Alexei Tsvelik (Ph. Lecheminant, A. Tsvelik).

}  We study the SU(2n), n>1 generalization of the S=1/2 spin ladder.

}  The phase diagram in the SU(2) case contains two disordered }  phases one of which is topological ( Haldane phase).

}  At phase boundaries there are Majorana zero modes.

}  In contrast the n>1 case has no topological phases, }  only valence bond solids. The issue of zero modes is open.

}  The excitation spectrum is very rich. There are regions of emergent enhanced symmetry.

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H = Pj , j+11,1 + Pj, j+1

2,2 + J⊥Pj , j1,2 + JXPj, j+1

1,1 Pj , j+12,2⎡

⎣ ⎢ ⎤ ⎦ ⎥

n∑

Where Pa,b is a permutation operator. The lattice model is just one of many models which have the same continuum limit as a sum of Wess-Zumino-Novikov-Witten models perturbed by relevant operators:

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H =W SU1(2n),g1[ ] +W SU1(2n),g2[ ] + λ1 d xTr(g1g2+ + H.c.)∫ +

λ2 d2x(Trg1Trg2+ + H.c.)∫

Where we kept only the most relevant operators with dimension d =2- 1/n.

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SnA = JR

A + JLA( ) + e2ikF nN A + e−2ikF nN A ,+ + ...,

NA = aTr(gτA ),SnASn+1

A = TR +TL + b(e2ikF nTrg+ H.c.) + ...

Where kF= p/2n, a,b are numerical constants, J are SU1(2n) Kac-Moody currents.

}  Conformal embedding:

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SU1(N) × SU1(N) ~ SU2(N) × ZN

Where the SU2(N) CFT has c=2(N2 -1)/(N+2) and ZN is the parafermionic CFT with c =2(N-1)/(N+2).

�

H =W SU2(2n);G[ ] +

H Z2n[ ] + (nλ1 /2π2) dx∫ Ψ1LΨ1R + H.c.( ) +

λ2 dxTr(Φadj∫ ) σ 2 +σ 2+( ).

�

HFZ = H Z2n[ ] + (nλ1 /2π2) dx∫ Ψ1LΨ1R + H.c.( )

The model decouples into the massless SU2(2n) and an integrable Fateev model of massive Z2n parafermions:

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σk, k =1,...2n −1;

σk = σ2n−k+ ,

dk =k(2n − k)4n(n +1)

.

are order parameters of the Z2n model. Their scaling dimensions are dk . Yk are chiral fields with conformal dimensions

�

k(2n − k)2n

,0⎛ ⎝ ⎜

⎞ ⎠ ⎟ , 0, k(2n − k)

2n⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,

}  The spectrum consists of massive kinks mass M ~ l1n interpolating between degenerate vacua s=1,3,…2n+1. The OPs expectation values

}  depend on the vacuum:

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< 0s |σ j | 0s >=sin π ( j +1)s

2n + 2⎡ ⎣ ⎢

⎤ ⎦ ⎥

sin πs2n + 2⎡ ⎣ ⎢

⎤ ⎦ ⎥ (M /4)d j eQ j .

j =1,...2n −1;

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H =W SU2(2n);G[ ] + λ2 dxTr(Φadj∫ ) < σ 2 +σ 2+( ) > .

We replace the Z2n OP by its expectation value, but it is not uniquely defined. The theory must choose the right vacuum to minimize the energy.

�

H =12∂µχ( )2 − λ1 cos( 6π χ) − λ2 cos( 2π /3χ) cos 8π /3 Φa −Φb( )[ ]

a>b∑ +

12∂µΦa( )2 .

a=1

6

∑

The sector of adjoint fields can be bosonized, the s OPs are nonlocal. The vacuum average

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< cos( 2π /3χ) > has different values for different vacua.

}  l1 >0, l2 >0 }  The coupling achieves }  Its maximum at the single

}  value of c field. The Sine-Gordon kinks confine.

l1 >0, l2 <0, The coupling achieves Its maximum at the two values of c field.

Cos c

Cos(c/3)

�

H =12∂µχ( )2 − λ1 cos( 6π χ) − λ2 cos( 2π /3χ) cos 8π /3 Φa −Φb( )[ ]

a>b∑ +

12∂µΦa( )2 .

a=1

6

∑ →

12∂µΦa( )2 .

a=1

6

∑ − < λ2 cos( 2π /3χ) > cos 8π /3 Φa −Φb( )[ ]a>b∑ =

i −Ra+∂xRa + La

+∂xLa( ) + γ 0 Ra+RaLa

+La( ) − γ1Ra+LaLb

+Rb .

The spectrum at each Z4 vacuum is like SU(6) Chiral Gross-Neveu model mj = M sin(pj/3), j=1,2,…5. The effects of the anisotropy? Bound states like in the Sine-Gordon?

�

Tr g1 ± g2( ) ~ TrG σ1 ± σ1+( )

We have to consider different cases. For example: A.  l1 <0, l2 <0. The ground state corresponds to s=1 when the coupling

�

η = λ2 < σ 2 +σ 2+( ) > . is maximal. Since

�

< σ1 >s=1=<σ1+ >s=1( )

we have <Trg1 > =<Trg2 > and 2kF VBS.

}  The phase diagram of the SU(2n) (n>1) ladder contains just Valence Bond Solids, - phases with spontaneously broken symmetry.

}  There are no disordered gapped phases as the Haldane and the Rung Singlet ones for the SU(2) ladder.

}  The excitation spectra are very rich: already for the SU(4) ladder we have SU(6) (for 2kF VBS) or SU(6)xSU(6) (4kF VBS) emergent symmetry

}  mj = M sin(pj/3), j=1,2,…5, }  M ~ (l2 )2 }  On top of these there may be }  a.) bound states of these particles }  b.) heavy kinks of the c –field (Fateev’s Z4 parafermions).