Pfaffian-like ground state for three-body hard-core bosons in one-dimensional lattices

Abstract

We propose a Pfaffian-like Ansatz for the ground state of bosons subject to 3-body infinite repulsive interactions in a 1D lattice. Our Ansatz consists of the symmetrization over all possible ways of distributing the particles in two identical Tonks-Girardeau gases. We support the quality of our Ansatz with numerical calculations and propose an experimental scheme based on mixtures of bosonic atoms and molecules in 1D optical lattices in which this Pfaffian-like state could be realized.Our findings may open the way for the creation of non-abelian anyons in 1D systems.Beyond bosons and fermions, and even in contrast to the fascinating abelian anyons (AA) [1], non-abelian anyons (NAA) [2] exhibit an exotic statistical behavior: If two different exchanges are performed consecutively among identical NAA, the final state of the system will depend on the order in which the two exchanges were made. NAA appeared first in the context of the fractional quantum Hall effect (FQHE) [2], as elementary excitations of exotic states like the Pfaffian state [3,4], the exact ground state of quantum Hall Hamiltonians with 3-body contact interactions. Recently, the possibility of a fault tolerant quantum computation based on NAA [5] has boosted the investigation of new models containing NAA [6], as well as the search for techniques for their detection and manipulation [7]. Meanwhile, the versatile and highly controllable atomic gases in optical lattices [8] have opened a door to the near future implementation of those models as well as for the artificial creation of non-Abelian gauge potentials [9].All actual models containing NAA are 2D models. The motivation of the present work is the foreseen possibility of creating NAA in one-dimension (1D). This long-term goal requires in the first place to define the concept of NAA, which is essentially 2D, in 1D. For abelian anyons (AA) this generalization has been already made by Haldane [10]. Within his generalized definition the spinon excitations of 1D Heisenberg antiferromagnets are classified as 1 2 -AA [10]. This classification becomes very natural through the connection between the 1D antiferromagnetic ground state (for a long-range interaction model, the Haldane-Shastry model [11]) and the Laughlin state [12] for bosons at ν = 1/2. In a similar way we anticipate that a connection can be established between quantum Hall models containing NAA and certain long-range 1D spin models exhibiting NAA within a generalized definition [13].Here, far from analyzing the above questions in general, our aim is to pave the way for the creation of exotic Pfaffian-like states in 1D systems, which we believe may serve as the basis to create NAA. We present a realistic 1D system whose ground state is very close to a Pfaffian-like state. The actual system we consider is that of bosonic atoms in a 1D lattice with infinite re-pulsive 3-body on-site interactions, which we call 3-hardcore bosons. Inspired by the form of the fractional quantum Hall Pfaffian state for bosons [4,14], we propose an Ansatz for the...

Low-energy effective theory for one-dimensional lattice bosons near integer filling

“…These results agree with those in Ref. 15. It supports our observation that the ground state laying on the SF-MI phase boundary is nothing but the Pfaffian-like state.…”

“…Similar to the previous discussion, the interacting lattice boson model in Ref. 15 can again be mapped to a spin-one chain in Eq. (2) with U = V = 0 and n 0 = 1.…”

“…That is, the lattice boson model studied in Ref. 15 is just a special case of the lattice spin-one Hamiltonian studied in the present paper. Therefore, the proposed gapless Pfaffian-like state should correspond to one of the gapless ground states described by the present continuum effective field theory.…”

“…Since our main concern is in discussing the phase transition among various insulating phases, where the local particle number fluctuations should not be strong, near an integer filling n 0 (i.e.,n ≃ n 0 ), one can truncate the bosonic Fock space to only three local states with particle numbers n 0 and n 0 ± 1. 9,15,18,19,20 Now the lattice boson model in Eq.…”

“…Recently, due to the possibility of a fault tolerant quantum computation based on non-abelian anyons, 12 people begin to search for models containing non-abelian anyons, 13 as well as for techniques for their detection and manipulation. 14 It is pointed out that bosonic atoms in a 1D lattice with infinite repulsive three-body on-site interactions aroundn = 1 have a ground state very close to a gapless Pfaffian-like state, 15 which may serve as the basis to create non-abelian anyons in 1D systems.…”