組合せ論的ベーテ仮説(Combinatorial Bethe ansatz)

東京大学 総合文化研究科 (Univ. Tokyo)

国場敦夫 (Atsuo Kuniba)

日本数学会 東北大学 21 Sep. 2007

“Unsere Method liefert also alle Losungen des Problems.”

· · · H. Bethe (1931)

“ . . . composition of our bijection with the Robinson-Schensted-Knuth correspondence may be viewed as a combinatorial version ofthe Bethe ansatz.”

· · · Kerov-Kirillov-Reshetikhin (1986)

1 dimensional Heisenberg chain

H =L∑

k=1

(σxkσx

k+1 + σykσy

k+1 + σzkσz

k+1 − 1)

σxk =

(0 11 0

)

k

, σyk =

(0 −ii 0

)

k

, σzk =

(1 00 −1

)

k

, (σaL+1 = σa

1)

H acts on

(C2)⊗L = ⊗(spin up or down)

Symmetry: [sl2, H] = 0.

In particular, H preserves N := ] of down spins.

Diagonalization of H. (Bethe 1931)

H|u1, . . . , uN〉 = E|u1, . . . , uN〉, E =N∑

j=1

−8

u2j + 1

,

if u1, . . . , uN satisfy Bethe equation

(uj + i

uj − i

)L

=N∏

k=1 (k 6=j)

uj − uk + 2i

uj − uk − 2i(j = 1, . . . , N).

|u1, . . . , uN〉 : Bethe vector

· contains N down spins

· symmetric under permutations of uj’s.

Example. Length L = 6 chain with N = 3 down spins.

Unknowns: u1, u2, u3

(u1 + i

u1 − i

)6

=(u1 − u2 + 2i)(u1 − u3 + 2i)

(u1 − u2 − 2i)(u1 − u3 − 2i),

(u2 + i

u2 − i

)6

=(u2 − u1 + 2i)(u2 − u3 + 2i)

(u2 − u1 − 2i)(u2 − u3 − 2i),

(u3 + i

u3 − i

)6

=(u3 − u1 + 2i)(u3 − u2 + 2i)

(u3 − u1 − 2i)(u3 − u2 − 2i).

How may solutions up to permutations of u1, u2 and u3?

We should have 5 vectors if Bethe ansatz is complete.

Reason:

· Bethe vector is highest weight vector with respect to sl2.

· Irreducible decomposition:

⊗6 = + 5 + 9 + 5

26 = 7 + 5 × 5 + 9 × 3 + 5 × 1

(N = 0) (N = 1) (N = 2) (N = 3)

General case: Kostka number K(L−N,N),(1L) =

(L

N

)−

(L

N − 1

)

• 0.859•• −0.859

•2.02i

••−2.02i

•−0.472 − i

•−0.472 + i

• −0.944

•−i

• •i

•0.472 − i

•0.472 + i

• 0.944

String hypothesis: Bethe roots {u1, . . . , uN} = collection of strings.

α• β−i• β+i• γ−2i• γ• γ+2i• · · ·1-string 2-string 3-string · · ·

with small distortions (α, β, γ, . . . ∈ R : string center).

Bethe equation + string hypothesis

↓ physicist’s argument

{Bethe roots} 1:1?←→ {Rigged configuration}

• 0.859•• −0.859

000

•2.02i

••−2.02i

0

•−0.472 − i

•−0.472 + i

• −0.944

00

•−i

• •i

01

•0.472 − i

•0.472 + i

• 0.944

02

Rigged configuration for sl2 (spin 12)⊗L

Young diagram = configuration, {ri} = rigging

···

jmj

r1

r2···rmj

···

0 ≤ r1 ≤ · · · ≤ rmj≤ pj

· · · (fermionic) selection rule

pj = L − 2∑

k≥1 min(j, k)mk

· · · vacancy number

] of rigged configurations =∑

{mi}

∏

i≥1

(pi + mi

mi

)

1 : 1?

