Quantum Thermal Hall Effect in a Time-Reversal-Symmetry-Broken Topological Superconductor in Two Dimensions: Approach from Bulk Calculations

Abstract

We discuss thermal transport of two-dimensional topological superconductors (TSCs) with broken time reversal symmetry, which are described by Bogoliubov-de Gennes (BdG) Hamiltonians. From the calculations of bulk quantities only, without refereeing to Majorana edge states, we show that the thermal Hall conductivity of two-dimensional TSCs in the lowtemperature limit is quantized in multiples of 1 2 πT 6 , which is exactly one half of the value of quantization in the case of the integer quantum Hall effect, and that this exact halfquantization is caused by the structure of the Nambu spinor and the particle-hole symmetry, which BdG Hamiltonians generally have. In the case of spinless chiral p-wave superconductors, this result is in perfect agreement with the argument based on the Ising conformal field theory with the central charge c = 1/2, which is an effective low-energy theory of the Majorana edge states. KEYWORDS: thermal Hall effect, topological superconductorIntroduction -Topological insulators (TIs) and superconductors (TSCs) are new quantum phases which are characterized by bulk topological invariants. It has been found that some of these invariants directly characterize the transport properties of the system. For example, the integer quantum Hall effect (IQHE) state is characterized by the topological number, the TKNN (or first Chern) number C 1 , and this invariant appears in the Hall conductivity σ xy in the low-temperature limit: σ xy = C 1 e 2 /h, which has been derived from the linear response theory. 1, 2 A similar idea is also applicable to topological superconductors without time reversal symmetry in two dimensions (i.e. class C and D in the topological periodic table 3 ), for which topological nontriviality is characterized by the nonzero TKNN number. However, in this case, it is not charge transport but thermal transport that characterizes the topological feature, since charge is not conserved, while energy is still a conserved quantity in supercon- * LETTERS ducting states. For instance, in the case of spinless chiral p-wave superconductors, which have the TKNN number C 1 = ±1, 4 Read and Green showed that by using the effective low-energy theory for Majorana edge states, the thermal Hall coefficient in the low-temperature limit is precisely given by c πT 6 (in the unit of k B = 1, = 1), where c = 1/2 is the central charge of the Ising conformal field theory (CFT) which describes the Majorana edge state. 5,6 Note that this value is one half of the value of IQHE states, which have edge modes described by the chiral Luttinger liquid theory, i.e. CFT with the central charge c = 1. 5 The Read-Green's

Duality in topological superconductors and topological ferromagnetic insulators in a honeycomb lattice

“…It is known that the thermal Hall conductivity of the topological superconductors in the low temperature limit is given by κ xy = C 2 πT 6 with the coefficient to the temperature T being quantized. [43][44][45] Here the appearance of half the Chern number is a reflection of the half-fermion nature for Majorana modes. Different topological phases shown in Fig.…”

Symmetry protected topological superfluid3He-B

“…The order parameter fluctuation modes contribute to the thermal conductivity through the vertex corrections [359], which may deviate κ H from the quantized value. We also notice that the thermal response of the 3 He-A in a thin film that is a time-reversal breaking topological superfluid is characterized by the Chern number [177].…”

Majorana Fermions and Topology in Superconductors

“…f (E) is the fermi distribution function. By using the Sommerfeld expansion, we obtain the following expression in the low-temperature limit: 197) κ tr xy = C 1 (0) 2…”

“…197) In fact, the bulk topological invariant, i.e. the first Chern number, can be directly related to the Hall conductivity via the expression for the currentcurrent correlation function.…”