Continuum limit of a staggered superspin chain...simplest case: 6v model with alternating shift in spectral parameter ˜one of integrable manifolds of square lattice Potts model [Temperley,

Continuum limit of a staggered superspin chain

Holger Frahm

Institut für Theoretische PhysikLeibniz Universität Hannover

MATRIX workshop

Integrability in low-dimensional quantum systemsCreswick, 13. July 2017

www.itp.uni-hannover.de

Inhmogeneities & staggering

class of Lax-operators to given R-matrix

commuting transfer matrices with isolated inhomogeneities

e.g. Kondo / Anderson impurity problems [Andrei, Furuya, Lowenstein (1983); Tsvelik, Wiegmann (1983)]

dynamical (boundary) impurities in lattice models[Andrei, Johannesson (1984); Bedürftig, Essler, HF (1997); HF, Zvyagin (1997); Zhou et al. (1999); HF, Slavnov (1999); . . . ]

integrable defects in continuum models (→ Corrigan)

Z2-staggered models

simplest case: 6v model with alternating shift in spectral parameter� one of integrable manifolds of square lattice Potts model[Temperley, Lieb (1971); Baxter, Kelland, Wu (1976)]

also: spin chains with n-neighbour interactions [Popkov, Zvyagin (1993); HF, Rödenbeck (1996)]

Holger Frahm: Staggered superspin chain | MATRIX, July 2017 2

Staggered superspin chains

Network models for disorder problems(e.g. QH plateau transition)

average over realizations by replica trick orsupersymmetry techniques

mapping to spectral problem for spin chains withalternating representations of super Lie algebra[Zirnbauer (1994); Gruzberg et al. (1999)]

� possible insights into critical properties at disorder driven quantum phasetransitions from vertex models with alternating representations of superalgebra as

Integrable ’network’ modelssimplest cases: finite dimensional irreps of sl(2|1) and deformation Uq[sl(2|1)]

three-dim. quark/antiquark (3 / 3̄) [Gade (1999); Essler, HF, Saleur (2005); HF, Martins (2011)]

four-dim. typical irrep [b, 12 ] and dual [−b,12 ] [HF, Martins (2012)]


Staggered Uq[sl(2|1)] superspin chainsCartan subalgebra generated by charge B, spin J3

Z2 graded basis: fermionic (2j3 odd) and bosonic (2j3 even) statesstaggered chain of 4-dim. typical reps ([b, 12 ]⊗ [−b,

12 ])

two parameters: b and deformation q = eiγ

special lines:– γb = π/4: self-dual under γb ↔ π2 − γb– b = 12 : projection onto 3⊗ 3̄ chain (degeneration into atypical irreps)

Hidden six-vertex models [HF, Martins (2012)]:

zero-charge sector of the (periodic) superspin chain contains staggeredsix-vertex model (shifts ±ibγ) with anti-periodic boundary conditionsspecial lines:

– γb = π/4: q-state Potts model / twisted Uq[D(2)2 ] quantum group

symmetry– b = 12 : spin-1 XXZ chain (fusion

12 ⊗

12 → 1)


Staggered Uq[sl(2|1)] superspin chainsPhase diagram [HF,Martins (2012)]

phases A1, A2: two (compact) bosonic degrees of freedom (spin and charge)

phases B (b > 12 ) and C (b ≥12 ):

in addition one compact degree of freedom (charge resp. spin) plus acontinuum of critical exponents[Essler, HF, Saleur (2005); Jacobsen, Saleur (2006); Ikhlef, Jacobsen, Saleur (2008); HF, Martins (2011&2012)]

this talk: b = 12 between A1 and C, i.e. afm 3⊗ 3̄ chainisotropic point γ = 0 [Essler,HF,Saleur (2005)], 0 < γ < π/2 [HF,Martins (2011); HF, Hobuß (2017)]


The 3⊗ 3̄ superspin chainYang-Baxter equations: Rjk ∈ End(Vj ⊗ Vk)

R(ω1,ω2)12 (λ)R(ω1,ω3)13 (λ+ µ)R

(ω2,ω3)23 (µ) = R

(ω2,ω3)23 (µ)R

(ω1,ω3)13 (λ+ µ)R

(ω1,ω2)12 (µ)

(ωj ∈ {3, 3̄} labels representation on space Vj)two commuting transfer matrices on space (3⊗ 3̄)⊗L

τ (3)(λ) = strA(G(3)(ϕ)R(3,3)A,2L(λ)R

(3,3̄)A,2L−1(λ− iγ)R

(3,3)A,2L−2(λ) . . .R

(3,3̄)A,1 (λ− iγ)

