Critical spin chains from modular invariancesemi-simple Lie groups. Weyl had a proof, but it used...

Critical spin chainsfrom

modular invariance

Eddy Ardonne

TPQM-ESI2014-09-12

Ville LahtinenTeresia Månsson

Juha Suorsa

PRB 89, 014409 (2014)&

in preparation

Mathematics meet PhysicsComplete reducibility of finite dimensional representations ofsemi-simple Lie groups.

Hendrik Casimir updates his former advisor Paul Ehrenfest in a letter:



H. Casimir to P. Ehrenfest (image: ESI - 2013)



H. Casimir to P. Ehrenfest (image: ESI - 2013)

Lieber Chefchen/Cheferl


Weyl had a proof, but it used analysis, and Casimir quotes Pauli in his letter:



Casimir: ‘But it’s unsatisfactory that one proves a purely algebraic theorem using a transcendental detour’




Quoting Pauli: ‘da sind the Mathematiker weinend umhergegangen’




Quoting Pauli: ‘da sind the Mathematiker weinend umhergegangen’

Casimir & B.L. v.d. Waerden give an algebraic proof, using a Casimir operator

Outline★ Low-energy description of 2-d topological phases: anyon models

★ Topological phase transitions in 2-d:• condensation• modular invariance

★ Analogue on the level of spin chains: Ising examples

★ Beyond condensation: parafermions

A little about anyon modelsAn anyon model consist of a set of particles C = {1, a, b, c, . . . , n}

‘vacuum’

Moore, Seiberg,....


‘vacuum’These particles can ‘fuse’ (like taking tensor products of spins)

a⇥ b =X

c�CNabcc b⇥ a = a⇥ b (a⇥ b)⇥ c = a⇥ (b⇥ c) a⇥ 1 = a

fusion coefficients

Moore, Seiberg,....



a⇥ b =X


fusion coefficients

Anyons are represented by their ‘worldlines’ a

Moore, Seiberg,....



a⇥ b =X


fusion coefficients


Fusion is represented as a b

c

Moore, Seiberg,....



a⇥ b =X


fusion coefficients



c

Twisting of a particle �a = e2�iha

a a= ✓a

Moore, Seiberg,....



a⇥ b =X


fusion coefficients



c


a a= ✓a

A boson has �b = 1 (hb 2 Z)

Moore, Seiberg,....



a⇥ b =X


fusion coefficients



c


a a= ✓a

A boson has �b = 1 (hb 2 Z)

Moore, Seiberg,....

Braiding of particlesa b

c

a b

c= Ra,b

c Ra,bc = ±e⇡i(hc�ha�hb)

Condensation in anyon modelsBais, Slingerland, 2009

Condensation amounts to identifying a boson with the vacuum

bosonb ⇠ 1



This has several consequences:boson

b ⇠ 1




b ⇠ 1

Anyons which ‘differ by a boson’ are identified a⇥ b = c =) a ⇠ c




b ⇠ 1


Anyons with non-trivial monodromy with the boson ‘draw strings in the condensate’ and are therefore ‘confined’




b ⇠ 1



Some of the remaning particles might ‘split’: a ⇠ a1 + a2




b ⇠ 1



a� a = 1+ b+ · · · a� a = 1+ 1+ · · ·

vacuum twice





b ⇠ 1




In CFT language, one condenses a boson by adding it to the chiral algebra, and in the end, one has constructed a new modular invariant partition function

Modular invariant partition functionsA conformal field theory splits in two pieces, a chiral and anti-chiral part.

To each chiral sector (primary field), one associates a ‘character’, describing the number of states in this sector

The constants aj are non-negative integers, and τ is the modular parameter, describing the shape of the torus (next slide).

��(q) = qh��c/24�a0q

0 + a1q1 + a2q

2 + · · ·�

q = e2⇡i⌧

Modular invariant partition functionsA conformal field theory splits in two pieces, a chiral and anti-chiral part.

To each chiral sector (primary field), one associates a ‘character’, describing the number of states in this sector

The constants aj are non-negative integers, and τ is the modular parameter, describing the shape of the torus (next slide).

