Physical combinatorics and TBA: paths, ( m, n ) systems and finitized - - PowerPoint PPT Presentation

Physical combinatorics and TBA: paths, (m, n) systems and finitized characters Giovanni Feverati

Laboratoire d’Annecy-le-Vieux de physique theorique

Paul A. Pearce

Department of Mathematics and Statistics University of Melbourne

• Nucl. Phys. B 663, 409-442 (2003), hep-th/0211185, hep-th/0211186

&. . . in preparation

Physical Combinatorics C ∩ IM ∩ CFT = Physical Combinatorics C IM CFT Phys. Comb.

Integrable RSOS Models with boundaries

Double Row Transfer Matrix:

D(N, u, ξ)σ,σ′ =

• τ1,...,τN

u ξ

1 σ1 σ σN−1 r r 1 σ′

1

σ′

2

σ′

N−1

r r 2 τ1 τ2 τN−1 τN u u u λ − u λ − u λ − u

fixed b.c.

✒ ✻

boundary interaction

Critical AL RSOS Face Weights:

(ABF 1984) ℓ1 ℓ2 ℓ3 ℓ4 u

= sin(λ − u) sin λ δℓ1,ℓ3 + sin u sin λ

• Sℓ1Sℓ3

Sℓ2Sℓ4 δℓ2,ℓ4 λ = π L + 1 ; Sℓ = sin ℓλ

”height” variables:

ℓj = 1, . . . , L

nearest neighbor sites:

|ℓi − ℓj| = 1

1 2 3 L

Boundary Weights:

(Behrend, Pearce 2001)

u ξ

r r r±1 =

• sin(r ± 1)λ

sin rλ sin(ξ ± u) sin(rλ + ξ ∓ u) sin2 λ

fusion:

Dq+1 ∼ DqD1 + Dq−1, ˜ dq ∼ Dq+1Dq−1

Integrability: YBE ⇒ [Dq(u), Dq′(v)] = 0 basis of eigenstates independent of u Analyticity: each entry of D1(u) is entire function (zeros); each entry of Dq(u) (q > 1) is the ratio of an entire function by some known function Periodicity:

u + π ≡ u

Functional equations

q = 1, . . . , L − 2 ˜ dq(u − λ 2)˜ dq(u + λ 2) =

• 1 + ˜

dq−1(u) 1 + ˜ dq+1(u)

• ˜

d0(u) = ˜ dL−1(u) = 0

• true for the eigenvalues of ˜

dq(u)

• solution given by the analytic properties in u ∈ C :

** L − 2 analyticity strips ** zeros of eigenvalues Dq(u), zeros of the numerical factors ** (with boundaries) additional numerical factors with zeros/poles

A4: zeros of the eigenvalues of D(u)

Two analyticity strips: periodicity u + π ≡ u;

λ = π/5 (1) − λ 2 < Re(u) < 3λ 2 , (2) 2λ < Re(u) < 4λ mi =

{number of 1-strings in strip i = 1, 2}

ni =

{number of 2-strings in strip i = 1, 2}

(m, n)−system:

• m1 + n1 = N+m2

2

m2 + n2 = m1

2

= ⇒ m1, m2, ∈ 2N

• m1 = 4

n1 = 2 n2 = 0 m2 =2

(1) (2)

−λ

2

−4λ

Relative order

Paths

paths on AL: only diagonal paths are permitted; initial and final point at height=1 paths on TL: diagonal paths are permitted; horizontal paths permitted at heigth=1; initial and final point at height=1; fixed shape rectangle L = ⌊ N

2 ⌋

1 2 3 4 5 L = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22= N

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ Local energy density (Baxter80, ABF84)

h(σj−1, σj, σj+1) =                         

if

σj+1 =σj−1

• ❅

❅ ❅ ❅

• 1

if

σj+1−σj−1 =±2

• ❅

❅ ❅ s s

1

if

(σj−1, σj, σj+1) = (1, 1, 1) 1

1 2

if

( σj−1,σj,σj+1)=(2, 1, 1) or (1, 1, 2) 1 ❅

❅

• 1-dim configurational sums

E(σ) = 1 2

N

• j=1

j h(σj−1, σj, σj+1)

Quasi-particles

(Warnaar 1995)

1 2 3 4 5 L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 N n4 n3 n2 n1 n0

. . . . . .

