Topological Insulators and Superconductors from D-branes

Realization of topological insulators (TIs) and superconductors (TSCs), such as the quantum spin Hall effect and the Z_2 topological insulator, in terms of D-branes in string theory is proposed. We establish a one-to-one correspondence between the K-theory classification of TIs/TSCs and D-brane charges. The string theory realization of TIs and TSCs comes naturally with gauge interactions, and the Wess-Zumino term of the D-branes gives rise to a gauge field theory of topological nature. This sheds light on TIs and TSCs beyond non-interacting systems, and the underlying topological field theory description thereof. In particular, our string theory realization includes the honeycomb lattice Kitaev model in two spatial dimensions, and its higher-dimensional extensions.


Introduction
A gapped state of quantum condensed matter is called topological phase when it supports stable gapless boundary modes, such as an edge or a surface state. The integer quantum Hall effect (QHE), which exists in d = 2 spatial dimensions and under a strong magnetic field, is the best known example of such a phase. The recent discovery of the quantum spin Hall effect (QSHE) in d = 2 and the Z 2 topological insulator in d = 3 [1][2][3][4][5][6][7][8] shows topological phases can exist even in d > 2 spatial dimensions, and can be protected by some discrete symmetries such as time-reversal symmetry (TRS, T), particle-hole symmetry (PHS, C), and chiral (or sublattice) symmetry (SLS, S).
For non-interacting fermions, an exhaustive classification of topological insulators (TIs) and superconductors (TSCs) is proposed in Refs. [9,10]: TIs/TSCs are classified in terms of spatial dimensions d and the 10 = 2 + 8 symmetry classes (two "complex" and eight "real" classes) ( Table 1). The ten symmetry classes are in one-to-one correspondence to the Riemannian symmetric spaces (without exceptional series) and, as pointed out in [10], they are equivalent to K-theory classifying spaces [11]. For example, the IQHE, QSHE, and Z 2 TI are a topologically non-trivial state belonging to class A (d = 2), AII (d = 2), and AII (d = 3), respectively.
The complete classification of non-interacting TIs and TSCs opens up a number of further questions, most interesting among which are interaction effects: Do non-interacting topological phases continue to exist in the presence of interactions? Can interactions give rise to novel topological phases other than non-interacting TIs/TSCs? What is a topological field theory underlying TIs/TSCs, which can potentially describe TIs/TSCs beyond non-interacting examples?, etc.
On the other hand, the ten-fold classification of TIs/TSCs reminds us of D-branes, which are fundamental objects in string theory, and are also classified by K-theory [12] ( Table 2) via the open string tachyon condensation [13]. It is then natural to speculate a possible connection between TIs/TSCs and of D-branes. In this paper, we propose a systematic construction of TIs/TSCs in terms of two D-branes (Dp-and Dq-branes), possibly with an orientifold plane (O-plane). Besides the appealing mathematical similarity between TIs/TSCs and D-branes, realizing TIs/TSCs in string theory has a number of merits, since string theory and D-branes are believed to be rich enough to reproduce many types of field theories and interactions in a fully consistent and UV complete way. Indeed, our string theory realizations of TIs/TSCs give rise to massive fermion spectra, which are in one-to-one correspondence with the ten-fold classification of TIs/TSCs, and come quite naturally with gauge interactions. These systems, while interacting, are all topologically stable, as protected by the K-theory charge of D-branes. We thus make a first step toward understanding interacting TIs/TSCs [14]. We are also separately preparing a regular paper with more details and expanded results in [15].
In Dp-Dq-systems, massive fermions arise as an open string excitation between the two D-branes. The distance between the branes corresponds to the mass of fermions. Open strings ending on the same D-branes give rise to a gauge field, which we call A µ (Dp) andÃ µ (Dq) with gauge group G andG, respectively, and couple to the fermions. These class\d 0 Table 1: Classification of topological insulators and superconductors [9,10]; d is the space dimension; the left-most column (A, AIII, . . ., CI) denotes the ten symmetry classes of fermionic Hamiltonians, which are characterized by the presence/absence of time-reversal (T), particle-hole (C), and chiral (or sublattice) (S) symmetries of different types denoted by ±1 in the right most three columns. The entries "Z", "Z 2 ", "2Z", and "0" represent the presence/absence of topological insulators and superconductors, and when they exist, types of these states (see Ref. [9] for detailed descriptions).
two gauge fields play different roles in our construction: The gauge field A µ "measures" K-theory charge of the Dq-brane, and in that sense it can be interpreted as an "external" gauge field. In this picture, the Dq-brane charge is identified with the topological (Ktheory) charge of TIs/TSCs. On the other hand,Ã µ is an internal degree of freedom on the Dq-brane. For example, in the integer/fractional QHE, the external gauge field is the electromagnetic U(1) gauge field, which measures the Hall conductivity, while the internal gauge field is the Chern-Simons (CS) gauge field describing the dynamics of the droplet itself.
The massive fermions can be integrated out, yielding the description of the topological phase in terms of the gauge fields. The resulting effective field theory comes with terms of topological nature, such as the CS or the θ-terms. In our string theory setup, they can be read off from the Wess-Zumino (WZ) action of the D-branes, by taking one of the D-branes as a background for the other. One can view these gauge-interacting TIs/TSCs from Dp-Dq-systems as an analogue of the projective (parton) construction of the (fractional) QHE [16]. Our string theory realization of TIs/TSCs sheds light on extending the projective construction of the QHE to more generic TIs/TSCs; it tells us what type of gauge field is "natural" to couple with fermions in topological phases, and guarantees the topological stability of the system. Table 2: Dp-brane charges from K-theory, classified by K(S 9−p ), KO(S 9−p ) and KSp(S 9−p ) [12]. A Z 2 charged Dp-brane with p even or p odd represents a non-BPS Dp-brane or a bound state of a Dp and an anti-Dp brane, respectively [13].

