Condensation properties of Bethe roots in the XXZ chain K. K. - - PowerPoint PPT Presentation

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The setting of the problem The main result Conclusion

Condensation properties of Bethe roots in the XXZ chain

• K. K. Kozlowski

CNRS, Laboratoire de Physique, ENS de Lyon.

25th of August 2016

• K. K. Kozlowski "On condensation properties of Bethe roots associated with the XXZ chain."

Math-ph:1508.05741

Recent Advances in Quantum Integrable Systems 2016, Genève

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion

Outline

1

The setting of the problem The particle-hole roots The condensation property

2

The main result Main steps of the proof for the ground state

3

Conclusion

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion The particle-hole roots The condensation property

The XXZ spin-1/2 chain

⊛ The XXZ spin-1/2 chain on hXXZ = ⊗L

a=1C2

HXXZ = J

L

∑

n=1

{ σx

nσx n+1 + σy nσy n+1 + cos(ζ)

( σz

nσz n+1 − id

)} , σn+L ≡ σn σα Pauli matrices, σα

n = id ⊗ · · · ⊗ id ⊗ σα ⊗ id ⊗ · · · ⊗ id.

L: length of circle, cos(ζ) anisotropy parameter. [HXXZ, Sz] = 0 with Sz = ∑L

a=1 σz a .

hXXZ = ⊕L

N=0h(N) XXZ

with h(N)

XXZ =

{ v ∈ hXXZ : Sz · v = (L − 2N) · v } , XXX (’31 Bethe), XXZ (’58 Orbach) quantum integrable by Bethe Ansatz Eigenvectors in h(N)

XXZ :

⇝ v(λ1, . . . , λN) ( sinh(iζ/2 − λa) sinh(iζ/2 + λa) )L ·

N

∏

b=1

       sinh(λa − λb + iζ) sinh(λb − λa + iζ)        = (−1)N+1 , a = 1, . . . , N . Eigenvalues HXXZ · v({λa}N

1 ) = E({λa}N 1 )v({λa}N 1 )

E({λa}N

1 ) = N

∑

a=1

e(λa) with e(λ) = −2J sin2(ζ) sinh(λ − iζ/2) sinh(λ + iζ/2)

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion The particle-hole roots The condensation property

The particle-hole excited states and the ground state

⊛ Distinguish solutions by taking logarithm ℓa ∈ Z, λa ∈ R, ϑ(λ | 1

2ζ) − 1

L

N

∑

a=1

ϑ(λa − λb | ζ) + N + 1 2L = ℓa L and ϑ(λ | η) = i 2π ln ( sinh(iη + λ) sinh(iη − λ) ) ⊛ ( ’38 Húlten) Ground state in h(N)

XXZ

ℓa = a and λa ∈ R ⊛ (’64 Griffiths , ’66 Yang,Yang ) Existence for all cos(ζ), uniqueness when −1 < cos(ζ) ≤ 0. ⊛ Real-valued particle-hole excitation, λa ∈ R ℓa = a for a ∈ [ [ 1 ; N ] ] \ {h1, . . . , hn} and ℓha = pa for a = 1, . . . n ⊛ (’64 Griffiths , ’83 Gaudin ) Existence for cos(ζ) ≥ 1, for some subsets of ℓa’s.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion The particle-hole roots The condensation property

Existence of particle-hole solutions

Proposition (’15 K ) The Log BAE with ℓa admit a real valued solution {λa}N

1 if for any J ⊂ [

[ 1 ; N ] ]: −r|J| < 1 L ∑

a∈J

( ℓa − N + 1 2 ) < r|J| with rm = m ( π − ζ 2π − N(π − 2ζ) 2πL ) + m2 π − 2ζ 2πL If −1 < cos(ζ) < 0, the condition is necessary and the solution is unique. ⊛ Particle-hole solutions exist for any h1 < · · · < hn and p1 < · · · < pn such that π − ζ π ( 1 2 − N − 1 L ) > pn − N L , p1 − 1 L > − π − ζ π ( 1 2 − N − 1 L ) and π − ζ π ( 1 2 − N L ) ≥ n L ♦ Existence follows by showing that the Yang-Yang action blows up at infinity. ♦ Necessariess and uniqueness follow from strict convexity. Lemma (’15 K ) If 0 ≤ N/L ≤ 1/2 − ϵ, the ground state roots {λa}N

