SLAC-PUB-16305 Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
Martin Claassen,1, 2 Ching Hua Lee,3 Ronny Thomale,4 Xiao-Liang Qi,3 and Thomas P. Devereaux2, ∗ 1Department of Applied Physics, Stanford University, CA 94305, USA 2Stanford Institute for Materials and Energy Sciences, SLAC & Stanford University, CA 94025, USA 3Department of Physics, Stanford University, CA 94305, USA 4Institute for Theoretical Physics and Astrophysics, University of W¨urzburg, D 97074 W¨urzburg We develop a first quantization description of fractional Chern insulators that is the dual of the conventional fractional quantum Hall (FQH) problem, with the roles of position and momentum interchanged. In this picture, FQH states are described by anisotropic FQH liquids forming in momentum-space Landau levels in a fluctuating magnetic field. The fundamental quantum geometry of the problem emerges from the interplay of single-body and interaction metrics, both of which act as momentum-space duals of the geometrical picture of the anisotropic FQH effect. We then present a novel broad class of ideal Chern insulator lattice models that act as duals of the isotropic FQH effect. The interacting problem is well-captured by Haldane pseudopotentials and affords a detailed microscopic understanding of the interplay of interactions and non-trivial quantum geometry.
PACS numbers: 73.43.-f, 71.10.Fd, 03.65.Vf
The effects of topology and quantum geometry in con- to describe interacting electrons on the lattice [50–55]. A densed matter physics have recently garnered immense resolution is crucial to provide a foundation for studies attention. With topological insulators and the quantum of non-Abelian phases [56–58] and to provide microscopic anomalous Hall effect constituting the well-understood insight that can ultimately drive experimental discovery. case of non-interacting or weakly-correlated electron sys- In this work, we develop a first quantization description tems [1–3], the recent theoretical discovery of the frac- of FQH states in FCIs that leads to an effective Hamil- tional quantum Hall (FQH) effect in nearly-flat bands tonian that is the dual of the usual FQH problem, with with non-trivial topology [4–8] poses deep questions re- the roles of position and momentum interchanged. In garding the confluence of strong interactions and non- this picture, FCI analogues of FQH states are described trivial quantum geometry. On one hand, an experimen- by anisotropic FQH liquids forming in momentum-space tal realization of such a fractional Chern insulator (FCI) Landau levels in a fluctuating magnetic field. The chal- could conceivably push relevant energy scales by an order lenge of understanding FQH states in FCIs reduces to of magnitude, paving the way to robust FQH signatures a variational problem of determining the deformation of [9–14]. On the other hand, the disparity of conventional the guiding-center orbitals due to the presence of the lat- Landau levels and flat bands with non-zero Chern num- tice, in analogy to Haldane’s geometrical picture of the ber C suggests a rich playground to realize novel states anisotropic FQH effect (FQHE) [59–61]. Guided by these with topological order that cannot be attained in a con- insights, we then present and provide examples of a broad ventional electron gas in a magnetic field [15, 16], while class of ideal FCI host lattice models that constitute FCI simultaneously presenting a profound challenge to under- analogs of the isotropic FQHE. These models afford a par- stand the underlying microscopics of strong interactions. ticularly simple description of the interacting low-energy Most of our current understanding regarding FCIs dynamics, acting as FQH parent Hamiltonians with emer- stems from exact diagonalization (ED) of small clusters gent guiding center and SU(C) symmetry. The effects of for C = 1 [17–30] and C > 1 [31–35], mutatis-mutandis quantum geometry and Berry curvature fluctuations are mappings of the Hilbert space of flat Chern bands to the analyzed in terms of Haldane pseudopotentials. lowest Landau level (LLL) [15, 36–40] or vice versa [16], Consider a 2D band insulator hosting an isolated and approximate long-wavelength projected density alge- fractionally-filled flat band with non-zero C, generically bra [41–48]. The latter approaches however treat exclu- ˆ described by an N-orbital Bloch Hamiltonian hk. In band sively the universal long-wavelength continuum limit of basis, the