Quantum Integrability of SuperSpin Chains From Discrete Hirota Dynamics

1
Quantum Integrability of Super-Spin Chains From
Discrete Hirota Dynamics
Recent Advances in Quantum Integrable Systems

• V. Kazakov (ENS, Paris)
• LAPTH, Annecy-le-Vieux, 13/09/07

with A.Sorin and A.Zabrodin, hep-th/0703147.
2
Motivation

• Classical and quantum integrability are
  intimately related (not only through the
  classical limit!)
• Quantization discretization
• Quantum spin chain Discrete classical
  Hirota dynamics
• We study SUSY spin chain via Hirota equation for
  fusion rules, with specific integrable boundary
  conditions.
• More general and more transparent with SUSY!
• An alternative to algebraic Bethe ansatz

Klumper,Pearce 92, Kuniba,Nakanishi,92


Krichever,Lupan,Wiegmann, Zabrodin97
Kulish,Sklianin80-85
3
Plan
R-matrix and Yang-Baxter eq. for SUSY spin chain
Bazhanov-Reshetikhin rel. Hirota eq. for fusion
Bazhanov,Reshetikhin90
SUSY boundary cond., Bäcklund transf.
undressing
Baxter TQ relations Hirota eq for Q-functions
(QQ relations)
Analyticity and SUSY nested Bethe ansatz Fusion
in quantum space examples, gl(11), gl(21)
4
sl(KM) super R-matrix
Vector irrep v in aux. space and arbitrary l
in quantum
For lv graded permutation
for even (odd) components
5
Yang-Baxter relation for R-matrix
ß''
ß''
u
ß'
a
ß'
?
?'
u
a
?''
?''

?'
ß
ß
?
v
a'
v
a'
a''
a''
0
0
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Monodromy Matrix and Transfer Matrix
ß2
ß1
ßN
u
l, ai

l
l
T ßi
u1
u2
uN
a2
? quantum space ?
a1
aN
auxiliary space

• Transfer matrix supertrace of monodromy
  matrix, in irrep l

polynomial of degree N

• Defines all conserved charges of
  (inhomogeneous)
• super spin chain.

How to calculate it?
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T-matrix Eigenvalues as Quantum Characters

• Conservation laws

• Bazhanov-Reshetikhin determinant formula

• Expresses

for general irrep ?
through
for the row
s
8
Hirota relation for rectangular tableaux
a
T(a,s,u) ?
s

• From BR formula, by Jakobi relation for det

T(u)
T (u)
T(u)
T (u1)T(u-1)
a
T (u)
representation plane (a,s)
s
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Hirota relation

• Direct consequence of Bazhanov-Reshetikhin
  quantum
• character formula.
• Hirota eq. integrable, Master equation of the
  soliton theory.
• The classical inverse scattering method can be
  applied.
• Boundary conditions specific to the model
  (R-matrix).
• Analyticity T-functions become polynomials
• of the same power NLength of spin chain,
• We use Hirota eq. to find all possible Baxters
  TQ relations and nested Bethe ansatz equations
  for superalgebras.

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SUSY Boundary Conditions Fat Hook
a
K
T(a,s,u)?0
s
M

• All super Young tableaux of sl(KM) live within
  this fat hook

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Bäcklund Transformation I (BT-I)
T(u1)
F(u)
T(u)
F(u)
F(u1)
T(u1)
? First Lax pair of linear problems for ,
equiv. to Hirota eq. ?
a
F(u)
s
T(u1)
T(u)
T(u1)
F(u)
F(u1)
F(u)
On the horizontal boundary one can put
F(K,s,u)0.
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Undressing by BT-I vertical move
a
K
K-1
s
M
T(a,s,u) TK,M (a,s,u) ? F(a,s,u)
TK-1,M(a,s,u)
Notation
sl(KM) sl(K-1M)
n
TK-1,M(a,s,u) also satisfies Hirota eq., but
with shifted B.C.
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Bäcklund Transformation - II
T(u)
F(u)
F(u1)
T(u)
T(u1)
F(u1)
? Second Lax pair of linear problems ?
a
s
F(u1)
F(u)
F(u1)
T(u)
T(u1)
T(u)
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Undressing by BT-II horizontal move
a
K
s
M
M-1
TK,M (a,s,u) ? F(a,s,u) TK,M-1 (a,s,u)
.?Tk,m (a,s,u)
sl(KM) sl(K-1M) sl(km)
0
n
General Nesting
n
n

• One can repeat this procedure until the full
  undressing K,M0 T0,0(u)1.

