The anisotropic Harper-Hofstadter-Mott model:
competition between condensation and magnetic fields
We derive the reciprocal cluster mean-field method to study the strongly-interacting bosonic Harper-Hofstadter-Mott model. The system exhibits a rich phase diagram featuring band insulating, striped superfluid, and supersolid phases. Furthermore, for finite hopping anisotropy we observe gapless uncondensed liquid phases at integer fillings, which are analyzed by exact diagonalization. The liquid phases at fillings exhibit the same band fillings as the fermionic integer quantum Hall effect, while the phase at is -symmetric with zero charge response. We discuss how these phases become gapped on a quasi-one-dimensional cylinder, leading to a quantized Hall response, which we characterize by introducing a suitable measure for non-trivial many-body topological properties. Incompressible metastable states at fractional filling are also observed, indicating competing fractional quantum Hall phases. The combination of reciprocal cluster mean-field and exact diagonalization yields a promising method to analyze the properties of bosonic lattice systems with non-trivial unit cells in the thermodynamic limit.
Since the discovery of the quantum Hall effect von Klitzing et al. (1980); Tsui et al. (1982); Laughlin (1983), the lattice geometry’s influence on charged particles in magnetic fields has been the subject of extensive research. Prototypical models such as the non-interacting Harper-Hofstadter model (HHm) Harper (1955); Hofstadter (1976) exhibit fractionalization of the Bloch bands with non-trivial topology (see Fig. 1), manifesting in quantum (spin) Hall phases Bernevig and Zhang (2006); Goldman et al. (2010); Aidelsburger et al. (2013). Ultracold atomic gases with artificial magnetic fields Madison et al. (2000); Abo-Shaeer et al. (2001); Bloch et al. (2008); Lin et al. (2009); Dalibard et al. (2011) enabled the experimental study of the non-interacting model Aidelsburger et al. (2013); Miyake et al. (2013); Atala et al. (2014); Aidelsburger et al. (2015), while the effect of strong interactions on the band properties remains an open problem. While heating processes in the regime of strong interactions still represent a problem for cold atom experiments with artificial magnetic fields, recent experimental progress gives hope that this can be controlled in the near future Kennedy et al. (2015); Tai et al. (2016).
For bosons in the HHm with local interaction, i.e. the Harper-Hofstadter-Mott model (HHMm), previous theoretical studies found fractional quantum Hall (fQH) phases, which have no counterpart in the continuum for strong fields Möller and Cooper (2009), using exact diagonalization (ED) Sørensen et al. (2005); Hafezi et al. (2007); Möller and Cooper (2009), composite fermion theory Möller and Cooper (2009), and the density matrix renormalization group (DMRG) on a cylinder He et al. (2017). Composite fermion studies also found evidence of a bosonic integer quantum Hall phase in bands with Chern number two Möller and Cooper (2015), also observed with ED Zeng et al. (2016) in the presence of next-neighbor hopping. In a recent DMRG study He et al. (2017) a bosonic integer quantum Hall groundstate was also observed in the standard HHMm at filling . However, the composite fermion approach is biased by the choice of the wavefunction Möller and Cooper (2009, 2015), while ED on small finite systems suffers from strong finite-size effects Sørensen et al. (2005); Hafezi et al. (2007); Zeng et al. (2016).
The issue becomes especially challenging when going to strong fluxes, where extrapolation to the thermodynamic limit is impossible Hafezi et al. (2008) and the overlap of the finite-size groundstate with Laughlin Sørensen et al. (2005); Hafezi et al. (2007) or composite fermion Möller and Cooper (2009) wavefunctions quickly decreases. DMRG, on the other hand, is restricted to cylinder geometries He et al. (2017, 2015); Motruk et al. (2016) and encounters huge numerical difficulties in the case of critical phases. Variational Gutzwiller mean field studies also found evidence of fQH phases Palmer et al. (2008); Umucalilar and Oktel (2007); Umucalilar and Mueller (2010), as well as striped vortex-lattice phases Palmer et al. (2008), but the variational basis is restricted by construction. The results of a recent cluster Gutzwiller mean field (CGMF) study Natu et al. (2016) are likewise hard to interpret since the method breaks the translational invariance and the topology of the system.