{ Bethe roots }

1 : 1?

{ irreducible sl2 components }

l 1 : 1 rigorously !

{ rigged configuration }

Theorem. (“alle Losungen”, Bethe’s fermionic formula 1931)

Kostka number K(L−N,N),(1L) =∑

{mi}

∏

i≥1

(pi + mi

mi

)

KKR theory. (Kerov-Kirillov-Reshetikhin 1986)

Canonical bijection and q-analogue of Bethe’s formula fromintegrable spin chain with sln symmetry.

{rigged configurations} KKR←→ {standard tableaux} RS←→ {highest paths}

000

1 3 52 4 6

121212

01 2 34 5 6

111222

00

1 3 42 5 6

121122

01

1 2 53 4 6

112212

02

1 2 43 5 6

112122

“composition of our bijection with the Robinson-Schensted-Knuthcorrespondence may be viewed as a combinatorial version of theBethe ansatz.”(KKR 1986)

KKR bijection for sln

{rigged configurations} 1:1←→ {highest paths}

µ(0)

02

3

µ(1)

10

µ(2)

0

µ(3)

←→ 111 122 22 13 23 4 3

“Bethe roots” “Bethe vectors”

• highest path = b1b2 . . . bL, bi = shape (µ(0)i ) semistandard tab.

(+ highest condition)

• rigged configuration: (µ, r) = (µ(0), (µ(1), r(1)), . . . , (µ(n−1), r(n−1)))

µ(0), µ(1), . . . , µ(n−1) : configuration

r(1), . . . , r(n−1) : rigging

}(+selection rule)

Example of KKR algorithm

00

1

0 021

0 01

0

01

0 11

0 1 ∅

∅ ∅ ∅ ∅ ∅ ∅∅

-2 -3

-1 -2 -3

-2 -1 -1

Top left rigged configurationKKR7−→ 11232132

• Lascoux-Schutzenberger charge on semistandard tableaux

↓ KKR

charge function c(µ, r)

c(µ, r) =1

2

n−1∑

a,b=1

Cab min(µ(a), µ(b)) − min(µ(0), µ(1)) +

n−1∑a=1

|r(a)|

(min(λ, µ) =

∑ij min(λi, µj), |r| =

∑i ri

(Cab) = Cartan matrix of sln

)

Corollary. (Fermionic formula for Kostka-Foulkes polynomial)

q∗Kλ,µ(0)(q) =∑

(µ,r)

qc(µ,r) =∑

µ

qc(µ,0)∏

a,i

[p

(a)i + m

(a)i

m(a)i

]

q

Summary so far

KKR bijection

{rigged configurations} 1:1←→ {highest paths}is a combinatorial analogue of the Bethe ansatz

{Bethe roots} −→ {Bethe vectors}that establishes the Fermionic formula.

∃ Conjectural fermionic formula for ∀affine Lie algebra

(Hatayama et al. 1999, 2002).

Box-ball system on ∞ lattice (Takahashi-Satsuma 1990)

· · · 11111111443221111111111111111111111111 · · ·· · · 11111111111114432211111111111111111111 · · ·· · · 11111111111111111144322111111111111111 · · ·

1 =empty box, 2, 3, 4 = color of balls

time evolution: T∞ = (move 2)(move 3)(move 4)

(move i) · Pick the leftmost ball with color i and move it to thenearest right empty box.

· Do the same for the other color i balls.

• soliton=consecutive balls i1 . . . ia with color i1 ≥ · · · ≥ ia ≥ 2.

• velocity=amplitude.

• Collisions of 2 solitons

· · · 11144322111143311111111111111111111111 · · ·· · · 11111111443221143311111111111111111111 · · ·· · · 11111111111114432243311111111111111111 · · ·· · · 11111111111111111132244433111111111111 · · ·· · · 11111111111111111111132211444331111111 · · ·· · · 11111111111111111111111132211114443311 · · ·

• Amplitudes are individually conserved.

