On singularities of dynamic response functions in the massless - - PowerPoint PPT Presentation

SLIDE 1

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion

On singularities of dynamic response functions in the massless regime of the XXZ chain

• K. K. Kozlowski

CNRS, Laboratoire de Physique, ENS de Lyon.

20th of July 2017

• K. K. Kozlowski "Form factors of bound states in the XXZ chain." J. Phys. A: Math. & Theor., 50, 184002, (2017).
• K. K. Kozlowski "On the thermodynamic limit of form factor expansions of dynamical correlation functions in the

massless regime of the XXZ spin-1/2 chain." math.-ph. 1706.09459

• K. K. Kozlowski " On singularities of dynamic response functions in the massless regime of the XXZ chain."

to appear.

Integrability in Gauge and String Theory 2017 ENS, Paris

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 2

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion

Outline

1

Introduction The dynamic response functions in the XXZ chain Universality and response functions

2

The edge singular behaviour of dynamic response functions Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations

3

The massless form factor expansion The main ingredients The overall strategy

4

Conclusion

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 3

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The dynamic response functions in the XXZ chain Universality and response functions

Dynamic response functions in the XXZ chain

⊛ The XXZ spin-1/2 chain on hXXZ = ⊗L

n=1C2, σα Pauli matrices

H =

L

∑

n=1

{ σx

nσx n+1 + σy nσy n+1 + ∆σz nσz n+1 − hσz n

} , σn+L ≡ σn ⊛ Massless regime −1 < ∆ < 1, 0 < h < hc ⇝ Ground state Ω. ⊛ Space-time evolution of operators σγ

m+1(t) = eimP+iHt · σγ 1 · e−itH−imP

⊛ Connected dynamical two-point function at zero temperature ⟨( σγ

1(t)

)†σγ

m+1

⟩

c =

lim

L→+∞

{( Ω, ( σγ

1(t)

)†σγ

m+1Ω

) −

• (

Ω, σγ

1Ω

)

• 2}

♦ Experiments ⇝ Dynamic Response Functions S (γ)(k, ω) = ∑

m∈Z

∫

R

⟨( σγ

1(t)

)†σγ

m+1

⟩

c · ei(ωt−km)

dt (2π)2

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 4

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The dynamic response functions in the XXZ chain Universality and response functions

Expermiental measurements on KCuF3

⊛ Bragg spectroscopy. ⊛ KCuF3 ⇝ XXX chain at h = 0. Observations for S (z)(k, ω) Dominant intensity delimited by viking helmet curves Large intensity concentrated on lower curves Decrease of intensity when approaching top curve

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 5

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The dynamic response functions in the XXZ chain Universality and response functions

Universal features of response functions

⊛ ’67 (Mahan), ’67 (Noziére, De Dominicus) Arguments for a power-law behaviour near edges. ⊛ ’80’s Luttinger liquid, CFT methods ⇝ predictions of a local behaviour at (k, ω) = (0, 0). ⊛ ’08 (Glazman, Kamenev, Khodas, Pustilnik) X-Ray edge singularities S (z)(k, ω) ≃ A (k) · Ξ (ω − εh(k)) · [ω − εh(k)]ϑ ⊛ ’08 (Glazman, Imambekov) Non-linear Luttinger liquid A (k) = (2π)2 Γ−1(µR + µL) [v(k) + vF]µR [vF − v(k)]µL ·

• F (z)(k)
• 2

, ϑ = µR + µL − 1 ⊛ ’08 (Glazman, Imambekov) ’08 (Cheianov, Pustilnik) ’09 (Affleck, Pereira, White) Bethe Ansatz spectrum ⇝ closed expressions for µR,L. ⊛ ’10 (Caux, Glazman, Imambekov, Shashi)

• F (z)(k)
• 2 ⇝ volume-renormalised form factor;
• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 6

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The dynamic response functions in the XXZ chain Universality and response functions