flat band of interest is spanned by Bloch states the FQH problem, whereas the presence and relevance ˆ |uki with dispersion hk |uki = |uki. If the band gap is of the lattice manifests itself in the short-wavelength larger than intra-band interactions, then the kinetic en- physics. This conundrum is highlighted by the zoo of FCI ergy is effectively quenched while momentum-dependent lattice models established so far, which display strongly orbital mixing for |uki gives rise to a non-trivial quantum varying proclivities to host stable FQH phases that do not geometry, expressed by a gauge field and a Riemann met- correlate well with simple measures such as “flatness” of N−1 ric on CP for the Bloch band, the Berry curvature band dispersion and Berry curvature. Ω(k) and Fubini-Study metric gµν (k): At its heart, the theoretical challenge stems from the fact that the immense success in describing the micro- µν Ω(k) = ∂k Aν (k) Aν (k) = −i huk|∂k uki (1) scopics of the conventional FQH effect resists a simple µ ν g (k) = 1 ∂ u [1 − |u ihu |] ∂ u + (µ ↔ ν) (2) description in second quantization [49] that is essential µν 2 kµ k k k kν k
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515. 2
Here, Aν (k) is the Berry connection, and C = rotational symmetry if present, whereas n = 0, ..., C − 1 1 R 2 2π BZ d k Ω(k). We use lattice constants a0 = ~ = 1. acts as a component index for C > 1 [66]. In real space, The first task at hand is to describe the guiding- MLL wave functions are radially-localized (Fig. 1(a-f)). center basis. In the case of an isotropic free elec- Importantly, (4) does not enter as physical confinement tron gas in a magnetic field, Laughlin constructed a se- in the infinite system, thus λ → 0 can be taken in the ries of incompressible FQH trial wave functions [62] for thermodynamic limit. However, λ 6= 0 enters as a proper odd-fraction filling factors, from a single-body basis of energy scale when considering finite-size droplets [67, 68]. radially-localized symmetric-gauge LLL wave functions Anisotropic case: Generically, the confinement metric 2 hr|mi ∼ zme−|r| /4, which are uniquely determined by ηµν can be expressed in terms of a complex vector ω that µν µ ν ν µ demanding that they are eigenstates of the angular mo- obeys η =√ω ¯ ω +ω ¯ ω . The isotropic limit becomes mentum operator. While angular momentum does not ω¯ = [1, i]>/ 2. Corresponding guiding center operators readily translate to the lattice, a key observation is that µ † µ such states are simultaneous eigenstates of a parabolic πˆ = Πˆ µ ω , πˆ =ω ¯ Πˆ µ (7) ˆ λ 2 confinement potential V (r) = 2 r projected to the LLL: P 0 0 ˆ obey commutation relations [ˆπ, πˆ†] = Ω when ∂ ωµ = 0, [ 0 |m ihm |] V (r) |mi = λ (m + 1) |mi. kµ m iϕ A natural way to adapt this construction to an FCI is which fixes a phase freedom ω → ωe . Substituting to consider anisotropic confinement on a lattice, as a tool (7) in (4) yields the confinement Hamiltonian in guiding- to determine the guiding-center basis: center language that determines the single-body basis: ˆ 1 µν † 1 µν V (r) = 2 λ xµη xν (3) V = λ πˆπˆ + 2 λ (η gµν − Ω) (8) Here, ηµν is a unimodular Galilean metric which a priori Comparison with Haldane’s construction [59] reveals that serves as a variational degree of freedom, constrained to ηµν is precisely the FCI momentum-dual of the guiding- retain the discrete rotational symmetries of the host lat- center metric of the anisotropic FQHE. The root cause tice, and r = m1a1 + m2a2, m1,2 ∈ Z indexes the unit for this duality can be inferred by noting that the con- cell with lattice vectors a1,2. Placing Vˆ on a L × L lat- ventional FQHE can in fact be formulated both in po- tice via appropriate long-distance regularization [63], the sition and momentum representation, with single-body low-energy dynamics follow from projection onto the flat dynamics and LLL wave functions form-invariant under ˆ P band with projector P = k |kihuk|, and taking L → ∞: interchange of complex coordinates x + iy and momenta kx + iky. This situation is drastically different in FCIs, > λ µν λ µν V = PˆVˆ (r)Pˆ = Πˆ µη Πˆ ν + η gνµ k ∈ BZ (4) where the discreteness of the lattice necessitates switch- 2 2 ing to momentum space in order to retain a first-quantized description in terms of continuous coordinates. where Πˆ µ are momentum-space analogues to the usual canonical momentum operators, with µ = x, y: Conceptually, the challenge of devising a microscopic description of FQH states in FCIs reduces to a variational ˆ ˆ ˆ Πµ = −i∂kµ + Aµ(k)[Πµ, Πν ] = −iµν Ω(k) (5) problem of determining the deformation of the guiding- center orbitals upon placing a FQH liquid on the lattice. Isotropic case: Physical insight may be gleaned by Given an appropriate choice of ηµν (k), any FCI can in identifying (4) with an electron in a magnetic field but in µν µν principle be captured by many-body trial ground states, momentum-space; η = δ is particularly instructive: constructed from the single-body eigenstates of (8): for
