# Visualising the connection between edge states and the mobility edge in adiabatic and non-adiabatic topological charge transport

###### Abstract

The ability to pump quantised amounts of charge is one of the hallmarks of topological materials. An archetypical example is Laughlin’s gauge argument for transporting an integer number of electrons between the edges of a quantum Hall cylinder upon insertion of a magnetic flux quantum. This is mathematically equivalent to the equally famous suggestion of Thouless’ that an integer number of electrons are pumped between two ends of a one-dimensional quantum wire upon sliding a charge-density wave over a single wave length. We use the correspondence between these descriptions to visualise the detailed dynamics of the electron flow during a single pumping cycle, which is difficult to do directly in the quantum Hall setup, because of the gauge freedom inherent to its description. We find a close correspondence between topological edge states and the mobility edges in charge-density wave, quantum Hall, and other topological systems. We illustrate this connection by describing an alternative, non-adiabatic mode of topological transport that displaces precisely the opposite amount of charge as compared to the adiabatic pump. We discuss possible experimental realisations in the context of ultra-cold atoms and photonic waveguide experiments.

###### pacs:

71.45.Lr,73.43.Cd,72.20.Ee,73.20.At## I Introduction

The experimental discovery of the integer quantum Hall effect (IQHE) in GaAs von Klitzing et al. (1980) revealed the first example of a two-dimensional () material that has an insulating bulk, but metallic edge states. The IQHE state is now understood to be part of a much larger class of topological insulators Bernevig et al. (2006); Koenig et al. (2007); Kane and Mele (2005a, b); Qi and Zhang (2010); Hasan and Kane (2010); Qi and Zhang (2011); Slager et al. (2013), which can be labelled by topological invariants such as the Chern number or spin-Chern number Thouless et al. (1982). Although these indices determine the number and types of states localised on the edges of a finite sample, they may be computed entirely from bulk quantities, and depend on the symmetries of the bulk Hamiltonians. Different topological classes emerge depending on whether time-reversal, particle-hole and chiral symmetries are present or not Altland and Zirnbauer (1997); Ryu et al. (2010), and may be further refined using lattice symmetries Kruthoff et al. (2017a, b); Po et al. (2017); Watanabe et al. (2018); Bradlyn et al. (2017).

A correspondence between the bulk topological invariants and the dynamics of edge states in the IQHE has been put forward by Laughlin Laughlin (1981); Halperin (1982a); Thouless et al. (1982); Halperin (1982b); Hatsugai (1993). It shows that adiabatically threading a single Aharonov-Bohm flux quantum through the interior of an IQHE cylinder Laughlin (1981); Halperin (1982a), results in a quantized number of electrons moving from one edge of the cylinder to the opposite. The amount of charge transported is given precisely by the bulk Chern number, defined as an integral over the Brillouin zone Thouless (1983). The resulting cyclic charge transfer may be regarded as a dynamical manifestation of the IQHE Gritsev and Polkovnikov (2012), and has been observed in a Corbino geometry Dolgopolov et al. (1992); Vorob’ev et al. (2004). More generally, it is an example of the type of quantized adiabatic particle transport, or topological charge pumping, first proposed by Thouless Thouless et al. (1982); Thouless (1983); Niu (1986); Xiao et al. (2010). Experimentally, topological pumps have been realized in cold-atom Nakajima et al. (2016); Lohse et al. (2016) and single-spin Ma et al. (2018) systems, and attracted attention in both the adiabatic Altshuler and Glazman (1999); Switkes et al. (1999) and non-adiabatic Moskalets and Buttiker (2002); Maruyama and Hatsugai (2009) regimes.

Despite its status as an archetype of topological transport, the detailed dynamics of exactly how electrons are transferred between edges as a quantum of flux is threaded through Laughlin’s IQHE cylinder remains difficult to visualize. The straightforward plotting of electronic wave functions for arbitrary flux values throughout the pumping cycle is hampered by the need to choose a gauge. In this paper, we circumvent this problem by using a well-known family of one-dimensional () charge-ordered systems that can be mathematically mapped onto the IQHE setup Flicker and van Wezel (2015). This allows us to directly visualize the topological transport as electrons flow from one edge of the system to the other, and thus clarify the nature of the transport in both the charge-ordered and IQHE systems.

