A Supplemental Material

Haldane-Hubbard Mott Insulator: From Tetrahedral Spin Crystal to Chiral Spin Liquid

Abstract

Motivated by cold atom experiments on Chern insulators, we study the honeycomb lattice Haldane-Hubbard Mott insulator of spin- fermions using exact diagonalization and density matrix renormalization group methods. We show that this model exhibits various chiral magnetic orders including a wide regime of triple-Q tetrahedral order. Incorporating third-neighbor hopping frustrates and ultimately melts this tetrahedral spin crystal. From analyzing the low energy spectrum, many-body Chern numbers, entanglement spectra, and modular matrices, we identify the molten state as a chiral spin liquid (CSL) with gapped semion excitations. We formulate and study the Chern-Simons-Higgs field theory of the exotic CSL-to-tetrahedral spin crystallization transition.

Electronic bands in crystals can display nontrivial topology, as exemplified by the recent discoveries of topological insulators König et al. (2007); Hsieh et al. (2008), Weyl semimetals Yang et al. (2015); Lv et al. (2015); Xu et al. (2015), and quantum anomalous Hall insulators (QAHIs) Chang et al. (2015); Bestwick et al. (2015). Interactions can dramatically modify this single-particle physics, for instance by rendering indistinguishable certain topologically distinct free-fermion phases Fidkowski and Kitaev (2010, 2011). An alternative outcome is the emergence of topological order Wen (1995), manifested by nontrivial ground state degeneracies depending on the lattice topology, as discovered in numerical studies of partially filled Chern bands which realize lattice fractional quantum Hall liquids Parameswaran et al. (2013); Bergholtz and Liu (2013). Interactions may also lead to charge localization, while the spin degrees of freedom display topological order. Finding even quasi-realistic models of such topological Mott insulators (TMIs) Pesin and Balents (2010); Bhattacharjee et al. (2012); Kargarian et al. (2012); Maciejko et al. (2014) is a crucial step towards identifying experimental candidates and understanding exotic quantum phase transitions out of TMIs.

In this Letter, we study interaction effects in the Haldane model Haldane (1988), a paradigmatic model of a QAHI on the two-dimensional (2D) honeycomb lattice. The Haldane model supports two bands with Chern numbers ; it has been realized in recent cold atom experiments Jotzu et al. (2014); Aidelsburger et al. (2015). We study the effect of strong Hubbard repulsion on spin- (i.e., two-component) fermions in the Haldane model, at a filling of one fermion per site, obtaining the following key results. (i) We establish that the effective spin model for the Haldane-Mott insulator exhibits a variety of chiral magnetic orders including a wide regime of tetrahedral order with large scalar spin chirality. Our results are obtained using exact diagonalization (ED) on cluster of up to spins. (ii) Incorporating third-neighbor hopping is shown to frustrate and ultimately melt the tetrahedral order. Our ED results in the liquid phase find a gapped, approximately two-fold degenerate ground state, with total many-body Chern number , suggesting that this state is a chiral spin liquid (CSL): the bosonic quantum Hall state with gapped semion excitations Anderson (1973); Kalmeyer and Laughlin (1987); Wen (1990a). We provide conclusive evidence for this using state-of-the-art density matrix renormalization group (DMRG) White (1992); McCulloch () computations on infinitely long cylinders with circumference up to lattice unit cells, computing entanglement spectra, quantum dimensions of all anyon types, and quasiparticle braiding properties via topological and matrices. This frustration-induced melting of tetrahedral order is a completely distinct mechanism to realize CSLs compared with previous studies, and allows us, for the first time, to identify the tetrahedral state as a ‘parent’ state for the CSL. (iii) Our ED results suggest a continuous phase transition between the tetrahedral state and the CSL. We formulate a Chern-Simons-Higgs field theory to describe this exotic spin crystallization transition out of the topologically ordered CSL.