λ 2 λ instance, given the lowest MLL wave function Ψ0(k) V = [−i∇ + A(k)] + tr g(k) k ∈ BZ (6) † 2 k 2 and ladder operatorsa ˆ that generate higher MLLs, the Laughlin state at ν reads Ψ ∼ Q (ˆa† − aˆ†)1/ν Ψ (k). Here, Ω(k) identifies with the magnetic field, and ν i 2 quantization. Analogous to the FQHE, density interac- |Ψm(r)| µν tions may be generalized via an interaction metric η˜ : 100 ˆ X X † ˆ µν 10- 1 HI = Vq ρˆqρˆ−q, ρˆq = cˆkαTq[˜η ]c ˆkα (11) q αk 10- 2 (a) (b) (c) µν 10- 3 µν qµη˜ ∂ where Tˆq[˜η ] = e kν enters as a metric-dependent momentum-space translation operator. In contrast to 10- 4 the FQHE on the plane, translation symmetry and pe- 10- 5 µν riodicity in q constrainη ˜ to SL(2, Z). Importantly, 10- 6 the guiding center metric is still a variational degree - 7 µν 10 of freedom if the deviation of (9) andη ˜ is signifi- (d) (e) (f) cant. To proceed, note that if the ideal droplet con- dition (10) is satisfied, then an exact operator identity µ [1 − |ukihuk|]ω ¯ ∂kµ |ukihuk| = 0 [43] entails that any op- − µ + µ erator of the form Oˆ = Λˆ (−i∂µω )Λˆ (−iω¯ ∂µ) with analytic functions Λ± can be projected to the flat band as PˆOˆPˆ> = Λˆ −(ˆπ)Λˆ +(ˆπ†). Consider thus a decomposi- tion ofη ˜µν =χ ¯µων + χµω¯ν with complex vector χ: if χ is momentum-independent, then the translation opera- µ µ q+ω ∂k q−ω¯ ∂k tor is precisely of this form: Tˆq = e µ e µ with (g) µ µ q± = qµχ¯ , qµχ — this case is considered in detail be- low [69]. Note that in the isotropic limitη ˜µν , ηµν = δµν , ˆ ˆ Tq reduces to the conventional translation√ operator Tq = FIG. 1. (color online). (a-f) Real-space lattice guiding-center iq·(−i∇ ) e k with q± = (qx ± iqy)/ 2, while HI is just the wave functions for the ideal isotropic C = 3 model, evalu- usual density interaction. While emphasis has thus far ated from (8). Insets depict analogous conventional LLL wave been placed on narrowing down to a suitable class of in- functions. (g) Berry curvature Ωk (inset) and associated MLL spectra for the FCI models discussed in the main text. m teractions, substantial progress has been made: projected is the guiding center index; dotted lines indicate ideal flat-Ω to the flat band, the joint dynamics of (3), (11) can now λCm spectrum m = 2π . The C-fold degeneracy of MLLs reflects be succinctly expressed in guiding-center language: the C-component basis for C > 1 models. The guiding-center structure remains robust in the presence of fluctuations of Ωk. X † X † † ˆ iq+(ˆπi−πˆj ) iq−(ˆπi −πˆj ) H = λ πˆiπˆi + Vqe e (12) i i model truncated to nearest-neighbor (NN) hoppings acts as a toy model for strongly-correlated states with conventional order. While a classification of such ideal FCI host lattice models remains an im- portant open task, note that simple physical models can emerge after truncation of irrelevant hoppings. For instance, hC=2 is well-described by a canonical d-wave lattice model hC=2(k) = d(k) · σ~ with d(k) = 0 > [t(cos kx + cos ky), t(cos kx − cos ky), t sin(kx) sin(ky)] with nearest- and next-nearest-neighbor hoppings. Analysis: In addition to the ideal isotropic models (14,15), we study both a conventional three-orbital C = 1 model on the square lattice [50] that does not satisfy (10), and an “optimized” variant with flat Berry curva- −6 ture (var(Ωk) ≈ 10 ), obtained via adiabatically adding FIG. 2. (color online). (a)-(c) Haldane pseudopotentials for symmetry-preserving hoppings up to 5th neighbor [63]. the C = 1 three-orbital and C > 1 ideal droplet models (on-site and NN interactions). Only odd (even) pseudopo- Fig. 2 depicts the pseudopotential decomposition for tentials determine interactions between same-species fermions on-site and NN repulsion, using the MLL basis of Fig. (bosons). Error bars quantify the residual center-of-mass M 1. The C = 2, 3 ideal lattice models display emergent variation, strongly suppressed for the ideal C = 2, 3 models guiding-center