Torsional Anomalies, Hall Viscosity, and Bulk-boundary Correspondence in Topological States

TL;DR

This work develops a bulk-boundary framework for topological viscoelastic responses in 2+1D Chern insulators by coupling massive Dirac fermions to gauge and frame fields with torsion. It shows that the Hall viscosity is generically divergent but its phase-to-phase difference is finite and governed by the mass gap, with the jump encoded by edge anomalies through anomaly inflow. The renormalized effective action in the nontrivial phase contains a chiral gravity piece, signaling a deep link between topological transport and gravitational Chern-Simons structures, and the edge theory exhibits covariant diffeomorphism and Lorentz anomalies with torsion that precisely account for the bulk response. The analysis uses multiple routes (Berry curvature, stress-stress correlators, and background field methods) and Pauli-Villars regularization to pin down universal differences, and it connects spectral flow of edge modes under torsion to the bulk transport, suggesting broader implications for holographic and higher dimensional generalizations.

Abstract

We study the transport properties of topological insulators, encoding them in a generating functional of gauge and gravitational sources. Much of our focus is on the simple example of a free massive Dirac fermion, the so-called Chern insulator, especially in 2+1 dimensions. In such cases, when parity and time-reversal symmetry are broken, it is necessary to consider the gravitational sources to include a frame and an independent spin connection with torsion. In 2+1 dimensions, the simplest parity-odd response is the Hall viscosity. We compute the Hall viscosity of the Chern insulator using a careful regularization scheme, and find that although the Hall viscosity is generally divergent, the difference in Hall viscosities of distinct topological phases is well-defined and determined by the mass gap. Furthermore, on a 1+1-dimensional edge between topological phases, the jump in the Hall viscosity across the interface is encoded, through familiar anomaly inflow mechanisms, in the structure of anomalies. In particular, we find new torsional contributions to the covariant diffeomorphism anomaly in 1+1 dimensions. Including parity-even contributions, we find that the renormalized generating functionals of the two topological phases differ by a chiral gravity action with a negative cosmological constant. This (non-dynamical) chiral gravity action and the corresponding physics of the interface theory is reminiscent of well-known properties of dynamical holographic gravitational systems. Finally, we consider some properties of spectral flow of the edge theory driven by torsional dislocations.

Figures (6)

• Figure 1: Fluid mechanics illustration of the viscous forces. A counter-clockwise rotating solid cylinder immersed in 2d liquid droplet with (a) non-zero shear viscosity (b) non-zero dissipationless viscosity. Note that the resulting forces (arrows outside cylinder) are tangent and perpendicular to the cylinder motion (arrows inside cylinder) respectively. The shear viscosity impedes the cylinder while the dissipationless viscosity pushes fluid toward or away from the cylinder depending on the rotation direction.
• Figure 2: (a) Reference state (hollow circles) and displaced state (solid circles) for an elastic medium. Displacement vectors for each site $n$ are denoted by $\mathbf{u}(x_n).$ Zoom-in shows frame field vectors $\mathbf{e}_1,\mathbf{e}_2$ in the reference state (aligned to crystal $x,y$-axes) and the displaced state (rotated with respect to crystal axes). (b) Edge dislocation representing the fundamental torsion lattice defect. An electron traveling the thick line surrounding the dislocation will be translated with respect to the same path in the reference state that does not enclose a dislocation. The Burgers vector is in the $y$-direction. (c) Disclination represented by a single triangular plaquette in a square lattice crystal. Gives rise to curvaturei.e. objects that travel around a disclination are rotated with respect to the reference-state path.
• Figure 3: Laughlin gauge argument for torsion: Thought experiment with an insertion of torsion flux i.e. a dislocation into cylindrical hole, equivalent to shrinking or enlarging the cylinder in the $y$-direction as a function of time. Non-zero dissipationless viscosity causes transfer of $p_y$-momentum in the $x$-direction, i.e. a momentum current perpendicular to time-dependent strain.
• Figure 4: (a) Energy spectrum from Eq. (\ref{['eq:chiralHgauge']}) at time $t=0.$ Right/left handed fermion spectra are represented by positively/negatively sloped lines. The filled/empty circles represent occupied/unoccupied states. (b) Energy spectrum at time $t=T$ where one flux quantum has been threaded through the spatial ring. The spectrum returns to itself but the state occupation changes. One electron has been added to the right movers, and one has been removed from the left movers.
• Figure 5: (a)Energy spectra for left (blue) and right (red) handed chiral fermions from Eq. \ref{['eq:torsionHam']} at $t=0$ (b) Energy spectra for $t>0$ assuming $b_L=b_R=b<0$ which gives an increase to the velocity for both branches of chiral fermions. Note that when compared to the $t=0$ case there are states that were occupied chiral fermion states that have been pushed past the cut-off scale and the states at $p=0$ are unchanged. During this process no states cross $E=0$ and there is not a conventional notion of spectral flow at low-energy. Both figures have the same momentum discretization spacing, but different velocities which leads to a different number of states within the cut-off window.