The full partition function is obtained by combining the chiral halves, and summing over the primary fields:

Zcft =X

j

|��j |2

��(q) = qh��c/24�a0q

0 + a1q1 + a2q

2 + · · ·�

q = e2⇡i⌧

Modular invariant partition functionsOne should be able to put the cft on the torus: partition function should be invariant under re-parametrization of the torus!


Shape of the torus is encoded by the modular parameter τ.The transformations S and T do not change the shape of the torus.

T : ⌧ ! ⌧ + 1

U : ⌧ ! ⌧/(⌧ + 1)

S : ⌧ ! �1/⌧

image: BYB

S = T�1UT�1

The partition function should be invariant under S and T!


Shape of the torus is encoded by the modular parameter τ.The transformations S and T do not change the shape of the torus.

T : ⌧ ! ⌧ + 1

U : ⌧ ! ⌧/(⌧ + 1)

S : ⌧ ! �1/⌧

image: BYB

S = T�1UT�1

Modular invariant partition functionsThe most general way to combine the chiral halves:

Zcft =X

i,j

ni,j��i�⇤�j

ni,j 2 Z�0


Zcft =X

i,j


ni,j 2 Z�0

Presence of the vacuum:

Invariance under T:

Invariance under S: the matrix (ni,j) commutes with the modular S-matrix, that diagonalizes the fusion rules.

ni,j 6= 0 ) h�i � h�j = 0 mod 1

n1,1 = 1


Zcft =X

i,j


ni,j 2 Z�0


Invariance under T:


ni,j 6= 0 ) h�i � h�j = 0 mod 1

n1,1 = 1

The so-called ‘diagonal invariant’ always exist: Zcft =X

j

|��j |2


Zcft =X

i,j


ni,j 2 Z�0


Invariance under T:


ni,j 6= 0 ) h�i � h�j = 0 mod 1

n1,1 = 1

The so-called ‘diagonal invariant’ always exist: Zcft =X

j

|��j |2

Finding all invariants is, in general, a hard task, but progress has been made (minimal models, su(2)k, su(3)k, parafermions...)

Cappelli et al., Gepner et al., ...

Example: Ising2 theoryThe Ising cft has three sectors: h1 = 0 , h� = 1/16 , h1 = 1/2�1,��,�

Example: Ising2 theoryThe Ising cft has three sectors: h1 = 0 , h� = 1/16 , h1 = 1/2

The Ising2 cft has nine sectors:�(1,1) ,�(1,�) ,�(1, ) ,�(�,1) ,�(�,�) ,�(�, ) ,�( ,1) ,�( ,�) ,�( , )

�1,��,�



Apart from the diagonal invariant, one also finds a block diagonal invariant:

Z = |�(1,1) + �( , )|2 + |�(1, ) + �( ,1)|2 + 2|�(�,�)|2

�1,��,�




Z = |�(1,1) + �( , )|2 + |�(1, ) + �( ,1)|2 + 2|�(�,�)|2

One sees identification of sectors: (1,1) ⇠ ( , ) (1, ) ⇠ ( ,1)

Confined sectors: (1,�) , ( ,�) , (�,1) , (�, )

Split sector: (�,�)

�1,��,�




Z = |�(1,1) + �( , )|2 + |�(1, ) + �( ,1)|2 + 2|�(�,�)|2

One sees identification of sectors: (1,1) ⇠ ( , ) (1, ) ⇠ ( ,1)

Confined sectors: (1,�) , ( ,�) , (�,1) , (�, )

Split sector: (�,�)

The resulting invariant describes the u(1)4 cft, and the construction amounts to the orbifold construction.

�1,��,�

Permutation invariantsIn some cases, one can permute some of the labels of the primary fields (anyons), without changing the fusion rules. Let π be such a permutation:

n⇡(a),⇡(b),⇡(c) = na,b,c S⇡(a),⇡(b) = Sa,b


n⇡(a),⇡(b),⇡(c) = na,b,c S⇡(a),⇡(b) = Sa,b

If π preserves the scaling dimension (mod 1), then one can construct new invariants from block-diagonal ones:

Z =X

a

|�a|2 �! Z⇡ =X

a

�a�⇤⇡(a)


n⇡(a),⇡(b),⇡(c) = na,b,c S⇡(a),⇡(b) = Sa,b

Z⇡ = |�(1,1)|2 + |�(�,�)|2 + |�( , )|2 + �(1,�)�⇤(�,1) + �(�,1)�

⇤(1,�)+

�(1, )�⇤( ,1) + �( ,1)�

⇤(1, ) + �( ,�)�

⇤(�, ) + �(�, )�

⇤( ,�)

Example in the Ising2 theory, starting from the diagonal invariant:


Z =X

a

|�a|2 �! Z⇡ =X

a

�a�⇤⇡(a)


n⇡(a),⇡(b),⇡(c) = na,b,c S⇡(a),⇡(b) = Sa,b

Z⇡ = |�(1,1)|2 + |�(�,�)|2 + |�( , )|2 + �(1,�)�⇤(�,1) + �(�,1)�

⇤(1,�)+

�(1, )�⇤( ,1) + �( ,1)�

⇤(1, ) + �( ,�)�

⇤(�, ) + �(�, )�

⇤( ,�)

Example in the Ising2 theory, starting from the diagonal invariant:

In this case, we have , but that’s not generic.Z⇡ = Z


Z =X

a

|�a|2 �! Z⇡ =X

a

�a�⇤⇡(a)

HTFI =L�1X

i=0

�z

i

+ �x

i

�x

i+1

Transverse field Ising model (critical)

�zj

0

12

L� 1

i

i+ 1

spin-1/2�x

i

�x

i+1

Note: we use periodic boundary conditions:crucial for our purposes!

�↵j+L ⌘ �↵

j

Symmetry: P =Y

i

�zi

HTFI =L�1X

i=0

�z

i

+ �x

i

�x

i+1

Transverse field Ising model (critical)

�zj

0

12

L� 1

i

i+ 1

spin-1/2�x

i

�x

i+1

Note: we use periodic boundary conditions:crucial for our purposes!

�↵j+L ⌘ �↵

j

Define fermionic levels (Jordan-Wigner):|�⇤ = |0⇤ |⇥⇤ = |1⇤

�zi = 1� 2c†i ci

ci =⇣Y

j<i

�zi

⌘�+i c†i =

⇣Y

j<i

�zi

⌘��i

Y

i

�zi = (�1)F

Lieb, Schultz, Mattis, (1961)Pfeuty, (1970)

Symmetry: P =Y

i

�zi

HTFI =L�1X

i=0

�z

i

+ �x

i

�x

i+1

=L�1X

j=0

(2c†j

cj

� 1)+

L�2X

j=0

(cj

� c†j

)(cj+1 + c†

j+1)+

� (�1)F (cL�1 � c†

L�1)(c0 + c†0)

Transverse field Ising model

0

12

L� 1

i

i+ 1

HTFI =L�1X

i=0

�z

i

+ �x

i

�x

i+1

=L�1X

j=0

(2c†j

cj

� 1)+

L�2X

j=0

(cj

� c†j

)(cj+1 + c†

j+1)+

� (�1)F (cL�1 � c†

L�1)(c0 + c†0)


0

12

L� 1

i

i+ 1

The parity of the number of fermions is conserved!The fermion boundary conditions depend on the symmetry sector:

For F even: anti-periodic boundary conditionsFor F odd: periodic boundary conditions


HTFI =

L�1X

i=0

⌅z

i

+ ⌅x

i

⌅x

i+1

=

X

k

⇥k

��†k

�k

� 1/2�

⇥k

= 2

r2� 2 cos

�2⇤k

L

�

0

12

L� 1

i

i+ 1

Momenta k: half integer for even F integer for odd F

Solution: go to k-space, and perform a diagonalize a 2x2 matrix (or, in general 2 L x 2 L if couplings are disordered).


HTFI =

L�1X

i=0

⌅z

i

+ ⌅x

i

⌅x

i+1

=

X

k

⇥k

��†k

�k

� 1/2�

⇥k

= 2

r2� 2 cos

�2⇤k

L

�

0

12

L� 1

i

i+ 1

Momenta k: half integer for even F integer for odd F

Solution: go to k-space, and perform a diagonalize a 2x2 matrix (or, in general 2 L x 2 L if couplings are disordered).

