Fractional topological insulators in three dimensions
Topological insulators can be generally defined by a topological field theory with an axion angle of or . In this work, we introduce the concept of fractional topological insulator defined by a fractional axion angle and show that it can be consistent with time reversal () invariance if ground state degeneracies are present. The fractional axion angle can be measured experimentally by the quantized fractional bulk magnetoelectric polarization , and a ‘halved’ fractional quantum Hall effect on the surface with Hall conductance of the form with odd. In the simplest of these states the electron behaves as a bound state of three fractionally charged ‘quarks’ coupled to a deconfined non-Abelian ‘color’ gauge field, where the fractional charge of the quarks changes the quantization condition of and allows fractional values consistent with -invariance.
pacs:73.43.-f, 75.80.+q, 71.27.+a, 11.15.-q
Most states of quantum matter are classified by the symmetries they break. However, topological states of quantum matter Qi and Zhang (2010) evade traditional symmetry-breaking classification schemes, and are rather described by topological field theories (TFT) in the low-energy limit. For the quantum Hall effect, the TFT is the dimensional Chern-Simons (CS) theory Zhang (1992) with coefficient given by the quantized Hall conductance. In the noninteracting limit, the integer quantized Hall (IQH) conductance in units of is given by the TKNN invariant D. J. Thouless et al. (1982) or first Chern number. In the presence of strong correlations, one can also observe the fractional quantum Hall effect (FQHE), where the Hall conductance is quantized in rational multiples of . In both cases however, these topological states can exist only in a strong magnetic field which breaks time reversal () symmetry.
More recently, -invariant topological insulators (TI) have been studied extensively Qi and Zhang (2010); Moore (2009); Has (). The TI state was first predicted theoretically in HgTe quantum wells, and observed experimentally Bernevig et al. (2006); M. König et al. (2007); Kane and Mele (2005); Bernevig and Zhang (2006) soon after. The theory of TI has been developed along two independent routes. Topological band theory identified topological invariants for noninteracting band insulators Kane and Mele (2005); Fu et al. (2007); Moore and Balents (2007). The TFT of -invariant insulators was first developed in dimensions, where the CS term is naturally -invariant Zhang and Hu (2001); B. A. Bernevig et al. (2002). Dimensional reduction then gives the TFT for TI in and dimensions Qi et al. (2008). The TFT is generally valid for interacting systems, and describes the experimentally measurable quantized magnetoelectric response. The coefficient of the topological term, the axion angle , is constrained to be either or by -invariance. The TFT has been further developed in Ref. Essin et al. (2009); Karch (2009). More recently, it has been shown that it reduces to the topological band theory in the noninteracting limit Wan ().
By analogy with the relation between the IQHE and FQHE, one is naturally led to the question whether there can exist a ‘fractional TI’. In dimensions, an explicit wavefunction for the fractional quantum spin Hall state was first proposed in Ref. Bernevig and Zhang (2006), and the edge theory was investigated in Ref. Levin and Stern (2009). The -invariant fractional topological state has also been constructed explicitly in dimensions Zhang and Hu (2001). Since -invariant TI form a dimensional ladder in , and dimensions Qi et al. (2008); A. P. Schnyder at al. (2008); Kitaev (2009), it is natural to investigate the -invariant TI in dimensions. Fractional states generally arise from strong interactions. Since topological band theory cannot describe such interactions, we formulate the general theory in terms of the TFT. The TI is generally described by the effective action where and are the electromagnetic fields Qi et al. (2008). Under periodic boundary conditions, the partition function and all physical quantities are invariant under shifts of by multiples of . Since is odd under , it appears that the only values of allowed by are or mod .