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

• ✉

✉ ✉ ✉ ✉ ✉ ✉ ✉

n-family

• L − 1 types of pure particles (pyramids): · · ·
• tower particle (TL case only)

na = {# of particles of type a = 0, 1, 2, . . ., L − 1} m-family: in sites where there is a straight line segment or in the middle of horizontal lines

• L − 2 types of dual particles (strings): • • • · · ·

ma = {# of dual particles of type a = 0, 1, 2, . . . , L − 2}

• Geometric packing constraints

N = n0 + 2(L − 1)nL−1 + . . . + 6n3 + 4n2 + 2n1 m1 = n0 + 2(L − 2)nL−1 + . . . + 4n3 + 2n2 m2 = n0 + 2(L − 3)nL−1 + . . . + 2n3

. . .

mL−2 = n0 + 2nL−1 m0 = n0 AL case: m0 = n0 = 0

• (m, n) system

ma + na = 1

2Nδ(a, 1) + 1 2 L−2

• b=1

Aa,bmb a = 1, 2, . . . , L − 2 Aa,b : AL−2 adjacency matrix ma + na = 1

2Nδ(a, 1) + 1 2 L−1

• b=1

Aa,bmb a = 0, 1, 2, . . ., L − 2 Aa,b : T ′

L−1 adjacency matrix

• Interaction: n-family particles can be sliced and diced and turned upside-down (geometric

packing constraints are respected).

General paths: interactions

Path = {non-interacting particles} + {complexes of overlapping particles} (c.f. Warnaar 1995)

1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅

• t

t t t t t t t t t t t t t baseline and maximum peak

Decomposition Algorithm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 3 4 5 6 7

• 1. Identify any tower particles and decorate them with dual-particles. These

tower particles automatically separate the path into non-overlapping complexes with respect to the initial baseline at height 1.

• 2. For each current baseline, separate the pure particles from the complexes

and decorate the pure particles. A complex with respect to the current baseline is any path that is not a pyramid.

• 3. For each current complex, identify the left-most and right-most global

maxima and connect these with a new baseline. The left-most and right-most global maxima may coincide and in this case no new baseline is drawn.

• 4. From each left (right) maxima, outline the profile of the complex moving

continuously down and to the left (right) drawing new baselines as

• needed. Decorate the sliced particles correponding to these maxima.
• 5. Stand on your head, identify all the current baselines and go to 2.

Energy-preserving bijection: RSOS paths ↔ strings

There is a natural energy-preserving bijection between RSOS paths and string patterns: a n-particle (m-particle) of type a at position j corresponds to a 2-string (1-string) in strip a; the relative order within a strip is preserved. The RSOS particle decomposition matches the pattern of the zeros of the transfer matrix eigenvalues.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 3 4 5

In the example, m1 = 10, m2 = 4, m3 = 2 and E = 50.

more data

In[98]:=

chiralPaths@1, 1, 5, 8D

1 2 3 4 5 6 7 8 9 10 2 3 4 5 1: E=0 m=80, 0, 0< n=85, 0, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 2: E=2 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 3: E=3 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 4: E=4 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________

PeriodicStatesMaster.nb 1

1 2 3 4 5 6 7 8 9 10 2 3 4 5 5: E=4 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6: E=5 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 7: E=6 m=84, 2, 0< n=82, 0, 1, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 8: E=6 m=82, 0, 0< n=83, 1, 0, 0< _____________________________________________

PeriodicStatesMaster.nb 2

1 2 3 4 5 6 7 8 9 10 2 3 4 5 9: E=7 m=84, 2, 0< n=82, 0, 1, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 10: E=8 m=84, 2, 0< n=82, 0, 1, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 11: E=8 m=84, 0, 0< n=81, 2, 0, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 12: E=9 m=84, 2, 0< n=82, 0, 1, 0< _____________________________________________

PeriodicStatesMaster.nb 3

1 2 3 4 5 6 7 8 9 10 2 3 4 5 13: E=10 m=84, 2, 0< n=82, 0, 1, 0< _____________________________________________ 1 2 3 4 5 6 7 8 9 10 2 3 4 5 14: E=12 m=86, 4, 2< n=81, 0, 0, 1< _____________________________________________

PeriodicStatesMaster.nb 4

Minimal Energy Configurations

For given particle/string content, the minimal energy occurs when the particles are ordered from largest to smallest and the 1-strings are all to the left of the 2-strings (creation energy). The minimal energy is

E = 1 4 m C m

where C = 2I − A = {Cartan matrix of AL−2}

Particle Dynamics:

relative order within a strip does matter!