The rendering of edge state dynamics additionally reveals the in-gap states to be connected to the so-called mobility edge, which is the critical state separating extended from localized states in the bulk of a disordered system Huckestein (1995); Basko et al. (2006). In the case of quantum Hall systems, the bulk extended states are known to be constricted to just a single energy in the centre of each (impurity-broadened) Landau level, as long as inter-Landau level scattering is negligible Huckestein (1995); Chalker (1987). Both edge states and the mobility edge are topological in nature Pruisken (2010), which we confirm by considering their common insensitivity to the presence of weak impurities. Together, the edge states and the mobility edges make up a single set of connected states winding around the electronic spectrum. We illustrate this connection of all the topological states by suggesting an alternative form of topological charge transport, based on a non-adiabatic variation of externally applied fields. The non-adiabatic charge pump transfers precisely the opposite amount of charge from its traditional adiabatic counterpart, and thus acts as an “anti-Thouless” pump. We suggest a possible experimental realisation in cold atomic gasses in an optical lattice.

## Ii Mapping between and

Harper’s equation, and more generally, a Mathieu equation Harper (1955); Bellissard and Simon (1982); Hofstadter (1976), can be interpreted both as a tight-binding model for the IQHE in , and as a family of mean-field descriptions for monatomic commensurate charge-density waves (CDW) in Peierls (1930); Flicker and van Wezel (2015). The filling of the charge-ordered system, and hence its CDW periodicity, for example, can be taken to correspond to the magnetic field strength perpendicular to the surface of a quantum Hall cylinder, which determines the filling of its Landau levels Flicker and van Wezel (2015); Hofstadter (1976). Under the same mapping, the phase of the CDW order parameter then corresponds to a flux threading the quantum Hall cylinder Laughlin (1981), while the spatial coordinate along the CDW chain is directly related to the spatial coordinate in the periodic direction of the cylinder. The mapping is indicated schematically in figure 1.

Upon varying the parameter adiabatically from to , the CDW slides along the chain over precisely one wave length. The flux threading the IQHE system is increased by one quantum under the same variation. When introducing edges, we thus expect to find adiabatic transport of a quantised number of electrons from one side of the system to the other in both cases. In this way, the mapping relates the quantized adiabatic particle transport (Thouless pumping Thouless (1983)) in charge-ordered systems to the topological transport between edge states upon insertion of a flux quantum in a quantum Hall cylinder (Laughlin’s gauge argument Laughlin (1981)).

The bulk of both the CDW and IQHE systems is insulating, and only the edge states cross the Fermi level. By solving Harper’s equation on a cylinder, the spectrum can be plotted as a function of momentum in the periodic direction. It consists of bulk bands and isolated topological edge states crossing the gaps between them. The edge states are protected, in the sense that the number of edge channels cannot be modified, as long as the bulk of the system remains gapped Laughlin (1981). In a CDW system the periodic direction is given by the mean-field value of the phase variable , as shown in figure 1. Each value of the phase then corresponds to a single realisation of a CDW on the chain, which may or may not host edge states. The combined spectrum of the family of CDWs containing all values of coincides with that of the IQHE cylinder.

To be concrete, consider a CDW on a finite chain of sites described by the Hamiltonian:

(1) |

Here, is the tunneling amplitude, is the strength of the nearest-neighbor Coulomb interaction between electrons, and is a random potential that describes the effect of impurities. The operator () creates (annihilates) a spinless electron at position , where is the lattice constant. The filling factor is a common fraction, so that is the total number of electrons in the system. As usual, we assume the system to be charge neutral in total, and ignore any ionic charges.

Using the mean-field Ansatz for the particle density, the interaction term becomes Flicker and van Wezel (2015). This Ansatz defines the propagation vector of the charge-density modulations, as well as the phase , which determines the position of the CDW with respect to the lattice. Substituting the interaction back into equation (1) yields the full mean-field Hamiltonian for the CDW chain. For periodic boundary conditions and , this form of the Hamiltonian will coincide with Hofstadter’s tight-binding description of the IQHE Hofstadter (1976), if we make the identifications , , and . Here, is the momentum in the CDW, and are the -components of momentum in the IQHE. The radial magnetic field in the IQHE cylinder determines the filling of its Landau levels. Under this mapping, adding flux along the central axis of the IQHE cylinder corresponds to changing the value of the phase in the CDW Laughlin (1981), and thus to sliding the charge-density wave along the chain.

The Hamiltonian itself may be realised as a mean-field description of charge order in chains of aligned orbitals within a three-dimensional material Grüner (1988), or as an effective description of cold atoms in an optical lattice Flicker and van Wezel (2015); Ha et al. (2015). Here, we focus first on the topological properties of the theoretical mean-field model, before discussing possible experimental probes of the various emerging modes of topological transport in more realistic settings.