and SU(C) symmetries, manifested in a due to emergent conservation of guiding center. (d) Log plot vanishing center-of-mass dependence of pseudopotentials of guiding-center and SU(C) symmetry breaking in the inter- (figs. 2b, 2c), and suppression of symmetry-breaking two- acting problem with matrix elements V n1...n4 , quantified as m1...m4 body interaction terms (fig. 2d). As anticipated from the ratio of the norm of ∆M = m + m − m − m (∆N = 1 2 3 4 (13), on-site repulsion results in a non-zero pseudopoten- n1 + n2 − n3 − n4 mod C component) symmetry-breaking and -preserving two-body matrix elements. The guiding-center in- tial only for V0, acting as an optimal (221)-Halperin state dex identifies with C4 symmetry. (e) Role of broken rotational [74] parent Hamiltonian for hardcore bosons, whilst not 0 symmetry on the lattice, depicted via leading-order Vm,m+4 of stabilizing fermionic FQH states [75]. The latter can be 0 0 generalized pseudopotentials Vmm0 = hm, M| H |m ,Mi. remedied by tuning NN repulsion to tune V1, V3. The controlled expansion in pseudopotentials is a key merit of this construction and highlights that, contrary to common µν µν µν η , η˜ = δ with corresponding Fubini-Study metric perception, the confluence of flat Berry curvature and lo- gxx − gyy, gxy = 0, tr g(k) = Ω(k) is uniquely satisfied by cal interaction does not stabilize a fermionic FQH liquid any Bloch state that can be written without normaliza- in an FCI. Conversely, the non-optimal C = 1 models vi- tion as a meromorphic function |u¯ i = u¯ . The k kx+iky olate (10) and do not pin the guiding-center metric to g. number of poles in the BZ defines C [73]; periodic bound- The guiding-center description (12) is incomplete, broad- ary conditions in k restricts u¯ to elliptic functions, kx+iky ening the effective interaction range with non-vanishing constrained to C ≥ 2. Skew-anisotropic guiding center and decaying Vm with substantial center-of-mass devia- metrics ensue from distortions of the lattice: for instance, tion even for a purely local interaction (fig. 2a) that per- a ’squeezed’ FQH liquid with ηxx = 1/ζ, ηyy = ζ follows sists even for uniform Berry curvature. We stress that from BZ strain deformations with kx → ζkx. this highlights the shortcomings of long-wavelength limit To illustrate our construction, we consider two multi- arguments [41–47] in predicting the microscopics of the orbital toy models on the square lattice for C = 2, 3 [76], FQHE on the lattice. with the guiding-center basis of MLLs depicted in fig. 1: In summary, we introduced a first-quantized descrip- > tion of FCIs, with the FQHE emerging in a picture of C=2 1, α ℘(k + ik ) u¯k = x y (14) anisotropic momentum-space Landau levels in a fluctuat- C=3 0 > u¯k = 1, β ℘(kx + iky), γ ℘ (kx + iky) (15) ing magnetic field. We presented a novel class of ideal FCI lattice models as duals of the isotropic FQHE and Here, ℘(z) is the Weierstrass elliptic function with periods demonstrated their optimality via an expansion of local 2π, 2πi and a second-order pole at the Γ point, and α = interactions into Haldane pseudopotentials which can be 5.77, β = −7.64, γ = 6.73 are band structure parameters, determined straightforwardly in first quantization. A pri- chosen to minimize Berry curvature fluctuations. mary goal of this work is to establish a deeper microscopic The corresponding Bloch Hamiltonian is not unique; understanding of the stabilization of FQH states in flat a possible definition with flat bands is hC(k) = Chern bands - the resulting interplay of topology and ge- C C C C 1 − u¯k u¯k / u¯k u¯k . In general, hC(k) exhibits ometry to determine long- and short-wavelength physics long-ranged but exponentially-decaying hopping terms on the lattice serves as a natural application of the for- [73] and acts as an artificial toy lattice model for FCI malism of the anisotropic FQHE. The results presented states with topological order, much like the Hubbard set a foundation for microscopic analysis of non-Abelian 5 phases on the lattice and extension to fractional topolog- [24] A. G. Grushin, J. Motruk, M. P. Zaletel, and F. Pollmann, ical insulators. arXiv:1407.6985 (2014). [25] Z. Liu, D. L. Kovrizhin, and E. J. Bergholtz, Phys. Rev. B 88, 081106 (2013). We acknowledge support from the U. S. Depart- [26] W.-W. Luo, W.-C. Chen, Y.-F. Wang, and C.