• Two body scattering:

Exchange of internal labels (colors) like quarks in hadrons

Phase shift

Collision of 3 solitons

· · · 11432114211113111111111111111 · · ·· · · 11111432142111311111111111111 · · ·· · · 11111111431422131111111111111 · · ·· · · 11111111114311423211111111111 · · ·· · · 11111111111143112143211111111 · · ·· · · 11111111111111431211143211111 · · ·· · · 11111111111111114132111143211 · · ·

· · · 11432111142113111111111111111 · · ·· · · 11111432111421311111111111111 · · ·· · · 11111111432114231111111111111 · · ·· · · 11111111111432124311111111111 · · ·· · · 11111111111111413243211111111 · · ·· · · 11111111111111141132143211111 · · ·· · · 11111111111111114111321143211 · · ·

Yang-Baxter relation is valid.

(Solitons in final state are independent of the order of collisions)

Double origin of integrability

(1) UltraDiscretization (UD) of soliton equations

• Key formula

limε→+0

ε log

(exp(

a

ε) + exp(

b

ε)

)= max(a, b)

(+, ×) −→ (max, +)

keeps distributive law:

AB + AC = A(B + C) → max(a + b, a + c) = a + max(b, c)

• UD of a discrete KdV equation turns out to be an evolution eq.of the 2-state box-ball system. (Tokihiro et al. 1996)

(2) Solvable lattice model at “temperature 0”

Time evolution pattern

· · · 1421131111111 · · ·· · · 1114213111111 · · ·· · · 1111142311111 · · ·· · · 1111111243111 · · ·· · · 1111111121431 · · ·

is a configuration of a 2D lattice model in statistical mechanics

11 11 11 11 11 11 11 11 1114 24 12 11 131 4 2 1 1 3 1 1 1 1 1 1 1

1 1 1 4 2 1 3 1 1 1 1 1 111 11 11 11 11 11 11 11 11 1114 24 12 13

1 1 1 1 1 4 2 3 1 1 1 1 111 11 11 11 11 11 11 11 11 1114 24 34 13

1 1 1 1 1 1 1 2 4 3 1 1 111 11 11 11 11 11 11 11 11 1112 14 34 13

1 1 1 1 1 1 1 1 2 1 4 3 1

• n-state box-ball system

= 2D solvable vertex model associated with quantum group

Uq(sln) at q = 0. (q ∼ temperature)

• row transfer matrix at q = 0

= deterministic map

= time evolution of box-ball system(forms a commuting family T1, T2, . . . T∞.)

• proper formulation by Kashiwara’s crystal base theory(Hatayama et al. 2000, Fukuda et al. 2000)

Some outcome.

• integrable cellular automata associated with affine Lie algebras

example: D(1)5 = so10 -automaton

· · · 11135322111122511111111111111111111111111111111 · · ·· · · 11111111353221122511111111111111111111111111111 · · ·· · · 11111111111113532222511111111111111111111111111 · · ·· · · 11111111111111111135311511111111111111111111111 · · ·· · · 11111111111111111111111411455111111111111111111 · · ·· · · 11111111111111111111111111422224551111111111111 · · ·· · · 11111111111111111111111111111422112245511111111 · · ·· · · 11111111111111111111111111111111422111122455111 · · ·

• particles and anti-particles undergo pair-creations/annihilations

• solitons and their scattering rules characterized bycrystal theory. (Hatayama-K-Okado-Takagi-Yamada 2002)

Nonlinear waves

Soliton equations

Classicalintegrable system

Inverse scattering method

−→UD Cellular automata

Box-ball systems

Ultradiscreteintegrable system

Combinatorial Bethe ansatz

←−0←q Lattice statistical models

Solvable vertex models

Quantumintegrable system

Bethe ansatz

Natural to invest KKR theory to box-ball systems.

