Home
Search
Collections
Journals
About
Contact us
My IOPscience
Topological quantum phase transitions and topological flat bands on the kagomé lattice
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 J. Phys.: Condens. Matter 24 305602 (http://iopscience.iop.org/0953-8984/24/30/305602) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 61.175.228.131 The article was downloaded on 07/07/2012 at 03:32
Please note that terms and conditions apply.
IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 305602 (8pp)
doi:10.1088/0953-8984/24/30/305602
Topological quantum phase transitions and topological flat bands on the kagom´e lattice Ru Liu1 , Wen-Chao Chen1 , Yi-Fei Wang1 and Chang-De Gong1,2 1
Center for Statistical and Theoretical Condensed Matter Physics, and Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China 2 National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China E-mail: yfwang
[email protected]
Received 22 March 2012, in final form 3 June 2012 Published 6 July 2012 Online at stacks.iop.org/JPhysCM/24/305602 Abstract We investigate the topological properties of the tight-binding electrons on the two-dimensional kagom´e lattice with two kinds of short-range hopping integral and two kinds of staggered magnetic flux. Considering the nearest-neighbor hopping (t1 ) with the staggered flux parameter φ1 and the next nearest-neighbor hopping (t2 ) with the staggered flux parameter φ2 , we demonstrate a series of topological quantum phase transitions and find some topological bands with high Chern numbers, when tuning one parameter (t2 or φ2 ) while the others are fixed. We have also found that, in some parameter regions, the system exhibits interesting topological flat bands with Chern number C = ±1 and a large gap above them, and the flatness ratio can reach a high value of about 170. (Some figures may appear in colour only in the online journal)
1. Introduction
noteworthy that topological flat bands (TFBs) [11–15] with nonzero Chern numbers have been proposed in some 2D lattice systems. The proposal of TFBs points out a possible new avenue to realize the fractional quantum Hall effect (FQHE) without Landau levels, as demonstrated in recent numerical works [13, 16–20]. In this paper, we investigate the possible topological quantum phase transitions (TQPTs) and topological flat bands (TFBs) on the kagom´e lattice with nearest-neighbor (NN) hopping integral t1 , next nearest-neighbor (NNN) hopping integral t2 and two kinds of staggered magnetic flux. Such a topological model might be realized in optical lattices by manipulating cold atoms [21–23], and recent pioneering experiments have actually created artificial staggered gauge fields [24–26]. Some interesting results have been found through explicit calculations of the energy spectrum, the density of states, the Hall conductance and the edge-state spectrum. Firstly, we have observed a series of TQPTs, with the corresponding Chern numbers of the three bands found to change from {−1, 0, +1} to {−1, +2, −1} and then to {+3, −2, −1} when tuning the parameter t2 from 0.0 to 1.0,
The study of topological quantum phase transitions and the topological states of matter has progressed significantly in the past few years. Topological states of matter are the quantum states which are distinguished from ordinary states by nontrivial topological properties. Such topological properties are intrinsically characterized by topologically invariant Chern numbers [1]. The well-known Haldane model [2] is a tight-binding model on the two-dimensional (2D) honeycomb lattice with staggered magnetic fluxes. There are two types of insulating state which are characterized by zero or nonzero Chern numbers. When only the lower topological band of the Haldane model is totally filled, it presents as an intriguing example of the integer quantum Hall effect (IQHE) without Landau levels. Topological bands (TBs) with nonzero Chern numbers have also been found in some other lattice models which are more or less similar to the original Haldane model, e.g. the checkerboard lattice [3–5], the kagom´e lattice [6, 7], the Lieb lattice [8] and the star lattice [9, 10]. Recently, it is 0953-8984/12/305602+08$33.00
1
c 2012 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 24 (2012) 305602
R Liu et al
Figure 1. The kagom´e-lattice model has both NN and NNN hopping integrals. Each NN bond has the phase ±φ1 , and each NNN bond has the phase ±φ2 . The signs of the phases are represented by the directions of the arrows.