The study of CSLs was rejuvenated by the construction of exact parent Hamiltonians Schroeter et al. (2007); Thomale et al. (2009), and recent works have found evidence for CSLs on the kagome He et al. (2014); Bauer et al. (2014); Gong et al. (2014); Kumar et al. (2014); Gong et al. (2015); Wietek et al. (); Zhu et al. (); Hu et al. (2015); Bieri et al. (2015); Kumar et al. (2015) and square lattices Nielsen et al. (2013); Poilblanc et al. (2015); Liu et al. (2016), and in certain Mott insulators Hermele et al. (2009) and coupled wire models Meng et al. (2015); Gorohovsky et al. (2015). Our work provides the first example of a CSL on the honeycomb lattice in a realistic model starting from fermions with on-site interactions. This is nontrivial since a symmetric spin-gapped phase on lattices with even number of spin- per unit cell is not guaranteed to have topological order Oshikawa (2000); Hastings (2005). Our work goes well beyond previous work on this model He et al. (2011a, b); Maciejko and Rüegg (2013); Hickey et al. (2015); Zheng et al. (2015), and studies of Gutzwiller projected Chern-insulator wavefunctions Zhang et al. (2011, 2012a) which did not consider microscopic models that support such ground states. The tetrahedral state we find here also occurs in certain triangular lattice Hubbard and Kondo models Martin and Batista (2008); Jiang et al. (2015), suggesting that such frustration-induced CSLs may appear in a wider class of models and materials.

Model. The Haldane-Hubbard model for spin- fermions shown in Fig. 1(a) is defined by the Hamiltonian

(1)

where and denote, respectively, first and second nearest neighbors, produces a flux pattern with a net zero flux per unit cell, and is the Hubbard repulsion. For , this model supports Chern bands for . At half-filling, this leads to a QAHI with per spin for small . At large and , the Chern bands strongly disperse, leading to a metal with but non-quantized Hickey et al. (2015).

For , degenerate perturbation theory in the Mott insulator MacDonald et al. (1988) with one fermion per site leads to the spin model

(2)

where is the scalar spin chirality operator. The sites in are labelled going anticlockwise around the small or big triangles of the honeycomb lattice. As shown in Fig. 1(a), the fluxes in are on small (green) triangles, and on large triangles which do (do not) enclose a lattice site. Classical magnetic ground states of this model, valid for , have been studied in Hickey et al. (2015); here, we resort to a numerical study for , retaining strong quantum fluctuations.

Figure 1: (Color online) (a) Haldane-Hubbard model showing short distance hopping amplitudes, plaquette fluxes, and Hubbard repulsion . (b) Phase diagram of for from ED on clusters with spins; color indicates the chirality on small triangles. White lines indicate phase boundaries, broadened to account for finite-size effects. In the hatched (blue) region we cannot sharply identify the phase in ED as Triad-I or II.

ED phase diagram. For , reduces to the - honeycomb lattice Heisenberg model, with . Previous work indicates that kills Néel order, leading to incommensurate spirals Mulder et al. (2010) for , and competing valence bond crystals for Fouet, J. B. et al. (2001); Albuquerque et al. (2011); Mosadeq et al. (2011). Here, we study the unexplored regime , using Lanczos ED on clusters up to spins, varying and for fixed which puts us in the Mott insulator Hickey et al. (2015). We focus on flux values , which reveals commensurate phases with large scalar spin chirality; restricting ourselves to this window of flux avoids incommensurate spiral orders Mulder et al. (2010); Hickey et al. (2015) expected at small , which have strong finite-size effects in ED. Below, we work in units where .