## Iii Visualising charge transfer between edge states

In Laughlin’s argument for quantised transport across the IQHE cylinder Laughlin (1981), the electromagnetic gauge structure plays a central role. Although helpful in establishing why the charge transferred between the ends of the cylinder must be integer, the presence of gauge freedom makes it hard to directly visualize the precise dynamics. We know and understand what happens upon insertion of a single flux quantum, but questions like at which value of the flux one edge state becomes unoccupied, and the opposite one occupied, or what the wave function looks like after insertion of only half a flux quantum, are difficult to answer in a gauge-independent fashion. For the CDW system, this problem does not exist. The process corresponding to a flux insertion is the sliding of the CDW by precisely one wave length, and this results in a precisely quantised amount of charge being transferred from one end of the chain to the other (Thouless pumping Thouless (1983)). The quantised conductivity is determined by the sum of Chern numbers for all occupied bands (in -space), as in the IQHE. However, in this case we can plot the charge distribution for both bulk and edge states for any value of , and hence visualize the topological transport as a continuous process.

As shown in figure 1, the spectrum and eigenstates can be computed for a specific choice of parameter values, including any value for the phase . As an example, we show the results for a period- CDW with , , and . We use this specific example throughout the paper, but note that taking other parameter choices does not qualitatively affect any of our results. Among the numerically obtained wave functions, bulk and edge states can be easily distinguished. The bulk wave functions for a CDW with open boundary conditions can be understood as products of plane waves (solutions of a CDW system with periodic boundary conditions), and the eigenstates of a particle in a box. The edge states, on the other hand, are exponentially localized on one side of the chain.

Upon varying , the highest occupied state changes from an edge state localized on one side to that on the opposite side. In the example of filling shown in figure 2, the highest state with at is an edge state localised at the right side of the chain (). In the region , the edge state is adiabatically transformed into a bulk state, until it emerges again as an edge state on the opposite side of the system. At , the highest occupied state discontinuously changes from being localised on the left side of the chain (), to being localised on the right. From the spectrum, it is clear that this behaviour stems from the two edge states crossing in energy at this point.

Notice that the states displayed in figure 2 are the highest occupied states at zero temperature for a given value of the phase variable . Starting from a phase value and adiabatically sliding the CDW forward, the system will in fact not stay in the instantaneous ground state. The two edge states crossing within the bulk gap are located at opposite edges of the chain in real space, and any matrix element of local operators that could assist in tunnelling across is exponentially small in the chain length. A state on one end of the system can therefore not simply jump to the other end in the way suggested by figure 2. For a sufficiently long CDW, adiabatic variation of causes the system to end up in an excited state at , with a high-energy edge state occupied and a lower-energy edge state empty. The system can only return to its instantaneous ground state, and the topological material can only function as an adiabatic charge pump, if the two edges of the CDW chain are connected to one another through some external coupling. In fact, in any experimental implementation of a topological quantum pump, one would indeed include a wire connecting the two sides of the system, and typically measure the current through the wire as the phase is being varied. Such a connection can be easily included in the simulation as a weak hopping element between the opposing sides of the chain. It allows the two edge states to interact, and turns their intersection in the spectrum into an avoided crossing by opening up a small energy gap. Adiabatic evolution, which retains the instantaneous ground state throughout a pumping cycle, is once again possible. Varying from to then results in the transfer of precisely one (the Chern number of the lowest band in -space Flicker and van Wezel (2015)) electron from one side of the system to the other, through the connecting wire, and thus realises the Thouless pump, or in the IQHE interpretation of the same system, Laughlin’s gauge argument.

## Iv The connection of edge states to the mobility edge

Inspecting the bulk wave functions in the spectrum of figure 1, two features stand out. First of all, as the edge state enters the bulk band, it does not simply disappear. Tracking for example the highest occupied state in figure 1, it is clear that the state, which was an edge state at , becomes a bulk state at . The bulk state is the particle in a box state with the highest available number of nodes, dressed with charge-density modulations. A state with the same number of nodes in fact already existed for , as part of the occupied bulk states. The edge state therefore does not evolve into a new bulk state as is adiabatically changed, but rather has an avoided crossing with an existing bulk state and takes over its character. The existing bulk state takes over the character of the edge state and is pushed down in energy in the process. It then has an avoided crossing with the next bulk state, and so on. This pattern is shown schematically in figure 3.

