Majorana Stripe Order on the Surface of a Three-Dimensional Topological Insulator
The effect of interactions in topological states is a topical issue that includes not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a vortex lattice in 2D topological superconductors. Under a special neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, we show that a minimal model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping to a compass model, we demonstrate that the interaction induces a Majorana stripe order spontaneously breaking translational and rotational symmetries. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization, beyond which Majorana cones appear in the excitation spectrum.
Topological states of matter have been the focus of research during the last decade Hasan and Kane (2010); Qi and Zhang (2011); Chiu et al. (2016). In this rapid development, bulk-boundary correspondence has been a central guiding principle, which predicts low-energy modes at the interface between topologically distinct states. The same principle also applies to topological defects (such as dislocations and superconducting vortices) in topological matter, because they can be regarded as generalized interfaces bordering on normal states Ran et al. (2009); Teo and Kane (2010); Juričić et al. (2012); Slager et al. (2012); Hughes et al. (2014); Teo and Hughes (2013); Slager et al. (2014); Teo and Hughes (2017). Of particular interest are Majorana zero modes (MZMs) predicted to emerge at vortices in 2D topological superconductors, because of their potential for quantum computation Kitaev (2003); Nayak et al. (2008); Kitaev (2006); Fu and Kane (2008); Beenakker (2013); Alicea (2012); Stanescu and Tewari (2013); Leijnse and Flensberg (2012); Elliott and Franz (2015); Das Sarma et al. (2015). Meanwhile, the idea of designing lattices of Majorana fermions out of MZMs is fascinating in its own right, because interaction between MZMs may lead to novel phases and critical phenomena Rahmani et al. (2015a, b); Milsted et al. (2015); Chiu et al. (2015a, b); Zhu and Franz (2016); Affleck et al. (2017); Hayata and Yamamoto (2017); Sannomiya and Katsura (shed).
In this Letter, we study a square lattice of interacting MZMs, which may emerge at vortices in 2D topological superfluid and superconductor Volovik (1999); Read and Green (2000); Ivanov (2001); Grosfeld and Stern (2006); Chiu et al. (2012); Biswas (2013), as was predicted for the phase of He and SrRuO Das Sarma et al. (2006); Chung et al. (2007); Jang et al. (2011). For definiteness, we consider a surface of a 3D strong topological insulator (TI) subject to superconducting proximity effect, the setup first proposed by Fu and Kane where the surface state resembles a spinless superconductor Fu and Kane (2008); see Refs. Xu et al., 2014, 2015 for recent experimental progress. When an Abrikosov vortex lattice is induced by a magnetic field, MZMs are expected to emerge at vortices Teo and Kane (2010), leading to a novel lattice of Majorana fermions at low energies. Here we assume additional conditions to stabilize a square vortex lattice as in LuNiBC De Wilde et al. (1997), such as strong fourfold lattice anisotropy. Besides being one of the simplest structures, the reason for considering the square lattice is that in the neutrality condition, which automatically leads to the strong coupling limit for Majorana modes Chiu et al. (2015a, b), a minimal Hamiltonian admits a faithful spin representation. This allows for employing a powerful quantum Monte Carlo (QMC) method Gubernatis et al. (2016) to investigate thermodynamic properties in an unbiased way. In particular, we find a novel “Majorana stripe” phase. We also present a duality transformation elucidating the nature of this phase. With this solid understanding, we extend our analysis away from neutrality by a mean-field (MF) theory. We find a quantum critical point induced by Majorana hybridization, beyond which Majorana cones appear in the excitation spectrum.
At the non-interacting level, the system is described by the Fu-Kane Hamiltonian Fu and Kane (2008) with being the Nambu spinor of the electronic operators () and
where () is the Pauli matrix in the spin (Nambu) basis, is the chemical potential, is the proximity-induced pair potential, and is velocity of the surface Dirac mode when . The distribution and the structure of vortices are encoded in and MZMs can be obtained by solving Eq. (1) Chiu et al. (2015a). The neutrality condition has a significant consequence on the emergent symmetry of an effective Hamiltonian of MZMs Chiu et al. (2015a, b). When satisfied, an artificial time-reversal symmetry ( is the complex conjugation) with emerges in addition to the particle-hole symmetry inherent to the Bogoliubov-de Gennes formalism. The consequence is the MZMs bound to vortices, , with the spin antiparallel to the magnetic field Chiu et al. (2015b). Because and is antiunitary, single-body hybridization is prohibited at between any pair of MZMs at and ’. For an interacting many-body system, this suggests that the neutrality condition automatically corresponds to the strong coupling limit for the Majorana modes. Assuming the simplest, quartic local interaction of the vortex Majorana modes on the square lattice, we consider a Hamiltonian Chiu et al. (2015a); Affleck et al. (2017) with
where is the Majorana fermion operator at site satisfying and and the summation runs over elementary plaquettes of the square lattice; – are four corners of a plaquette, , , and , with and the primitive lattice vectors [Fig. 1(a)].