Exact results

⊛ ’79 (Beck,Bonner,Müller) Closed formula for S (z)(k, ω) for XX (∆ = 0) ∀h. ⊛ ’95 (Jimbo,Miwa) Closed formula for S (γ)(k, ω) for XXX, h=0. ⊛ ’98-’00 (Bougourzi, Couture,Kacir, Fledderjohann,Karbach,Müller,Mütter,.... ) Analysis of 2 and 4 spinon contributions ⊛ ’00-’07 (Biegel,Karback,Müller,Sato,Shiroishi,Takahashi,Caux, Hagemans,Maillet,... ) Numerics & Bethe Ansatz ⊛ ’12 (Caux,Konno,Sorrel,Weston) 2 spinon for XXZ and h = 0 ⊛ ’12 (Kitanine,K.,Maillet,Slavnov,Terras) Formal saddle-point approach to form factor series in NLSM.

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 7

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The dynamic response functions in the XXZ chain Universality and response functions

The open problems

Questions Can one carry out an ab inicio, exact, calculation of the DRF? Can one access to the predicted universal features for an integrable model? Is the non-linear Luttinger picture complete? Can one bring the description of this universality to a singularity theory?

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 8

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations

The two-particle-hole excitation thresholds

⊛ Form factor series representation for DRF S (γ)(k, ω) = ∑

n∈S

S (γ)

n

(k, ω)

Pmin π - pF π pF + π Pmax 2 π k ω 2 pF 2 (π - pF ) Ch 1 Ch 2 Cp 1 Cp 2 Cp 3 Cp 4 Cp-2 h Cp-h 1 Cp-h 2

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 9

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations

The vicinity of a hole threshold

(k, ω) configuration close to the hole excitation line (P0, E0) = (pF − t0 , −e(t0)) with t0 ∈ ] −pF ; pF [ . ⊛ The hole threshold

S (z)

h

(

P0, E0 + δω

)

= Ξ (δω) · [δω]∆(h) [v(t0) + vF]δ(h)

+ [vF − v(t0)]δ(h) −

·

• F (z)

h

(t0)

• 2

Γ(δ(h)

+ + δ(h) − )

( 1 + O ( [δω]1−0+ )) + S (zz)

h;reg(δω) .

⊛ v(t0) = e′(t0): velocity of the hole at t0 vF: velocity excitations on Fremi boundary. ⊛ Edge exponent: ∆(h) = δ(h)

+

+ δ(h)

−

− 1 ⊛ δ(h)

± : microscopic shift of right(+)/left(-) Fermi boundary due to excitation.

• F (z)

h

(t0)

• 2 =

lim

L→+∞

       ( L 2π )δ(h)

+ +δ(h) − +1

• (

Ω, σz

1Υt0

)

• 2
• Ω
• 2 ·
• Υt0
• 2

      

rrrrrrrrrrrrrr rrrrs ❝ ❫

−pF pF t0 ⋆ ground state Ω ⋆ excitation Υt0 { E0 = −e(t0) P0 = pF − t0

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 10

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations

The vicinity of a particle threshold

(k, ω) configuration close to the particle excitation line (P0, E0) = (k0 − pF , e(k0)) with k0 ∈ ] pF ; Km [ . ⊛ vF < v(k) on ] pF ; Km [ and v(Km) = vF ⊛ The particle threshold

S (zz) p

(

P0, E0 + δω

)

= [δω]∆(p) Γ(−∆(p)) ·

• F (z)

p (k0)

• 2

[v(k0) + vF ]δ(p) + [vF − v(k0)]δ(p) − × { Ξ(δω) sin[πδ(p) + ] π + Ξ(−δω) sin[πδ(p) − ] π }( 1 + O ( [δω]1−0+ )) + S (zz) p;reg(δω) .

