Fractional quantum Hall effect in the absence of Landau levels
It has been well-known that topological phenomena with fractional excitations, i.e., the fractional quantum Hall effect (FQHE) Tsui1982 () will emerge when electrons move in Landau levels. In this letter, we report the discovery of the FQHE in the absence of Landau levels in an interacting fermion model. The non-interacting part of our Hamiltonian is the recently proposed topologically nontrivial flat band model on the checkerboard lattice sun (). In the presence of nearest-neighboring repulsion (), we find that at filling, the Fermi-liquid state is unstable towards FQHE. At filling, however, a next-nearest-neighboring repulsion is needed for the occurrence of the FQHE when is not too strong. We demonstrate the characteristic features of these novel states and determine the phase diagram correspondingly.
As one of the most significant discoveries in modern condensed matter physics, FQHE Tsui1982 () has attracted intense theoretical and experimental studies in the past three decades. The wavefunctions proposed by LaughlinLaughlin1983 (); Haldane1983 (); Halperin1984 () first explained FQHE by introducing fractionally charged quasi-particles. Later, the ideas of flux attachment and the composite Fermi-liquid theoryJain1989 () have provided us a rather simple and deep understanding for the nature of the FQHE. Among many of its interesting and unique properties, two of the most striking features of FQHE are: (1) fractionalization, where quasi-particle excitations carry fractional quantum numbers and even fractional statistics of the constituent particles, and (2) topological degeneracy, where the number of degenerate ground states responds nontrivially to the changing of the topology of the underlying manifold. These two phenomena are the central ideas of the topological ordering Wen1990 (). In addition, on the potential application side, they are also the key components in the studies of topological quantum computation Nayak2008 ().
Since 1988, great efforts have been made to study quantum Hall effects in lattice models without Landau levelsHaldane1988 (). The first such an example is the theoretical model proposed by Haldane Haldane1988 (), where it was demonstrated that in a half-filled honeycomb lattice, an integer quantum Hall state can be stabilized upon the introduction of imaginary hoppings. This study was brought to the forefront again recently due to a major breakthrough, in which a brand new class of topological states of matter was discovered, known as the time-reversal invariant Z topological insulators [See Refs. Hasan2010 (); Qi2010 () and references therein]. From the topological point of the view, both Haldane’s model and Z topological insulators can be considered as generalizations of the integer quantum Hall effect. Opposite to the FQHE, they don’t support fractional excitations and the ground states here have no topological degeneracy.
For fractional states, fractional Z topological insulators are found to be theoretically possible Levin2009 (). However, it is highly unclear how to realize such kind of states on lattice models. This difficulty originates from the strong coupling nature of realizing fractionalized states. In both the integer quantum Hall effect and Z topological insulators, all their essential topological properties can be understood within a noninteracting picture. However, for FQHE, interaction effects are expected to play the vital role in stabilizing these fractionalized topological states. In fact, without interactions, a fractional quantum Hall system will become a Fermi liquid due to the fractional filling factor.
Most recently, a series of models with topologically-nontrivial nearly-flat band models have been proposed Tang2010 (); Neupert2010 (); sun (). These lattice models have topologically nontrivial bands, similar to Haldane’s model and Z topological insulators, and their bandwidth can be tuned to much smaller than the band gap, resulting in a nearly-flat band structure. In particular, based on the mechanism of quadratic band touching Tang2010 (); sun (), a large class of flat band models have been explicitly obtained with the ratio of the band gap over bandwidth reaching the high value of . Due to the strong analogy between these nearly-flat bands and the Landau levels, it was conjectured that in these models, FQHE (or fractional topological insulators) can be stabilized in the presence of repulsive interactions. However, the conjecture is challenged by competing orders, e.g. the charge-density wave, and thus the fate of these systems are unclear. More importantly, the flux attachment picture Jain1989 () may also break down here and it is very interesting to examine the nature of the corresponding emergent fractionalized quantum state if it can be realized in such a model without Landau levels.
In this Letter, based on exact calculations of finite-size systems, we report the discovery of FQHE at filling factor and in models with topologically-nontrivial nearly-flat bands sun (). The existence of the FQHE are confirmed by using two independent methods, both of which are well-established and have been widely adopted in the study for the traditional FQHE. We have studied both the characteristic low-energy spectrum and the topologically-invariant Chern numbers Thouless1982 (); Niu1985 (); Arovas1988 (); Huo1992 (); Sheng2003 (); Sheng2005a () of the low-energy states. Here the first method detects directly the topological degeneracy and the other is related with the phenomenon of fractionalization.
We consider the following Hamiltonian on the checkerboard lattice:
where describe the short-range hoppings in the two-band checker-board-lattice model defined in Ref.sun (). is the on-site fermion particle number operator. The nearest-neighboring (NN) and next-nearest-neighboring (NNN) bonds are represented by and , respectively.