Away from the neutrality condition, the hybridization terms are allowed. In this case, we consider with
which has a gauge-invariant flux per plaquette because of the underlying vortices Chiu et al. (2015a). Equation (3) preserves the full symmetry of the square lattice (e.g., the translation in the direction is accompanied by a gauge transformation). By continuity, we expect for small . In other words, the Fermi level is expected to serve as an experimental control of Chiu et al. (2015a). We assume below unless otherwise mentioned.
We start from the many-body Majorana Hamiltonian in the neutrality condition, [Eq. (2)]. Assuming a periodic (open) boundary condition in the () direction, we map it to a quantum spin model by using a 2D Jordan-Wigner (JW) transformation. We define a spinless complex fermion by introducing an artificial (but arbitrary) pairing convention as in Fig. 1(b), where (cross) is the position of a pair combining and . Assuming the “column-major” index for pairs in Fig. 1(b), the transformation is and , where () are the Pauli matrices. The plaquette interaction becomes
with , where the number of pairs involved in the interaction is two and four, respectively [Fig. 1(c)]. The string factor does not appear in either case because any plaquette operator comprises products of two Majorana operators at neighboring sites in the same column, and such composite objects behave as bosons. We thus obtain
with , which combines the Ising coupling on the horizontal bonds and a transverse four-spin term associated with plaquettes () of spins [Fig. 1(b)].
Remarkably, in the spin representation (5), we can apply a bosonic QMC method Gubernatis et al. (2016) to study the thermodynamic properties of MZMs unbiasedly without a negative sign problem. Specifically, the directed-loop algorithm Syljuåsen and Sandvik (2002); Alet et al. (2005) in the basis is used in our work. To reduce finite-size effects, we use a trick of fictitious MZMs to simulate the lattice of Majorana fermions comprising an even number of plaquettes in the direction SM (). We investigated the spin lattices of up to , which corresponds to MZMs.
Figure 2(a) shows the specific heat . In addition to the broad peak around temperature , it exhibits a size-dependent sharp anomaly at , indicating a second-order transition. This observation points to a symmetry breaking phase at low , which contradicts with the previous conjecture of a topological quantum critical point Chiu et al. (2015a). However, any kind of long-range order of is impossible at because of the gauge-like 1D symmetries for ; in the spin language, the string operator is a conserved quantity, which flips eigenvalues of all spins in the horizontal () chain with . As known as a generalized Elitzur’s theorem Batista and Nussinov (2005), these symmetries reduce the effective dimensionality of the order parameter field from 2D to 1D. Hence, the corresponding 1D physics may explain the broad peak of at high , but not the transition itself.
To elucidate the nature of the low- phase and the transition, we show that (hence, ) is dual to two decoupled copies of a square-lattice quantum compass model Nussinov and van den Brink (2015), which was investigated in depth in various contexts Mishra et al. (2004); Dorier et al. (2005); Chen et al. (2007); Tanaka and Ishihara (2007); Wenzel and Janke (2008); Orús et al. (2009); Xu and Moore (2004, 2005); Nussinov and Fradkin (2005); Vidal et al. (2009); Nasu et al. (2017). Explicitly, we first define spins at the midpoint of every horizontal link. With the “row-major” site ordering in Fig. 3(a), the first transformation is , with , by which the and terms become an effective magnetic field and a four-spin term for spins, respectively. We find that the new four-spin term does not mix spins in even and odd columns, e.g., [Fig. 3(d)]. Consequently, the dual Hamiltonian is composed of decoupled even and odd components as with
To complete the mapping, we introduce spins at the midpoint of each vertical link for spins, such that , with , where is the column-major ordering for spins [Fig. 3(b)]. This transformation preserves the decoupling of and , transforming each into the quantum compass model on a square lattice with an enlarged unit cell [see Figs. 3(c)–3(e)],
The total Hamiltonian is .