Close inspection of the wave functions in the spectrum indeed shows a cascade of avoided crossings, witnessed by a remainder of an exponential localisation that is visible in bulk states undergoing the avoided crossings. These continue until the edge state reaches precisely the middle of the bulk band. There, it emerges again for an extended range of , in the form of another well-known topologically special state, the mobility edge Huckestein (1995). This isolated critical state was to be expected in the centre of the bulk band, because the extended bulk states generic to disordered system are known to be constricted to just a single energy precisely in the centre of each (impurity-broadened) Landau level in the case of the quantum Hall effect Chalker (1987). Just like the mobility edge was formed from a left edge state at , it goes through a second series of avoided crossing and reemerges as a right edge state at . The edge states are of the form , with negative or postive for left or right localised states respectively. The mobility edge naturally connects these two states, and has the same shape of wave function, with . This special, totally delocalised wave function is thus a plane wave connecting both sides of the sample. Within the CDW, the plane wave is again dressed by charge modulations, as shown in figure 4. Note that for even , it is possible to have two orthogonal, delocalized wave functions. We consider chains with odd here, so that a single mobility edge connects to edge states in both bulk gaps.

The topological nature of the mobility edge only really becomes apparent upon adding a random impurity potential . Notice that in contrast to the IQHE, where the presence of weak impurities is necessary to observe the quantisation of the Hall conductivity, in the CDW system topological transport may be observed even with only the Coulomb interaction localising charges within the unit cell. Nevertheless, the presence of weak impurities (modeled here with ) may be considered, and their effect is much the same as in the IQHE, localising electronic states at specific locations in the chain. The difference between the mobility edge and other bulk states is now immediately obvious. As shown in figure 4, typical bulk states are severely affected by the impurity potential, being amplified and suppressed at random locations. The mobility edge on the other hand, retains a more or less constant amplitude along the entire length of the chain, as long as the impurity strength is weak. This clearly indicates the topological nature of the mobility edge, and hence its connection with the topological edge states.

## V Non-adiabatic quantised particle transport

The connection between edge states and mobility edge may be illustrated by considering a non-adiabatic mode of quantised transport, which exists in addition to the well-known adiabatic Thouless pump. Starting for example with the (red) left edge state just below occupied, an induced non-adiabatic evolution that jumps all avoided crossings with both bulk and edge states would result in the highest occupied state going around the full spectrum as shown schematically in figure 3. As is increased by , the highest occupied state has traversed all topologically special states in the entire spectrum, and returns to its initial configuration. Stopping the non-adiabatic evolution after is increased by , however, the highest occupied state will have moved from its initial left edge state to the right edge state in the lowest bulk gap. In the presence of a wire connecting both ends of the chain, we can then either adiabatically increase by another , or let the system spontaneously relax to its ground state. Both will lead to the excited state going from right to left through the wire. That is, a current is carried through the connecting wire in the direction opposite to that of the usual adiabatic topological transport. This non-adiabatic transfer of charge with the same magnitude but in the opposite direction of the usual Thouless pumping, is made possible by the edge states and mobility edges forming a connected set of states winding throughout the electronic spectrum.

Generalising the procedure to systems of any filling, the integer number of charges transferred by the non-adiabatic mode of transport will equal , with the total Chern number of all occupied bands, the exact opposite of the electrons conveyed in the usual Thouless pump. In this sense, it acts as a sort of anti-Thouless pump. Notice that hybrid protocols employing both adiabatic and non-adiabatic driving can accomplish the same thing in a much simpler fashion. In such protocols, however, the highest occupied state is in non-topological, localised bulk states for part of the pumping cycle. The purely non-adiabatic process suggested here, in which the highest occupied state jumps all avoided crossings that it encounters, is special because it depends entirely on the special nature and connectivity of the topological edge states and mobility edge states.

The non-adiabatic topological charge pump presented here is intended purely as an illustration of the connection between different types topological states. Nevertheless, non-adiabatic or hybrid protocols may in some aspects be preferable to purely adiabatic ones in practical implementations of topological pumping. As has been recently pointed out, the presence of a spectrum of unoccupied states imposes much more stringent conditions for achieving adiabaticity than just the time scale of the driving being longer than the inverse gap size Lychkovskiy et al. (2017, 2018); Polkovnikov and Gritsev (2008). Non-adiabatic driving across an avoided crossing on the other hand, can always be accomplished by sufficiently fast changes of the driving parameters, as long as the overlap between initial and target states is large. We confirmed that this is the case in the model CDW system by achieving an almost complete transfer of occupation across avoided crossings upon ramping up the driving speed.