In this paper, we show that there exist -invariant insulating states in dimensions with quantized in non-integer, rational multiples of of the form with odd integers. The magnetoelectric polarization is defined by the response equation , where is an applied magnetic field and is the induced electric polarization. Such a fractionalized bulk topological quantum number leads to a fractional quantum Hall conductance of on the surface of the fractional TI. In contrast to the usual QHE in dimensions, the surface QHE does not necessarily exhibit edge states and thus cannot be directly probed by transport measurements. Alternatively, it can only be experimentally observed through probes which couple to each surface separately, such as magneto-optical Kerr and Faraday rotation Qi et al. (2008); Karch (2009). Generically, a slab of fractional TI can have different fractional Hall conductance on the top and bottom surfaces, which can be determined separately by combined Kerr and Faraday measurements, independent of non-universal properties of the material J. Maciejko et al. (2010). Our approach is inspired by the composite particle, or projective construction of FQH states Zhang (1992); Jain (1989); Wen (); Bar (); Levin and Fisher (2009). The idea is to decompose the electron with charge into fractionally charged, fermionic ‘partons’, which have a dynamics of their own. One considers the case that the partons form a known topological state, say a topological band insulator. When the partons are recombined to form the physical electrons, a new topological state of electrons emerges. In the FQH case for example, the Laughlin state can be obtained by splitting each electron into partons of charge . Each parton fills the lowest Landau level and forms a noninteracting IQH state. Ignoring the exponential factors, the parton wavefunction is the Slater determinant IQH wavefunction , and the electron wavefunction is obtained by gluing three partons together, which leads to the Laughlin wavefunction . Similarly, in 3+1 dimensions one can construct an interacting many-body wavefunction by gluing partons which are in a topological band insulator state. The parton ground state wavefunction is a Slater determinant describing the ground state of a noninteracting TI Hamiltonian such as the lattice Dirac model Qi et al. (2008), with , the position and spin coordinates of the partons. The electron wavefunction is obtained by requiring the coordinates of all partons forming the same electron to be the same Jain (1989),
Equation (1) is the -dimensional generalization of the Laughlin wavefunction, and serves as a trial wavefunction for the simplest fractional TI phases we propose.
More generally, we can consider different ‘flavors’ of partons, with partons of each flavor . This decomposition has to satisfy two basic rules. First, to preserve the fermionic nature of the electron, the total number of partons per electron must be odd (Fig. 1a),
Second, if is the (fractional) charge of partons of flavor , the total charge of the partons must add up to the electron charge ,
For instance, the Laughlin state described above corresponds to , , and , which satisfies both conditions. Here we consider that partons of each flavor condense in a (generally different) noninteracting -invariant TI state with axion angle . This is the analog of having partons condense in various IQH states in the FQH construction. Finally, the partons have to be bound together to yield physical electrons. As we will see, this can be done by coupling partons of flavor to a gauge field, which can be interpreted as a ‘color field’ where partons of flavor come in colors. Since the TI analog of the Laughlin state will involve three partons coupled to a gauge field in dimensions, we dub our partons ‘quarks’ by analogy with quantum chromodynamics (QCD).
To obtain a more systematical understanding of the fractional TI, we now deduce its effective gauge theory by way of a gedanken experiment. We consider subjecting a noninteracting TI to strong electron-electron interactions, and start with the simplest case of with odd. The electron being split into quarks of charge , the electron operator will be written as a product of quark operators , . However, the quark operators act in a Hilbert space which is larger than the physical electron Hilbert space. We need to remove those states of the quark Hilbert space which are not invariant under unitary transformations which leave the electron operator unchanged, i.e. transformations with quarks in the representation. The projection onto the electron Hilbert space can therefore be implemented by coupling the quarks to a gauge field with a coupling constant . Outside the fractional TI, we expect the system to be in the confined phase, in analogy to quark confinement in QCD, which has only singlet excitations in its low-energy spectrum, i.e. gauge-invariant ‘baryons’. Quarks of a given flavor within the baryon are antisymmetric in their color indices; Fermi statistics then implies that their spins are aligned. In a relativistic theory this would imply that in the theory the baryon has spin . In nonrelativistic lattice models this is not a concern, but even within the context of relativistic continuum field theories one can obtain composite spin- electrons for .