The above duality transformation provides a useful insight into the problem of . The most crucial input is that the compass model with is known to undergo a “nematic” transition in the Ising universality class at Mishra et al. (2004); Tanaka and Ishihara (2007); Wenzel and Janke (2008). For , while any spin-spin correlation function such as and is short-ranged, the reflection symmetry () in the spin (real) space is spontaneously broken, which can be detected by a directional order parameter Nussinov and van den Brink (2015). To translate this back to the language of Majorana fermions, first we note that the even-odd decomposition () corresponds to the geometrical checkerboard decomposition of . Defining and as composed of quartic interactions in one sublattice of the checkerboard decomposition () and its complement (), respectively [see Fig. 4(a)], we find and . Here, corresponds to or and does to the other. Indeed, there is one-to-one correspondence between the quartic Majorana plaquette operators and the bond operators in the compass model; each Ising-like bond interaction in (7) corresponds to a plaquette term that it graphically overlaps, as illustrated in Fig. 4(a). In this sense, the nematic order quantified by corresponds to a spontaneous energy density modulation associated with the plaquette interaction . As shown in Fig. 4(b), the even-odd decoupling implies that the energy-density wave order emerges in the two sublattices and independently ( symmetry breaking), resulting in fourfold degenerate ground states modulo 1D symmetries.
We confirm this Majorana stripe order by evaluating the order parameter by QMC in the -spin representation. Figure 2(b) shows with , where the summation runs over either even or odd columns, is a proper normalization SM (), and
We find that becomes nearly size-independent at low , confirming the Majorana stripe order. The nonmonotonic -dependence for small is suggested to be a finite-size effect due to the open boundary condition in the direction. Figure 2(c) shows the Binder parameter , which exhibits crossing for different at , in agreement with the temperature associated with the divergent peak in .
Finally, we address the effect of hybridization (3) on the Majorana stripe phase. The finite-temperature Ising transition also implies a first-order transition line in the extended - phase diagram [Fig. 5(a)], where nonzero explicitly breaks the translational symmetry; see Fig. 1(c). Since the QMC method cannot be applied to due to the sign problem, we employ the MF approximation to examine the discontinuous transition at . Figure 5(b) shows the order parameter as a function of for varying values of hybridization . A clear jump of at for small indicates that the discontinuous transition persists even in the presence of a weak hybridization. This in turn implies that the finite- transition remains stable for small , although the induced coupling between and subsystems may modify the universality class Ashkin and Teller (1943). As increases, the jump, , vanishes at a critical point [Fig. 5(d)]. The MF band structure of Majorana fermions at is shown in Fig. 5(c) for . The spectrum in the stripe phase () is gapped with SM (), where the Chern number is 0 SM (). The gap is reduced with increasing and vanishes for . Assuming that the critical temperature , our result suggests as . Our calculation thus points to the existence of a quantum critical point characterized by gapless Majorana fermion excitations for .
In summary, the square-lattice Majorana Hamiltonian , which may have an experimental realization in the hybrid of a 3D strong TI and a superconductor, induces a stripe order that spontaneously breaks the translational and rotational symmetries in the strong-coupling regime . Our QMC simulations as well as the duality mapping (via the JW transformation) provide a solid confirmation of this phenomenon in the strong-coupling limit. We note that Affleck et al. also investigated the same model recently, where it was suggested the quantum phase transition belongs to a supersymmetric universality class Affleck et al. (2017). Our unbiased approach coming from the strong coupling is complementary to their weak-coupling instability analysis. In fact, lifts the 1D gauge-like symmetries, reducing the Majorana stripe state to the dimerized state found by Affleck et al. using a MF treatment similar to ours Affleck et al. (2017). We hope that our work will trigger an experimental effort in the search for intriguing phase transitions in the system of interacting Majorana modes.
Acknowledgements.We are grateful to Sharmistha Sahoo for useful discussions at the early stage of this project, and to Cristian Batista for valuable discussions. The numerical simulations in this work in part utilized the facilities of the Supercomputer Center, ISSP, the University of Tokyo. Y.K. acknowledges support by JSPS Grants-in-Aid for Scientific Research under Grant No. JP16H02206. A.F. acknowledges support by JSPS Grants-in-Aid for Scientific Research under Grant No. 15K05141. J.C.Y.T. is supported by the NSF under Grant No. DMR-1653535.
