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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
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