with the fluxes φ1 and φ2 being fixed. Secondly, we have also observed TQPTs with the Chern numbers of the three bands found to change from {−1, +2, −1} to {+1, −2, +1} and then to {−1, 0, +1} when tuning the flux φ2 from 0 to 2π , with the parameter t2 and the flux φ1 being fixed. Thirdly, we have found that in some parameter regions the system exhibits interesting TFBs with the Chern number C = ±1 and a large gap above them, and the flatness ratio (the ratio of the band gap over the band width) can reach a high value of about 170.
hhi,jii
In the following we set t1 = 1 as the unit of energy. After Fourier transformation and numerical diagonalization of the Hamiltonian (1), the zero-temperature Hall conductance (HC) is given by the Kubo formula [1] ie2 h¯ X X A E E nk
mk
hnk|vx |mkihmk|vy |nki − hnk|vy |mkihmk|vx |nki (Emk − Enk )2
(n)
It is well-known that the kagom´e lattice has three sublattices, thus it has three bands when only the NN hopping t1 is considered. The top band has the energy E = 2, and the lower two bands contact at the energy E = −1 √ with two Dirac points at K = (2π/3, 0) and K 0 = (π/3, π/ 3) [6, 28]. Previously, when the NN hopping integral t1 and the staggered flux φ1 were considered [6], the three bands were found to have the Chern numbers {±1, 0, ∓1} respectively, thus the top and bottom bands are topological bands. Here we also consider the NNN hopping integral t2 and flux φ2 , and are interested in the possible topological quantum phase transitions (TQPTs) when tuning one parameter (t2 or φ2 ) while the others are fixed. We consider an example where φ1 = 0.30π is fixed and φ2 = 0 while tuning t2 from 0.0 to 1.0, and a systematic evolution of the IQHE plateaus appears (figure 2). At t2 = 0.0, the DOS has three peaks, and among those peaks there are two energy gaps and the Hall conductance exhibits two quantized plateaus both with σH = −(e2 /h). Thus we can label the Chern numbers of the three bands as {−1, 0, +1} respectively (see figure 2(a)), the three bands are separated respectively. At t2 = 0.064, the upper two bands merge together and form a pseudogap; this is the critical point where the first TQPT takes place with a quantized jump from the σH = −(e2 /h) IQHE plateau to the σH = +(e2 /h) IQHE plateau when the lower two bands are totally filled; after the first TQPT, the three bands have the Chern numbers {−1, +2, −1} (see figure 2(b)). When t2 is further increased to 0.20, the three bands separate again with the Chern numbers {−1, +2, −1} (see figure 2(c)). At t2 = 0.40, the lower two bands have a much narrower band gap, and the three bands still have the Chern numbers {−1, +2, −1} (see figure 2(d)). At t2 = 0.59, the lower two bands merge together and form a pseudogap (see figure 2(e)); this is close to the critical point where the second TQPT takes place with a quantized jump from the σH = −(e2 /h) IQHE plateau to the σH = +3(e2 /h) IQHE plateau when only the lowest band is filled; after the second TQPT, the three bands have the Chern numbers {+3, −2, −1}. When t2 = 1.0, the
The kagom´e-lattice model has both NN hopping integrals t1 and NNN hopping integrals t2 , and two kinds of staggered flux as shown in figure 1. This model is similar to the spin-polarized version of the kagom´e-lattice model with spin–orbit couplings [11]. Each NN hopping bond carries the phase ±φ1 , and the signs of the phases are represented by the directions of the arrows. Each elementary triangular plaquette with NN bonds has the flux 3φ1 , and each elementary hexagonal plaquette with NN bonds has the flux −6φ1 . In addition, for the NNN hoppings, each larger triangular plaquette has the additional flux 3φ2 . Each lattice point is connected by four NN bonds and four NNN bonds. The Hamiltonian of this kagom´e-lattice model is given by X X H= t1 (eiφ1 c†i cj + H.c.) + t2 (eiφ2 c†i cj + H.c.). (1)
σH (E) =
P
3. Topological quantum phase transitions
2. Model and formulation
hi,ji
(n) En
