Hall viscosity and putative quantum Hall states without positive-definite K-matrix

Abstract

We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.

Figures (13)

• Figure 1: Torus representation on the two-dimensional complex plane and in its embedding in three dimensions. While in the first case it amounts to the quotient space obtained after the identification of opposite sides of the unit cell, in the second case the periodicity is imposed from the beginning on the surface of the "doughnut". On the complex plane we identify the coordinates along the real axis and the straight line generated by the complex number $\tau$ as $x$ and $y$.
• Figure 2: Action of creation operators $a^\dagger$ and $b^\dagger$ on the semiclassical trajectories of electrons. The first moves between different Landau levels, which measures the cyclotron radius. The second span the orbital degeneracy in the energetic spectrum, which moves the guiding center.
• Figure 3: The classification of the space of possible K-matrices of the form abc, i.e., $K=\left(accb\right)$, with $a,b,c>0$. Region I contains the positive definite K-matrices, which can also be modelled with CFT techniques. Regions II has two positive eigenvalues, but one of the filling fractions is negative. Region three has one negative eigenvalue and one negative filling fraction, $\nu_j$. Both regions II and III are unphysical. Region IV has one negative eigenvalue, but both filling fractions are positive and can, therefore, be physical. The CFT approach gives holomorphic CoM pieces in regions I and II whereas the monodromy matching only does the same in region I. On the other hand, the CFT cannot produce a CoM piece in region IV which the monodromy matching can.
• Figure 4: The structure of the zeros of the Laughlin CoM term, $\mathcal{F}$, for a) eigenstates of $T_1$ and b) eigenstates of $T_2$. In both cases the zeros of $\mathcal{F}$ sit on a one-dimensional line, in the direction indicated by the translation of $T_j$. The action of $T_j$ move each zero onto the next one, mapping the state onto itself. The other operator $T_{\neq j}$ will move the row of zeros in the orthogonal direction, leading to a new state (with a different quantum number).
• Figure 5: (left) The modular plane, parameterized by $\tau$. In grey, the fundamental domain bounded by $|\tau|>1$ and $|\tau_1|<\frac{1}{2}$. Special points in the place are marked as a guide to the eye. (right) Berry flux $\varphi$ on a plaquette of area $\varepsilon^2$ with a vertex placed in $\tau$. The arrows indicate the direction of the circuitation on the path that encloses the square area.