Conformal field theory: spectrum is described in the following way:

✏i = E0L+2⇡v

L

�� c

12+ hl + hr + nl + nr

�


Conformal field theory: ‘towers’ of states with: 0

p

2 p3 p2 2 p

k

1

2

3

4

5

6E

Transverse Field Ising model, L=16

�E = 1 �k =2�

L



p

2 p3 p2 2 p

k

1

2

3

4

5

6E


�E = 1 �k =2�

LPrimary fields: 1, σ, ψ



p

2 p3 p2 2 p

k

1

2

3

4

5

6E


�E = 1 �k =2�


Ising conformal field theory: h1 = 0 h� =1

16h =

1

2



p

2 p3 p2 2 p

k

1

2

3

4

5

6E


�E = 1 �k =2�


Ising conformal field theory: h1 = 0 h� =1

16h =

1

2

F: even F: odd

CFT sectors, Ising caseRelation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries):

sym. sector P = (�1)

Fboundary condition fields

1 A 1, �1 P �

TFI, next nearest neighbor interaction

0

12

L� 1

i

i+ 1

3

Spectrum is the ‘product’ of two spectra of the TFI model with L/2

H(2)TFI =

L�1X

i=0

�z

i

+ �x

i

�x

i+2

TFI, next nearest neighbor interaction

0

12

L� 1

i

i+ 1

3

Spectrum is the ‘product’ of two spectra of the TFI model with L/2

H(2)TFI =

L�1X

i=0

�z

i

+ �x

i

�x

i+2

Both the number of fermions on the even and the odd sites is conserved modulo two

Pe

=Y

i,even

�z

i

= (�1)Fe Po

=Y

i,odd

�z

i

= (�1)FoSymmetries:

CFT sectorsRelation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries):

HTFI

H(2)TFI

(Pe

,Po

) (BCe

,BCo

) fields(1, 1) (A,A) (1,1), (1, ), ( ,1), ( , )(1,�1) (A,P ) (1,�), ( ,�)(�1, 1) (P,A) (�,1), (�, )(�1,�1) (P, P ) (�,�)

sym. sector P boundary condition fields

1 A 1, �1 P �

We now change our model, by adding a ‘boundary term’, that changes the boundary condition of one chain, depending on the symmetry sector of the other.

Adding a boundary term

0

12

L� 1

i

i+ 1

3H

(2)TFI =

L�1X

i=0

�z

i

+ �x

i

�x

i+2

HBoundary

=�Po

� 1��x

L�2

�x

0

+�Pe

� 1��x

L�1

�x

1

What is this new model ? H2

TFI

+HBoundary

CFT sectorsRelation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries): H

(2)TFI

(Pe

,Po

) (BCe

,BCo

) fields(1, 1) (A,A) (1,1), (1, ), ( ,1), ( , )(1,�1) (A,P ) (1,�), ( ,�)(�1, 1) (P,A) (�,1), (�, )(�1,�1) (P, P ) (�,�)

H2

TFI

+HBoundary

(Pe

,Po

) (BCe

,BCo

) fields(1, 1) (A,A) (1,1), (1, ), ( ,1), ( , )(1,�1) (P, P ) (�,�)(�1, 1) (P, P ) (�,�)(�1,�1) (A,A) (1,1), (1, ), ( ,1), ( , )

The new model has u(1)4 critical behaviour, i.e., the other modular invariant in the Ising2 theory. This is the critical behaviour of the XY chain!

TFI2 v.s. XY chainOne can explicitly relate the TFI2 to the XY chain:

⇣L�1X

i=0

�z

i

+ �x

i

�x

i+2

⌘+

�Po

� 1��x

L�2�x

0 +�Pe

� 1��x

L�1�x

1 =

X

i

⌧xi

⌧xi+1 + ⌧y

i

⌧yi+1

H(2)

TFI

+HBoundary

= HXY


⇣L�1X

i=0

�z

i

+ �x

i

�x

i+2

⌘+

�Po

� 1��x

L�2�x

0 +�Pe

� 1��x

L�1�x

1 =

X

i

⌧xi

⌧xi+1 + ⌧y

i

⌧yi+1

H(2)

TFI

+HBoundary

= HXY

⌧z2j = �y

2j�y

2j+1

⌧x2j =Y

i2j

�x

i

⌧z2j+1 = �x

2j�x

2j+1

⌧x2j+1 =Y

i�2j+1

�y

i

One uses a modified version of the transformation for open chains (used, f.i., by D. Fisher, but dating back to the 70’s:


⇣L�1X

i=0

�z

i

+ �x

i

�x

i+2

⌘+

�Po

� 1��x

L�2�x

0 +�Pe

� 1��x

L�1�x

1 =

X

i

⌧xi

⌧xi+1 + ⌧y

i

⌧yi+1

H(2)

TFI

+HBoundary

= HXY

⌧z2j = �y

2j�y

2j+1

⌧x2j =Y

i2j

�x

i

⌧z2j+1 = �x

2j�x

2j+1

⌧x2j+1 =Y

i�2j+1

�y

i

One uses a modified version of the transformation for open chains (used, f.i., by D. Fisher, but dating back to the 70’s:

So, by changing boundary conditions, one can change spin chains, such that one realizes CFT that is a different modular invariant of the original one!

Can we construct interesting chains?Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant:

Ising

n �! so(n)1


Ising

n �! so(n)1

In particular, so(3)1 = su(2)2


Ising

n �! so(n)1


Chains with su(2)2 critical points are known: in S=1 chains (BA solvable), or with long-range interactions (Greiter et al., Quella et al., Sierra et al., Tu)


Ising

n �! so(n)1



We start with three decoupled Ising chains, and add the appropriate ‘boundary’ term. After a spin-transformation, we obtain

Hsu(2)2 =X

i

⌧xi

⌧xi+1 + ⌧y

i

⌧zi+1⌧

y

i+2


Ising

n �! so(n)1



We start with three decoupled Ising chains, and add the appropriate ‘boundary’ term. After a spin-transformation, we obtain

Hsu(2)2 =X

i

⌧xi

⌧xi+1 + ⌧y

i

⌧zi+1⌧

y

i+2

Generalization to arbitrary so(n)1 critical chains is straightforward

Can we construct interesting chains?

This hamiltonian can be solved by Jordan-Wigner, in terms of a single fermion. The critical point gx=gy is described by su(2)2:

Hsu(2)2 =X

i

gx

⌧xi

⌧xi+1 + g

y

⌧yi

⌧zi+1⌧

y

i+2

Can we construct interesting chains?

This hamiltonian can be solved by Jordan-Wigner, in terms of a single fermion. The critical point gx=gy is described by su(2)2:

Hsu(2)2 =X

i

gx

⌧xi

⌧xi+1 + g

y

⌧yi

⌧zi+1⌧

y

i+2

0 π/6 π/3 π/2K

0

1

2

3

Ener

gy

1 - sectorσ - sectorψ - sector

Spectrum of the so(3)1 chain - L=36

5

2 2

4

1

1 12 2

6 6 6 6

1 1

5 52 24 45 5 54 4 4

6

6 6

6

6

6 6

612 12

Hsu(2)2 =

X

k

✏k��†k�k � 1/2

�

✏k = 2

q2 + 2 cos

�6k⇡/L

�

Going beyond condensationTo go beyond condensation transitions, we consider the 3-state Potts chain (compare: Fendley & Qi’s talks).

Z =

0

@1 0 00 ! 00 0 !2

1

A X =

0

@0 1 00 0 11 0 0

1

A ! = e2⇡i/3

X3 = 1 Z3 = 1 XZ = !ZX

H3�Potts

= �X

i

XiX†i+1

+ Zi + h.c.

Going beyond condensationTo go beyond condensation transitions, we consider the 3-state Potts chain (compare: Fendley & Qi’s talks).

The 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5.Field content:Scaling dimensions:

{1, 1, 2, ⌧0, ⌧1, ⌧2}h1 = 0 , h 1,2 = 2/3 , h⌧0 = 2/5 , h⌧1,2 = 1/15

Z =

0

@1 0 00 ! 00 0 !2

1

A X =

0

@0 1 00 0 11 0 0

1

A ! = e2⇡i/3

X3 = 1 Z3 = 1 XZ = !ZX

H3�Potts

= �X

i

XiX†i+1

+ Zi + h.c.

