Geometry, topology and symmetry in strongly correlated materialsRahul Roy, UCLA

Research OverviewTopological phenomena in strongly correlated systems are poorly understood due to the fact that they lie outside the conventional paradigm of symmetry breaking; nevertheless, they hold great potential in engineering new devices and quantum computation by virtue of their novel electronic properties. To advance our understanding of these systems, I focus on the following areas of research:

Recent publications[1] T. S. Jackson, G. Möller, R. Roy, Geometric stability of topological lattice phases, arXiv:1408.0843.[2] Akshay Kumar, Rahul Roy, S. L. Sondhi, Generalizing Quantum Hall Ferromagnetism to Fractional Chern Bands, arXiv:1407.6000.[3] Suk Bum Chung, Rahul Roy, Hall conductivity in the normal and superconducting phases of the Rashba system with Zeeman field, arXiv:1407.3883.[4] Fenner Harper, Steven H. Simon, Rahul Roy, Perturbative Approach to Flat Chern Bands in the Hofstadter Model, Phys. Rev. B 90, 075104 (2014).[5] Rahul Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).[6] S. A. Parameswaran, R. Roy, S. L. Sondhi, Fractional Quantum Hall Physics in Topological Flat Bands, Comptes Rendus Physique 14, 816 (2013).[7] Rahul Roy, Space group symmetries and low lying excitations of many-body systems at integer fillings, arXiv:1212.2944.[8] S. A. Parameswaran, R. Roy, S. L. Sondhi, Fractional Chern Insulators and the W-Infinity Algebra, Phys. Rev. B 85, 241308(R) (2012).[9] Rahul Roy, Topological pumps and adiabatic cycles, arXiv:1104.1979.

Band geometry of fractional Chern insulators

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Topological order and space group symmetries

Theorems due to Oshikawa and Hastings extending previous work of Lieb, Schultz and Mattis tell us that at fractional fillings, without spontaneous symmetry breaking, a gap can arise only in conjunction with topological order. In [7] we show that this is the case even at certain integer fillings of systems with nonsymmorphic space group symmetries.

3050 M. R. SALEHPOUR AND S. SATPATHY 41

cell, while hexagonal diamond has four. The hexagonalstructure, therefore, has twice as many bands as the cubicstructure at any k point in the BZ. Consequently, there isa two-to-one mapping between the two BZ's. Two kpoints in the fcc zone are mapped onto the same k pointin the hexagonal zone. For instance, both I f„and Lf„are mapped onto the I point of the hexagonal zone.Similarly, both Xf„and Lf„are mapped onto a point onthe line U of the hexagonal zone, two-thirds of the dis-tance away from M, as shown in Fig. 2. We note, in-cidentally, that the eight equivalent Lf„corresponding towave vectors along the bond directions in cubic diamondare no longer equivalent in the hexagonal structure. Thewave vectors k corresponding to Lf„are along the bonddirections. Two of these along the c axis, denoted Lf„,map onto the I' point, whereas the rest (six) along otherbond directions, denoted L)„, map onto a point on theline U. Notice also how the hf„ line maps onto the hex-agonal zone. This line is especially important since, as iswell known, the conduction-band minimum of cubic dia-mond occurs along hf„close to the Xf„point.

III. RESULTSThe calculated electronic band structures for the cubic

and the hexagonal diamond are shown in Fig. 3. The cu-bic diamond bands have been "folded" into the hexago-nal zone as discussed above. Our cubic diamond bands,although presented here in a nonconventional fashion for

direct comparison with the hexagonal bands, are in excel-lent agreement with those obtained by earlier authors.The LDA band structure of cubic diamond has been

calculated by earlier authors by using a variety ofmethods. The calculated valence-band width and thelowest gap are, respectively, 20.44 and 6.33 eV (LCAO), "21.36 and 5.66 eV (LAPW), ' 21.28 and 5.5 eV (LMTO-ASA), ' and 21.45 and 5.4 eV (plane-wave pseudopoten-tial). ' These compare very well with our results of 21.2and 5.5 eV, respectively, for the valence-band width andthe lowest gap. The experimental values are 24.2 eV(Ref. 15) and 7.3 eV (Ref. 16), respectively. Thediscrepancy between the LDA results and the experimen-tal values is typical of the local-density calculations.Our calculated indirect 6 gap for cubic diamond is

about 3.8 eV. In our calculation we have omitted the"combined-correction term, "which would add about 0.3eV (Refs. 13 and 17) to the b, gap, making it equal to 4.1eV, which is the correct local-density result. This term,which corrects for the errors introduced by the ASA aswell as for the omission of the higher-angular-momentumorbitals in the basis set, is expected to be roughly thesame for states in the gap region for the cubic and thehexagonal structures. Therefore omission of this termdoes not alter the key results presented in this paper.There is a considerable similarity between the cubic

and the hexagonal bands, as seen from Fig. 3. In theLMTO theory, the Hamiltonian Hk may be written interms of the structure constants SI, and a k-independent