As shown in Fig.1(b), we find that the phase diagram contains four magnetically ordered phases — Néel, tetrahedral and triad-I/II orders — which are also observed in the classical phase diagram Hickey et al. (2015). (i) The Néel order on the honeycomb lattice is translationally invariant, with ferromagnetic order on each sublattice and a single structure factor peak at the point of the hexagonal Brillouin zone. (ii) The tetrahedral order has an -site magnetic unit cell, with spins pointing toward the four corners of a tetrahedron and structure factor peaks at the three points. It is a so-called “regular magnetic order”, respecting all lattice symmetries modulo global spin rotations. (iii)/(iv) Triad-I/II both have -site magnetic unit cells, with three spins on each sublattice forming a cone and structure factor peaks at the and points. They can be thought of as umbrella states on each triangular sublattice, with their common axis being parallel in the triad-I case and anti-parallel in the triad-II. This yields a net ferromagnetic moment in triad-I and a net staggered moment in triad-II.

We identify these magnetic orders within ED, on clusters with up to spins, through a careful analysis of the low energy spectrum, extracting quantum numbers of the quasi-degenerate joint states, i.e., the ‘Anderson tower’, in each total spin sector, whose energies collapse onto the ground state as leading to spontaneous symmetry breaking in the thermodynamic limit Bernu et al. (1992, 1994) (see Supplemental Material (61)). The phase boundaries in Fig.1(b) are determined (61) by dips in the ground state fidelity which signal quantum phase transitions Zanardi and Paunković (2006), where is a tuning parameter (here, or ). We substantiate this by studying changes in the finite-size singlet () and triplet () gaps, , and reorganization of the low energy spectrum. Our results are in contrast to slave-rotor mean field theory of the Haldane Mott insulator He et al. (2011a, b), in which the ground state is a CSL which simply inherits the band topology of the underlying QAHI.

Figure 2: (Color online) (a) Phase diagram of at and , keeping the additional term induced by . Background shows ground state chirality on small-. Using ED and DMRG (at indicated points), we find a window of CSL with topological order. (b) Topological robustness of the CSL ground states upon threading flux through one hole of the torus. Energy spectrum as a function of boundary phase is shown for sites, , and . (c) Energy spectrum for cluster, with states labelled by total spin and Brillouin zone momenta shown in the inset. We find approximate two-fold ground state degeneracy with total Chern number .

Melting tetrahedral order. The tetrahedral state is a “regular magnetic state” Messio et al. (2011) which respects all lattice symmetries in its -invariant correlations. Given its large scalar spin chirality, it is tempting to speculate that quantum disordering this state might lead to a CSL. We thus modify the Haldane model in order to frustrate the tetrahedral order. We notice that the tetrahedral state has spins on opposite vertices of the honeycomb hexagon aligned ferromagnetically. Thus incorporating third-neighbor hopping will lead to an additional exchange interactions in , i.e., the Heisenberg exchange which will inevitably frustrate tetrahedral order, as well as additional chiral interactions. Below, we present extensive results retaining only since keeping all chiral terms induced by significantly increases the computational complexity; we have explicitly checked that these additional terms induce very small quantitative differences in the ED spectra, and only slightly shift the phase boundaries in the phase diagram (see Supplemental Material (61)).

One key signature of a CSL is a nonzero spin gap and two-fold ground state degeneracy on the torus. We thus look for regimes where the lowest excited state is a spin-singlet whose energy gap becomes smaller with system size, while the triplet gap remains nonzero. Fig. 2(a) shows the ED phase diagram as we vary , where we find a candidate CSL regime. Here, we have fixed , at which the coefficient of on the large- vanishes, enormously simplifying the numerics.

Fig. 2(c) shows a representative ED spectrum on an torus at . We find an approximate two-fold ground state degeneracy, both states being spin singlets with crystal momentum as expected for a honeycomb lattice CSL, and a spin gap . Threading flux through one hole of the torus (see Fig. 2(b)), we find the two-fold ground state manifold does not with mix with higher excited states, demonstrating that the ground state degeneracy is of topological origin. We have computed the many-body Chern numbers using twisted boundary conditions on the two ground states , since two ground states have the same momentum and thus do not cross. However, only the total Chern number of this degenerate manifold is meaningful in the thermodynamic limit; we find . These results provide strong evidence that melts tetrahedral order, leading to a bosonic Laughlin liquid. Our ED results delineate a regime at , see Fig. 2(a), which we identify as a CSL candidate.