To overcome these problems, we develop a reciprocal cluster mean field (RCMF) method, directly defined in the thermodynamic limit, which preserves the topology of the lattice, and yields excellent agreement with numerically exact results for the Bose-Hubbard model. Further, we introduce an observable for the measure of topological properties in the presence of interactions.
We systematically map out the phase diagram of the strongly interacting HHMm as a function of the chemical potential and the hopping anisotropy. The phase diagram features band insulating, striped superfluid, and supersolid phases. At integer fillings we further observe highly anisotropic gapless uncondensed liquid phases, which are analyzed using exact diagonalization. For fractional filling, we find incompressible metastable states, indicating competing fQH phases. We define the respective order parameters, and present spatially resolved density, condensate-density, and current patterns. Finally, we discuss how on an infinite cylinder with a single unit-cell in the -direction the liquid phases become gapped and show a quantized Hall response to the adiabatic insertion of a magnetic flux.
This paper is organized as follows. In Sec. II the HHm and HHMm are discussed, and the method for measuring non-trivial topological properties is introduced. The RCMF method is derived and discussed in Sec. III, while the results for the HHMm are presented and discussed in Sec. IV. Finally, Sec. V is devoted to conclusions.
ii.1 Harper-Hofstadter model
To facilitate the discussion for the strongly-interacting system, we first review the non-interacting HHm on the square lattice. The Hamiltonian is given by
with hopping amplitudes and annihilation (creation) operators . Each plaquette is pierced by a flux such that a phase is picked up when going around it, as illustrated in Fig. 1a. For the unit cell can be chosen as sites in the -direction.
Eq. (1) is diagonalized by the transform , where and () are the momenta in ()-direction. For even the Hamiltonian reduces to , with
For , used below, the system has three isolated topologically non-trivial bands, see Fig 1b. Here we use the notation of Ref. Aidelsburger et al., 2015 where the central (super)band contains twice the number of states as compared to the two other bands. For a discussion of how the hopping anisotropy affects the bandstructure, see Appendix A.
The Hamiltonian and Eq. (2) can be rewritten in the compact notation
where is a vector of scalars and is a vector of operators
The operator fully determines the momentum dependence of the non-interacting system, and we can apply the concept of parallel transport Qi et al. (2008). The local Berry curvature at the point is proportional to the rotation of the unit-vector
under an infinitesimal momentum shift. In fact, if shows a non-trivial winding under transport on a closed path through the Brillouin zone, the Berry-curvature cannot be continuously deformed to a trivial one and the system is topologically non-trivial. The Chern number of the th band is given by the number and direction of closed loops of , i.e.
where is the solid angle subtended by when taking the expectation value with respect to the single-particle eigenstates of the th band and sweeping the momenta through the Brillouin zone. This is shown in Fig. 1b. For the lowest band and are shown while varies only slightly: performs one anti-clockwise loop, corresponding to a Chern number of . Equivalently, for the central band performs a double clockwise loop (), while the highest band again has .
The connection between the winding of and the Hall conductivity can be seen from the example of the integer quantum Hall effect, i.e. the lowest band being completely filled with non-interacting fermions. Adding a magnetic flux piercing a torus of size in -direction can be achieved by transforming the hopping amplitudes as for hopping processes in , while taking the complex-conjugate in the opposite direction. The effect of this transform on the Hamiltonian (5) amounts to
which is manifested in a translation of the vector with respect to the case without flux at each momentum , i.e.
where is the many-body groundstate under the flux .
The sole effect of is therefore a transform of the many-body groundstate such that at each momentum Eq. (7) is fulfilled, resulting in a rotation of . Inserting a flux of yields the transform , , and therefore . Adding a magnetic flux of is equivalent to translating the manybody groundstate by one site in the -direction.
If the lowest band is completely filled, the total number of particles on the torus is . Therefore adiabatically inserting a flux of results in particles being translated by one site in -direction, or equivalently a total number of particles being transported once around the periodic boundary in the -direction. Consequently, adiabatically inserting a flux of results in a quantized total transverse transport of a single particle around the periodic boundary.
ii.2 Harper-Hofstadter-Mott model
We proceed with the study of the HHMm with interaction , chemical potential , and magnetic flux ,
in the hard-core limit .
