Theory of interacting topological crystalline insulators

Abstract

We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z 8 . In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of U (1) × Z 2 symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries. The prediction and observation of topological crystalline insulators (TCIs) in the SnTe material class has expanded the scope of topological matter and gained wide interest [1][2][3][4][5]. These TCIs possess topological surface states that are protected by mirror symmetry of the rocksalt crystal and become gapped under symmetry-breaking structural distortions [6][7][8][9]. These surface states are predicted to exhibit a plethora of novel phenomena ranging from large quantum anomalous Hall conductance [1,10,11] to strain-induced pseudo-Landau levels and superconductivity [12], which are currently under intensive study [13][14][15].According to band theory, TCIs protected by mirror symmetry are classified by an integer topological invariant, the mirror Chern number [16]. However, recent theoretical breakthroughs [17][18][19][20][21][22][23][24] have found that the classifications of interacting systems are markedly different from noninteracting systems in various classes of topological insulators and superconductors protected by internal symmetries [25]. This raises the open question about the classification of interacting TCIs protected by spatial symmetries. On the experimental side, a growing body of interaction-driven phenomena has been found in existing TCI materials, including spontaneous surface structural transition and gap generation [6][7][8] and anomalous bulk band inversion [26]. Moreover, new TCI materials have been predicted in transition-metal oxides [27,28] and heavy fermion compounds [29,30], where strong electron interactions are expected.Motivated by these theoretical and experimental developments, in this work we study the effect of electron interactions in mirror-symmetric TCIs. Our main result is that interactions reduce the classification of three-dimensional (3D) TCIs from Z in the noninteracting case to Z 8 . We obtain this result by introducing a "domain wall" construction of interacting surface states of 3D TCIs, which exploits the nonlocal nature of mirror symmetry. This construction builds on interacting edge states of two-dimensional (2D) TCIs or U (1) × Z 2 symmetryprotected topological (SPT) phases, which...

Tomonaga-Luttinger liquid and localization in Weyl semimetals

“…Remarkably, it implies a field-induced dimensional reduction [22] that will strongly enhance correlations hence the advent of the (quasi-) 1D system without electron quasiparticle excitations. This connects to the long-lasting search or application of the Tomonaga-Luttinger liquid (TLL) physics [23], including semiconductor quantum wires [24,25], singlewalled carbon nanotubes [26,27], edge states in fractional quantum Hall states [28,29] and 2D topological insulators [30,31], and so on.Because of the large cyclotron gap, it is expected and confirmed that the Dirac/Weyl semimetals can be driven to the quantum limit at lower magnetic fields than semiconductors [20]. Due to the instability from electron correlations, one possibility is the gap-opening or dynamical mass generation [32] in the nominally massless semimetal as density waves are formed [33].…”

“…Remarkably, it implies a field-induced dimensional reduction [22] that will strongly enhance correlations hence the advent of the (quasi-) 1D system without electron quasiparticle excitations. This connects to the long-lasting search or application of the Tomonaga-Luttinger liquid (TLL) physics [23], including semiconductor quantum wires [24,25], singlewalled carbon nanotubes [26,27], edge states in fractional quantum Hall states [28,29] and 2D topological insulators [30,31], and so on.…”

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

“…Since then, there have been several works which discuss the breakdown of the noninteracting classification in the presence of interactions [14][15][16][17][18][19][20][21][22][23][24][25].…”

Bulk characterization of topological crystalline insulators: Stability under interactions and relations to symmetry enriched U (1) quantum spin liquids

“…An important question then is whether the nontrivial TCIs predicted by band theories are stable under interactions, or, more precisely, whether these band-theory-based nontrivial TCIs can be smoothly connected to a trivial insulator once strong interactions are switched on (still in the presence of the relevant symmetries). Very recently, the stability of various such TCIs under interactions has been considered and some of them are shown to be unstable, by studying the anomalous surface properties of the TCIs, studying certain dimensionally reduced versions of the TCIs, or studying the formal field theories of the TCIs [20][21][22][23][24][25][26][27][28][29][30][31]. However, an important and interesting question has remained unanswered: how to characterize nontrivial interacting TCIs directly by their bulk properties in a physical way?…”