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Boundary conditions..
a
Tsuboi97
Qk,0(uam)Q0,m(u-a-m)(-1)m(a-k)
Qk,m(ua)
Qk,0(usk) Q0,m(u-s-k)
k
Tk,m (a,s,u)? 0
s
m
Qk,m(u-s)

• B.C. respect Hirota equation.

Qk,m(u)?j (u-uj )

• B.C. defined through Baxters Q-functions

k1,,K m1,M
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Undressing along a zigzag path (Kac-Dynkin
diagram)
undressing (nesting) plane (k,m)
k
(K,0)
9
(K,M)
At each (k,m)-vertex there is a Qk,m(u)
8
6
7
3
4
5
particle-hole duality
Tsuboi98
2
m
0
(0,M)
1
Analyticity

• By construction T(u,a,s) TK,M(u,a,s),
• Qk,m (u) and Tk,m(u,a,s) are polynomials
  in u.

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Strategy

• Express T-functions through Q-functions.
• Find Q-functions from analyticity
  (polynomiality).
• This gives Nested Bethe Ansatz

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Generalized Baxters T-Q Relations

• Diff. operator encoding all Ts for symmetric
  irreps

• From Hirota eq.

where
are shift operators on (k,m) plane.
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Generalized Baxters T-Q Relations
k
(K,0)
9
(K,M)
n1
8
x
n2
6
7
4
5
3
2
m
0
1
(0,M)
V.K.,Sorin,Zabrodin07
- coordinate on (k,m) plane
- unit vector in the direction of shift
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Hirota eq. for Baxters Q-functions
(Q-Q relations)
k1,m
k1,m1
Zero curvature cond. for shift operators
k,m
k,m1
V.K.,Sorin,Zabrodin07
Q (u2)
Q (u)
Q (u2)
k
Q (u)
Q (u)
Q (u2)
m
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Bethe Ansatz Equations along a zigzag path

• BAEs follow from zeroes of various terms in
  Hirota QQ relation

and Cartan matrix along the zigzag path
1, if
where
-1, if
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Higher irreps and fusion in quantum space
V.K.,Sorin,Zabrodin07
quantum
M-m
M
l'km
K-k
µkm
µKM
K
lkm
auxiliary
arbitrary polynomial
A relevant subsequent work
Ragoucy, Satta07
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gl(11) algebra
a
a-1
Q1,0(ua1)Q0,1(u-a-1) (-1)
Q1,1(ua)
Q1,0(us1) Q0,1(u-s-1)
s
Q1,1(u-s)

• We reproduce BAEs and Baxters TQ relations,
  including
• the typical irreps with continuous labels, in
  accordance with

Fendley,Intriligator92
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sl(21) algebra
a
Q2,0(ua1)Q0,1(u-a-1) (-1)a-2
Q2,1(ua)
Q2,0(us2) Q0,1(u-s-2)
T2,1 (1,s,u)
s
Q2,1(u-s)

• We reproduce BAEs and Baxters TQ relations,
  including
• the irreps with continuous labels, in
  accordance with

• Frahm,Pfanmüller96
• .
• Bazhanov,Tsuboi07

• Related to Beiserts su(22) S-matrix.

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Example Baxter and Bethe equations for sl(21)
with Kac-Dynkin diagram
Generating functional for antisymmetric irreps
T-matrix eigenvalue in fundamental irrep
Bethe ansatz equations
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Applications and Problems

• Generalizations noncompact irreps, mixed
  (covariantcontravariant) irreps, so(MK),
  sp(MK) algebras.
• Non-standard R-matrices, like Hubbard or
• su(22) S-matrix in AdS/CFT, should be also
• described by Hirota equation with different
  B.C.
• A powerful tool for constructing and studying
  supersymmetric spin chains and 2d integrable
  field theories, including classical limits.
• An alternative to the algebraic Bethe
  ansatz.

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k
K
(K,M)
m
(0,0)
M
28
s
µ(a,s)
a
µ\µ(a,s)