Figure 3: (Color online) Entanglement spectrum (rescaled and shifted) of the reduced density matrix for half an infinite cylinder (with circumference unit cells) computed for the ground states (left panel) and (right panel) of the effective spin model at . Vertical axes show entanglement energies defined as , where are the eigenvalues of . The transverse momenta of the corresponding eigenvectors of are shown on horizontal axes, separately for every tower labeled by quantum number. The number of close lying states with the same momenta in a given sector is shown in red.

DMRG results. To further confirm the existence of CSL, we investigate the model with additional terms generated by non-zero , using DMRG McCulloch (), on a cylinder of infinite length with circumference up to unit cells. The characterization of a topologically ordered phase is achieved by: (i) identifying the conformal field theory (CFT) that describes gapless edge excitations via the “entanglement spectrum” Li and Haldane (2008), and (ii) computing topological and matrices that contain information about bulk anyon excitations Wen (1990a); Zhang et al. (2012a); (65); Cincio and Vidal (2013); Zaletel et al. (2013); Zhu et al. (2013). Simulations were performed for , and four different values of marked by red dots on the phase diagram in Fig.2(a), keeping only the additional exchange term. We present detailed results below for one point ; we obtain similar results at the other three points. We also performed simulations on smaller width cylinders (upto ) keeping and all additional chiral terms from having in , obtaining similar results.

Randomly initialized DMRG finds two ground states, , with well-defined anyon flux threading inside the cylinder Cincio and Vidal (2013). Fig. 3 shows the entanglement spectrum of the reduced density matrix for half an infinite cylinder computed for both ground states. Studying these spectra, we can extract universal information about possible gapless boundary excitations, as if the system had an actual, physical edge Li and Haldane (2008); Chandran et al. (2011); Dubail et al. (2012); Swingle and Senthil (2012); Qi et al. (2012). The spectra are seen to be consistent with corresponding sectors of the chiral Wess-Zumino-Witten CFT Wen (1990b). is associated with the identity primary operator and its Kac-Moody descendants. The computed degeneracy pattern in every tower (labeled by ) is seen to follow the expected partition numbers (1–1–2–3–5–7–…) Di Francesco et al. (1997). corresponds to the chiral boson vertex operator and its descendants.

The ground states on an infinite cylinder may be used to mimic grounds states on a torus by means of cutting and reconnecting matrix-product states of Cincio and Vidal (2013); Zaletel et al. (2013). Every such ground state has a well-defined anyon flux threading inside the torus. The topological and matrices of the emergent anyons can be extracted Zhang et al. (2012b) from the overlaps , where denotes clockwise rotation of a torus. For , we find

(3)
(4)

in excellent agreement with the exact and matrices of a chiral semion anyon model, and . The combined DMRG results thus provide an unambiguous identification of the phase as a CSL.

Spin crystallization transition. Our ED results show that the chirality and ground state fidelity vary smoothly going from the tetrahedral state into the CSL. This suggests that the two phases might be separated by an exotic critical point since the tetrahedral state is topologically trivial but breaks spin symmetry while the CSL has topological order and no broken symmetries. A powerful route to accessing such exotic transitions is via fractionalizing the spins Senthil et al. (2004). We formulate our theory in terms of spin- bosonic spinons minimally coupled to an Abelian level Chern-Simons (CS) gauge field. In the CSL, integrating out gapped spinons results in a CS topological field theory. The lowest energy excitations are gapped spinons, which carry unit gauge charge and bind -flux, converting them into semions. On the tetrahedral side, spinon condensation produces magnetic order, destroying topological order via the Higgs mechanism.