)τ (3̄)(λ) = strA

(G(3̄)(ϕ)R(3̄,3)A,2L(λ+ iγ)R

(3̄,3̄)A,2L−1(λ)R

(3̄,3)A,2L−2(λ+ iγ) . . .R

(3̄,3̄)A,1 (λ)

)twist matrices G(ω)(ϕ = α + π) = exp

(2iαJ

(ω)3

)ϕ = π (α = 0): periodic boundary conditions for fermions and bosons

ϕ = 0 (α = π): strA → trA, i.e. periodic b.c. for bosons, antiperiodic b.c. forfermions

correspond to Ramond / Neveu-Schwarz sector in the continuum limit

generic ϕ: allows to study spectral flow between these sectors


Local integrals of motion

Local integrals of motion generated by double row transfer matrix

τ(λ) = τ (3)(λ)τ (3̄)(λ)

e.g. Hamiltonian

H = i ∂∂λ

ln τ(λ)∣∣∣λ=0

nested algebraic Bethe ansatz: eigenvalues Λ(λ) = Λ3(λ)Λ3̄(λ) of τ(λ) are

parametrized by complex roots {λ(1)j }N1j=1 and {λ

(2)j }

N2j=1 of[

sinh(λ(1)j + iγ)

sinh(λ(1)j − iγ)

]L= −eiϕ

N2∏k=1

sinh(λ(1)j − λ

(2)k + iγ)

sinh(λ(1)j − λ

(2)k − iγ)

, j = 1, . . . ,N1 , (1)

[sinh(λ

(2)j + iγ)

sinh(λ(2)j − iγ)

]L= −eiϕ

N1∏k=1

sinh(λ(2)j − λ

(1)k + iγ)

sinh(λ(2)j − λ

(1)k − iγ)

, j = 1, . . . ,N2 . (2)

in sector with charge b = (N1 − N2)/2 and spin j3 = L− (N1 + N2)/2.


The XXZ spin-1 subsector

solutions with {λ(1)j }N1j=1 ≡ {λ

(2)j }

N2j=1 (i.e. charge b = 0):

Bethe eqs. reduce to those of XXZ spin-1 chain (twist ϕ+ π),CFT is known: Ising ⊗ U(1) Kac-Moodyallows for identification of effective scaling dimensions in finite size spectrumof the superspin chain:

X eff(m,w)(ϕ) = −1

4δm+w∈2Z +

m2

2k+

k

2

(w +

ϕ

π

)2, k =

π

π − 2γ

m: quantum number j3 of the corresponding state in the superspin chainw : vorticity of that state


The XXZ spin-1 subsector

solutions with {λ(1)j }N1j=1 ≡ {λ

(2)j }

N2j=1 (i.e. charge b = 0):

Bethe eqs. reduce to those of XXZ spin-1 chain (twist ϕ+ π),CFT is known: Ising ⊗ U(1) Kac-Moodyallows for identification of effective scaling dimensions in finite size spectrumof the superspin chain:

X eff(m,w)(ϕ) = −1

4δm+w∈2Z +

m2

2k+

k

2

(w +

ϕ

π

)2, k =

π

π − 2γ

m: quantum number j3 of the corresponding state in the superspin chainw : vorticity of that state

NB: X eff(m,w)=(0,−1) vanishes identically in the Ramond sector (ϕ = π) of the

superspin chain, sl(2|1) singlet (b, j3) = (0, 0)!highly degenerate Bethe root configuration λ

(a)j ≡ 0, a = 1, 2, j = 1..L

ground state (c = 0) of the periodic superspin chain (and related spin-1 XXZmodel with anti-periodic boundary conditions) for anisotropies γ ∈ {0, π4 }.singlet under an exact lattice supersymmetry of the XXZ spin-1 chain [Hagendorf 15]


More zero charge states

string classification of Bethe roots

XXZ-1 sector: low energy states from cc. pairs with Im(λ(1,2)) = ±iγ/2strange 2-strings: pairs of rapidities λ(1) = (λ(2))∗

• type +: λ(1) = λ0 + iγ/2, λ(2) = λ0 − iγ/2 for real λ0• type −: λ(1) = λ0 − iγ/2, λ(2) = λ0 + iγ/2low energy states parametrized by N± strings of type ±, ∆N = N+−N− 6= 0:charge b = 0, spin j3 = L− (N+ + N−)

found to exist for (m + w) even, effective scaling dimensions are

X = X eff(m,w) + O

((∆N

ln L

)2)