To realise the proposed illustration of a non-adiabatic pumping cycle in experiments, several techniques for imaging the real-space structure of CDW systems can be used. Scanning tunneling microscopy (STM) for example, can directly observe the amplitude and phase of charge-density modulations induced by a CDW Snijders et al. (2006); Brun et al. (2010); Soumyanarayanan et al. (2013); Ugeda et al. (2016), as well as charge excess at the end of a chain Kraus et al. (2012). More straightforward experimental realisations of quantised transport protocols, however, use ultracold atoms in an optical lattice or photons in a waveguide array. Using ultracold atoms, the mean-field Hamiltonian can be constructed by projecting a circular optical dipole lattice through a microscope objective Ha et al. (2015); Lohse et al. (2018). Introducing a weak link between two particular sites in the lattice allows them to act as edges, while the interaction term can be implemented by superimposing a rotating lattice with larger spacing Ha et al. (2015). Variations of the mean-field CDW phase are then implemented by shifting the phase of the rotating lattice with respect to the system edge. After a full driving cycle, the quantised build-up of charge at the edges can be directly observed using the microscope. Photons in a waveguide array are an alternative way of directly simulating the CDW Hamiltonian Kraus et al. (2012). Variations of the mean-field CDW phase are then implemented by appropriate variatons of the index of refraction. The quantised transport of charge can be observed by injecting photons into the waveguide at one edge and observing the intensity distribution in the waveguides after various propagation distances. The required values for experimental parameters in both types of experiments can all be realistically obtained with existing technology.

## Vi Conclusions

We visualize adiabatic and non-adiabatic topological charge transport in a charge-ordered system, which is well-known to map precisely onto the tight-binding model for the IQHE in the Landau gauge used by Laughlin to explain topological transport in an IQHE cylinder. The absence of an electromagnetic field in the CDW, and hence of the need to make a gauge choice, enables us to directly compare plots of wave functions at different values of the CDW phase. The topological transport and dynamics of electrons in the CDW chain found here, thus give direct insight into the detailed motion of electronic states in the IQHE as well.

The visualisation of the topological transport brings to the fore an explicit connection between edge states localised at the ends of the chain or cylinder, and the mobility edges residing in the very centre of their bulk states. To illustrate the connection between these two types of topological states, localised and extended, we formulate a purely non-adiabatic protocol that nonetheless results in quantised, topological transport of charge. The number of electrons pumped in a single cycle is precisely equal to that of the well-known adiabatic pump, but they flow in the opposite direction, creating a sort of “anti-Thouless” pump.

We also point out experiments on ultra cold atoms in an optical lattice, as well as in photonic waveguide experiments, which can test the proposed connection between edge states and mobility edges, as well as the non-adiabatic pumping cycle, using realistic parameter values.

### vi.1 Acknowledgments

This work is part of the Delta Institute for Theoretical Physics (DITP) consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). JvW acknowledges support from a VIDI grant financed by the Netherlands Organization for Scientific Research (NWO). This project has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) (Grant agreement No. 615117 QuantStro).

