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Position-Momentum Duality and Fractional Quantum Hall Effect In 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- ^ 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 ^ juki with dispersion hk juki = juki. 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 juki 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 hukj@k uki (1) scopics of the conventional FQH effect resists a simple µ ν g (k) = 1 @ u [1 − ju ihu j] @ 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 hrjmi ∼ zme−|rj =4, which are uniquely determined by ηµν can be expressed in terms of a complex vector ! that µν µ ν ν µ demanding that they are eigenstates of the angular mo- obeys η =p! ¯ ! +! ¯ ! . The isotropic limit becomes mentum operator. While angular momentum does not !¯ = [1; i]>= 2. Corresponding guiding center operators readily translate to the lattice, a key observation is that µ y µ such states are simultaneous eigenstates of a parabolic π^ = Π^ µ ! ; π^ =! ¯ Π^ µ (7) ^ λ 2 confinement potential V (r) = 2 r projected to the LLL: P 0 0 ^ obey commutation relations [^π; π^y] = Ω when @ !µ = 0, [ 0 jm ihm j] V (r) jmi = λ (m + 1) jmi. kµ m i' A natural way to adapt this construction to an FCI is which fixes a phase freedom ! ! !e . 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: ^ 1 µν y 1 µν V (r) = 2 λ xµη xν (3) V = λ π^π^ + 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 2 Z indexes the unit for this duality can be inferred by noting that the con- cell with lattice vectors a1;2. Placing V^ 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 ^ P band with projector P = k jkihukj, and taking L ! 1: interchange of complex coordinates x + iy and momenta kx + iky. This situation is drastically different in FCIs, > λ µν λ µν V = P^V^ (r)P^ = Π^ µη Π^ ν + η gνµ k 2 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 Π^ 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 ^ ^ ^ 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 = [−ir + A(k)] + tr g(k) k 2 BZ (6) y 2 k 2 and ladder operatorsa ^ that generate higher MLLs, the Laughlin state at ν reads Ψ ∼ Q (^ay − a^y)1/ν Ψ (k).
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