In contrast to the non-interacting case, for a finite interacting system the Berry curvature is defined with respect to boundary twisting angles Hafezi et al. (2008), i.e.,
where is the Berry connection, is the many-body groundstate, and , and are twisting angles of the boundary conditions in - and -direction, respectively (i.e. , where is a translation by the system size in -, and -direction, respectively).
The twisted boundary conditions can be implemented in the same way as the magnetic flux discussed in Sec. II.1 by transforming the hopping as and for hopping processes in , and -direction, respectively, while taking the complex-conjugate in the opposite directions. The interaction and chemical potential terms in Eq. (8) remain unchanged. The only effect of adding the twisting angles to the infinite system is, as in Sec. II.1,
In other words, if is the momentum-space translation operator which transforms each momentum as , and , we have
For the Berry-curvature
we therefore have
The Berry curvature is therefore fully determined by the response of the periodic-boundary many-body groundstate to a translation in momentum.
If we define as the projector onto the Hilbert space of hard-core bosons (where multiple occupancy in position space is forbidden), the interacting many-body Hamiltonian [Eq. (8)] can be written as
with particle-number operator . The full momentum dependence of the hard-core bosons is therefore contained in the term . Furthermore, for any hard-core boson many-body eigenstate we have . As in the non-interacting case therefore a non-trivial winding of in momentum space indicates a non-trivial topology of the many-body groundstate. It should be emphasized that this measure is different from summing over the individual single-particle Chern numbers of the occupied bands, since no projection onto non-interacting bands is involved.
For a further discussion of the measurement of topological properties with the -vector, see Appendix D.
Iii Reciprocal Cluster Mean Field
The previously employed CGMF method Lühmann (2013); Natu et al. (2016) breaks translational invariance by applying the mean-field decoupling approximation only to the hopping-terms at the boundary of the cluster, while the hopping terms within the cluster are treated exactly. The simplest case where this can be observed is when the symmetry-breaking field is zero, reducing the lattice to a set of decoupled clusters with open boundaries. This violation of translational invariance breaks the symmetries of the dispersion and thereby its topological properties. In order to mitigate such artifacts we develop a mean-field decoupling based on the concept of momentum coarse-graining, introduced in the context of the dynamical cluster approximation Maier et al. (2005).
We term this method as “reciprocal cluster mean field” (RCMF). It crucially preserves both the translational invariance and the topology of the system. For topologically trivial translationally-invariant systems it yields more accurate results than previous mean-field methods (see Appendix F). It is well-suited for cases where the underlying symmetries of the dispersion are indispensable to understand the physical properties, such as, e.g., topological insulators. For benchmarks of the method on the hard-core Bose-Hubbard model and the chiral ladder with artificial magnetic fields Hügel and Paredes (2014); Piraud et al. (2015), see Appendix F.
To illustrate our procedure let us first start from a general non-interacting hopping Hamiltonian
with hopping amplitudes in position-space and dispersion in reciprocal space. As in the dynamical cluster approximation Maier et al. (2005), the main idea of RCMF consists in projecting the lattice system onto a lattice of clusters (later we will take , but the method is also well-defined for finite systems). Each cluster is spanned by the internal cluster coordinates and , such that we can decompose the position coordinates and on the lattice into
where and are inter-cluster coordinates. In the same way the momenta in and -direction – and , respectively – are decomposed as
where and are the cluster momenta in reciprocal space. Through a partial Fourier transform, the creation and anihilation operators in reciprocal space can be written in the mixed representation
where annihilates a boson with cluster-momenta and on the cluster located at Maier et al. (2005).
The central idea of the momentum coarse-graining consists of projecting the dispersion of the lattice onto the clusters in reciprocal space. This can be done by a partial Fourier transform of the dispersion onto the subspace of cluster-local hopping processes, giving the intra-cluster dispersion as
representing hopping processes within the cluster, while the remainder represents all other hopping processes between different clusters Maier et al. (2005).
Now we can decompose the Hamiltonian of Eq. (10) into
where, using Eq. (11), the part is cluster-local,
while contains the coupling between different clusters
where in the second line we introduced the mixed representation of ,
Our goal is to derive an effective Hamiltonian which is cluster local through a mean-field decoupling approximation of . To this end we decompose the creation/annihilation operators into their static expectation values and fluctuations, i.e.
