arXiv:1104.0494 · Physical Review Letters

Effective Field Theory for the SO ( n ) Bilinear-Biquadratic Spin Chain

Hong-Hao Tu, Román Orús

Published Aug 9, 2011• 25 citations

Abstract

We present a low-energy effective field theory to describe the SO(n) bilinear-biquadratic spin chain. We start with n=6 and construct the effective theory by using six Majorana fermions. After determining various correlation functions, we characterize the phases and establish the relation between the effective theories for SO(6) and SO(5). Together with the known results for n=3 and 4, a reduction mechanism is proposed to understand the ground state for arbitrary SO(n). Also, we provide a generalization of the Lieb-Schultz-Mattis theorem for SO(n). The implications of our results for entanglement and correlation functions are discussed.

DOI: 10.1103/physrevlett.107.077204

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arXiv:1104.0494 · Physical Review Letters

Effective Field Theory for the SO ( n ) Bilinear-Biquadratic Spin Chain

Hong-Hao Tu, Román Orús

Published Aug 9, 2011• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.107.077204

Related papers

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Oct 23, 19970 citations

Here we investigate the phase diagram of the SO(

n

) bilinear-biquadratic quantum spin chain by studying the global quantum correlations of the ground state. We consider the cases of

n=3,\phantom{\rule{0.28em}{0ex}}4

, and 5 and focus on the geometric entanglement in the thermodynamic limit. Apart from capturing all the known phase transitions for these cases, our analysis shows a number of distinctive behaviors in the phase diagrams which we conjecture to be general and valid for arbitrary

n

. In particular, we provide an intuitive argument in favor of an infinite entanglement length in the system at a purely biquadratic point. Our results are also compared to those of other methods, such as fidelity diagrams.

{}_{1}

universality class.

We introduce a class of exactly solvable

SO(n)

symmetric Hamiltonians with matrix product ground states. For an odd $n\ensuremath{\ge}3$ case, the ground state is a translational invariant Haldane gap spin liquid state; while for an even $n\ensuremath{\ge}4$ case, the ground state is a spontaneously dimerized state with twofold degeneracy. In the matrix product ground states for both cases, we identify a hidden antiferromagnetic order, which is characterized by nonlocal string order parameters. The ground-state phase diagram of a generalized

SO(n)

symmetric bilinear-biquadratic model is discussed.

Using a generalized Jordan-Wigner transformation combined with the defining representation of the SO(

N

) spin, we map the SO(

N

) bilinear-biquadratic (BLBQ) spin chain into the

N

-color bosonic particle model. We find that, when the Jordan-Wigner transformation disentangles the symmetry-protected topological entanglement, this bosonic model becomes negative-sign-free in the context of quantum Monte Carlo simulation. For the SO(3) case, moreover, the Kennedy-Tasaki transformation for the

S=1

BLBQ chain, which is also a topological disentangler, derives the same bosonic model through the dimer-R bases. We present the temperature dependence of the energy, entropy, and string order parameter for the SO(

N=3,4,5

) BLBQ chains by a world-line Monte Carlo simulation for the

N

-color bosonic particle model.

_1

WZW model (or equivalently 5 massless Majorana fermions) deformed by symmetry breaking terms that nonetheless leave the theory critical at T=0. The SO(5) currents of the theory comprise the charge and spin currents and linear combinations of the so-called pi operators (S.C. Zhang, Science 275, 1089 (1997)) which are local in terms both of the original fermions and those of the effective theory. Using bosonisation we obtain the asymptotic behaviour of all correlation functions. We find that the 5 component ``superspin'' vector has power law correlations at T=0; other fermion bilinears have exponentially decaying correlations and the corresponding tendencies are suppressed. Conformal field theory also allows us to obtain the energies, quantum numbers, and degeneracies of the low-lying states and fit them into deformed SO(5) multiplets.