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Appendix A Trick of fictitious Majorana zero mode
Our derivation of in the main text assumes an even number of rows in the square lattice of Majorana fermions. Because of the open boundary in the direction, this means there are an odd number of plaquettes in this direction. However, since the low phase induces a modulation of energy density associated with the plaquette interaction , a particular type of the modulation is favored in such a setup, even if there are an even number of plaquettes in the direction. To avoid a resulting finite-size effect in the QMC simulation, we consider the lattice with an even number of plaquettes in the direction, hence, an odd number of rows of Majorana fermions [Fig. S1(a)].
A faithful spin representation can be derived also in this case. To combine the Majorana fermions in pairs properly, we add an additional row of fictitious MZMs as shown in Fig. S1(b). These fictitious modes are neither coupled to the rest of the system nor hybridizing within themselves. Therefore, they simply contribute to a constant to the free energy. We then apply the JW transformation in the same way as described in the main text. The only difference in the final form of is that the horizontal coupling for spins in the top row of the lattice is absent in the present case [Fig. S1(c)], because it would correspond to an interaction involving the fictitious modes, which do not exist. Finally, Fig. S1(d) illustrates each component of the order parameter of the Majorana stripe state,
where and are the size of the lattice of spins in the horizontal and vertical directions, respectively.
Appendix B Mean field theory
Here we present a zero-temperature mean-field calculation of the Majorana stripe phase. To consider the more general situations, we introduce two coupling constants and for plaquettes on the even and odd-numbered rows, respectively, of the square lattice. The difference serves as a symmetry-breaking field. As a result of this explicit breaking of translation symmetry along the vertical direction, the unit cell is doubled. We thus label the Majorana fermions as , where is the sublattice index and , with being integers, denotes the Bravais lattice points (Fig. S2). The interacting Majorana fermions are described by the following Hamiltonian
We first consider the most general mean-field decouplings of the quartic terms assuming no further breaking of the translation symmetry. Direct numerical calculation nonetheless shows that the diagonal term vanishes identically in the self-consistent solution. We thus consider the following nonzero mean-field averages:
The resultant mean-field Hamiltonian is
We next perform Fourier transform , where is the number of unit cells and the basis vectors are . After introducing a basis vector the mean-field Hamiltonian can be expressed as , where the matrix is given by
Here are the Pauli matrices, and we have used the relation obtained from numerical solutions to simplify the expression. The mean-field spectrum is obtained by solving the above Hamiltonian self-consistently with Eq. (S3).
Figure S3 shows the order parameters as a function of the ratio with a fixed for two different hybridization constant and . The stripe order parameter defined in the main text is given by . For small hopping [Fig. S3(a)], the stripe order remains finite as the system approaches the symmetric limit . This result indicates the existence of a zero-temperature first-order transition at , discussed in the main text. As the hopping increases, the discontinuity in at is also reduced and eventually vanishes when . The mean-field band structure of Majorana fermions is shown in Fig. S4 for various ratios of with other parameters and . Expressing , the eigenenergy of the mean-field Hamiltonian is given by , and always appears in pairs. The energy gap corresponds to the minimum of occurs at , , , and . In the symmetric point , the spectral gap is related to the stripe order parameter
The energy gap as a function of the hybridization is shown in Fig. 5 of the main text. Importantly, the closing of the gap for gives rise to a critical state with low-energy gapless Majorana fermions; see Fig. S4.
Starting from large hybridization (focusing on the symmetric case ), the Majorana stripe order occurs through a quantum phase transition that gaps out the Majorana nodal points upon reducing the hopping . A natural question is whether the gapped mean-field band structure is topologically nontrivial. To answer this question, we compute the Chern number of the bands explicitly for . We first define a unit-length vector . The spectral Chern number is given by
For simplicity, we introduce coefficients such that ; these coefficients can be easily obtained by comparing with Eq. (S5). In our case, the integrand evaluates to . Consequently, the Chern number is zero, indicating a topologically trivial gapped phase. This result can also be understood by noting that the -vector maps the Brillouin zone to a unit sphere, and the Chern number is simply the winding number of this mapping. In the gapped phase (say ), the neighborhood around the original Majorana nodes and is mapped to the north hemisphere, while that around the other two nodes is mapped to the south hemisphere; see Fig. S5. Within each hemisphere, the winding number is determined by the in-plane vorticity of the two nodes. In our case, the two Majorana nodes within the same hemisphere have opposite vorticity , hence the net winding number is zero.