Complex case
Let us start with the most familiar example of the QHE (class A in d = 2). We fix the value of p to be p = 5 by T-duality, and consider a D5-brane in type IIB string theory which extends in the x 0,1,2,3,4,5 directions in ten-dimensional space-time. We take the Dq-brane with q = 5 in the x 0,1,2,6,7,8 directions (Table 4). By T-duality, this setup is equivalent to the D3-D7 system studied in [19][20][21]. Since the number of Neumann-Dirichlet (ND) directions is six, open string excitations between the D5-branes give rise to two Majorana fermions (Mj) [= one two-component Dirac fermion (Di), ψ] and no bosons. The distance between the D-branes in x 9 direction (∆x 9 ) is proportional to the mass m of the fermions. The low-energy effective theory is schematically summarized by the effective Lagrangian in the (2 + 1)-dimensional common direction of the two D5-branes, Integrating the massive fermions yields the CS terms k 4π A ∧ dA and k 4π Ã ∧ dÃ with k = ±1/2 (parity anomaly). The Hall conductivity is read off from the CS term for A µ as σ xy = k/(2π). Alternatively, the presence of the CS terms can be read off from the WZ action of either one of D5-branes, e.g., for the external gauge field A µ , where C 2 is the RR 2-form from the Dq-brane. When we change the sign of m by passing the Dq-brane through the Dp-brane, the value of k jumps from ±1/2 to ∓1/2. If we instead put N f Dq-branes, we have N f copies of massive Dirac fermions ψ i which couple with U(N f ) gauge fields A µ andÃ µ (when all Dq are coincident). This brane construction can be extended to other even space dimensions d = 2n by considering D5-Dq systems with q = 5, 7 (Table 4). This setup gives rise to the fermion spectrum consisting of one Dirac fermion per Dq-brane, and the CS terms of the form ∝ k A ∧ F n . All of these brane configurations are identified with class A TIs, which are characterized by the absence of any discrete symmetries. Now let us turn to AIII, which is characterized by the presence of SLS. We argue that SLS is equivalent to an invariance of the brane configurations under inversion of a coordinate in the Dirichlet-Dirichlet (DD) directions of open strings between the D5and Dq-branes. One way to realize this is to assume two DD directions, say, x 1 and x 9 , and impose x 1 = 0. Indeed such a configuration is obtained by taking T-dual of class A configurations (Table 4). Again, the fermion spectrum consists of two Majorana fermions (=one Dirac fermion) for all dimensions, and the mass of the fermion is, again, proportional to ∆x 9 .
In our setup in general, the number of Neumann-Neumann (NN) directions #NN is equal to the space-time dimensions d+1 of TIs/TSCs. On the other hand, #DD represents the number of possible mass deformations: it is one if there is no SLS (class A), while it is two in the presence of SLS. Finally, #ND is determined by #NN and #DD via the 0 1 2 3 4 5 6 7 8 9 d