Dynamics of box-ball system in terms of rigged configuration

t = 0: 111122221111133211431111111111111111111111111111

t = 1: 111111112222111133214311111111111111111111111111

t = 2: 111111111111222211133243111111111111111111111111

t = 3: 111111111111111122221132433111111111111111111111

t = 4: 111111111111111111112221322433111111111111111111

t = 5: 111111111111111111111112211322433211111111111111

t = 6: 111111111111111111111111122111322143321111111111

t = 7: 111111111111111111111111111221111322114332111111

(148)

µ(0)

4t6 + 3t

11 + 2t

µ(1) µ(2)

10

µ(3)

0

configuration · · · conserved quantity (action variable)

rigging · · · linear flow (angle variable)

KKR bijection · · · direct/inverse scattering map (separation of variables)

Theorem. (KO-Sakamoto-TY 2006)

• KKR theory (a version of combinatorial Bethe ansatz) is theinverse scattering scheme of the box-ball system on ∞ lattice.

• Solution of the initial value probelm.

KKR theory box-ball system crystal theory

rigged configuration scattering data ⊗(affine crystals)

KKR bijection direct/inverse scattering composition of R

Ultradiscrete tau function

rigged configuration: (µ, r) = (µ(0), (µ(1), r(1)), . . . , (µ(n−1), r(n−1)))

charge function: c(µ, r) = 12

n−1∑

a,b=1

Cab min(µ(a), µ(b))−min(µ(0), µ(1))+

n−1∑a=1

|r(a)|

UD tau function :

τk,i = − min(ν,s)

{c(ν, s) + |ν(i)|} (k≥1, 1≤ i≤n)

min(ν,s)

: over all subsets (ν, s) ⊆ (µ, r) with ν(0) = (µ(0)1 , . . . , µ

(0)k ).

µ(0)

(114)

µ(1)

02

3

µ(2)

10

µ(3)

0

Theorem. (KSY 2007) τk,i admits characterizations:

(1) UD of tau functions for KP hierarchy

(2) “corner transfer matrix” of box-ball system

(1) τk,i = limε→0

ε log 〈i| exp(∑

j

cjψ(pj)ψ∗(qj)

)|i〉

(cj, pj, qj) ←→ strings in rigged configuration

Corollary of (1). (UD Hirota bilinear eq.)

τk,i−1 + τk−1,i = max(τk,i + τk−1,i−1, τk−1,i−1 + τk,i − µ(0)k )

τk,i = τk,i associated with T∞((µ, r))

(2) Suppose b1b2 · · · bLKKR←→ (µ, r) −→ {τk,i}. Then,

b1 b2 · · · bk

τk,i = ] of balls in

?

box-ball system

time evolution T∞

· · · “analogue” of Baxter’s corner transfer matrix

UD Hirota bilinear = eq. of motion of box-ball system.

Corollary of (1)&(2).

• piecewise linear formula for KKR map: (µ, r)KKR7−→ b1 . . . bL

semistandard tab. bk = (

xk,1︷ ︸︸ ︷1 . . . 1 . . .

xk,n︷ ︸︸ ︷n . . . n) is specified by

xk,i = τk,i − τk−1,i − τk,i−1 + τk−1,i−1.

• general N soliton solution of box-ball system

Bethe ansatz Corner transfer matrix

main combinatorial object rigged configurationcharge (energy)

in affine crystal

role in box-ball system action-angle variable tau function

dynamics linear bilinear

Periodic box-ball system (sl2 spin 12, size L)

evolution under T2 evolution under T3

1 1 2 1 1 1 2 2 2 1 1 1 2 2 1 1 2 1 1 1 2 2 2 1 1 1 2 22 2 1 2 1 1 1 1 2 2 2 1 1 1 2 2 1 2 1 1 1 1 1 2 2 2 1 11 1 2 1 2 2 1 1 1 1 2 2 2 1 1 1 2 1 2 2 2 1 1 1 1 1 2 22 1 1 2 1 1 2 2 1 1 1 1 2 2 2 2 1 2 1 1 1 2 2 2 1 1 1 12 2 2 1 2 1 1 1 2 2 1 1 1 1 1 1 2 1 2 2 1 1 1 1 2 2 2 11 1 2 2 1 2 2 1 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 1 1 1 21 1 1 1 2 1 2 2 2 1 1 1 2 2 1 1 2 1 2 2 1 1 2 2 2 1 1 12 2 1 1 1 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 1 1 1 2 2 2

commuting family of time evolutions T1, T2, . . ..

