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Zhou Wen-yan, Wu Ya-jie, Kou Su-Peng. Bogoliubov excitations in a Bose–Hubbard model on a hyperhoneycomb lattice. Chinese Physics B, 2018, 27(5): 050302  
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Bogoliubov excitations in a Bose–Hubbard model on a hyperhoneycomb lattice
Zhou Wen-yan1, Wu Ya-jie2, Kou Su-Peng1, †
Department of Physics, Beijing Normal University, Beijing 100875, China
School of Science, Xi’an Technological University, Xi’an 710032, China

 

† Corresponding author. E-mail: spkou@bnu.edu.cn

Abstract

We study the topological properties of Bogoliubov excitation modes in a Bose–Hubbard model of three-dimensional (3D) hyperhoneycomb lattices. For the non-interacting case, there exist nodal loop excitations in the Bloch bands. As the on-site repulsive interaction increases, the system is first driven into the superfluid phase and then into the Mott-insulator phase. In both phases, the excitation bands exhibit robust nodal-loop structures and bosonic surface states. From a topology point of view, these nodal-loop excitation modes may be viewed as a permanent fingerprint left in the Bloch bands.

PACS: 03.75.Kk;67.25.dj;37.10.Jk
Keyword:hyperhoneycomb lattice;Mott-superfluid transition;Bogoliubov excitations;nodal-loop
1. Introduction

Owing to the existence of emergent Dirac fermions, the two-dimensional (2D) honeycomb lattice has attracted the considerable attention of researchers in condensed matter physics. Recently, the 2D honeycomb lattice was materialized in graphene, silicene and other related materials. Naturally, there is a three-dimensional (3D) extension to the honeycomb lattice, it has been addressed in a number of works in both theory and experiments, such as hyperhoneycomb[1–3] and stripyhoneycomb lattices,[3] which have been realized in β- and γ- , loop-nodal semimetal[4,5] and topological insulators,[4] nodal ring,[6,7] and Weyl spinons[7] of interacting spin systems. In particular, Ezawa proposed a wide class of 3D honeycomb lattices constructed by two building blocks and ,[8] which subsequently facilitated the research.

According to the correspondence between bulk and edge, the energy bands of excited bosonic modes with topological structure always give rise to topologically protected edge modes. In order to understand these new exotic quantum states, there are growing efforts in searching for the topology of bosonic modes in interacting bosons on optical lattices,[9,10] photonic systems,[11,12] magnonic excitations,[13,14] phononic excitations,[15–17] polaritonic excitations,[18,19] etc. Accordingly, the topology of magnetic excitations has been widely studied recently, including magnon Chern insulators,[15,20] Weyl magnons,[21,22] magnon nodal-line semimetals[23,24] and Dirac magnons.[25] Here, we will focus on the issue of whether there are excitations with a loop structure in the 3D Bose–Hubbard extension.

In this paper, we study the properties of excitation modes in the Bose–Hubbard extension of the Hamiltonian on hyperhoneycomb lattices and offer an insight into a Bose–Einstein condensate (BEC) on the optical lattice in cold atoms. As the on-site repulsive interaction increases, the system is first driven into the superfluid (SF) phase and then into the Mott-insulator (MI) phase. We find that there exist excitations with nodal line structures on the hyperhoneycomb lattices in both SF and MI phases. Therefore, the topological properties of the excitation modes in the Bose–Hubbard model on the hyperhoneycomb lattices may be viewed as a permanent fingerprint in the Bloch bands.

The paper is organized as follows. In Section 2, we introduce the extended Bose–Hubbard model on the 3D hyperhoneycomb lattice, and then give the band structure for its non-interacting case. In Section 3, we present the phase diagram consisting of superfluid and Mott-insulator phases. In Section 4, the excitation spectra for bosonic superfluids are calculated by means of the Gross–Pitaevskii (GP) theory. In Section 5, we show the excitation spectra in the Mott-insulator phase. Finally, we conclude our discussions in Section 6.