Real case
To realize eight "real" symmetry classes in string theory, we need to implement TRS and PHS. To preserve PHS, we require the internal gauge fieldÃ µ is not independent of its complex conjugate. This is the same as the orientation (Ω) projection in string theory.
To realize TRS, we recall that it can be viewed as a product of PHS and SLS [9]. As SLS can be imposed as a parity symmetry in string theory, we can interpret TRS as the orientifold projection. Let us start with class C and D, which are characterized by the presence of PHS but lack of TRS. We take an Ω projection of the class A setup. Note that there are two types of Ω projections, represented by two types of O9-plane, i.e., O9 − (orthogonal) and O9 + (symplectic). While only O9 − leads to supersymmetric type I string theory, here we consider both because the T-dual of O9 + is equivalent to Op + planes (p ≤ 8) in type II string theory. We realize class C and class D TSCs by considering a D5-brane which extends in the x 0,1,2,3,4,5 directions, in the presence of an O9-plane. As before, we put a Dq-brane with q = d + 3 so that there are d + 1 common directions (Table 5). For class AII and AI, characterized by the presence of TRS but lack of PHS, we take the orientifold projection which leads to an O8-plane in type IIA theory [22]. By choosing (p, q) = (4, d + 4), we obtain the brane configuration given in the third table in Table 5. Though the D-brane charges with an Op-plane for p ≤ 8 are classified by KR-theory, the same result can be obtained from KO-theory via T-duality for our purpose [22]. Finally, the remaining four classes, CII, BDI, CI and DIII can be obtained by taking DD directions to be two instead of one. SLS is imposed by requiring x 9 = 0 for all of these classes. These are O8-or O9-projection of the class AIII setup (Table 5).
For these D-brane configurations, we chose a Dp-brane to be a standard one with the integer K-theory charge so that it is regarded as the background (bulk) material itself. Then, we find that the K-theory charge of the Dq-branes (shown in the last two columns in Table 5) agrees precisely with the corresponding classification of TIs/TSCs (Table 1). Moreover, the fermion content of these string theory realizations (denoted in the last two columns in Table 5 either by "N f Mj" or "N f Di" with N f an integer) can be compared with the Dirac representative of TIs/TSCs constructed in Ref. [9]. Indeed, they agree completely. It is also interesting to note that, for the Dirac representative of TIs/TSCs, the momentum dependence of the projection operator, which is one of key ingredients in the classification of TIs/TSCs [9], looks quite similar to the spatial profile of the tachyon field in string theory in real space [12].
We now describe the field theory content of the Dp-Dq systems charges in more details. First, for the Z TIs/TSCs on the diagonal in Table 1 ("primary series"), the internal gauge group is O(1) = Z 2 . In particular, for class D in d = 2, our string theory realization corresponds to (a proper supersymmetric generalization of) the honeycomb lattice Kitaev model in the weak pairing (non-Abelian) phase [17]. Similarly, for class DIII in d = 3, it corresponds to an interacting bosonic model on the diamond lattice [18].
Even though the ten-dimensional string theories in the bulk are supersymmetric, our brane setups are not in general. When #ND = 4 with Z charge, they exceptionally preserve a quarter of supersymmetries. In general, when #ND = 4, there exist massive bosons in addition to the massive fermions. This happens when d = 4 for A, C, AI, AII, and when d = 3 for AIII, CII, CI and DIII. Since #NN > 4 for all the other branes systems, we only have fermions from open strings between the Dp-and Dq-branes and there are no tachyons. Edge state Dq Dp Dp Figure 1: A boundary of a TI/TSC as a brane intersection. Table 6: Dp-Dq systems in the presence of an O-plane; "d ≥ 2" means the constraint of possible spatial dimensions d in the brane systems due to the existence of open string tachyons. "Chiral" denotes the existence of chiral fermions and is interpreted as boundary (edge) states. The horizontal direction is shifted by a T-duality in the NN direction.
We can take the T-duality further in NN directions. However, this lead to theories with different properties than TIs/TSCs ( Table 6). Note that we have succeeded to realize all TIs/TSCs in space dimensions d ≤ 4.

Boundary of TIs/TSCs
A defining property of TIs/TSCs is the appearance of stable gapless degrees of freedom, when the system is terminated by a (d − 1)-dimensional boundary. In our brane construction, the sample boundary can be constructed by bending the Dp-brane toward the Dq-brane, to create an intersection between these branes (Fig. 1). This leads to a positiondependent fermion mass, which changes its sign at the intersection. This increases #ND by two and the correct number of massless fermions appears at the intersection.

Conclusions
The main conclusion of this paper is that there is a one-to-one correspondence between the tachyon-free Dp-Dq systems and the ten classes of topological insulators in d ≤ 4 dimensions. Indeed, we explicitly constructed the corresponding ten classes of Dp-Dq brane configurations in superstring theory. Two out of ten are realized in type II string theory without orientifolds, while the other eight require orientifolds. The K-theory charges of the Dq-branes agree with that of the topological insulators.
One may wonder if there are other tachyon-free Dp-Dq systems which have not been considered in this paper. However, it turns out that their low-energy theories just correspond to multiple copies of the ten classes of topological insulators (for details refer to [15]).
Since a topological insulator has a mass gap in the bulk, the distance between the Dp and Dq is taken to be non-vanishing in its corresponding D-brane system. When we discuss the boundary of a TI/TSC, however, the Dp-brane is bent toward the Dq-brane and thus Dp and Dq are intersecting with each other. Therefore massless fermions appear at the intersection and they are identified with the boundary modes (or edge modes).
We can also consider holographic descriptions of these systems by extending the constructions in [20,21] in principle, though it is not possible to take the large-N limit of Z 2 charged D-branes.