⊛ v(k0) = e′(k0): velocity of the particle at k0 ⊛ Edge exponent: ∆(p) = δ(p)

+

+ δ(p)

−

− 1

• F (z)

h

(t0)

• 2 =

lim

L→+∞

       ( L 2π )δ(p)

+ +δ(p) − +1

• (

Ω, σz

1Υk0

)

• 2
• Ω
• 2 ·
• Υk0
• 2

      

rrrrrrrrrrrrrr s ❝ ❫

−pF k0 pF ⋆ ground state Ω ⋆ excitation Υk0 { E0 = e(k0) P0 = k0 − pF

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 11

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations

Beyond the Luttinger liquid: Multi particle-hole thresholds

⊛ Particle and holes may have equal velocities v(k) = v(t(k)) t : ] Km ; KM [ → ] −pF ; pF [ (k, ω) configuration close to an equal-velocity particle-hole excitations

(P0, E0) = ( npk0 − nht(k0) + (nh − np)pF , npe(k0) − npe(t(k0)) ) with k0 ∈ ] Km ; KM [ .

⊛ The particle threshold

S (zz)

ph

(

P0, E0 + δω

)

=

• F (z)

p

(k0)

• 2

√ np − nht′(k0) ·

(

−2π v′

1(k0)

) np−1

2

·

(

2π v′

1(t(k0))

) nh

2

[δω]∆(ph) Γ(−∆(p)) [v(k0) + vF]δ(ph)

+

[vF − v(k0)]δ(ph)

−

× {Ξ(δω) π sin [ π ( δ(ph)

+

+δ(ph)

−

)] + Ξ(−δω) π sin [ π

2(np +nh −3)

]} · ( 1+O ( [δω]

1 2 ln |δω|

)) + S (zz)

ph;reg(δω)

⊛ Edge exponent: ∆(ph) = δ(ph)

+

+ δ(ph)

−

+

np+nh−3 2

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 12

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The necessary ingredients

⊛ Form factor expansion ( Ω, ( σγ

1(t)

)†σγ

m+1Ω

) = ∑

Υ

exp { im PΥ\Ω − it EΥ\Ω } ·

• (

Υ, σγ

1Ω

)

• 2

⊛ Building blocks of the formula Excited states Υ; Relative excitation momentum PΥ\Ω and energy EΥ\Ω; Form factors of local operators

• (

Υ, σγ

1Ω

)

• 2;

" OK from the Bethe Ansatz ’58-99 Orbach, Gaudin,McCoy,Wu, Korepin, Slavnov, Kitanine, Maillet,Terras,Tarasov,Varchenko, ⊛ Reasonable assumptions for ⟨( σγ

1(t)

)†σγ

m+1

⟩ = lim

L→+∞

{( Ω, ( σγ

1(t)

)†σγ

m+1Ω

)} Only excited states s.t. lim

L→+∞

{ EΥ\Ω } < +∞ contribute; Only leading L → +∞ of each summand contributes to lim

L→+∞.

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 13

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The finite energy excited states

⊛ Ω ground state ⇝ N real Bethe roots condensing on [ −q ; q ] ⊛ Υ parametrised by R = {{ λ(p)

a

}n(tot)

p

1

∪ { λ(h)

a

}n(tot)

h

1

} ∪ {{ ν(r)

a

}nr

a=1

}

r∈Nst

n(tot)

h

= n(tot)

p

+ ∑

r∈Nst

rnr n(tot)

h

∈ { 0, . . . , L

1+0+ 2

} solution to higher level Bethe Ansatz equation

• ξ1

( λ(p)

a

| RΥ ) =

2π L ma ,

• ξ1

( λ(h)

a

| RΥ ) =

2π L m′ a

and

• ξr

( ν(r)

a

| RΥ ) =

2π L d(r) a

driven by counting functions solution of NLIE.

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 14

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The excitation energy and momentum

• EΥ\Ω = E(RΥ) + O(L−1)

and

• PΥ\Ω = P(RΥ) + πsΥ + O(L−1)

E(R) =

n(tot)

p

∑

a=1

ε1 ( λ(p)

a

) −

n(tot)

h

∑

a=1

ε1(λ(h)

a ) +

∑

r∈Nst nr

∑

a=1

εr ( ν(r)

a

) P(R) =

n(tot)

p

∑

a=1

p1 ( λ(p)

a

) −

n(tot)

h

∑

a=1

p1(λ(h)

a ) +

∑

r∈Nst nr

∑

a=1

pr ( ν(r)

a

) εr(λ) > cr > 0 on R + δriπ/2, for r ∈ Nst; ε1 > 0 on { R + i π

2

} ∪ { R \ [ −q ; q ] } and ε1 < 0 on ] −q ; q [ massless branches ε1(±q) = 0

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 15

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

Splitting into massive and massless branches

⊛ Partition into massive and massless branches { λ(p)

a

}n(tot)

p

1

= Υ(p)