The effect of the NN and NNN interactions are summarized in the phase diagrams Figs. 1(a) and (b). At filling factor 1/3, the FQHE emerges with the turn on of a relatively small . Interestingly, the FQHE phase remains robust at the large limit where particles avoid each other at NN sites and can only be destroyed by an intermediate . At filling factor 1/5, the FQHE occurs in the most region of the parameter space as long as the NNN repulsion exceeds a critical value , whose value drops to zero for larger . The observed FQHE states are characterized by (i) nearly -fold ( and for filling factors and respectively) degenerating ground states with the momentum quantum numbers of these states related to each other by a unit momentum translation of each particle as an emergent symmetry, (ii) a finite spectrum gap separates the ground state manifold (GSM) from the low-energy excited states with a magnitude dependent on the interaction strengths and , and (iii) the GSM carries a unit total Chern number as a topological invariant protected by the spectrum gap, resulting in a fractional effect of quantum for each energy level in the GSM. We identify the quantum phase transition based on the spectrum gap collapsing. As shown in Fig. 1, the Fermi-liquid phase with gapless excitations is found for relatively strong NNN interaction at 1/3 filling and for relatively weak NNN interaction at 1/5 filling. This observation is consistent with Haldane’s pseudopotential theoryHaldane1983 () and previous studies on ordinary FQHE, where a FQHE state is found to be sensitive to interactions and other microscopic details (e.g. the thickness of the 2D electron gas Peterson2008 ()).
Here we present the studies of the low energy spectrum. Consider a system of unit cells ( sites) with twisted boundary conditions: , where is the translation operator with representing the and directions, respectively. Note that the filling factor is defined as the ratio of the number of particles () over the number of unit cells (). In the absence of impurities the total momentum of the many-body state is a conserved quantity and thus the Hamiltonian can be diagonalized in each momentum sector for systems with to sites (depending on filling factors). We consider periodic boundary condition () first. Fig. 2(a) illustrates the evolution of the low energy spectrum with changing for and ( and ) at particle filling factor . Here, the NN hopping strength () sun () is set to unity. We denote the momentum of a state by using two integers as shown in Fig. 2. For vanishing NN interaction (the bottom panel of Fig.2(a)), the ground state has while no particular structure is observed in other sectors. For a weak interaction , we find an interesting change in the spectrum. There are two states with momenta ( and ), which have lowered their energies substantially. For a stronger interaction, the energies of the three states with and form a nearly-degenerate GSMWen1990 () at . In the mean time, a sizable spectrum gap opens up, separating the GSM from the other excited states as shown in the top panel of Fig.2(a) for . The obtained three-fold ground state near degeneracy and a robust spectrum gap are the characteristic features of the 1/3 FQHE phase, which emerge with the onset of the NN repulsion . By increasing the NNN repulsion to a certain critical value , we have observed the collapsing of the spectrum gap, which determines the boundary of the 1/3 FQHE phase, as shown in Fig. 1(a). Further evidence of the FQHE based upon topological quantization will be presented later in Figs. 3.
In Fig. 2(b), we present the formation of the FQHE by showing the energy spectrum at particle filling factor with increasing at for a system with ( and ). From the bottom panel to the top panel, five states with momenta () form the nearly degenerate GSM with the increase of , while a large spectrum gap is formed at . The same feature of the energy spectrum is observed for the whole regime of the 1/5 FQHE above the critical line shown in the phase diagram Fig. 1(b) while drops to near zero for larger .
We note that there is a small energy difference between the states in the GSM in both filling factors. This is a finite-size effecttao (); Sheng2003 () as each of these states has to fit into the lattice structure. The finite-size effect is substantially smaller for 1/5 FQHE comparing to the 1/3 case due to the lower particle density. Interestingly, for all cases that we have checked for different system sizes (), the members of the GSM are always related to each other through a momentum space translation as an emerging symmetry of the system. Namely, if is the momentum quantum number for a state in the GSM, then another state in the GSM can be found in the momentum sector (modulo ). This relation of the quantum numbers of the GSM demonstrates the correlation between the real space and momentum space in a manner precisely resembling the FQHE in a uniform magnetic field. For the case where the particle number is integer multiples of both and (e.g., and at 1/3 and 1/5 fillings, respectively), all the states of the GSM are indeed observed to fall into the same momentum sector as expected.
To further establish the existence of the FQHE here, we study the topological property of the GSM through numerically inserting flux into the system using the generalized boundary phases. As first realized by Thouless and co-workers Thouless1982 (); Niu1985 (); Arovas1988 (), a topological quantity of the wavefunction, known as the first Chern number, distinguishes the quantum Hall states from other topological trivial states. In particular, it has been known that one can detect fractionalization phenomenon in FQHE by examining Chern number through inserting flux into the system Huo1992 (); Sheng2003 (); Sheng2005a (). The results of this calculation are shown in Fig. 3. With details presented in the Methods part, the total Berry phase as a function of should be a linear function with slope and for the FQHE with filling factors and , respectively. This agrees very well with the observation shown in Fig. 3. On the other hand, for the Fermi-liquid phase, strong fluctuations and nonuniversal behaviors of the Berry phase are found, suggesting the absence of topological quantization.