The product of two Z3 parafermion theories: no bosons or fermions!But, there are 16 modular invariants, giving two different partition functions (not obtainable as orbifolds):

Going beyond condensationThe 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5.Field content:Scaling dimensions:

{1, 1, 2, ⌧0, ⌧1, ⌧2}h1 = 0 , h 1,2 = 2/3 , h⌧0 = 2/5 , h⌧1,2 = 1/15

The product of two Z3 parafermion theories: no bosons or fermions!But, there are 16 modular invariants, giving two different partition functions (not obtainable as orbifolds):

Going beyond condensationThe 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5.Field content:Scaling dimensions:

{1, 1, 2, ⌧0, ⌧1, ⌧2}h1 = 0 , h 1,2 = 2/3 , h⌧0 = 2/5 , h⌧1,2 = 1/15

hl + hr ‘degenaracy’0 1

2/15 44/15 44/5 214/15 44/3 422/15 88/5 132/15 48/3 4

ZDiagonal

hl + hr ‘degenaracy’0 1

4/15 44/5 214/15 417/5 84/3 422/15 88/5 18/3 4

ZPermutation

Going beyond condensationThe Potts chain realizing the (Z3 parafermion)2 theory is simply:

H(2)

3�Potts

= �X

i

XiX†i+2

+ Zi + h.c.


H(2)

3�Potts

= �X

i

XiX†i+2

+ Zi + h.c.

The appropriate chain for the other invariant reads

H(2,perm)

3�Potts

=�⇣X

i

XiX†i+2

+ Zi + h.c.⌘

�� Y

i,even

Z†i � 1

�X†

L�1

X1

�� Y

i,odd

Zi � 1�X†

L�2

X0

+ h.c.

=�⇣X

i

X̃iX̃†i+1

+ Z̃iZ̃i+1

+ h.c.⌘


H(2)

3�Potts

= �X

i

XiX†i+2

+ Zi + h.c.


H(2,perm)

3�Potts

= �⇣X

i

XiX†i+1

+ ZiZi+1

+ h.c.⌘


H(2)

3�Potts

= �X

i

XiX†i+2

+ Zi + h.c.


H(2,perm)

3�Potts

= �⇣X

i

XiX†i+1

+ ZiZi+1

+ h.c.⌘

This model is closely related to the ‘quantum torus chain’ Qin et al.

HQTC =

⇣X

i

cos(✓)XiX†i+1 + sin(✓)ZiZ

†i+1 + h.c.

⌘


H(2)

3�Potts

= �X

i

XiX†i+2

+ Zi + h.c.


H(2,perm)

3�Potts

= �⇣X

i

XiX†i+1

+ ZiZi+1

+ h.c.⌘

This model is closely related to the ‘quantum torus chain’ Qin et al.

HQTC =

⇣X

i

cos(✓)XiX†i+1 + sin(✓)ZiZ

†i+1 + h.c.

⌘

By coupling three 3-state Potts chains, one can again condense a boson:

H(3,cond)3�Potts

= �⇣X

i

XiX†i+1

+ ZiZi+1

Zi+2

+ h.c.⌘

ConclusionsWe can construct interesting spin-chains in analogy with 2d topological condensation transitions as well as modular invariance

Construction works for Jordan-Wigner solvable models, ‘BA’ solvable models, and non-integrable models.

Latter category (non discussed here): S=1 Blume-Capel model, giving a N=1 susy cft, (A,E) exceptional modular invariant.

Open questions:

Can we do this without coupling several chains together (4-state Potts?)How general is this method?Can we learn something about the modular invariant partition functions?

Study of the phase diagrams of the new models is underway

ConclusionsWe can construct interesting spin-chains in analogy with 2d topological condensation transitions as well as modular invariance

Construction works for Jordan-Wigner solvable models, ‘BA’ solvable models, and non-integrable models.

Latter category (non discussed here): S=1 Blume-Capel model, giving a N=1 susy cft, (A,E) exceptional modular invariant.

Open questions:

Can we do this without coupling several chains together (4-state Potts?)How general is this method?Can we learn something about the modular invariant partition functions?

Study of the phase diagrams of the new models is underway

Thank

you for

your at

tention!

Critical spin chains from modular invariancesemi-simple Lie groups. Weyl had a proof, but it used...

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Transcript of Critical spin chains from modular invariancesemi-simple Lie groups. Weyl had a proof, but it used...