Inside the fractional TI, electron-electron interactions translate into complicated interactions among quarks. We consider the case that these interactions lead the quarks to condense at low energies into a noninteracting -invariant TI state with axion angle , and that the non-Abelian gauge field enters a deconfined phase dec (). We now show that such a phase is a fractional TI. A low-energy effective Lagrangian for can be conjectured in the form
where is the gauge covariant derivative, and is the single-particle Hamiltonian for a -invariant TI with axion angle . represents weak -invariant residual interactions which do not destabilize the gapped TI phase, and can thus be safely ignored. The kinetic Yang-Mills Lagrangian for is generally present but not explicitly written.
Since the quarks are in a gapped TI phase, they can be integrated out to yield an effective Lagrangian for the gauge fields Qi et al. (2008),
where is the trace in the representation of , and are the and field strengths, respectively, and the electromagnetic response is governed by an effective axion angle
Equation (Fractional topological insulators in three dimensions) is obtained by replacing the ‘electron’ field strength in the topological term for noninteracting TI Qi et al. (2008) by the total ‘quark’ field strength . Note that the crossed terms of the form vanish due to the tracelessness of the gauge field. More generally, can be obtained from the Adler-Bell-Jackiw anomaly, since corresponds to the phase of the quark mass Qi et al. (2008). In principle, the effective theory can also be obtained for quarks in a trivial insulator state with . However, such a state is adiabatically connected to a trivial vacuum with , so that it is a trivial insulator in the bulk, although a fractional can still be obtained due to pure surface effects. Since the focus of the present work is a fractional TI state with nontrivial bulk, we always consider quarks with in the following.
We are now faced with our initial question of whether the effective theory (Fractional topological insulators in three dimensions),(6) breaks -invariance. According to the first term in Eq. (Fractional topological insulators in three dimensions), -invariance would require to be quantized in integer multiples of if the minimal electric charge was Witten (1995). However, the minimal charge in our theory is , i.e. that of the quarks. Therefore, has to be quantized in integer multiples of . On the other hand, the second term in Eq. (Fractional topological insulators in three dimensions) requires to be quantized in integer multiples of lan (), which means by Eq. (6) that has to be quantized in units of . This latter constraint is consistent with, but stronger than, the former mon (), and the values of allowed by -invariance are thus correctly given by Eq. (6).
Equations (Fractional topological insulators in three dimensions) and (6) constitute a TFT which, precisely because it is topological, is insensitive to small -invariant perturbations and defines a new stable phase of matter, the -invariant fractional TI in dimensions. The effective theory can also be derived in the multi-flavor case , with satisfying rules (2) and (3). Considering that quarks of flavor form a noninteracting TI with axion angle and integrating them out yields an effective Lagrangian in the form of (Fractional topological insulators in three dimensions), but with gauge group . Here is the overall gauge transformation of the electron operator. The electromagnetic axion angle is given by . When is odd for each flavor, one can show that with , odd integers.
Important physical properties of the fractional TI can be read off from Eq. (Fractional topological insulators in three dimensions). The surface of the fractional TI is an axion domain wall with the axion angle jumping from in the fractional TI to in the vacuum. Such a domain wall has a surface QHE with surface Hall conductance Qi et al. (2008). Therefore, the surface Hall conductance of the fractional TI has the general form
For example, in the simplest single-flavor case with in Eq. (Fractional topological insulators in three dimensions), we have with an odd integer, corresponding to half of a FQH Laughlin state. The more general result (7) corresponds to half of a generic Abelian FQH state FQH (); Jain (1989).