+ ∪

{ ν(1)

a

}n1

1 ∪ Υ(p) −

and { λ(h)

a

}n(tot)

h

1

= Υ(h)

+ ∪

{ µa }nh

1 ∪ Υ(h) −

⊛ massless d ( Υ(p/h)

±

, ±q) ≤ δ vs. massive d ({ ν(1)

a

}n1

1 ∪

{ µa }nh

1 , ±q

) ≥ δ ⊛ Subordinate partition of variables R = Y ∪

υ=±

{ Υ(p)

υ

∪ Υ(h)

υ

} Umklapp deficiencies ℓυ = nυ

p − nυ h

macroscopic variables of the massive excitations Y = {{ µa }nh

1 ;

{{ ν(r)

a

}nr

a=1

}

r∈N ;

{ ℓυ }} with N = Nst ∪ {1} microscopic variables of the massless excitations Υ(p/h)

±

; ⊛ Effective reduction: P(R) − 1 v · E(R) = U (Y, v) + O(δ) v = m t U (Y, v) = ∑

r∈N nr

∑

a=1

{ pr ( ν(r)

a

) − 1 v εr ( ν(r)

a

)} −

nh

∑

a=1

{ pr(µa) − 1 v εr(µa) } + ∑

υ=±

υℓυpF

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 16

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The large volume behaviour of form factors

’16 K. ⊛ ’91 Slavnov , ’09-’10 Kitanine, K., Maillet, Slavnov, Terras particle-hole form factors ⊛ Uniform large-L asymptotics

• (

Υ, σγ

aΩ

)

• 2 =
• F (γ)(

RΥ ) · ( 1 + O (

ln L L

)) ⊛

• F (γ)(R) holomorphic function of R
• F (γ)(R) = :

∏

υ∈{±}

{( 2π

L

)ϑ2

υ(Y)Fυ(Y | Zυ

Y)

} : F (γ)(Y) · ( 1 + O (

ln L L

+ δ ln δ ))

nh

∏

a=1

{

L 2π

ξ′

1(µa | Y)

} · ∏

r∈N nr

∏

a=1

{

L 2π

ξ′

r

( ν(r)

a

| Y )} ⊛ Massive form factor density F (γ)(Y) ⊛ Massless form factor density Fυ(Y | Zυ

Y) with reparametrised massless variables

Zυ

Y =

{{ υ [ L

2π

ξ1(µ | Y) − Nυ ]}

µ∈Υ(p)

υ

; { υ [ Nυ − L

2π

ξ1(µ | Y) ] − 1 }

µ∈Υ(h)

υ

} ⇝ Fυ is responsible for IR renormalisation.

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 17

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

Towards the massless form factor expansion I

⊛ Assumptions on relevant contributions ⟨( σγ

1(t)

)†σγ

m+1(t)

⟩ = lim

L→+∞

{

• C (γ)

1

(m, t) }

• C (γ)

1

(m, t) = (−1)msγ ∑

n∈Stot

∑

RΥ

• F (γ)(

RΥ ) · eimP(RΥ)−itE(RΥ) ⊛ Constraint n(tot)

h

= n(tot)

p

+ ∑

r∈Nst

rnr ⊛ Recast as using multidimensional residues

• C (γ)

1 (m, t) = (−1)msγ ∑ n∈Stot

∏

r∈Nst

      

• Γ(r)

tot

Dnr

r ν(r)

      ·

• Γ(p)

tot ∗Γ(h) tot

D

n(tot) p 1 λ(p) ·D n(tot) h 1 λ(h) ·

H(R)· F (γ)(R)·eimP(RΥ)−itE(RΥ)

⊛ Γ(p)

tot ∗ Γ(h) tot , Γ(r) tot sub-manifolds "surrounding" all solutions to HLBAE

⊛ measures Dnr

r ν = n

∏

a=1

{ L ξ′

r(νa | R)

eiL

ξr (νa|R) − 1

} · dnr ν nr!