(strong log. corrections)

Figure: finite size spectrum for γ = π/3,

the (m,w) = (0,−1) zero energy state and(1,−1) states with ∆N = 0, 1, 2, 3

0 0.25 0.5

1/ln(L)

-0.1

0

0.1

0.2

LE

/2π

vF

XXZ, L odd

∆N=1, L even

∆N=2, L odd

∆N=3, L even



string classification of Bethe roots

XXZ-1 sector: low energy states from cc. pairs with Im(λ(1,2)) = ±iγ/2strange 2-strings: pairs of rapidities λ(1) = (λ(2))∗

• type +: λ(1) = λ0 + iγ/2, λ(2) = λ0 − iγ/2 for real λ0• type −: λ(1) = λ0 − iγ/2, λ(2) = λ0 + iγ/2low energy states parametrized by N± strings of type ±, ∆N = N+−N− 6= 0:charge b = 0, spin j3 = L− (N+ + N−)found to exist for (m + w) even, effective scaling dimensions are

X = X eff(m,w) + O

((∆N

ln L

)2)

(strong log. corrections)

Figure: finite size spectrum for γ = π/3,

the (m,w) = (0,−1) zero energy state and(1,−1) states with ∆N = 0, 1, 2, 3

0 0.25 0.5

1/ln(L)

-0.1

0

0.1

0.2

LE

/2π

vF

XXZ, L odd

∆N=1, L even

∆N=2, L odd

∆N=3, L even



emergent continuous spectrum of conformal weights in the thermodynamic limit!

Proposal for isotropic model (γ = 0) [Essler, HF, Saleur (2005); Saleur, Schomerus (2007)]

lattice model flows to SU(2|1) WZNW model at level 1modular invariant partition function built from characters of generic class Iirreps of the affine ŝl(2|1)k=1: spin j = 12 , charge b ∈ C, conformal weighthR = j

2−b2k+1 =

18 −

b2

2 — now lower bound!

physical spectrum requires analytical continuation of charge b → iβ� continua starting at observed eff. dimensions (m + w even)caveat: “discrete levels” (m + w odd), e.g. the sl(2|1) singlet with X eff = 0,are not captured . . . (artefact of lattice model? / normalization of part. fct.?)



Questions:

which continuous quantum number (≡ eigenvalue of operator commutingwith transfer matrix) determines the log. amplitudes? (related to(∆N = N+ − N−)/ ln L)completeness of “discrete levels” in the CFT?

role of boundary conditions (spectral flow NS ↔ R)?

Easier to analyze in the anisotropic model (additional parameter)!


Integrals odd under 3̄↔ 3Observation: τ(λ) is even under interchange 3̄↔ 3: eigenvalues of configurationswith N± (±)-strings and N∓ (±)-strings are degenerate.Consider integrals generated by [Ikhlef, Jacobsen, Saleur (2012)]

τo(λ) = τ(3)(λ)

[τ (3̄)(λ)

](−1)e.g. “quasi-momentum”

K = γ2π(π − 2γ)

ln

(τ (3)(λ)

[τ (3̄)(λ)

]−1) ∣∣∣λ=0

.

q-momentum of single real Bethe root is imaginary

single strange string of type ± with rapidities Re(λ(1,2)) = λ contributes

k+(λ) = −k−(λ) = ln(

cosh(2λ)− cos(3γ)cosh(2λ)− cos(γ)

)renormalized in distribution of ±-strings (as in low-E states) to

q+(λ) = −q−(λ) = − ln coshπλ

γ+ C1 λ+ C0 .


Quasi-momentum

low lying excitations above ∆N = 0 states in the (m,w) = (1,−1) sector

2 3 4 5 6 7 8ln(L)

0

1

2

3

4∆

Ν /

Κ

γ = π / 40

γ = 9 π / 40

γ = 15 π / 40

2 3 4 5 6 7 8ln(L)

0

1

2

3

4

∆ Ν

/ Κ

γ = π/40

γ = 9π / 40

γ = 15π / 40

Ramond (ϕ = π) Neveu-Schwarz (ϕ = 0)

linear dependence of ∆N/K on ln L!


