Boundary central charge from bulk odd viscosity: Chiral superfluids

Abstract

We derive a low energy effective field theory for chiral superfluids, which accounts for both spontaneous symmetry breaking and fermionic ground-state topology. Using the theory, we show that the odd (or Hall) viscosity tensor, at small wave-vector, contains a dependence on the chiral central charge c of the boundary degrees of freedom, as well as additional non-universal contributions. We identify related bulk observables which allow for a bulk measurement of c. In Galilean invariant superfluids, only the particle current and density responses to strain and electromagnetic fields are required. To complement our results, the effective theory is benchmarked against a perturbative computation within a canonical microscopic model. < l a t e x i t s h a 1 _ b a s e 6 4 = " r L w / X U c T o a 2 Z e K G I 9 H 4 s i a p 2 4 C g = " > A A A B 6 n i c b V B N S w M x E J 2 t X 7 V + V T 1 6 C R b B U 9 k V Q b 0 V v X i s 4 N p C u 5 R s m m 1 D k 2 x I s k J Z + h e 8 e F D x 6 i / y 5 r 8 x 2 + 5 B q w 8 G H u / N M D M v V p w Z 6 / t f X m V l d W 1 9 o 7 p Z 2 9 r e 2 d 2 r 7 x 8 8 m D T T h I Y k 5 a n u x t h Q z i Q N L b O c d p W m W M S c d u L J T e F 3 H q k 2 L J X 3 d q p o J P B I s o Q R b A u p r w w b 1 B t + 0 5 8 D / S V B S R p Q o j 2 o f / a H K c k E l Z Z w b E w v 8 J W N c q w t I 5 z O a v 3 M U I X J B I 9 o z 1 G J B T V R P r 9 1 h k 6 c M k R J q l 1 J i + b q z 4 k c C 2 O m I n a d A t u x W f Y K 8 T + v l 9 n k M s q Z V J m l k i S w M x E J 2 t X 7 V + V T 1 6 C R b B U 9 k V Q b 0 V v X i s 4 N p C u 5 R s m m 1 D k 2 x I s k J Z + h e 8 e F D x 6 i / y 5 r 8 x 2 + 5 B q w 8 G H u / N M D M v V p w Z 6 / t f X m V l d W 1 9 o 7 p Z 2 9 r e 2 d 2 r 7 x 8 8 m D T T h I Y k 5 a n u x t h Q z i Q N L b O c d p W m W M S c d u L J T e F 3 H q k 2 L J X 3 d q p o J P B I s o Q R b A u p r w w b 1 B t + 0 5 8 D / S V B S R p Q o j 2 o f / a H K c k E l Z Z w b E w v 8 J W N c q w t I 5 z O a v 3 M U I X J B I 9 o z 1 G J B T V R P r 9 1 h k 6 c M k R J q l 1 J i + b q z 4 k c C 2 O m I n a d A t u x W f Y K 8 T + v l 9 n k M s q Z V J m l k i S w M x E J 2 t X 7 V + V T 1 6 C R b B U 9 k V Q b 0 V v X i s 4 N p C u 5 R s m m 1 D k 2 x I s k J Z + h e 8 e F D x 6 i / y 5 r 8 x 2 + 5 B q w 8 G H u / N M D M v V p w Z 6 / t f X m V l d W 1 9 o 7 p Z 2 9 r e 2 d 2 r 7 x 8 8 m D T T h I Y k 5 a n u x t h Q z i Q N L b O c d p W m W M S c d u L J T e F 3 H q k 2 L J X 3 d q p o J P B I s o Q R b A u p r w w b 1 B t + 0 5 8 D / S V B S R p Q o j 2 o f / a H K c k E l Z Z w b E w v 8 J W N c q w t I 5 z O a v 3 M U I X J B I 9 o z 1 G J B T V R P r 9 1 h k 6 c M k R J q l 1 J i + b q z 4 k c C 2 O m I n a d A t u x W f Y K 8 T + v l 9 n k M s q Z V J m l k i S w M x E J 2 t X 7 V + V T 1 6 C R b B U 9 k V Q b 0 V v X i s 4 N p C u 5 R s m m 1 D k 2 x I s k J Z + h e 8 e F D x 6 i / y 5 r 8 x 2 + 5 B q w 8 G H u / N M D M v V p w Z 6 / t f X m V l d W 1 9 o 7 p Z 2 9 r e 2 d 2 r 7 x 8 8 m D T T h I Y k 5 a n u x t h Q z i Q N L b O c d p W m W M S c d u L J T e F 3 H q k 2 L J X 3 d q p o J P B I ...

Hall viscosity and conductivity of two-dimensional chiral superconductors

“…Our definition Eq. (3.11) of the viscosity coefficients from η follows [34]. In particular, comparison with Refs.…”

“…In contrast to the neutral chiral superfluid, where e = 0, the energy gap of the plasmon excitation ensures that the Hall conductivity always vanishes as q 2 in the limit ω, q → 0. The Hall viscosity tensor of an isotropic P T -invariant system is given by [34] η o (ω, q) = η (1) 2) and is fixed by the two independent coefficients η (1) o (ω, q 2 ) and η (2) o (ω, q 2 ), where σ a b = σ a ⊗ σ b − σ b ⊗ σ a , a, b = 0, x, z, are anti-symmetrized tensor products of symmetric Pauli matrices, see Section 3. The physical content of Eq.…”

“…Much like for the conductivity, the dissipationless 7 odd viscosity is a signature of the breaking of time-reversal symmetry. It has been shown in [34] that, in an isotropic system that is symmetric under the combination of parity and time-reversal symmetries (P T symmetry) the Hall viscosity tensor is fixed by only two independent components η (1) o and η (2)…”

“…Recently, observable signatures of the Hall viscosity have been vigorously studied in classical and quantum fluids both theoretically [18][19][20][21][22][23][24][25][26][27][28][29][30][31] and experimentally [32,33]. In a two-dimensional isotropic system that is invariant under the combined P T symmetry the odd viscosity tensor reduces to two independent components [34], in this paper to be denoted η (1) o (ω, q 2 ) and η (2) o (ω, q 2 ), respectively. If the Hall viscosity tensor is regular in the limit q = 0, only the component η (1) o survives as q → 0 [15,16].…”

“…As a result, at q = 0 the Hall viscosity coefficient η (1) o does not depend on the topology of the fermionic ground state and thus cannot be used as a diagnostics of topological superconductivity that is characterized by protected chiral Majorana edge modes. It was shown however in [34] that the q 2 dependence of the Hall viscosity tensor contains information about the chiral central charge of the boundary theory, which is determined by the topology of the fermionic ground state.…”