The fractional axion angle and the associated surface Hall conductance (7) are properties of the bulk topology. It is important to distinguish them from a TI with and where the surface Dirac fermions form a FQH state ran (). In a noninteracting TI with for example, both the axion domain wall and the surface FQH state contribute to ,
with an allowed filling fraction for a FQH state in dimensions. For Abelian FQH states is odd, hence the surface Hall conductance has the same general form as for the fractional TI [Eq. (7)]. As the simplest example, the Laughlin state with leads to (Fig. 1c, right) which is the same as for a genuine fractional TI with bulk (Fig. 1c, left). However, the bulk topology is very different in both cases. Therefore, surface measurements are not sufficient to determine the bulk topology and bulk measurements of are needed. One such measurement would consist in embedding a monopole with magnetic charge inside the fractional TI (Fig. 1d) and measuring its electric charge induced by the Witten effect Witten (1979); X.-L. Qi et al. (2009); ros ().
Another possible ‘experiment’ is to measure the ground state degeneracy (GSD) on topologically nontrivial spatial 3-manifolds. Consider for example a fractional TI on a manifold with a Riemann surface of genus and a bounded interval, where is the sample ‘thickness’ and the two copies of (at each end of ) are the two bounding surfaces. We first discuss contributions to the GSD arising solely from the boundary, and comment on bulk contributions later on. A noninteracting TI with a Laughlin state deposited on both surfaces is described by two independent CS theories Zhang (1992); top () and has a GSD of ( for ) on each surface for a total GSD of . The situation is different for a genuine fractional TI with (Fig. 1c, left). To study the GSD we set the external electromagnetic fields to zero in Eq. (Fractional topological insulators in three dimensions) and consider the internal -term. Assuming that the system stays gapped as we take the limit of zero thickness where the gauge fields on both surfaces become identified, the system is described by a single CS theory on where the level is the sum of the contributions from both surfaces. If on both surfaces goes to the same value outside the TI, then and there is no GSD. If on one side and on the other, we have a CS theory with GSD top (). Another example is the solid torus . The unique boundary is a 2-torus , hence a Laughlin state deposited on it has GSD from the quantum mechanics of Wilson lines along the two non-contractible loops on top (). However, since one of these loops extends into the bulk of the solid torus and is thus contractible, the -term contributes no GSD to a genuine fractional TI. However, in addition to the boundary contributions of the -term to the GSD, the gauge theory in the bulk can have a nontrivial GSD even in the absence of boundaries. For instance, the deconfined phase of gauge theory has a GSD of on Sato (2008). The total GSD has in general both bulk and boundary contributions, and depends on the details of the gauge group.
It should be noted that several distinct fractional TI states can correspond to the same . For instance, can correspond to a theory with , or to a , theory with and . This is analogous to the well-known fact that topological orders in FQH states cannot be characterized by the Hall conductance alone top (). The property of GSD provides a finer classification of fractional TI. Moreover, all the states discussed so far can be considered as -dimensional generalization of Abelian FQH states. In principle, more generic non-Abelian fractional TI states with exotic even-denominator ‘halved’ non-Abelian FQH states on their surface can also be constructed using more generic parton decompositions, which correspond to effective theories with gauge groups other than Bar ().
In conclusion, we have shown that fractional TI states in dimensions with a quantized fractional bulk magnetoelectric polarization and a ‘halved’ odd-denominator surface FQHE are fully consistent with -invariance, and can in principle be realized in strongly correlated systems with strong spin-orbit coupling.
We thank M. Barkeshli, M. P. A. Fisher, M. Freedman, S. Kivelson, Z. Wang, X.-G. Wen, S. Yaida, and H. Yao for helpful discussions. J.M. is supported by the Stanford Graduate Program, X.L.Q is supported by Microsoft Research Station Q, A.K. is supported in part by DOE grant DE-FG02-96ER40956, and S.C.Z. is supported by the NSF under grant numbers DMR-0904264.
After this paper was originally posted, fractional TI were further investigated by B. Swingle et al. swi (). We acknowledge the insightful discussion with the authors M. Barkeshli, J. McGreevy, and T. Senthil of Ref. swi, which was very valuable for modifications to our original draft.
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