To construct the field theory for the matter sector, we imagine bosonic spinons with spins polarized along the local Zeeman axes of the underlying tetrahedral order. Adiabatic spinon transport around closed loops on the honeycomb lattice then produces nontrivial Berry phases; we find -flux around hexagonal loops and -flux around triangular plaquettes. Even if long wavelength quantum fluctuations disorder the tetrahedral state, so these Zeeman fields average to zero, we expect the local spin chirality and hence the local fluxes to persist. Diagonalizing this spinon Hofstadter Hamiltonian on the honeycomb lattice, we find equivalent dispersion minima located, for our gauge choice, at and (; the three points of the BZ). We thus study the action , where

(5)

describes bosonic spinons minimally coupled to the CS gauge field, while captures spinon interactions,

(6)

Latin indices label the modes at (), the notation implies all modes are different, and there is an implicit sum on Greek indices which label spin or space-time. We defined and . and respectively list all quartic interactions and important sixth order terms, consistent with momentum conservation, global symmetry, and local gauge invariance. are forward-scattering interactions, are backscattering terms, and is an Umklapp process. encodes broken time-reversal symmetry. At mean field level, with dominant , we find leads to the CSL, while tuning leads to a confining Higgs phase with . For , we get simultaneous condensation at all . The tetrahedral state emerges via a continuous transition for subdominant terms (see Supplemental Material (61)). Our construction of the field theory for the CSL-tetrahedral transition relies on a nontrivial flux pattern for the spinons, hinting at ‘crystal symmetry fractionalization’ Essin and Hermele (2014) in the CSL.

Summary. Using ED and DMRG, we have shown that the Haldane-Hubbard Mott insulator supports unusual chiral magnetic orders, while third-neighbor hopping induces a CSL with topological order. We have argued that this CSL descends from a ‘parent’ tetrahedral state and constructed a CS-Higgs theory for this exotic spin-crystallization transition. Recent work has shown that the kagome lattice admits only a single invariant symmetry enriched CSL Zaletel et al. (); Cincio and Qi (). However, the honeycomb lattice may admit multiple CSLs with distinct crystal symmetry fractionalization patterns. Future research directions include nailing down the precise nature of this CSL Wang et al. (2015); Zaletel et al. (); Qi and Fu (2015); Zaletel et al. (); Cincio and Qi (), and relating this CSL to Gutzwiller projected wavefunctions Zhang et al. (2011, 2012a). Another outstanding issue is fluctuation effects on the CS-Higgs transition proposed here, and in related U(1) symmetric bosonic quantum Hall to charge density-wave insulator transitions Barkeshli et al. (2015).

Acknowledgments. We thank R. Desbuquois, K. Hwang, G. Jotzu, C. Laumann, S. Sachdev, A. Thomson, S. Whitsitt, and D.N. Sheng for useful discussions. CH and AP acknowledge support from NSERC of Canada. Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. This work also made use of the facilities of N8 HPC Centre of Excellence, provided and funded by the N8 consortium and EPSRC (Grant No.EP/K000225/1). The Centre is co-ordinated by the Universities of Leeds and Manchester. L.C. acknowledges support by the John Templeton Foundation. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Appendix A Supplemental Material

Appendix S2 Exact diagonalization spectra for magnetically ordered states with

Exact diagonalization (ED) on system sizes of up to spins is used to construct the phase diagram of the Haldane-Hubbard Mott insulator, with fixed and varying and flux . The magnetic orders present can be identified by analysing the quantum numbers of the low-lying states in each total spin sector of the ED energy spectrum, the so-called ‘quasi-degenerate joint states’ (QDJS), or ‘Anderson tower’. These states collapse onto the ground state as leading to a spontaneous symmetry broken ground state in the thermodynamic limit.