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(a) Diamond crystal structure (an example of a nonsymmorphic space group) showing one possible screw transformation (blue spiral): this is the combination of a translation and rotation around the red-green axis. (b) Band structure of diamond [data reproduced from W. Saslow, T. Bergstresser, and M. Cohen, PRL 16, 354 (1966)]. Band touchings are indicated with red circles; the ∆5 band is multiply degenerate. At the X point, these degeneracies are exact and protected by the crystal symmetry group.

Ruby lattice model[Hu, Kargarian and Fiete, PRB 84, 155116 (2011)]

Kagome lattice model[Tang, Mei and Wen, PRL 106, 236802 (2011)]

Haldane model[F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).]

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1.) Developing a unified theory of the fractional quantum Hall effect and fractional Chern insulators. By incorporating aspects of the quantum geometry of the single particle Hilbert space in these systems, I have developed new analytic and numerical tools which aid the search for more experimentally accessible realizations of quantum Hall phenomena.

2.) The role of crystal symmetries in topological phenomena. Using topological arguments, I’ve improved constraints on symmetry enforced degeneracies and complete the classification of short-range entangled topological states with crystal symmetries.

3.) The exploration of topological phenomena in periodically driven systems. I plan to characterize these topologically robust phenomena, study the effects of interactions, the physical consequences of the classification, and connect these with experiments such as those on the fractional Josephson effect.

Plots of band geometry for couplings which minimize curvature fluctuations and maximize the many-body gap in several FCI models, respectively. Axes of ellipses are proportional to the eigenvectors of the quantum metric g(k). Ellipse color is given by the relative deviation of Berry curvature B(k) from its BZ-averaged value.

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Physics 4, 46 (2011)

FIG. 1: A colored Hofstadter butterfly: This figure representsthe phase diagram of Bloch electrons in a uniform magneticfield. The horizontal axis indicates the chemical potentialand the vertical axis the flux through the system. Each colorcorresponds to a distinct topological phase with a particularquantized value of the Hall conductance. (Credit: Yosi Avron)

netic field with zero average, and the Hall conductanceagain equals the Chern number of the band.

The three recent papers [1–3] take up this developmentand address the next logical question: Can the FQHE,canonically a property of interacting electrons in a frac-tionally filled Landau level, also be separated from theweak lattice and uniform magnetic field limit? More pre-cisely, they ask the following: If it is true for independentelectrons that a filled Chern band is equivalent to a filledLandau level, then is it also true for interacting electronsthat a fractionally filled Chern band is equivalent to afractionally filled Landau level?

A Landau level involves a set of exactly degeneratesingle-particle states and thus, at a fractional filling,the kinetic energy alone does not select a ground state,but instead, it falls to the interactions to force the is-sue. By contrast, a Chern band typically will have asignificant dispersion that will select a unique kinetic-energy-dominated ground state at reasonable interactionstrengths, as it does in all metals. Recognizing this, allthree papers devote considerable e�ort to constructinglattice models with nearly flat (degenerate) Chern bands.Neupert et al.[2] construct a flattened version of Hal-dane’s model on a square lattice. They note that whilea fully flattened model requires the inclusion of electronhopping over arbitrarily large distances, the hopping am-plitudes decrease exponentially, which allows a relativelyflat band to be constructed by keeping a small set of hop-ping amplitudes. The relevant flatness parameter, which

should be large for the e�ects of interactions to be impor-tant, is the ratio of the band gap (which sets a bound onthe strength of the interactions one can safely include)to the bandwidth and they show how to get this num-ber up to seven with just second-neighbor interactions.Similarly, Tang et al.[1], and Sun et al.[3] construct mod-els on the kagome and checker-board lattices, which alsoexhibit large values of the flatness parameter.