2. Bose–Hubbard model in the hyperhoneycomb lattice

We consider bosonic atoms on a simplest 3D lattice, i.e., the hyperhoneycomb lattice. See the illustrations in Fig 1. A hyperhoneycomb lattice is a three-dimensional system. The unit cell contains two building blocks (four atoms) and each atom is connected by three coplanar bonds spaced by 120°. Each unit cell contains four atoms, of which atoms 1–3 are in the xz plane, and atoms 2–4 are in the yz plane.[5,8] The Bose–Hubbard model on the hyperhoneycomb lattice is then written as

where represent the nearest neighbor sites, is the hopping parameter along the z-axis, and are other hopping parameters. In this paper, we set . ai is the bosonic annihilation (creation) operator at the lattice site i. μ is the chemical potential, and U describes the on-site interaction between the bosons.

Fig. 1. (color online) (a) The building blocks of a hyperhoneycomb with four atoms per unit cell. They are located at , , , and each atom has three neighbors connected by 120° bonds. Atoms 1 and 2 form the vertical building block, atoms 3 and 4 constitute the horizontal building block. The two blocks are connected by bonds along the z-direction. (b) The simplest 3D generalization of the hyperhoneycomb lattice. The four-site unit cell and primitive translation vectors are indicated, i.e., , , .

We first perform a Fourier transformation on the Hamiltonian by using , where N is the number of sites of the sublattice α, and α =1,2,3,4 denote the four atoms in a unit cell. Then we obtain the Hamiltonian in the momentum space for the non-interacting case with U = 0,

where
, and with . At zero temperature, the bosons condense into the lowest energy mode at the momentum point . The energy line with forms a nodal loop with and in the momentum space. Therefore, this system exhibits nodal-loop excitations in the Bloch bands.

3. Mott–superfluid transition

As the on-site repulsive interaction increases, the system will be driven from the SF phase into MI phase. In the strong coupling limit, i.e., , the bosons tend to localize to form an MI.

By the mean-field approach, which is capable of describing the Mott insulating phase, we introduce the superfluid order parameter as .[26,27] Then the term is written as

where j denotes the coordinate of lattice site iʼs nearest neighbor cell. We assume , and . The effective Hamiltonian is given by , where
In the occupation-number basis, the odd powers of the expansion of energy are always zero. Therefore, the zero-order ground state energy is given by
where , and n 1–n 4 are the particle numbers on sublattices 1–4, respectively. The second-order perturbation of energy takes the following form:
where or or or . We readily obtain the ground-state energy in terms of the order parameter ψ
where coefficients a0, a21, a22, b21, b22, c1, c2, c3, c4, c5, and c6 are

By means of the usual Landau procedure for second-order phase transitions, the points corresponding to the secondary derivative coefficients to superfluid order being zero form the phase boundary between the superfluid phase and the MI phase. Therefore, to derive the solutions of phase transition, we should consider the second-order derivation[28] to determinate , where At the point the derivative is obtained as

For the case of , the insulating phase is no longer stable and a phase transition occurs. One can obtain the function of phase-transition line (we chose )
where the subscripts ± denote the two halves of the Mott insulating regions in phase space. Figure 2 shows a plot of Eq. (21) for Correspondingly, we can find the critical point of the smallest on-site interaction

Fig. 2. The phase diagram of the Bose–Hubbard model on a 3D hyperhoneycomb lattice by the second-order perturbation theory. The vertical and horizontal axes represent the dimensionless parameters of the chemical and the interaction strength U/t.

By applying the Landau theory and the second-order perturbation theory of phase transitions, we obtain the phase diagram shown in Fig. 2. It shows that the phase transition occurs at for the MI(1,1,1,1) lobe with the particle number configuration .

4. Excitation modes for the superfluid

The excitation spectra are related to the collective modes of the BEC. In this section, by means of the Gross–Pitaevskii (GP) theory, we discuss excitations from the condensate ground state of the Bose–Hubbard model on the hyperhoneycomb lattice.