1 are bounded |λa| ≤ Λ.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion The particle-hole roots The condensation property

The thermodynamic limit for the ground state

⊛ Thermodynamic limit of observables in fixed magnetisation sector N/L → D ∈ [ 0 ; 1/2 ]: ♦ Ground state per site energy 1

L N

∑

a=1

e(λa) ⊛ One assumes that the Bethe roots condense on [ −q ; q ] with some density ρ(∗ | q): λa+1 − λa ≃ 1 Lρ(λa | q) 1 L

N

∑

a=1

e(λa) ≃

q

∫

−q

e(s)ρ(s | q) · ds + · · · ⊛ Easy to characterise (ρ(µ | q), q) if one assumes that the roots densify. ( ’38 Húlten , ’64 Griffiths , ’66 Yang,Yang ) ρ(λ | Q) +

Q

∫

−Q

ϑ′(λ − µ | ζ)ρ(µ | Q)dµ = ϑ′(λ | ζ

2)

and D =

q

∫

−q

ρ(λ | q)dλ ⊛ (ρ(µ | q), q) is the unique solution (’66 Yang,Yang ). ♦ Densification used in thermodynamic limit of correlation functions, 1/L corrections to GS and low-lying excitations, ...

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

The main result

⊛ (’09 Dorlas, Samsonov) proof of condensation of ground state roots for −1 < cos(ζ) ≤ 0. ♦ Use of convex analysis on spaces of probability measures. ⊛ Theorem (’15 K ) Let {λa} be any n particle-hole solution, n ≤ C, D ∈ [ 0 ; 1/2 ]. For any bounded-Lipschitz f it holds 1 L

N

∑

a=1

f(λa) −→

N,L→∞ N/L→D q

∫

−q

f(s)ρ(s | q) · ds There exists L0, such that for any such choice of ℓa, the Log BAE solution is unique when L ≥ L0.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

The counting function

⊛ Counting function for ground state roots {λa}N

1

• ξ(ω) = ϑ(ω | ζ

2 ) − 1

L

N

∑

a=1

ϑ(ω − λa | ζ) + N + 1 2L so that

• ξ(λa) = a

L ♦ Characterise ξ by a non-linear integral equation ⇝ AE for ξ ⇝ control on roots (’85 De Vega, Woynarovich , ’90 Batchelor, Klümper , ’91 Batchelor, Klümper, Pearce , ’91 Destri, De Vega ) ⊛ Main working assumption roots are bounded in L: −Λ ≤ λa ≤ Λ

• ξ′ > c on [ −2Λ ; 2Λ ];
• r
• ξ′ > c on [ −2Λ ; 2Λ ];

a priori control on growth of roots with L and local behaviour of ξ′ at these roots. The form taken by the NLIE depends on these assumptions.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

The convergence to first order

N/L → D < 1/2;

• ξ is a sequence in L of holomorphic functions on

{ |ℜ(z)| ≤ 2Λ |ℑ(z)| ≤ ζ/4 } ;

• ξ(ω)
• ≤ B for ω ∈

{ |ℜ(z)| ≤ 2Λ |ℑ(z)| ≤ ζ/4 } ; Montel theorem: ξe → ξe holomorphic on { |ℜ(z)| ≤ 2Λ |ℑ(z)| ≤ ζ/4 } ; Show that ξe = ξ0 for any extracted sequence, ξ0(ω) =

ω

∫ ρ(s | q) · ds + D

2 ;

ξ′

e changes sign on [ −Λ ; Λ ] a finite number of times ⇝ NLIE and AE from NLIE;

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

ξ′

e > 0 on [ −Λ ; Λ ]: the contour

b b

1 2L N + 1/2 L 1 2L + iα N + 1/2 L + iα 1 2L − iα N + 1/2 L − iα

• Γ+
• Γ−
• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

ξ′

e > 0 on [ −Λ ; Λ ]

• ξe(λ) = ϑ

( λ | 1

2 ζ

) + N + 1 2L −

• C

ϑ(λ − µ | ζ) ξ′

e(µ)

e2iπL

ξe(µ) − 1

· dµ with

• C =

ξ−1

e

( Γ+ ∪ Γ−) ♦ qR = ξ−1

e

( 1

2L (N + 1)