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 18

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Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The strategy of analysis

♦ Deform the sub-manifolds into regions where measures trivialise when L → +∞; Γ(p)

tot ∗ Γ(h) tot , Γ(r) tot

֒→ ( Γ(p)

tot ∗ Γ(h) tot

)

+, (Γ(r) + )nr

♦ Partitions ( Γ(p)

tot ∗ Γ(h) tot

)

+ into "massive" & "massless" parts

♦ IR renormalisation of the "massless" parts ♦ Take term-wise L → +∞ limit

⟨( σγ

1(t)

)†σγ

m+1

⟩ = (−1)msγ ∑

n∈S

∏

r∈N

       ∫

Cnr r,δ

dnr ν(r) nr!        · ∫

Cnh h,δ

dnh µ nh! · F (γ)( Y ) · eimU (Y,v) ∏

υ=±

[−imυ]ϑ2

υ(Y)

( 1 + r(Y, mυ, δ) ) .

⊛ Left right moving combinations mυ = υm − vFt ⊛ remainder r(Y, mυ, δ) = O ( δ ln δ + ∑

υ=±

{ δ2|mυ| + δ| ln |mυ|| + e−|mυ|δ}) ⊛ δ = C (|m+| + |m−|)1− τ

2

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

Dynamic response functions

⊛ Change variables to the momentum representation k (r)

a

= pr(ν(r)

a ), ta = p1(µa)

K = ( t, k (r1), · · · , k (r|N|), ℓ+, ℓ− ) ⊛ Take Fourier transform (preservation of the remainder) S (γ)(k, ω) = ∑

n∈S

S (γ)

n

(k, ω)

S (γ)

n

(k, ω) = ∫

J (ϵ) h

dnh t · ∏

r∈N

{ ∫

J (ϵ) r

dnr k (r) } · F (γ)(K) ∑

s∈Z

∏

υ=±

{ Ξ (

• yυ(K; s)

) · [

• yυ(K; s)

]∆υ(K)} · ( 1 + rϵ(K; s) )

⊛ ϵ-regularised real intervals J (ϵ)

h

= [−pF + ϵ; pF − ϵ] , J (ϵ)

1

= [pF + ϵ; p(1)

+ − ϵ]

, J (ϵ)

r

= [p(r)

− ; p(r) + ]

⊛ edge singularity functions yυ(K; s) = ω − E(K) + υvF[k − P(K) + 2πs] ⊛ Momentum and energy E(K) = ∑

r∈N nr

∑

a=1

er(k (r)

a ) − nh

∑

a=1

e1(µa) , P(K) = ∑

r∈N nr

∑

a=1

k (r)

a

+ pF ∑

υ=±

υℓυ + πsγ −

nh

∑

a=1

ta ⊛ Remainder rϵ(K; s) = O ( ∑

υ=±[

yυ(K; s)]1−τ )

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion The main ingredients The overall strategy

The Edge singular behaviour

⊛ Singularities ⇝ points (k, ω) where K → (

• y+(K; s),

y− ( K; s )) is not a manifold at (0,0). ⇝ one dimensional case ⇝ Equal velocity case ( ≡ local extremum of E(K) at P(K) = P0 fixed) ⊛ The strategy Split the integral into a vicinity of singular behaviour and regular Carry out local analysis close to the singularity

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ

SLIDE 21

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion

Conclusion and perspectives

Review of the results

" Explicit construction of a massless form factor expansion; " Confirms predictions of non-linear Luttinger-liquid structure (hole, particle, strings); " New effects beyond the non-linear Luttinger-liquid; " long-time & large-distance asymptotics of dynamical two-point functions; " Phenomenological form of correlators for the Luttinger liquid universality class; " universality ≡ singularity structure of form factors & classical saddle-point calculation;

Further developments

⊛ Models for c 1. ⊛ Develop a saddle-point/singularity based classification of universality classes.

• K. K. Kozlowski

On singularities of dynamic response functions in the massless regime of the XXZ