For weak interaction strength , the system will be in the superfluid phase. We use the following mean-field approximation:[29]

By replacing by in the Hamiltonian in Eq. (1), the following energy function is obtained
Here, we consider the homogeneous case that refers to the situation where the system has the periodicity of the lattice. The ground state wave function of the BEC can be well described by the GP equation
So we can obtain the equations
After considering the quantum fluctuations of the condensate, the order parameters become[30]
where A, B, C, D, E, F, G, H are small complex parameters, is the position of the lattice site, and k is the reciprocal lattice vector. After plugging these wave functions into the GP function, and keeping the linear terms of A, B, C, D, E, F, G, H, we obtain
The chemical potential is . Next, we simplify Eqs. (27)–(34) to an algebraic equation of the form
where
with and . The energy spectrum can be obtained by calculating the eigenvalues of H. The results are shown in Fig. 3. From the figure, one can see that there is a nodal loop in the excitation bands when . The results indicate that the loop structure has no significant change as the interaction strength increases in the SF phase.

Fig. 3. (color online) (a) The energy spectra of excitations with fixed and . (b) The contour plot of the energy spectra of the third band from the bottom up of the quasiparticle-excitation modes with fixed and , the nodal loop structure is presented by the blue loop.

In order to study the topological properties of the excitation modes of quasiparticles, we calculate the Berry phases[31,32] for paths in momentum space that encircle the nodal line[33]

where is the wave-function. We divide the path into Nj intervals with as[34]
After direct numerical calculations, for a closed loop encircling the nodal lines, we find the Berry phase , which indicates that the flat surface states exist.[35]

In addition, we apply the GP theory to study the edge modes of SF on the hyperhoneycomb lattice, where the system is finite along the z direction and the momentum kx and ky remain good quantum numbers. For certainty, we show the band structures of surface excitations on the zigzag edge of the SF in Fig. 4. It shows that there are partial flat bands appearing in the outer region of the loop structure when both edges are terminated by the zigzag edge, which is similar to the case of the fermionic system.[8]

Fig. 4. (color online) (a) Band structures of a slab with finite width along the z-direction. Here we set Un/t = 0.6. In particular, the edge states are partial flat bands. (b) Band structures of a slab with finite width along the z-direction and . Here we set Un/t = 0.6. The edge states are indicated by the red lines.
5. Excitation modes for the MI phase

In this section, we calculate the quasiparticle and quasihole dispersions using a functional integral formalism.[26] We first define complex functions and . The partition function of the system is written as a function integral

with , where is the Boltzman constant. The action is given by

To decouple the hopping terms, we rewrite the action by using the Hubbard–Stratonovich transformation as in Ref. [26]. Then the effective action is explicitly expressed as

where and ψ are the order parameter fields. After integrating out the complex fields and , we can obtain the effective action up to the second order as
with
The zero-order action goes to zero near the phase transformation point. So we can readily obtain the effective action near the phase transformation point in momentum space by the Fourier transformation
with
where , , , and

By substituting , we can obtain a function of real energies

The quasiparticle and quasihole dispersions are given by
with
where

We present the energy spectra of excitation modes in the MI phase in Fig. 5. It shows that there also exists a nodal-loop structure in the excitation spectra in the MI phase, and the Mott gap becomes large as the interaction strength increases, which is similar to the case in conventional Bose–Hubbard models.

Fig. 5. (color online) (a) The energy spectra of quasiparticle and quasihole excitations with fixed and at the MI–SF transition point. (b) The energy spectra of quasiparticle and quasihole excitations with fixed and in the MI(1,1,1,1) phase.
6. Discussion and conclusion

In this paper, starting from the Bose–Hubbard extension of the Hamiltonian on hyperhoneycomb lattices, we study the properties of the excitation modes. In particular, we find that there exist excitations with nodal loop structures on hyperhoneycomb lattices in both SF and MI phases. Therefore, the topological properties of the excitation modes in the Bose–Hubbard model on hyperhoneycomb lattices may be viewed as a permanent fingerprint in the Bloch bands.

Finally, we discuss its possible realization in cold atoms. The Hamiltonian of the hyperhoneycomb lattice in Eq. (2) on 3D optical lattices has been proposed by using laser-assisted tunneling in Ref. [36]. By using the Feshbach resonance technique, the interaction between the particles can be tuned readily. As a result, this work may provide us with a possible scheme to realize Bogoliubov excitations with nodal loop structures in a bosonic system.

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