) and

• qL =

ξ−1

e

( 1

2L

) ;

• ξsym =

ξe − (N + 1)/(2L)

• ξsym(λ) +
• qR

∫

• qL

θ′(λ − µ) ξsym(µ) · dµ = ϑ ( λ | 1

2ζ

) − N 2L [ θ(λ − qR) + θ(λ − qL) ] − ∑

ϵ=±

• Γϵ

θ′(λ − ξ−1

e (s))

• ξ′

e

( ξ−1

e (s)

) ln [ 1 − e2iπϵLs] · ds 2iπL ♦ L → +∞ ξsym → ξsym = ξe − D/2 and || ξ−1

e

− ξ−1

e ||L∞(ξ(O)) ≤ C · ||

ξe − ξe||L∞(O′).            ξsym(λ) +

qR

∫

qL

θ′(λ − µ)ξsym(µ) · dµ = ϑ ( λ | 1

2ζ

) − D 2 [ θ(λ − qR) + θ(λ − qL) ] ξsym(qR) = −ξsym(qL) = D/2 ⊛ The problem has a unique solution ξe = ξ0 and qR = −qL = q.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

ξ′

e changes sign on [ −Λ ; Λ ]: the definitions ξ′

e(z(k)) = 0, κ(k) = sgn

( ξ′

|]z(k);z(k+1)[

) ; J(±) = ∪

k : κ(k)=±1 ]z (k); z (k+1)[;

e2iπL

ξe( q(k)

R/L ) = −1,

• q (k)

R

− z(k+1) maximal and −z(k) + q (k)

L

minimal; X = { x ∈ [−Λ; Λ] : e2iπ

ξe(x) =; 1

} , X(in) = X ∩ {

r

∪

k=0

[ q(k)

L ;

q(k)

R ]

} , X(out) = X \ X(in); Y = {λa}N

1 , Y(in) = Y ∩ X(in), Y(out) = Y \ Y(in);

b b b b b b

−Λ z(0) z(1)

• q(k−1)

R

z(k) δ 2δ

• q(k)

L

κ(k) ξ′

e > 0

J(κ(k))

• q(k)

R

z(k+1) z(r) z(r+1) Λ

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion Main steps of the proof for the ground state

ξ′

e changes sign on [ −Λ ; Λ ]: the analysis

• ξ′

e(λ) +

{ ∫

J(+)

− ∫

J(−)

} θ′(λ − µ) ξ′

e(µ) · dµ = ϑ′(

λ | 1

2ζ

) − ϕout(λ) + ϕin(λ) +

r

∑

k=0

r(k)[ ξe ](λ) ⊛ Driving terms :

• ϕin(λ) = 1

L ∑

x∈X(in)\Y(in)

θ′(λ − x) ≥ 0 and

• ϕout(λ) = 1

L ∑

y∈Y(out)

θ′(λ − y) = O(δ) ⊛ Remainder r(k)[ ξ](λ) = −κ(k){ z(k) ∫

• q (k)

L

+

• q (k)

R

∫

z(k+1)

} θ′(λ − µ) ξ′

e(µ) · dµ +

∑

ϵ=± ϵ

∫

• Γ (k)

ϵ

θ′( λ − ξ−1

e (z))

e−2iπϵLz − 1 · dz = O ( δ + 1 L ) ♦ Invert operators id + θ′

J(+) and then id − [RJ(+)]J(−)

• ξ′

e(λ) =

( id + [RJ(+)]J(−) )[ P′

J(+) −

ψout + ψin +

r

∑

k=0

• r (k)[

ξe] ] (λ) ♦ positivity of ( id + [ RJ(+) ]

J(−)

) [ ψin] and then L → +∞ and δ → 0+ ξ′

e(λ) ≥ P′ J(+)(λ) > 0

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain

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The setting of the problem The main result Conclusion

Conclusion

Review of the results

" Condensation of particle-hole Bethe roots irrespectively of anisotropy; " proof of existence and uniqueness of solutions; " method works also for complex solutions (proof of existence of strings); " closes the proof of numerous results relative to the thermodynamics.

• K. K. Kozlowski

Condensation properties of Bethe roots in the XXZ chain