T∞ = “ball moving” procedure (Yoshihara et al. 2003).

proper formulation · · · crystals and combinatorial R.

• Action-angle variables

Any path b is expressed as

b = T d1 (b+) (T1 : cyclic shift, b+ : highest path, d ∈ Z),

where (d, b+) is not unique.

But anyway let (µ, I) be the rigged configuration for b+ :

Iµg

Iµg−1

Iµ1

KKR

b+ 7−→

¾ µ1-

···¾ µg−1

-

¾ µg-

µ = (µ1, . . . , µg), I = (Iµ1, . . . , Iµg).

(For simplicity we assume µ1 > µ2 > · · · > µg

pi = L − 2∑

j∈µ min(i, j) : vacancy number

)

Lemma.

• µ and (I + dh1)/AZg are unique, where

hl = (min(l, i))i∈µ ∈ Zg, A =(δijpi + 2 min(i, j)

)i,j∈µ

.

• µ is invariant under {Tl} (action variable).

Define

P(µ) := {paths whose action variable = µ}: iso-level set

J (µ) := Zg/AZg : set of angle variables

Φ(b) := (I + dh1)/AZg : P(µ) −→ J (µ)

Theorem. (KT-Takenouchi 2006)

Φ : P(µ) → J (µ) is a bijection.

P(µ)Φ−→ J (µ)

Tl

yyTl

P(µ)Φ−→ J (µ)

is commutative, where Tl(J) = J + hl on J (µ)

Nonlinear dynamics becomes straight motions in

J (µ) = Zg/AZg,

which is an ultradiscrete analogue of Jacobi variety.

Solution of initial value problem (inverse method)

22121111222111direct scattering Φ

- 36

2

?

linear flow T 10003

30032006

1002

mod AZ3

54

10

inverse scattering Φ−1

¾

?

12211122111122(answer)

T 10003

Riemann theta (with pure imaginary period matrix) :

ϑ(z) :=∑

n∈Zg

exp(−

tnAn/2 + tnz

ε

)

UD Riemann theta (z ∈ Rg):

Θ(z) := limε→+0

ε log ϑ(z) = − minn∈Zg

{tnAn/2 + tnz}

Theorem. (KS 2006)

Action-angle variable (µ, I) 7→ path b1b2 . . . bL ∈ {1, 2}L is given by

bk = 1 + Θ(J − kh1

) − Θ(J − (k−1)h1

)

− Θ(J − kh1 + h∞

)+ Θ

(J − (k−1)h1 + h∞

),

with J = I − t(pµ1, . . . , pµg)/2.

Inverse UD: double difference of Θ −→ double ratio of ϑ

b(k, t) =ϑ

(J + th∞ − kh1

)ϑ

(J + (t + 1)h∞ − (k−1)h1

)

ϑ(J + th∞ − (k−1)h1

)ϑ

(J + (t + 1)h∞ − kh1

).

Same structure as the quasi-periodic solution of theKdV/Toda eq. by Date-Tanaka and Kac-Moerbeke (1976).

Two soliton state with amplitudes 6 and 2.System size L = 170, duration 0 ≤ t ≤ 70.

Remarks.

• Origin of UD period matrix A:Uq(sl2) Bethe equation at q = 0 (K-Nakanishi 2000)

• General case:UD Riemann theta with rational characteristics.

• J (µ) from Jacobian of tropical hyperelliptic curve.(Inoue-Takenawa 2007)

“I haven’t a slightest idea of what people did with it.”

· · · H. Bethe