The standard procedure of the mean-field decoupling approximation consists of neglecting quadratic fluctuations. Furthermore, we assume translational invariance between the different clusters, i.e. that the condensate is independent of the cluster location
As derived in detail in Appendix E, this approach reduces a general system with local interactions and Hamiltonian
into a set of identical cluster local systems with effective mean-field Hamiltonian,
The symmetry breaking field is given by
and the effective hopping amplitudes are defined as
Within the RCMF approach the effective free energy of the lattice system is given by
where is the free energy of the cluster local Hamiltonian of Eq. (16). Note that Eq. (19) is consistent with the standard lattice free-energy within the single-site mean-field approximation Hügel et al. (2016). In fact, requiring stationarity in the symmetry breaking field ,
taking into account Eq. (17), reproduces the standard mean-field self-consistency condition
Here, means taking the expectation value with respect to the mean-field Hamiltonian [Eq. (16)].
In our RCMF approach the HHMm [Eq. (8)] is reduced to an effective cluster Hamiltonian, for details see Appendix G. For a comparison of our RCMF results with exact diagonalization at zero hopping anisotropy, see Appendix C. In Fig. 2 we present the groundstate phase diagram in terms of the chemical potential and the hopping-anisotropy , where . The phases at densities and are related by a -transformation consisting of a particle-hole transform combined with complex-conjugation (see Appendix B). The symmetry around the axis corresponds to gauge invariance, since and can be exchanged in combination with a lattice-rotation of . At and we find topologically trivial band insulators (BI). Below we discuss the other resulting phases in more detail.
iv.1 Condensed phases
At moderate values of we observe superfluid phases with striped density and condensate density modulation. For this is a vertically striped superfluid (VS-SF), with vertically striped density distribution and condensate-density distribution , as shown in Fig. 3 together with the spatially resolved particle current . The net current is zero, as expected for an infinite system. Locally, however, there are chiral currents around two plaquettes in the horizontal direction. We therefore introduce the striped-superfluid order parameter , where is the groundstate expectation value of the current in () direction. For the superfluid phase is horizontally striped (HS-SF), with the patterns of Fig. 3 rotated by compared to the VS-SF. Since at the system is invariant under a -rotation, for the VS-SF and HS-SF undergo phase separation.
At and for low anisotropy we find a supersolid phase (SS) with lower free energy than the striped phases. The density distributions and spontaneously break translational invariance, having a period larger than the unit cell (see Fig. 3). A similar spontaneous breaking of translational invariance has already been observed in the staggered-flux bosonic Harper-Mott model Möller and Cooper (2010) and the bosonic Hofstadter model on a dice lattice Möller and Cooper (2012), and has recently been measured experimentally in spin-orbit coupled Bose-Einstein condensates Li et al. (2017). The SS exhibits chiral currents around single plaquettes, with position-dependent amplitudes, as captured by the order parameter . In all phases, at the density distribution is homogeneous, , while remains modulated.
The phase transition between the striped superfluids and the SS phase is characterized by a kink in the average condensate density , see Figures 4a and 4b. For , the striped superfluid order parameter is only zero at (where the lattice is a set of trivial one-dimensional chains), exhibiting a kink at the phase transition to the SS, where also becomes non-zero.
iv.2 Uncondensed phases
At density (Fig. 4a) and stronger anisotropy we find a phase with zero condensate density (). In Figs. 4c and 4d we show a sweep in for , where we observe plateaus with zero , zero current, and homogeneous density distribtution , with fillings . In these phases, since , the RCMF Hamiltonian of Eq. (16) reduces to a finite torus without any external variational parameter. In order to further analyze these phases we therefore turn to ED using twisted boundary conditions in order to analyze finite-size effects (see Sec. II.2 and Appendix C). If the phases are gapped, one expects the manybody gap to stay finite for all twisting angles , while in gapless phases the groundstate mixes with excited states.
As can be seen in Fig. 5 for the groundstate remains gapped with respect to boundary twisting in the -direction with , while it mixes with the excited states for twisting in the -direction. For the behavior is reversed. This is also consistent with the correlations decreasing exponentially to zero as a function of for , while staying finite throughout the system for (see Appendix C). In contrast to the two-dimensional Bose-Hubbard model without magnetic flux, which in the superfluid groundstate always shows condensation as long as both hopping amplitudes are finite Schönmeier-Kromer and Pollet (2014), the HHMm therefore shows a transition at finite to a highly anisotropic uncondensed gapless liquid. The fact that these phases are adiabatically connected to the one-dimensional limit ( or ), where hard-core bosons are in a superfluid phase, as well as the highly anisotropic correlations , possibly point to an unconventional one-dimensional superfluid order.
