Neutral excitations of quantum Hall states: A density matrix renormalization group study

We use the dynamical structure factors of the quantum Hall states at $\nu =1/3$ and $1/2$ in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low-energy properties. At $\nu =1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasiparticle. At $\nu =1/2$, we find low-energy modes with linear dispersion and the static structure factor $\overline{s}\left(q\right)\sim {\left(q\ell \right)}^3$ in the limit $q\ell \to 0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.

Figure 1

The initial configuration in the TDVP algorithm. A finite segment containing $N_{\varphi }$ orbitals is embedded in the infinite quasi-1D ground state. We apply the operator $n_0$ at the mid-point to construct the excited state $|\psi \rangle$ and time-evolve it using TDVP in the region $L_t$. This region is dynamically expanded in a light cone as the disturbance spreads over time.

Figure 2

The neutral excitation spectrum of $\nu =1/3$ FQH state on an infinite plane. The red curve is the magnetoroton mode composed of a bound pair of a fractionally charged quasihole and quasielectron. It has a minimum at $q=q_{\min }$ and $\omega ={\omega }_{\min .}$. The cyan region contains the two-roton continuum, and other states such as multiple rotons, with a minimum energy equal to twice the energy of the roton minimum.

Figure 3

(a) The dynamical structure factor of $\nu =1/3$ Laughlin FQH state at $L_y=10\ell$ for $V_1$ Haldane pseudopotential. A sharp roton mode is visible at $0.7\lesssim q_x\ell \lesssim 2.6$ with the roton minimum at $q_x\ell =q_{x,\min }\ell =L_y/6\ell$. The energy of the roton mode is periodic under $q_x\ell \to q_x\ell +L_y/3\ell$. The two-roton continuum can be observed at around $q_x\ell =L_y/3\ell$. The dashed lime curve shows the sum of energies of two independent rotons each with a momentum $q_x\ell /2$. The dashed blue curve is the energy obtained from the single-mode approximation (SMA). (b) The dynamical structure factor at small $q_x\ell$. The sharp roton mode becomes overdamped as it enters the two-roton continuum at $q_x\ell \approx 0.7$.

Figure 4

(a) The dynamical structure factor of $\nu =1/3$ Laughlin FQH state at $L_y=9\ell$ for Gaussian Coulomb potential with $\xi =20\ell$. Generally, the features are similar to Fig. 3. The roton mode becomes overdamped at $q_x\ell \lesssim 0.6$ is as seen in (b).

Figure 5

The Fermi sea of CFs splits into discrete wires in an infinite cylinder geometry at $k_y=n\kappa$ where $n$ is an integer if the number of wires is odd and half-integer if the number of wires is even. Also, $\kappa =2\pi /L_y$. (a) The unique scatterings between the eight Fermi points for the case of 4 wires (horizontal lines). (b) $\overline{s}\left(\mathit{q}\right)$ calculated in iDMRG for $q_y/\kappa =0,1,2,3$ at $L_y=14\ell$. It contains singularities at wave vectors consistent with (a).

Figure 6

The case of CF Fermi sea composed of two wires. (a) The dynamical structure factor $\overline{s}(q_x,\omega )$ of the $\nu =1/2$ state at $L_y=7.2\ell$ for Gaussian Coulomb interaction (Eq. (6)) with $\xi =20$. There is a single low-energy mode at small $q_x$ and a fan corresponding to the particle-hole excitations near $q_x\ell =2K_x^F\ell =L_y/4\ell =1.8$. The dashed lime curve shows that the velocities of the two features are equal. (b) The low-energy mode at small $q_x\ell$ and the linear dispersion (inset) with a velocity $u_0=0.29e^2$, where $e^2$ represents the strength of the Coulomb interaction. (c) $\overline{s}(2K_x^F,\omega )$ vs $\omega$ on a log-log scale, (d) Real part of the density-density correlation function $\overline{D}(q_x,\omega )$ at $q_x\ell =0.25$, and (e) static structure factor $\overline{s}\left(q_x\right)/{\left(q_x\ell \right)}^3$ for different bond dimensions. Dashed curves in (c)–(e) show agreement between the theory and the data with a Luttinger parameter $K_{-}$ equal to 1.33. In (e), the static structure factor is converged at $q_x\ell <1$ and it crosses over from ${\left(q_x\ell \right)}^3$ to ${\left(q_x\ell \right)}^4$ at $q_x\ell \sim 0.1$ suggesting a gap at small wave vectors. The gap is estimated to be ${\omega }_g=0.01e^2/\ell$ within SMA.