-D. Gong, ment of Energy, Office of Basic Energy Science, Divi- Phys. Rev. B 88, 161109 (2013). sion of Materials Science and Engineering under Con- [27] Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. tract No. DE-AC02-76SF00515. C.H.L. is supported B 89, 155113 (2014). [28] Y.-H. Wu, J. K. Jain, and K. Sun, Phys. Rev. B 86, by a fellowship from the Agency of Science, Technology 165129 (2012). and Research of Singapore. R.T. is supported by the [29] T. Scaffidi, and G. M¨oller,Phys. Rev. Lett. 109, 246805 European Research Council through ERC-StG-336012- (2012). TOPOLECTRICS. X.L.Q. is supported by the David & [30] Z. Liu, and E. J. Bergholtz, Phys. Rev. B 87, 035306 Lucile Packard Foundation. (2013). [31] Z. Liu, E. J. Bergholtz, H. Fan, and A. M. L¨auchli, Phys. Rev. Lett. 109, 186805 (2012). [32] Y.-F. Wang, H. Yao, Z.-C. Gu, C.-D. Gong, and D. N. Sheng, Phys. Rev. Lett. 108, 126805 (2012). [33] A. Sterdyniak, C. Repellin, B. A. Bernevig, and N. Reg- ∗ Author to whom correspondence should be addressed to: nault, Phys. Rev. B 87, 205137 (2013). M. C. ([email protected]) [34] Z. Liu, E. J. Bergholtz, and E. Kapit, Phys. Rev. B 88, [1] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). 205101 (2013). [2] M. Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 [35] Y.-F. Wang, H. Yao, C.-D. Gong, and D. N. Sheng, Phys. (2010). Rev. B 86, 201101 (2012). [3] X. L. Qi, and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 [36] X.-L. Qi, Phys Rev. Lett. 107, 126803 (2011). (2011). [37] C. H. Lee, R. Thomale, and X.-L. Qi, Phys. Rev. B 88, [4] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. 035101 (2013). Rev. Lett. 106, 236804 (2011). [38] C. H. Lee, and X.-L. Qi, Phys. Rev. B 90, 085103 (2014). [5] D.N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature [39] C.-M. Jian, and X.-L. Qi, Phys. Rev. B 88, 165134 (2013). Commun. 2, 389 (2011). [40] Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. [6] N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 B 86, 085129 (2012). (2011). [41] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Phys. [7] E. J. Bergholtz, and Z. Liu, Int. J. Mod. Phys. B 27, Rev. B 85, 241308(R) (2012). 1330017 (2013). [42] B. Estienne, N. Regnault, and B. A. Bernevig, Phys. Rev. [8] S. A. Parameswaran, R. Roy, and S. L. Sondhi, B 86, 241104(R) (2012). arXiv:1302.6606 (2013). [43] R. Roy, Phys. Rev. B 90, 165139 (2014). [9] D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto, [44] M. O. Goerbig, Eur. Phys. J. B 85, 15 (2012). Nat. Commun. 2, 596 (2011). [45] E. Dobardzi´c,M. V. Milovanovi´c,and N. Regnault, Phys. [10] J. W. F. Venderbos, S. Kourtis, J. van den Brink, and M. Rev. B 88, 115117 (2013). Daghofer, Phys. Rev. Lett. 108, 126405 (2012). [46] C. Repellin, T. Neupert, Z. Papic, and N. Regnault, Phys. [11] S. Kourtis, J. W. F. Venderbos, and M. Daghofer, Phys. Rev. B 90, 045114 (2014). Rev. B 86, 235118 (2012). [47] G. Murthy and R. Shankar, Phys. Rev. B 86, 195146 [12] N. R. Cooper, and R. Moessner, Phys. Rev. Lett. 109, (2012). 215302 (2012). [48] T. S. Jackson, G. M¨oller,and R. Roy, arXiv:1408.0843 [13] Z. Liu, Z.-F. Wang, J.-W. Mei, Y.-S. Wu, and F. Liu, (2014). Phys. Rev. Lett. 110, 106804 (2013). ´ [49] G. Ortiz, Z. Nussinov, J. Dukelsky, and A. Seidel, Phys. [14] A. G. Grushin, A. G´omez-Le´on,and T. Neupert, Phys. Rev. B 88, 165303 (2013). Rev. Lett. 112, 156801 (2014). [50] K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Phys. Rev. [15] M. Barkeshli, and X.-L. Qi, Phys. Rev. X 2, 031013 Lett. 106, 236803 (2011). (2012). [51] E. Tang, J.-W. Mei, and X.-G. Wen, Phys. Rev. Lett. 106, [16] Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. 236802 (2011). Lett. 110, 106802 (2013). [52] X. Hu, M. Kargarian, and G. A. Fiete, Phys. Rev. B 84, [17] Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. N. Sheng, 155116 (2011). Phys. Rev. Lett. 107, 146803 (2011). [53] F. Wang, and Y. Ran, Phys. Rev. B 84, 241103(R) (2011). [18] Y.-L. Wu, B. A. Bernevig, and N. Regnault, Phys. Rev. [54] S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, Phys. Rev. B 85, 075116 (2012). B 86, 241112 (2012). [19] B. A. Bernevig, and N. Regnault, Phys. Rev. B 85, 075128 [55] M. Trescher, and E. J. Bergholtz, Phys. Rev. B 86, (2012). 