As a function of the hopping anisotropy, the liquid phases occur where the lowest band is particularily flat either in - or -direction, suppressing condensation in the minima of the dispersion (see Fig. 6a and Appendix A). While the system has zero current everywhere (due to the periodic boundaries), a signature of the response of the liquid to the magnetic field is found by analyzing current-current correlations (see Appendix G.2 for details), resulting in two counter-propagating currents which cancel each other, shown in Fig. 6b.
In Fig. 6c we show the projection of the groundstate onto the three non-interacting bands , , and , for the same parameters as in Figs. 4c and 4d. At the lowest band shows unit filling. As shown in Fig. 6a, this phase appears in the same regions of as the integer quantum Hall plateau of non-interacting spinless fermions (see Appendix A)). At the holes show unit filling in the lowest band, due to the transform (see Appendix B).
As can be seen in Fig. 6d, the vector shows the same behavior as for the lowest non-interacting band in all three liquid phases (shown for ), in contrast to the trivial BI at and the one-dimensional superfluid at or . For , as in the case of non-interacting fermions discussed in Sec. II.1, this winding indicates the transverse transport of a single particle if a magnetic flux of is inserted. At , the transverse transport consists of a single hole. This is consistent with the band fillings in Fig. 6c and the transform discussed in Appendix B, i.e. the reversal of the Hall conductivity . As these phases are gapless in the two-dimensional thermodynamic limit, the quantization of the Hall conductivity is not topologically protected by edge modes and therefore sensitive to disorder, as is the case for metallic Fermi-liquid-like phases of hard-core bosons Zeng et al. (2016); Read (1998); Sheng et al. (2003).
By contrast, in the case of a cylindrical geometry, i.e. and , the response to the twisted boundaries in -direction while , shown in Figs. 5a and 5c, indicates that all three plateaus are gapped. As a function of the vector shows a complete loop and appears to be robust against local perturbations (see Appendix D). What the nature of the phases at is in such a quasi-one-dimensional setup remains to be investigated: the non-trivial winding indicates a gapped phase with odd Hall conductivity, which is expected to show intrinsic topological order and fractional quasiparticle excitations for bosons in two dimensions Senthil and Levin (2013); Lu (2017). The non-degenerate groundstate we observe (which for bosons is only expected at even Hall conductivities) apparently is at odds with this prediction. However, the argument of Ref. Senthil and Levin, 2013 relies on the fact that the quasiparticle excitations need to behave as fermions under exchange in such an odd Hall conductivity phase. In the cylinder, however, we approach the one-dimensional limit, where hard-core bosons naturally behave as free fermions also in the absence of fractionalization. This possibly explains why we observe an ED groundstate which remains gapped and non-degenerate for all accessible system sizes (see Appendix C).
At (), the Hamiltonian of Eq. (8) is -symmetric. It directly follows that . This is consistent with the bands being equally filled with particles and holes (see Fig. 6b), resulting in a zero net Hall conductivity (see Appendix B). In two-dimensional systems such a -symmetric phase is expected to be topologically trivial Ryu et al. (2010); Chen et al. (2012); Lu and Vishwanath (2012), in line with the gaplessness observed in Figs. 5c and 5d.
On a cylinder, however, this phase is gapped as the one-dimensional limit is approached, where -symmetric phases can have a non-zero topological invariant for non-interacting fermions Ryu et al. (2010). The non-trivial winding implies the quantized transport of a single particle-hole pair under the adiabatic insertion of a flux of , resulting in a total zero Hall conductivity. Wether this particle-hole transport is a consequence of topologically protected edge modes on the cylinder remains to be investigated on larger system sizes, possibly using DMRG He et al. (2017, 2015); Motruk et al. (2016).
Whereas away from integer fillings the groundstate is always symmetry-broken, it is always possible within RCMF to find (metastable) stationary solutions with zero symmetry-breaking field () and therefore , as shown in Fig. 4d. While at large hopping anisotropy fractional fillings are largely supressed, at low anisotropy the solution shows plateaus at any filling commensurate with the cluster, i.e. with integer , as shown in Fig. 7c.