Quasi-momentum

low lying excitations above ∆N = 0 states in the (m,w) = (1,−1) sector

2 3 4 5 6 7 8ln(L)

0

1

2

3

4∆

Ν /

Κ

γ = π / 40

γ = 9 π / 40

γ = 15 π / 40

2 3 4 5 6 7 8ln(L)

0

1

2

3

4

∆ Ν

/ Κ

γ = π/40

γ = 9π / 40

γ = 15π / 40

Ramond (ϕ = π) Neveu-Schwarz (ϕ = 0)

Conjecture for subleading contribution to the scaling dimensions in terms of thequasi momentum K based on finite size data ∆N = 1, . . . , 13, and L up to 2000:

X eff(1,−1)(K ) = Xeff(1,−1) +

π2

4

π − 2γπ + 2γ

(∆N

ln(L/L0)

)2= X eff(1,−1) +

2k

(k − 1)2K 2 , k =

π

π − 2γ


Quasi-momentum

Quasi-momentum K ≡ continuous quantum number for the continuum of states!

Problem: scale of K cannot be fixed – any other dependence on level k ispossible!

possible solutions:

compare density of states in the continuum against prediction for a candidateCFT (which??):used in the staggered six vertex model → SL(2,R)/U(1) σ-modelidentify possible discrete part of conformal spectrum:non-normalizable states (not in Hilbert space of the periodic chain) mayappear (i.e. become normalizable) under adiabatic change of twist ϕ

similar as for a(2)2 model (19-vertex Izergin-Korepin model, regime III)

[Vernier,Jacobsen,Saleur (2014)]

here: ϕ = 0 . . . π maps states in Neveu-Schwarz sector to states in Ramondsector — spectral flow between two related theories


Spectral flow – XXZ-1 subsector

0 π/2 πϕ

-0.2

-0.1

0

0.1

X

X for m+w evenX for m+w odd

(0,0)

(2,0)

(1,-1)

(0,-1)

ϕ = 0 (Neveu-Schwarz): X eff,NS(0,0) = −14 (singlet g.s. of periodic XXZ-1 chain)

ϕ = π (Ramond): X eff,R(0,−1) = 0 (E = 0 state of the antiper. XXZ-1 chain)

X eff,R(1,−1): ground state for π/4 < γ (here we use γ = 17π/40)

under twist: X eff(m,w)(φ) = Xeff(m,w)(φ = 0) +

k2

(w + φπ

)2Holger Frahm: Staggered superspin chain | MATRIX, July 2017 15

Spectral flow

0 π/2 πϕ

-0.2

-0.1

0

0.1

X

X for m+w evenX for m+w odd

(0,0)

(2,0)

(1,-1)

(0,-1)

zero-charge levels in the superspin chain

m + w even: X(m,w) is lower edge of continuum of scaling dimensions(as L→∞)m + w odd: discrete level


Spectral flow

0 ϕc π/2 πϕc,2ϕ

-0.2

-0.1

0

0.1

X

finite size dataX for m+w evenX for m+w oddconjecture

ϕ = 0 (Neveu-Schwarz)

odd L: lowest level has N± = (L± 1)/2 (±)-strings: ∆N = ±1further excitations: (m,w) = (0, 0), ∆N = ±3,±5, . . ., gaps ' (ln(L))−2

(m,w) = (2, 0), another continuum. . .


Spectral flow


-0.2

-0.1

0

0.1

X


ϕ = π (Ramond)

discrete E = 0 state ∼ singlet g.s. of antiperiodic XXZ-1 spin chainabove X eff,R(1,−1): Bethe roots are (±)-strings, lower edge of continuum


Spectral flow


-0.2

-0.1

0

0.1

X


finite twist ϕ = 0 . . .:

compact (XXZ-1) part: X eff(m,w)(φ) = Xeff(m,w)(φ = 0) +

k2

(w + φπ

)2lowest (|∆N| = 1) level X ∗(0,0)(ϕ) splits off continuum at finite twistϕc = π/k!completely different ϕ-dependence afterwards!


Spectral flow


-0.2

-0.1

0

0.1

X


finite twist ϕ = 0 . . . π:

another change at ϕc,2 = π − π(k−1)k(2k−1)as ϕ→ π (Ramond sector): flows to a discrete level X ∗ = 14(2k−1) =

14π−2γπ+2γ

root config. for X ∗: 2-strings on either level, not strange strings!


Spectral flow

X ∗(0,0)(ϕ) =

X eff(0,0)(ϕ) for |ϕ| < ϕcX eff(0,0)(ϕ)−

2k−1(k−1)2

(12 −

k2

∣∣ϕπ

∣∣)2 for ϕc ≤ |ϕ| ≤ ϕc,2k2

(1− ϕπ

)2+ 14

12k−1 for ϕc,2 ≤ ϕ ≤ π

increasing the twist further gives X ∗(0,0)(ϕ) = X∗(0,0)(2π − ϕ)

finite size scaling of the discrete level X ∗ = 14(2k−1) in the Ramond sector has

strong f.s. corrections (unlike XXZ-1 levels)

similar behaviour is found for lowest states from other continua, e.g.X ∗(1,−1)(ϕ) for |ϕ− π| >

2πk .

quasi-momentum?