As stated in the main text, we find that the phase diagram contains four magnetically ordered phases - Néel, tetrahedral and triad-I/II orders. In Fig. S4 we present example spectra for these four phases for a site cluster which has the full point group symmetry of the lattice . In this case the QDJS can be characterised by their momenta and irreducible representation (IR) of . Properties of the phases include:
The Néel order is collinear and translationally invariant, with QDJS with momentum at the point and energy scaling linearly with as expected for quantum rotor excitations.
The tetrahedral order is non-coplanar with QDJS with momentum at the and points and large chirality on small triangles. The triad-I order is non-coplanar and has a net ferromagnetic moment (with the ground state lying in a sector with ), with QDJS at the point and the points as expected.
The triad-II order is similar in many respects to the triad-I but with a net anti-ferromagnetic moment and oppositely signed chirality on big triangles.

Figure S4: Example ED energy spectra for the four magnetically ordered phases described in the main text for a spin cluster.

Appendix S3 Ground state fidelity exact diagonlization results

The phase boundaries were determined by analysing dips in the ground state fidelity, with a tuning parameter, as well as changes in the low energy spectrum, the finite-size singlet () and triplet () gaps and the scalar spin chirality on big and small traingles. In Fig. S5 we show the ground state fidelity as a function of for and site torus geometries at . The sharp dips mark the transition from the Néel to the tetrahedral state.

Figure S5: Ground state fidelity results, , for and site torus geometries at .

Appendix S4 Comparison between ED results for for the full model versus simplified model with only

With , we have found that there is robust magnetically ordered states in the Mott insulating phase of the Haldane-Hubbard model. With , we showed that a CSL phase emerges. In Fig. S6 we show the phase diagram, at , for (a) the case presented in the main text in which only the additional third-neighbor Heisenberg term is considered, and (b) the case in which all of the additional terms are considered, i.e., the Heisenberg term as well as the additional chiral terms, . In Fig. S7 we show an example of the energy spectrum for both cases at . We see that keeping all of the terms results in only very small shifts in the phase boundaries, showing that it is really the Heisenberg exchange that is the driving force behind melting the tetrahedral order and getting the CSL phase.

To reduce the computational complexity of the ED/DMRG computations on the largest system sizes, we have retained only this Heisenberg term in the key results presented in the main text. However we have also done DMRG computations (on infinite cylinders with widths up to ) at the four points marked in Fig. S6(b) retaining all the extra chiral interactions, and confirmed that the CSL phase is robust.

Figure S6: ED phase diagram at and for a spin cluster keeping (a) only the additional Heisenberg exchange and (b) all additional terms, Heisenberg and chiral , generated by adding third-neighbor hopping with amplitude . DMRG computations have been carried out on the four points marked in each phase diagram with all of them showing the expected signatures of a CSL. Energy spectra for the blue points are shown below in Fig. S7.
Figure S7: Example ED energy spectra for the blue points marked in Fig. S6 for a spin cluster with (a) only the additional Heisenberg exchange and (b) all additional terms, Heisenberg and chiral , generated by adding third-neighbor hopping with amplitude . The two lowest lying states in both cases carry total Chern number .

Appendix S5 Field Theory of the Spin Crystallization Transition

In the main text we constructed a field theory of spin- bosonic spinons minimally coupled to an Abelian level Chern-Simons (CS) gauge field to describe a continuous CSL-tetrahedral transition. The action is , with

(S7)
(S8)

where Latin indices label the modes at (), implies all modes are different, there is an implicit sum on Greek indices which label spin or space-time, and we have defined and .

At mean field level, we drop all gradient terms. With dominant and with , we find leads to the CSL with , while tuning leads to a transition into a confining Higgs phase with . The tetrahedral state emerges for subdominant terms . Fig. S8 illustrates a concrete example of such a transition, with the square of the tetrahedral order parameter, plotted as a function of at , , and . It exhibits clear linear scaling as expected for the square of a mean-field order parameter.

Figure S8: Mean field computation of the squared order parameter for the tetrahedral state within the Chern-Simons-Higgs field theory, for parameter values in the action mentioned in the text, showing a continuous mean field transition at .

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