As mean-field approaches such as RCMF tend to overestimate the stability of symmetry-broken phases, we critically analyze the difference in free energy between the groundstate and the metastable plateaus in Fig. 7. As can be seen, this quantity shows local minima at integer () and half-integer () fillings, while quarter fillings correspond to local maxima. This is consistent with the argument that without long-range interactions it costs a negligible energy to compress the to the Laughlin liquid Hafezi et al. (2007). The energy difference is particularily low in the vicinity of the liquid phases indicating that these plateaus might extend to lower values of hopping anisotropy. Furthermore, at low anisotropy (where the lowest band is particlularily flat, see Fig. 6c and Appendix A) the metastable plateau at (and ) is very close in free energy to the groundstate. This plateau has been shown to correspond to a fQH phase in ED Sørensen et al. (2005); Hafezi et al. (2007); Möller and Cooper (2009), variational Gutzwiller mean field Umucalilar and Mueller (2010), and DMRG He et al. (2017) studies. As shown in Fig. 7c, at zero anisotropy, the condensate fraction of the groundstate shows a local minimum at both and , further indicating that it might converge to zero with increasing cluster size. These are also the fillings where at zero anisotropy there is the largest discrepancy between RCMF and ED results on small finite systems (see Appendix C). It should however be noted that when assuming a cylinder geometry, RCMF observes both a and plateau, in agreement with ED. To conclude, there are regions of the phase diagram, where the symmetry-broken groundstate and the metastable plateaus are too close in free energy to dismiss finite size effects. We denote these regions (identified by the condition of the groundstate energy) as dashed areas in the phase diagram of Fig. 2.
We derived the reciprocal cluster mean field method and applied it to the groundstate phase diagram of hard-core bosons in the Harper-Hofstadter-Mott model at flux . The bosons exhibit band insulating, striped superfluid, and supersolid phases. At finite anisotropy and integer filling we further found anisotropic gapless uncondensed liquid groundstates characterized by a non-trivial winding of the newly introduced vector . We further analyzed the properties of these phases using exact diagonalization. At fillings () this corresponds to integer particle (hole) filling of the lowest band, while the phase is symmetric with zero Hall response. We also observed metastable fractional quantum Hall phases predicted by other methods Sørensen et al. (2005); Hafezi et al. (2007); Möller and Cooper (2009); Umucalilar and Mueller (2010); He et al. (2017), which do not correspond to the groundstate (most likely due to finite-size effects), but are very close in free energy. Finally, we discussed how the liquid phases at integer fillings become gapped on a cylinder with just one unit-cell in the -direction and show a quantized Hall response to the adiabatic insertion of a magnetic flux. These properties, which are not expected for the full two-dimensional system, seem inherent to the quasi-one-dimensional nature of the cylinder geometry and need to be further investigated on larger system sizes. The combination of reciprocal cluster mean-field and exact diagonalization provides a promising venue for the numerical simulation of bosonic lattice systems with larger unit cells in the thermodynamic limit.
Acknowledgements.The authors would like to thank I. Bloch, H. P. Büchler, N. R. Cooper, F. Grusdt, A. Hayward, Y.-C. He, F. Heidrich-Meisner, F. Kozarski, A. M. Läuchli, K. Liu, M. Lohse, G. Möller, T. Pfeffer, M. Piraud, F. Pollmann, K. Sun, and R. O. Umucalilar, C.Wang, and N. Yao for valuable input. We especially thank A. M. Läuchli and F. Pollmann for sharing unpublished data and critically analyzing the results at and Y.-C. He for pointing out symmetry constraints in the uncondensed phases. DH and LP are supported by FP7/ERC Starting Grant No. 306897 and FP7/Marie-Curie Grant No. 321918, HS and PW by FP7/ERC starting grant No. 278023.
Appendix A Anisotropic Harper-Hofstadter model
The HHm can be solved by diagonalizing the Hamiltonian of Eq. (2), yielding three topologically non-trivial bands (see Fig 1b). For the gauge used in this work, the non-trivial topology arises in -direction, while in -direction the dispersion has a trivial cosine-shape. Both the topology and the four minima of the dispersion are independent of the anisotropy between the hopping amplitudes and . The bandwidths of the three bands, on the other hand, are affected by the ratio .