E = 14

1 2 3 4 5 6 7 8 9 10 2 3 4

E = 15

1 2 3 4 5 6 7 8 9 10 2 3 4

E = 16

1 2 3 4 5 6 7 8 9 10 2 3 4

E = 6

1 2 3 4 5 6 7 8 2 3 4

E = 7

1 2 3 4 5 6 7 8 2 3 4

E = 8

1 2 3 4 5 6 7 8 2 3 4

q-Binomials

m + n m

• q

=

n

• I1=0

I1

• I2=0

· · ·

Im−1

• Im=0

qI1+...+Im =          (q)m+n (q)m (q)n , m, n ≥ 0 0,

• therwise

q-factorial: (q)m =

m

• j=1

(1 − qj)

Example:

4

2

• q = 1 + q + 2q2 + q3 + q4

→ → → →

s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s

1 q 2q2 q3 q4 + + + +

• ✒

❅ ❅ ❅ ❅ ❘ Integer and non-negative quantum numbers

I(a)

j

= {number of 2-strings above the 1-string labelled j in strip a}

Energy eigenstate characterised by

I(a) = (I(a)

1 , I(a) 2 , . . . , I(a) ma)

Within one strip

na I(a)

1

I(a)

2

· · · I(a)

ma 0

Finitized fermionic characters

minimal energy configuration + “excitation energies (q-binomials)” = spectrum of E (lattice fermionic gas) Generating function qE:

χ(N) (q) = q−c/24

(m,n)

q

1 4 mT C m

L−2

• a=1

ma + na ma

• q

(m, n)-system ma + na = 1

2Nδ(a, 1) + 1 2 L−2

• b=1

Aa,bmb a = 1, 2, . . . , L − 2 ma , na bounded by N; in particular

N 2 m1 m2 . . . mL−2 0

Finite versus finitized Limit N → ∞, with N even, produces the CFT vacuum fermionic character

χ0(q) = q−c/24

• ma=0 (mod 2)

m1,m2,...,mL−2≥0

q

1 4 mT C m

(q)m1

L−2

• a=2

1

2(ma−1 + ma+1)

ma

• q

AL model TBA equations

Scaling energy:

E = − ∞

−∞

dy π2e−y log(1 + ˆ d1(y)) + 2 π

m1

• k=1

e−y(1)

k

TBA equations:

q = 1, . . . , L − 2 log ˆ dq(x) = −4e−xδq,1 +

L−2

• j=1

Aq,j

mj

• k=1

log tanh[1

2(x − y(j) k )] + L−2

• j=1

Aq,j ∞

−∞

dy 2π log(1 + ˆ dj(y)) cosh(x − y)

Counting function

Ψq(x) = i log ˆ dq(x − i1 2π) = 4e−xδq,1 + i

L−2

• j=1

Aq,j

mj

• k=1

log tanh(x − y(j)

k

2 − iπ 4 ) − ∞

−∞

dy 2π log(1 + ˆ dj(y)) sinh(x − y)

Quantization conditions (auxiliary equations)

Ψq(y(q)

k ) =

• 1 + 2(I(q)

k

+ mq − k)

• π

Solution of TBA

Energy eigenvalues can be worked out exactly

E = − 1 24

• 1 −

6 L(L + 1)

• + 1

4mT C m +

L−2

• a=1

ma

• k=1

I(a)

k

na I(a)

1

I(a)

2

. . . I(a)

mq 0

(for a = 1 , n1 = ∞) qE gives the vacuum sector fermionic character χ0(q) = q−c/24

• ma=0 (mod 2)

m1,m2,...,mL−2≥0

q

1 4 mT Cm

(q)m1

L−2

• a=2

1

2(ma−1 + ma+1)

ma

• q

TBA, integrals of motion and particle interpretation

... ricapitoliamo ...

• transfer matrix of integrable lattice model
• paths, quasi-particles and (m, n)-system
• energy-preserving bijection: RSOS paths ←

→ zeros of transfer matrix eigenvalues; paths encode

the classification of states

• finitized fermionic characters, lattice gas interpretation
• continuum scaling limit, thermodynamic Bethe Ansatz equations
• conformal partition function from TBA
• equivalence between Virasoro characters and 1-d configurational sums demonstrated in a lattice

approach working entirely at criticality, with open boundary conditions

• extension to other bound. cond. including periodic case, massive and massless cases
• extension to other models (non-unitary, parafermions ...)