Spectral flow - quasi-momentum

0 π/2ϕc ϕc,2ϕ

0

1

2

Im(K

)

L=3L=27conjecture for Im(K)

0.0

0.1

0.2

Re(

K)

twist ϕ = 0 . . . ϕcreal quasi-momentum K of ∆N = 1 state (in continuum)→ Bethe root config. contains only strange stringsfinite size scaling: K ∼ ∆Nln L vanishes for L→∞cf. staggered 6v model / Euclidean black hole: SL(2,R) affine primaries fromcontinuous series with spin j = − 12 + iK [Ikhlef,Jacobsen,Saleur (2012)]


Spectral flow - quasi-momentum

0 π/2ϕc ϕc,2ϕ

0

1

2

Im(K

)

L=3L=27conjecture for Im(K)

0.0

0.1

0.2

Re(

K)

twist ϕ = ϕc . . . πimaginary quasi-momentum with linear ϕ-dependence

K∗(ϕ) = i

(kϕ

2π− 1

2

), ϕc ≤ ϕ ≤ ϕc,2

small corrections to scaling (already seen for L = 3)K∗(ϕ) = i(k − 1)2/(2k − 1) for ϕ > ϕc,2


Continuum limit

Euclidean black hole CFT – infinite cigar shaped target space

spectrum of conformal dimensions in mini-superspace limit k →∞:

∆jm = −2j(j + 1)

k+

m2

2k, j ∈ −1

2+ iR , m ∈ Z

spin j ∼ momentum in non-compact direction of the cigar,m angular momentum in compact direction

CFT: need to include principal discrete representations (real j) of SL(2,R) asKac-Moody primaries in the black hole coset theory [Maldacena,Ooguri (2001)]

normalizable states in parent WZNW model require j ≤ − 12 (� finite twistneeded in lattice model ϕ ≥ ϕc [Ribault,Schomerus (2004); Vernier,Jacobsen,Saleur (2014)])non-negative conformal weights � unitarity bound, e.g. j ≥ −(k − 1)/2 inthe bosonic SL(2,R)/U(1) coset at level k [Hanany,Prezas,Troost (2002)]

Consistent with

j = − 12 + iK


Continuum limit

Candidate CFTs:

bosonic SL(2,R)/U(1) coset would give

X ∗(0,0)(ϕ) = Xeff(0,0)(ϕ)−

2

k − 2

(1

2− k

2

∣∣∣ϕπ

∣∣∣)2 , −k − 12≤ j ≤ −1

2

supersymmetric Kazama-Suzuki SL(2,R)/U(1) coset model

. . .− 2k

(1

2− k

2

∣∣∣ϕπ

∣∣∣)2 , −k + 12≤ j ≤ −1

2

rather than

. . .− 2k − 1(k − 1)2

(1

2− k

2

∣∣∣ϕπ

∣∣∣)2 , −k2≤ j∗ < j ≤ −1

2

observed in the twisted 3⊗ 3̄ chain.


Summary & Outlook

finite size analysis of a staggered Uq[sl(2|1)] superspin chain with alternatingquark (3) and antiquark (3̄) representations

continuous spectrum of conformal dimensions can be labelled by realeigenvalues of quasi momentum K representing a family of commutingtransfer matrices which are odd under the interchange 3↔ 3̄� continuous quantum number for non-compact degree of freedomspectral flow analyzed through twisted boundary conditions interpolatingbetween Neveu-Schwarz and Ramond sector: appearance of discrete statesin the continuum limit, labelled by imaginary quasi-momentum K

flow of continuum states ↔ discrete states

CFT?

F. H. L. Essler, HF, H. Saleur: Nucl. Phys. B 712 (2005) 513; cond-mat/0501197HF and M. J. Martins: Nucl. Phys. B 847 (2011) 220; arXiv:1012.1753

HF and K. Hobuß: J. Phys. A 50 (2017) 294002; arXiv:1703.08054


IntroductionStaggered superspin chain modelSummary & Outlook

Documents

Continuum limit of a staggered superspin chain...simplest case: 6v model with alternating shift in spectral parameter ˜one of integrable manifolds of square lattice Potts model [Temperley,