241111(R) (2012). [20] T. Liu, C. Repellin, B. A. Bernevig, and N. Regnault, [56] G. Moore, and N. Read, Nucl. Phys. B 360, 362 (1991). Phys. Rev. B 87, 205136 (2013). [57] N. Read, and E. Rezayi, Phys. Rev. B 59, 8084 (1999). [21] A. M. L¨auchli, Z. Liu, E. J. Bergholtz, and R. Moessner, [58] E. Ardonne, and K. Schoutens, Phys. Rev. Lett. 82, 5096 Phys. Rev. Lett. 111, 126802 (2013). (1999). [22] A. G. Grushin, T. Neupert, C. Chamon, and C. Mudry, [59] F. D. M. Haldane, Phys. Rev. Lett. 107, 116801 (2011). Phys. Rev. B 86, 205125 (2012). [60] B. Yang, Z. Papi´c,E. H. Rezayi, R. N. Bhatt, and F. D. [23] S. Kourtis, T. Neupert, C. Chamon, and C. Mudry, Phys. M. Haldane, Phys. Rev. B 85, 165318 (2012). Rev. Lett. 112, 126806 (2014). 6 [61] R.-Z. Qiu, F. D. M. Haldane, X. Wan, K. Yang, and S. [68] T. Scaffidi, and S. H. Simon, Phys. Rev. B 90, 115132 Yi, Phys. Rev. B 85, 115308 (2012). (2014). [62] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). [69] The general case follows from deformation of µ µ q+ω ∂k q−ω¯ ∂k [63] See Supplemental Material at [URL will be inserted by e µ e µ and functions q±(q) to satisfy publisher], which includes Refs. [50, 64]. ˆ µν Tq |q→0 ≈ 1 − iqµη˜ (−i∂kν ) whilst ensuring translation [64] V. Fock, Z. Phys. 47, 446 (1928); C. G. Darwin, Math. symmetry of (11). Proc. Cambridge Philos. Soc. 27, 86 (1930); P. A. Maksym [70] Equivalently, two-body eigenstates ofπ ˆ† πˆ |m˜ i = and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990). rel rel |m˜ i display perfect center-of-mass degeneracy [65] F. D. M. Haldane, and E. H. Rezayi, Phys. Rev. B 31, m˜ † h † i 2529(R) (1985). πˆrelπˆrel |m, Mi ∼ m |m, Mi only for πˆrel, πˆcm → 0. [66] Both the conventional and dual QH problems can be de- [71] Weak breaking of SU(C) results from the non- scribed in terms of a pair of harmonic oscillators. The commutation of left- and right-handed oscillators in the FCI component index oscillator has no analogue in the MLL problem for fluctuating magnetic field. conventional FQHE; conversely, FQH ’left-handed’ (Lan- [72] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983). dau level) degrees of freedom have no analogues in FCIs. [73] C.-M. Jian, Z.-C. Gu, and X.-L. Qi, Phys. Stat. Sol. RRL [67] The energy penalty of a steep confinement potential is 7, 154 (2013). u 0 u u approximately hm|(r/r0) |m i ∼ (C/r0) m , hence a fi- [74] B. I. Halperin, Helv. Phys. Acta 56, 75 (1983). nite geometry with a certain filling ν may be selected by [75] Even-n intra-component pseudopotentials Vn do not con- truncating at a maximum guiding center index mmax. tribute in the case of fermions due to Pauli exclusion. [76] Details will be discussed in a forthcoming publication. 7 Supplementary Material Momentum-Space Landau Levels As described in the main text, the guiding-center basis in a FCI can be understood in terms of momentum-space Landau levels (MLLs). We seek the low-energy momentum-space dynamics induced by a real-space confinement ˆ λ µν µν potential V (r) = 2 xµη xν . The confinement metric η can be expressed as an Nd-fold superposition of reciprocal µν PNd µ ν lattice vectors η = i ηidi di , where di = m1b1 + m2b2 and m1, m2 ∈ Z. A minimum of three basis vectors di is required to reproduce the three independent components of η. Regularization of the long-distance behavior of the confinement potential on an L × L lattice is not unique; a simple choice reads: N λ L2 Xd Vˆ = η [1 − cos(d · r/L)] (16) reg 2 (2π)2 i i i The regularized form of the confinement potential is used for numerical results presented in the main text. Choosing a sufficiently large lattice, one can solve for the guiding-center basis as eigenstates of H˜0 + Vˆreg, where care must be taken to ensure that the band gap of H˜0 is much larger than the depth of the confinement potential. The lattice dimensions are taken to be 160x160 sites for numerical results in this work. As is canonical in numerical studies of FCIs, we assume that the energy scale of interactions in the flat band is much greater than its band width and work 1 with a modified lattice Hamiltonian H˜0 = Hˆ0 to eliminate residual dispersion k in the flat band. This step is not k necessary for the ideal models of equations (14), (15) in the main text, as they intrinsically exhibit a perfectly flat Chern band. The low-energy dynamics follow from projecting Vˆreg onto the reduced Hilbert space built from the set of Bloch states |uki spanning the flat band, and taking the thermodynamic limit: λ X 0 Hˆ = lim |kihuk| Vregˆ |uk0 ihk | L→∞ 2 kk0∈BZ 2 Nd λL X X di di = lim |kihuk| ηi 2 |ukihk| − uk+ di k + − uk− di k − L→∞ 4 L L L L k∈BZ i Nd Z λ X 2 = η d2k |kihu | (−id · ∇ ) |u ihk| 2 i k i k k i BZ N ! λ Z Xd = d2k η dµdν |ki (−i∂ )(−i∂ ) + iA ∂ + iA ∂ − (hu |∂ ∂ |u i) hk| 2 i i i µ ν µ ν ν µ k µ ν k i λ Z = d2k |ki [(−i∂ + A ) ηµν (−i∂ + A ) + ηµν g ] hk| (17) 2 µ µ ν ν µν µν where Aµ = −i huk|∂µ|uki is the Berry connection, Ω = ∂µAν is the Berry curvature, and the Fubini-Study metric gµν is defined as: 1 g = h∂ u |∂ u i − h∂ u |u i hu |∂ u i + (µ ↔ ν) (18) µν 2 µ k ν k µ k k k ν k µν µν The last line of equation 17 uses the fact that the metric tensor η is symmetric, hence η µν = 0. In terms of operators Πˆ µ = −i∂µ + Aµ, one finally arrives at the first-quantized Hamiltonian (4) described in the main text. πC > A smooth gauge choice separates A(k) = [−ky, kx] + Afluct(k) into a topological contribution and fluctu- ABZ ations. C 6= 0 then stipulates that wave functions acquire twisted boundary conditions ψnm(k + m1b1 + m2b2) = C −i (m1a2−m2a1)k e 4π ψnm(k) that constrain the intra-MLL index to 0 ... C −1, giving rise to the MLL spectrum described in the main text. As a side note, residual dispersion in the Chern band can be understood as a momentum-space confinement potential, giving rise to a Darwin-Fock eigenbasis [64] on the torus, with the ratio of confinement λ and bandwidth entering as a variational degree of freedom when building the single-body basis. Ideal Droplet Condition – Two-Band Models The ideal droplet condition 2pdet g(k) = Ω(k) is automatically satisfied by any two-band FCI model. To see this, consider a generic two-band Bloch Hamiltonian h(k) = (k) dˆ(k) · σ~ parameterized by unit vector dˆ(k) and dispersion 8 ±(k), where σ~ are the Pauli matrices in orbital basis. The Berry curvature and Fubini-Study metric read: 1 ∂dˆ(k) ∂dˆ(k) Ω(k) = dˆ(k) · × (19) 2 ∂kx ∂ky ˆ ! ˆ ! 1 ∂d(k) ˆ ˆ ∂d(k) gµν (k) = × d(k) · d(k) × (20) 4 ∂kµ ∂kν The determinant of the Fubini-Study metric follows from application of quadruple-product identities and satisfies 2pdet g(k) = Ω(k). Haldane Pseudopotentials This section discusses the derivation of Haldane pseudopotentials for momentum-space Landau levels. The starting point is the two-body interaction in first quantization that follows from equation (12) in the main text. The guiding- center basis can be found via numerical solution of the single-body problem. Given the lowest MLL state |Ψ0i and a h †i (ˆa†)m set of ladder operatorsa ˆ,a ˆ†, aˆ , aˆ = 1 generating the m-th MLL |mi = √ |Ψ i, one can define two-body states i i m! 0 † † !m † † !M 1 aˆ1 − aˆ2 aˆ1 +a ˆ2 |m, Mi = √ √ √ |Ψ0i (21) m!M! 2 2 The Haldane pseudopotentials follow * + X iq (ˆπ −πˆ ) iq (ˆπ†−πˆ†) + 1 2 − 1 2 VmM = m, M Vqe e m, M (22) q and can be evaluated numerically, with the residual center-of-mass dependence studied in the main text. To recover the limit of flat Berry curvature, note that the guiding center operators can be written in terms of the ladder operators p πˆi =a ˆi/kB with kB = 2π/C the magnetic wave vector for a square lattice, which allows for an algebraic evaluation of the two-body matrix elements: Z 2 D 0 0E V mM = d q Vq mM Vˆ m M m0M 0 0 Z 1 m † † m 2 iq+(ˆa −aˆ )/kB iq−(ˆa −aˆ )/kB † † = d q Vq √ 0 aˆ − aˆ e 1 2 e 1 2 aˆ − aˆ 0 δMM 0 m!m0! 1 2 1 2 √ √ 0 Z l l 0 0 2 1 X ( 2iq+/kB) ( 2iq−/kB) D m+l †m +l E = d q Vq √ 0 aˆ aˆ 0 δMM 0 0 l!l0! m!m ! ll0 √ √ Z l m−m0+l 2 1 X ( 2iq+/kB) ( 2iq−/kB) = d q Vq √ (m + l)!δMM 0 0 l!(m − m0 + l)! m!m ! l √ 2 m−m0 0 2¯q Z r ( 2iq−/kB) 1F1 m + 1; m − m + 1, − 2 2 m! kB = d q V δ 0 (23) q m0! (m − m0)! MM µ µ where 1F1(a; b; z) is the Kummer confluent hypergeometric function. Here q+ =χ ¯ qµ, q− = χ qµ, andq ¯ = q−q+ = µ ν qµχ¯ χ qν . While the center of mass guiding center dependence is eliminated, lack of continuous rotational symmetry on the lattice does not forbid subdominant scattering between states with different relative guiding center indices m, m0 unless the parent interaction is purely local Vq → V . Instead, CN symmetry can provide weaker selection rules with 0 0 Vmm0 6= 0 only for√ m − m mod N = 0. The conventional Haldane pseudopotentials follow from imposing m = m , q± = (qx ± iqy)/ 2, and expressing the hypergeometric function in terms of a Laguerre polynomial: 2 Z q 2 2 2 −q /kB Vm = d q VqLm 2 e (24) kB which is the formula quoted in the main text. 9 Elliptic Function Models The main text describes a class of ideal FCI models which can be defined from unnormalized Bloch states u˜kx+iky written as elliptic functions. To see this, consider a generic N-band model with unnormalized Bloch state written in terms of complex momenta > |u˜ki = [φ1(z, z¯), φ2(z, z¯), . . . , φN (z, z¯)] (25) where z, z¯ = kx±iky. The corresponding