We further examine the phase at strong coupling limit for lower filling factor , where 1/5 FQHE demonstrates less sensitivity to either large or . We show the lowest 20 eigenvalues as a function of in Fig. 4, while we always set for simplicity. At small , we see the flatness of the spectrum in consistent with a Fermi liquid phase with small energy dispersion. As we increase , all the lowest five states remain nearly degenerate, while higher energy states jump a step up making a robust gap between them and the GSM. In fact, the 1/5 FQHE persists into infinite and limit as we have checked by projecting out the configurations with double or more occupancy between a site and all its NN and NNN sites. Physically, this can be understood as the particle at lower filling has enough phase space within the lower Hubbard band, and thus the FQHE remains intact. It would be very interesting to establish a variational state for the FQHE on the flatband model, which will be investigated in the future.
Finally, it is important to emphasize that the fractional topological phases we found are very stable and the same effect survives even if the hopping strengths are tuned by an amount of . On the experimental side, it is known that checker-board lattice we studied can be found in condensed matter systems (e.g. the thing films of LiVO, MgTiO, CdReO, etc), and the imaginary hopping terms we required can be induced by spin orbit coupling and spontaneous symmetry breaking Sun2009 (). In addition, this lattice model also has the potential to be realized in optical lattice system using ultra-cold atomic gases, in which the tuning of the parameters are much easier compared with condensed matter systems. Based on these observations, we conclude that there is no fundamental challenge preventing the experimental realization of these novel fractional topological states, but further investigation is still needed in order to discover the best experimental candidates.
The Chern number of a many-body state can be obtained as:
where the closed path integral is along the boundary of a unit cell with and , respectively. The Chern number is also the Berry phase (in units of ) accumulated for such a state when the boundary phase evolves along the closed path. Equation (2) can also be reformulated as an area integral over the unit cell , where is the Berry curvature. To determine the Chern number accurately Huo1992 (); Sheng2003 (); Sheng2005a (), we divide the boundary phase unit cell into about to meshes. The curvature is then given by the Berry phase of each mesh divided by the area of the mesh. The Chern number is obtained by summing up the Berry phases of all the meshes.
We find that the curvature is in general a very smooth function of inside FQHE regime. For an example, the ground state total Berry phase sums up to , slightly away from the 1/3 quantization for a system with , and at filling. Physically, as we start from one state with momentum in the GSM, it evolves to another state with a different momentum (), when the boundary phase along () direction is increased from to . Thus, with the insertion of a flux, states evolve to each other within the GSM. We observe that only the total Berry phase of the GSM is precisely quantized to and the total Chern number for all different choices of parameters inside either the 1/3 or 1/5 FQHE regime of Fig. 1.
As we move across the phase boundary from the FQHE state into the Fermi-liquid phase, there is no well defined nearly-degenerate GSM or spectrum gap, and the Berry curvature in general shows an order of magnitude bigger fluctuations. The obtained total Chern integer varies with system parameters (e.g., and ). In order to illustrate this feature, we start from the lowest-energy eigenstate and continuously increase the boundary phases for three periods, which allows the first state to evolve into other states and eventually return back to itself. In Fig. 3, we plot the accumulated total Berry phase as a function of the ratio of the total meshes included over the total number of meshes in each period. For the system in the 1/3 FQHE phase with , and , the total Berry phase follows a straight line in all three periods, well fitted by , indicating a nearly perfect linear law of the Berry phase to the area in the phase space with a deviation around 10%. In the Fermi liquid phase with , and , we see step-like jumps of the total Berry phase, with a magnitude in the order of , in sharp contrast to the linear law in the FQHE phase. The total Chern number for the Fermi-liquid state sums up to three, indicating the decorrelation between three states. Different integer values for the Chern number are found in this region (including negative ones) with changing system parameters, demonstrating a measurable fluctuating Hall conductance if particles are charged. For 1/5 FQHE state, by following up the Berry phase of five periods for the ground state as shown in Fig. 3(b), we observe the same linear law with a slope of and the total Chern number is quantized to one. Interestingly, a negative integer Chern number for Fermi liquid for the parameter and is observed confirming the nonuniversal nature of the topological number for such a gapless system. We conjecture that the Fermi liquid phase may be unstable towards Anderson localization especially at lower filling factors, similarly to the conventional FQHE systemsSheng2003 ().
This work is supported by DOE Office of Basic Energy Sciences under grant DE-FG02-06ER46305 (DNS), the NSF grant DMR-0958596 (for instrument), the State Key Program for Basic Researches of China under Grant Nos. 2009CB929504 and 2007CB925104, and the NSFC 10874066 and 11074110 (LS). ZCG is supported in part by the NSF Grant No. NSFPHY05-51164. KS acknowledges the support from JQI-NSF-PFC, AFOSR-MURI, ARO-DARPA-OLE, and ARO-MURI.
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