## References

- von Klitzing et al. (1980) K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
- Bernevig et al. (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).
- Koenig et al. (2007) M. Koenig, S. Wiedmann, C. Bruene, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
- Kane and Mele (2005a) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005a).
- Kane and Mele (2005b) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005b).
- Qi and Zhang (2010) X.-L. Qi and S.-C. Zhang, Phys. Today 63, 33 (2010).
- Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
- Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
- Slager et al. (2013) R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, Nature Physics 9, 98 (2013).
- Thouless et al. (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
- Altland and Zirnbauer (1997) A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
- Ryu et al. (2010) S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, New J. Phys. 12, 065010 (2010).
- Kruthoff et al. (2017a) J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Phys. Rev. X 7, 041069 (2017a).
- Kruthoff et al. (2017b) J. Kruthoff, J. de Boer, and J. van Wezel, arxiv , 1711.04769 (2017b).
- Po et al. (2017) H. C. Po, A. Vishwanath, and H. Watanabe, Nature Commun. 8, 50 (2017).
- Watanabe et al. (2018) H. Watanabe, H. C. Po, and A. Vishwanath, Science Adv. 4, eaat8685 (2018).
- Bradlyn et al. (2017) B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature 547, 298 (2017).
- Laughlin (1981) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
- Halperin (1982a) B. I. Halperin, Phys. Rev. B 25, 2185 (1982a).
- Halperin (1982b) B. I. Halperin, Phys. Rev. B 25, 2185 (1982b).
- Hatsugai (1993) Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
- Thouless (1983) D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
- Gritsev and Polkovnikov (2012) V. Gritsev and A. Polkovnikov, PNAS 109, 6457 (2012).
- Dolgopolov et al. (1992) V. T. Dolgopolov, A. A. Shashkin, S. I. Zhitenev, N. B. Dorozhkin, and K. von Klitzing, Phys. Rev. B 46, 12560 (1992).
- Vorob’ev et al. (2004) A. B. Vorob’ev, V. Y. Prinz, Y. S. Yukecheva, and A. I. Toropov, Physica E 23, 171 (2004).
- Niu (1986) Q. Niu, Phys. Rev. B 34, 5093 (1986).
- Xiao et al. (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
- Nakajima et al. (2016) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Nat. Phys. 12, 296 (2016).
- Lohse et al. (2016) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, Nat. Phys. 12, 350 (2016).
- Ma et al. (2018) W. Ma, L. Zhou, Q. Zhang, M. Li, C. Cheng, J. Geng, X. Rong, F. Shi, J. Gong, and J. Du, Phys. Rev. Lett. 120, 120501 (2018).
- Altshuler and Glazman (1999) B. L. Altshuler and L. I. Glazman, Science 283, 1864 (1999).
- Switkes et al. (1999) M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, Science 283, 1905 (1999).
- Moskalets and Buttiker (2002) M. Moskalets and M. Buttiker, Phys. Rev. B 66, 205320 (2002).
- Maruyama and Hatsugai (2009) I. Maruyama and Y. Hatsugai, J. Phys.: Conf. Ser. 150, 022055 (2009).
- Flicker and van Wezel (2015) F. Flicker and J. van Wezel, EPL (Europhysics Letters) 111, 37008 (2015).
- Huckestein (1995) B. Huckestein, Rev. Mod. Phys. 67, 357 (1995).
- Basko et al. (2006) D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys 321, 1126 (2006).
- Chalker (1987) J. T. Chalker, J. Phys. C 20, L493 (1987).
- Pruisken (2010) A. M. M. Pruisken, Int. J. Mod. Phys. B 24, 1895 (2010).
- Harper (1955) P. G. Harper, Proc. Phys. Soc. Lond. A 68, 874 (1955).
- Bellissard and Simon (1982) J. Bellissard and B. Simon, J. Funct. Anal. 48, 408 (1982).
- Hofstadter (1976) D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
- Peierls (1930) R. Peierls, Ann. d. Phys. Leipzig 4, 121 (1930).
- Grüner (1988) G. Grüner, Rev. Mod. Phys. 60, 1129 (1988).
- Ha et al. (2015) L.-C. Ha, L. W. Clark, C. V. Parker, B. M. Anderson, and C. Chin, Phys. Rev. Lett. 114, 055301 (2015).
- Lychkovskiy et al. (2017) O. Lychkovskiy, O. Gamayun, and V. Cheianov, Phys. Rev. Lett. 119, 200401 (2017).
- Lychkovskiy et al. (2018) O. Lychkovskiy, O. Gamayun, and V. Cheianov, AIP Conf. Proc. 1936, 020024 (2018).
- Polkovnikov and Gritsev (2008) A. Polkovnikov and V. Gritsev, Nature Phys. 4, 477 (2008).
- Snijders et al. (2006) P. C. Snijders, S. Rogge, and H. H. Weitering, Phys. Rev. Lett. 96, 076801 (2006).
- Brun et al. (2010) C. Brun, Z.-Z. Wang, P. Monceau, and S. Brazovskii, Phys. Rev. Lett. 104, 256403 (2010).
- Soumyanarayanan et al. (2013) A. Soumyanarayanan, M. M. Yee, Y. He, J. van Wezel, D. J. Rahn, K. Rossnagel, E. W. Hudson, M. R. Norman, and J. E. Hoffman, PNAS 110, 1623 (2013).
- Ugeda et al. (2016) M. M. Ugeda, A. J. Bradley, Y. Zhang, S. Onishi, Y. Chen, W. Ruan, C. Ojeda-Aristizabal, H. Ryu, M. T. Edmonds, H.-Z. Tsai, A. Riss, S.-K. Mo, D. Lee, A. Zettl, Z. Hussain, Z.-X. Shen, and M. F. Crommie, Nat. Phys. 12, 92 (2016).
- Kraus et al. (2012) Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
- Lohse et al. (2018) M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, Nature 553, 55 (2018).