Berry curvature and Fubini-Study metric may be evaluated straight-forwardly: N " 2 2 # X |φm∂zφn − φn∂zφm| |φm∂z¯φn − φn∂z¯φm| Ω = 2 − (26) |φ|4 |φ|4 n>m N " 2 2 # X |φm∂zφn − φn∂zφm| |φm∂z¯φn − φn∂z¯φm| tr g = 2 + (27) |φ|4 |φ|4 n>m N ? X (φm∂zφn − φn∂zφm)(φm∂z¯φn − φn∂z¯φm) + c.c. g − g = 2 (28) xx yy |φ|4 n>m N ? X (φm∂zφn − φn∂zφm)(φm∂z¯φn − φn∂z¯φm) − c.c. g = i (29) xy |φ|4 n>m 1 det g = (tr g)2 − (g − g )2 − 4g2 (30) 4 xx yy xy √ One can see by inspection that the conditions Ω = tr g = 2 det g and gxx − gyy = gxy = 0 are satisfied if and only if ∂z¯φn = 0. This constrains the unnormalized Bloch state to meromorphic functions in kx + iky. The corresponding droplet confinement metric is just the Euclidean metric as described in the main text. Generally, one can consider an arbitrary static lattice√ deformation defined by transformation k −→ U · k. As such a transformation preserves both Ω and det g, Ω = 2 det g remains satisfied, whereas the confinement metric changes to deform the shape of the droplet accordingly. Three-Orbital C = 1 Model – Flat Berry Curvature Optimization To study the effects of Berry curvature fluctuations, we introduced in the main text a C = 1 model with flat Berry curvature. This model is adiabatically connected to the three-orbital model by Sun et al. [50] by adding longer-ranged hopping terms to minimize Berry curvature fluctuations. We start from the original lattice Hamiltonian −2tdd(cos kx + cos ky) + δ 2itpd sin kx 2itpd sin ky cˆd,k X h † † † i Hˆ = cˆ cˆ cˆ −2it sin k 2t cos k − 2t0 cos k i∆ cˆ d,k px,k py ,k pd x pp x pp y px,k 0 k −2itpd sin ky −i∆ 2tpp cos ky − 2tpp cos kx cˆ py ,k (31) for a three-orbital model on the square lattice with dx2−y2 , px, py orbitals per site. This model exhibits a C4 rotational symmetry and mirror symmetries with symmetry operators 1 0 0 1 0 0 ˆ ˆ ˆ ˆ ˆ UC4 = 0 0 1 · Rπ/2 UM = 0 1 0 · Ix · K (32) 0 −1 0 0 0 −1 ˆ ˆ ˆ where Rπ/2, Ix and K are π/2 rotation, x-reflection and complex conjugation operators, respectively. To proceed, we write down the most general model Hamiltonian with up to 5th neighbor hoppings between orbitals that is invariant under the symmetry operations above. The parameter vector v of hopping amplitudes serve as the variational parameter of the numerical minimization of Berry curvature fluctuations using cost function Z 1 2 K[v] = d2k Ω(k, v) − (33) 2π 10 The minimization procedure uses the original 3-orbital model parameters as starting values for v. Furthermore, care has to be taken to ensure that the band gap remains open. We quote here the resulting model Hamiltonian that is used in the main text: hdd hdx hdy ˆ ? Hoptimized = hdx hxx hxy (34) ? ? hdy hxy hyy with hdd = 0.203 cos(kx − 2ky) − 0.0443 cos(2kx − 2ky) + 2.75 cos(kx − ky) + 0.203 cos(2kx − ky) + 2.75 cos(kx + ky) + 0.203 cos(2 kx + ky) + 0.203 cos(kx + 2ky) − 0.0443 cos(2kx + 2ky) + 4.13 cos(kx) − 0.663 cos(2kx) + 4.13 cos(ky) − 0.663 cos(2ky) − 0.76 (35) hdx = −(0.0372 − 0.1273i) sin(kx − 2ky) + (0.0391 − 0.0043i) sin(2kx − 2 ky) − (0.18 − 1.46i) sin(kx − ky) + (0.0439 − 0.0177i) sin(2kx − ky) + (0.18 + 1.46i) sin(kx + ky) − (0.0439 + 0.0177i) sin(2 kx + ky) + (0.0372 + 0.1273i) sin(kx + 2ky) − (0.0391 + 0.0043i) sin(2 kx + 2ky) + 1.01i sin(kx) − 0.374i sin(2kx) + 3.10 sin(ky) + 0.0351 sin(2ky) (36) hdy = (0.0439 + 0.0177i) sin(kx − 2ky) + (0.0391 + 0.0043i) sin(2kx − 2 ky) − (0.18 + 1.46i) sin(kx − ky) − (0.0372 + 0.1273i) sin(2kx − ky) − (0.18 − 1.46i) sin(kx + ky) − (0.0372 − 0.1273i) sin(2 kx + ky) + (0.0439 − 0.0177i) sin(kx + 2ky) + (0.0391 − 0.0043i) sin(2 kx + 2ky) − 3.10 sin(kx) − 0.0351 sin(2kx) + 1.01i sin (ky) − 0.374i sin(2ky) (37) hxx = −0.0912 cos(kx − 2ky) + 0.0912 cos(2kx − 1ky) + 0.0912 cos(2 kx + ky) − 0.0912 cos(kx + 2ky) + 0.156 cos(kx) + 0.0472 cos(2 kx) − 0.156 cos(ky) − 0.0472 cos(2ky) (38) hxy = (0.0808 + 0.0062i) cos(kx − 2ky) + (0.0273 + 0.0191i) cos(2kx − 2 ky) + (0.627 − 0.847i) cos(kx − 1ky) + (0.0808 + 0.0062i) cos(2kx − 1 ky) − (0.627 + 0.847i) cos(kx + ky) − (0.0808 − 0.0062i) cos(2 kx + ky) − (0.0808 − 0.0062i) cos(kx + 2ky) − (0.0273 − 0.0191i) cos(2 kx + 2ky) − 0.921i cos(kx) − 0.0454i cos(2kx) − 0.921i cos (ky) − 0.0454i cos(2ky) − 1.07i (39) hyy = 0.0912 cos(kx − 2ky) − 0.0912 cos(2kx − 1ky) − 0.0912 cos(2 kx + ky) + 0.0912 cos(kx + 2ky) − 0.156 cos(kx) − 0.0472 cos(2 kx) + 0.156 cos(ky) + 0.0472 cos(2ky) (40) which yields a measure of Berry curvature fluctuations Z 1 2 (2π)2 d2k Ω(k, v) − = 5.37 × 10−7 (41) 2π