Analytic exposition of the graviton modes in fractional quantum Hall effects and its physical implications

Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this paper, we derive a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and the thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin $\nu =1/5$ and the Gaffnian $\nu =2/5$ phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a “phase transition” even when the ground state is invariant. We discuss the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the “hollow-core” modes, as well as their experimental consequences.

Figure 1

Hierarchy of the null spaces of different FQH states. The null space of different model Hamiltonians can be organized as a hierarchical structure in the full Hilbert space. Meanwhile the density modes or more precisely, the graviton modes (denoted by the stars with the corresponding colors) constructed from model ground states live in the next larger null space. For example, the graviton modes of Laughlin-$1/3$ phase (orange star) live in the Haffnian null space (green circle), which contains the ground state and all the quasihole states of the Haffnian model Hamiltonian. It is efficient to verify this structure with the characteristic tensor formalism proposed in this paper.

Figure 2

Structure of the ${\overline{d}}_{n_1{n}_2}$ coefficients of the Laughlin-$1/3$ state. Here gray coefficients are zero because of the fermionic statistics. Bold coefficients are zero because of the vanishing ground-state energy (contributions from different three-body model Hamiltonians are denoted by different colors). Black coefficients are unknown and not necessarily zero. Note that here we omit the bars over the ${\overline{d}}_{n_1{n}_2}$ coefficients for simplicity.

Figure 3

Spectrum of the toy Hamiltonian ${\widehat{H}}_L$ with respect to 6 electrons and 26 orbitals. (a) As the color bar shows, the red dots depict the spectrum of the density modes and the hollow-core modes are denoted by blue dots. The color of each dot is determined by calculating their collective overlap with all the density modes in the Laughlin-$1/3$ null space ${\mathcal{H}}_{\text{L-1/3}}$, or all the hollow-core modes in the complementary space ${\overline{\mathcal{H}}}_{\text{L-1/3}}$. As $\lambda$ increases, though the ground state (denoted by dark red) stays invariant, the low-lying states show a clear cross over behavior and transform from density modes to hollow-core modes. (b) Illustrates the structure of the Hilbert space and the relationship between the states and the model Hamiltonians. The Laughlin-$1/5$ null space ${\mathcal{H}}_{\text{L-1/5}}$ (dark red circle) punished by neither ${\widehat{V}}_1^{\text{2bdy}}$ nor ${\widehat{V}}_3^{\text{2bdy}}$ is the sub-space of ${\mathcal{H}}_{\text{L-1/3}}$ (only punished by ${\widehat{V}}_3^{\text{2bdy}}$). Meanwhile there exist the hollow-core modes (blue circle) only punished by ${\widehat{V}}_1^{\text{2bdy}}$ in ${\overline{\mathcal{H}}}_{\text{L-1/3}}$. All of the other states are punished by both ${\widehat{V}}_1^{\text{2bdy}}$ and ${\widehat{V}}_3^{\text{2bdy}}$, living in the remaining part of ${\overline{\mathcal{H}}}_{\text{L-1/3}}$.

Figure 4

Spectra of the toy Hamiltonian ${\widehat{H}}_L$ diagonalized in different Hilbert spaces. The case with $\lambda =0.05$ is shown in (a) and $\lambda =0.95$ in (b). In the left panel, one can clear observe the transition of the low-lying states when $\lambda$ increases from 0.05 to 0.95. These states have different nature as explained in Fig. 5. In the right panel, we zoom in on the spectra to the range $E\in [0,0.5]$ and mark the states living in the Haffnian null space with green squares and other states with blue dots. As expected, by using the CF picture, both the density modes and the hollow-core modes live in the Haffnian null space. Note that the lowest angular momentum of the hollow-core modes is $L_{\text{min}}=4$ as can be seen in (b). Furthermore, the quantized energy of the hollow-core modes (especially when $\lambda \to 0$) can be understood by using the root configurations in Eq. (51).

Figure 5

Nature of the low-lying states with different model Hamiltonian in the composite fermion picture. (a) The Laughlin-$1/5$ state of the electrons can be reinterpreted as the Laughlin-$1/3$ state of CFs consisted of one electron and two fluxes. (b) The graviton modes can be understood as the excitations of CFs in the lowest CF level. (c) The hollow-core modes are created by exciting CFs to the second CF level, which still live in the Gaffnian null space.

Figure 6

Spectrum of the toy Hamiltonian ${\widehat{H}}_G$ with respect to 10 electrons and 22 orbitals. (a) Shows the spectra of the model Hamiltonian ${\widehat{H}}_G$ in Eq. (52). The low-lying states are density modes even when $\lambda$ is quite large, and the absence of hollow core modes is different from the Laughlin-$1/5$ state in Fig. 3. (b) Illustrates the structure of the Hilbert space and the relationship between the states and the model Hamiltonians. All the states in the complementary space of the Moore-Read null space are punished by both ${\widehat{V}}_3^{\text{3body}}$ and ${\widehat{V}}_5^{\text{3body}}$.

Figure 7

Spectra of the toy Hamiltonian ${\widehat{H}}_G$ diagonalized in the full Hilbert spaces. The spectrum with $\lambda =0.05$ is shown in the left panel and $\lambda =0.95$ in the right panel. As expected, when $\lambda =0.05$ all the low-lying states (density modes) live within the Moore-Read null space ${\mathcal{H}}_{\text{MR}}$. Meanwhile when $\lambda =0.95$, all the states except the ground state are in the complement of ${\mathcal{H}}_{\text{MR}}$ within the LLL, denoted by ${\overline{\mathcal{H}}}_{\text{MR}}$.

Figure 8

Finite size scaling of the structure factor coefficients of different states. (a) The structure factor expansion coefficients of the Laughlin-$1/3$ and the Laughlin-$1/5$ state. According to the orthogonality of Laguerre polynomials, $d^m$ of the Laughlin-$1/m$ state is equal to the expectation value of the model Hamiltonian ${\widehat{V}}_m^{\text{2bdy}}$ (thus $c^m=1$) acting on this state. Thus the numerical results shown here provide the value of the corresponding dimensionless coefficients despite with the dimension of energy, which is true for (b) as well. (b) The expectation value of ${\widehat{V}}_7^{\text{3bdy}}$ with respect to the Gaffnian state ($\approx 0.04$ in the thermodynamic limit), denoted by $d_{\text{3bdy}}^7$, is significantly smaller than other coefficients in the plot, where the expectation value of ${\widehat{V}}_5^{\text{3bdy}}$ with respect to the Moore-Read state is denoted by $d_{\text{3bdy}}^5$ ($\approx 1.6$ in the thermodynamic limit).