In order to analyze this, we introduce the quantities and for the lowest band, where is the bandwidth in -direction, i.e.
is the dispersion of the lowest band, and corresponds to taking the maximum with respect to and , respectively. The bandwidth in -direction, is defined analogously. As shown in the upper panel of Fig. 6c, for the lowest band is particularily flat in the -direction (), while for it is particularily flat in the -direction (). Note that this is not to be confused with the ”flatness” of the bands that typically supports fractional quantum Hall effects, which would consist in (in fact this quantity is low in the region of low anisotropy). Instead, having only or will result simply in supressing the condensation of bosons in the minima of the dispersion.
Another quantity affected by the anisotropy is the gap between the lowest and the central band. The simplest many-body problem where this plays a role is the case of spinless non-interacting fermions, which exhibit an integer quantum Hall phase for integer filling of the lowest band, i.e. if the chemical potential lies within the (anisotropy-dependent) gap, see Fig. 5c.
Appendix B Charge conjugation relations of hard-core bosons
For hard-core bosons a particle-hole transform (i.e. simultaneous and ) is equivalent to an inversion of the flux in the Hamiltonian of Eq. (8), i.e. . A direct consequence of this, is that the manybody groundstates at densities and are related by the operation, where is the particle-hole transform, and the complex conjugation operator. This implies that the Hall conductivity is anti-symmetric under the transform Lindner et al. (2010), i.e.
This effect is known as the charge conjugation symmetry of hard-core bosons Lindner et al. (2010).
Appendix C Comparison with exact diagonalisation
In Fig. 8a we compare our RCMF results with ED results on finite systems. The ED system differs from the solution of RCMF by a renormalization of the hopping according to Eq. (32), resulting in a shift in chemical potential of the plateaus. We present a sweep of the density in chemical potential without hopping anisotropy (i.e. ). As can be seen the only regions where we see a large discrepancy with respect to ED are around fillings and . These are the fillings where the metastable plateaus are particularly close in energy to the symmetry-broken groundstate (see Fig. 7).
We further compare the ED results with RCMF results on a cylinder with just sites and periodic boundary conditions in -direction. This can easily been done by modyfying the coarse-graining procedure of Eq. (12), which is now only integrated over . This results in a new cluster-hopping
As can be seen in Fig. 8a, in this case also RCMF shows a fractional plateau at and a plateau at , indicating that at zero anisotropy these phases are much more robust in the cylinder geometry than they are on the infinite square lattice.
Apart from the response to twisted boundaries discussed in Sec. IV.2, the anisotropic gapless nature of the two-dimensional uncondensed phases can also be observed in the scaling of the many-body gap as a function of while keeping fixed. As shown in Fig. 8b the manybody gaps remain essentially constant if is (sufficiently) smaller than , while it decreases in a non-monotonous way if is larger than . If the same scaling is done in -direction the situation is reversed. This also implies that on the cylinder these phases are gapped for . The same behavior can also be observed in the correlations in a system with and , shown in Fig. 8c, which quickly drop to zero as a function of for , indicating a gapped phase on the cylinder. For on the other hand, the correlations stay finite throughout the whole system hinting at the anisotropic gapless nature of the phase in two dimensions (and on the cylinder if is sufficiently large).
Appendix D Topological properties
Since RCMF does not give direct access to the many-body groundstate of the infinite lattice, nor to dynamical quantities, there is no way to directly compute the many-body Chern number of the system. Instead, we make use of the properties of the lattice to indirectly measure the topology of the groundstate using the vector introduced in Sec. II. It should be noted that the winding of is independent of the basis, since the geometric angle of a vector remains the same under an axis rotation , if is not an odd multiple of .
In our RCMF approach Eq. (19) reduces to in the absence of symmetry-breaking. Computing in the phases with by taking expectation values for the discrete momentum values of the cluster ( and ) then is equivalent to taking the same expectation values with respect to the infinite lattice. By looking at the values of at these discrete momenta and extrapolating its rotation on the infinite lattice, we are thereby able to measure the topology of the infinite lattice in a way that is not limited by finite-size effects. This is shown in Fig. 9a, where the vector is compared in a and periodic system, respectively, for filling and , yielding excellent agreement. In Figs. 9b and 9c we show the precession of for filling on a system, and