−1,1

• 2

Γ(α+ + α−) (v − vF )α+ (v + vF )α− δω 2π α++α−−1 , Exponents: α+ = Z 2 + 1 2Z + φ(q, λ) 2 , α− = Z 2 − 1 2Z + φ(−q, λ) 2 .

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• N. Kitanine

Form factor approach

Other models

The necessary conditions to apply this method:

• Finite Fermi zone.
• Cauchy determinant extraction

Form factor = Cauchy determinant × Smooth part Can be checked for a very large class of quantum integrable models. Open problems: Contribution from the bound states, h = 0, explicit results for the amplitudes, 3-point functions... Beyond integrability? Cauchy determinant of rapidities is always there, but can be difficult to compute the shift function.

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