With a flat Chern band in hand, Neupert et al. in-troduce interactions and study the system at a fractionalfilling of 1/3 through numerical computation on a mod-estly sized system. They find two of the classic signa-tures of the 1/3 FQHE state: a fractional quantum Hallconductance that was close to the filling fraction, and anontrivial ground-state degeneracy with periodic bound-ary conditions. As a test, they vary the band structurecontinuously to a topologically trivial band and find thatthese features go away. In a related piece of unpublishedwork, another group finds similar results at fillings of 1/3and 1/5[6]. Altogether, this work o�ers strong evidencethat fractionally filled Chern bands do indeed exhibit theFQHE.

This is perhaps a good place to note that on a lat-tice the distinction between having a net magnetic fieldand not having it at all is not as sharp as it may seem.Essentially, it is always possible to stick a full flux quan-tum through some subset of loops on the lattice to shiftthe average magnetic field without a�ecting the actualphysics. From this perspective, the physics in these flat-band models has a family resemblance to earlier studiesof lattice versions of the FQHE [7, 8] with uniform mag-netic fields. In this earlier work, the authors studied afixed filling factor while varying the flux per plaquettefrom small values and large unit cells, where the stan-dard Landau level description holds, to somewhat largerflux values and smaller unit cells, where that descriptionbroke down. As they were able to change this parame-ter without any evidence of encountering a phase tran-sition, the latter limit constituted an observation of theFQHE in the presence of strong lattice e�ects. Need-less to say, a more analytic approach can be expectedto clarify this possible equivalence between this earlier“Hofstadter” and the current “Haldane” versions of theFQHE.

The present work also leaves open several other inter-esting questions: Can analytic expressions for the wavefunctions for the ground states and elementary excita-tions of the FQHE in the lattice models be found, andcan they be related to those for the continuum FQHE?Can these FQHE states be realized in materials via thisroute at high temperatures, as speculated by Tang etal.[1]? Further afield, while an experimental example ofa Chern insulator (an insulator with filled bands with anonzero net Chern number, such as the Haldane model)has yet to be found, at least one example of the relatedtwo-dimensional topological insulators with time-reversalsymmetry has been found. The band structures in suchtopological insulators also have nontrivial topology and

DOI: 10.1103/Physics.4.46

URL: http://link.aps.org/doi/10.1103/Physics.4.46

c• 2011 American Physical Society

Experimental plots of longitudinal and transverse conductance in the integer quantum Hall effect (left) and fractional quantum Hall effect (right). Plateaus occur at rational multiples of h/e².

A colored Hofstadter butterfly: This figure represents the phase diagram of Bloch electrons in a uniform magnetic field. The horizontal axis indicates the chemical potential and the vertical axis the flux through the system. Each color corresponds to a distinct topological phase with a particular quantized value of the Hall conductance. In [4] this model was studied perturbatively from the point of view of Landau level mixing. (Image credit: Yosi Avron)

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Numerically computed many-body gaps for FQHE-like states in the kagome and ruby lattice models, evaluated on isosurfaces of constant Berry curvature fluctuations in coupling space. This isolates the effect of the band metric on the gap.

In [5] I showed that the curvature and metric are related by the following inequalities:

The quantum Hall effect (QHE) describes topologically quantized transport in two-dimensional electron gases in a transverse magnetic field. The theory of the QHE is built largely around the special properties of single-particle states in a magnetic field (Landau levels). This is particularly true of the F(ractional)QHE, where the construction of model wave functions with built-in analyticity forced by a restriction to the lowest Landau level has played an extremely important role in the theoretical development.

which are relevant for determining the gap to excitations of a FQHE-like state in the single mode approximation.

Numerically computed many-body gaps for FQHE-like states (dot color) in the above lattice models as a function of Berry curvature fluctuation (σc) and averaged metric trace inequality (<T>). Obtaining the most stable phases requires conditions on the curvature and the metric.

A flurry of interest in the field was set off by the recent insight that FQHE-like phases of matter may also arise in topologically nontrivial insulators with partially filled flat bands, or fractional Chern insulators (FCIs). The attractiveness of FCIs stems from the fact that the bandgap ∆ (which plays the same role as the Landau level spacing in the FQHE) may be set without the use of a large external magnetic field, the strength of which (∼ 10 Tesla) is one of the limiting factors in the semiconductor FQHE.

FCI phenomena constitute a non-trivial and poorly-understood generalization of the FQHE, since the Landau level Hamiltonian occupies a unique, highly symmetric point in the space of single-particle Hamiltonians. We have recently conjectured that an approach using Hilbert space geometry might prove the way for an unified theory which naturally accounts for the differences of the single particle Hamiltonians due to lattice effects.

The Hilbert space of bands admits a Fubini-Study metric

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Group members: Tom Jackson, David Bauer, Dominic Reiss