Figure 7

The case of CF Fermi sea composed of three wires. (a) The dynamical structure factor $\overline{s}(q_x,\omega )$ of the $\nu =1/2$ state at $L_y=9.1\ell$ for Gaussian Coulomb interaction [Eq. (6)] with $\xi =20$. There are two gapless degrees of freedom that produce the fans near $q_x\ell =1.325$ and $q_x\ell =1.9$. At small $q_x\ell$, we observe one gapless mode that has a velocity equal to the fan at $q_x=1.325$ (dashed lime curve). There is no spectral weight at the velocity of the second fan at $q_x\ell =1.9$ (dashed cyan curve) in agreement with the quasi-1D CF theory. (b) The single gapless mode at small $q_x\ell$. The inset shows a fit to a linear dispersion with a quadratic correction with velocity $u_{-}=0.19e^2$. Another mode is visible at $q_x\ell \approx 0.5$, however, it disappears at long wavelengths. (c) A comparison of the numerically computed dynamical structure factor and the theoretical prediction with $K_{-}=0.91$ at the first fan, i.e., $q_x\ell =2K_{x,-}^F=1.325$. (d) The static structure factor $\overline{s}\left(q_x\right)/{\left(q_x\ell \right)}^3$ vs $q_x\ell$ for various bond dimensions $\chi$. As shown in the inset, the point of the crossover between $\overline{s}\left(q_x\right)\sim {\left(q_x\ell \right)}^3$ and $\overline{s}\left(q_x\right)\sim {\left(q_x\ell \right)}^4$ goes to $q\ell =0$ as $\chi \to \infty$.

Figure 8

Additional figures for Laughlin state at $\nu =1/3, L_y=10\ell$ in the presence of $V_1$ Haldane pseudopotential interaction. (a) The connected correlation function defined in Eq. (30) (b) The excess bipartite von Neumann entanglement entropy $\mathrm{\Delta }S$ generated during time evolution. It is obtained by computing the von Neumann entanglement entropy of the time-evolved excited state when the cylinder is cut into two halves to the left of the given site and then subtracting the ground state value. (c) The error $\mathcal{E}\left(t\right)$ vs time. The blue and the green curve correspond to the excited states obtained by applying the occupation number operator $n_k$ on the second and third orbitals in the unit cell, respectively. (d) The comparison of the static structure factor obtained from the ground state vs. the integral of the dynamical structure factor over $\omega >0$. (e) First moment of the dynamical structure factor $\langle \omega \rangle$ vs. SMA.

Figure 9

Additional figures for CFL state with a Fermi sea composed of two wires, $L_y=7.2\ell$ in the presence of Coulomb interaction. (a) The connected correlation function, (b) the excess bipartite von Neumann entanglement entropy, (c) The error $\mathcal{E}\left(t\right)$ vs time. (d) The static structure factor obtained from the ground state vs the integral of the dynamical structure factor over $\omega >0$. (e) First moment of the dynamical structure factor $\langle \omega \rangle$ vs SMA.

Figure 10

(a) The dynamical structure factor $\overline{s}(q_x,\omega )$ of the $\nu =1/2$ state at $L_y=7.2\ell$ for Gaussian Coulomb interaction (6) with $\xi =20$. The results are similar to Fig. 6. (b) The low-energy mode at small $q_x\ell$ and the inset shows the linear dispersion with a velocity $u_{-}=0.43V_1\ell$ where $V_1$ represents the strength of the interaction. (c) $\overline{s}(2K_x^F,\omega )$ vs $\omega$ on a log-log scale. (d) Real part of the density-density correlation function $\overline{D}(q_x,\omega )$ at $q_x\ell =0.25$, and (e) static structure factor $\overline{s}\left(q_x\right)/{\left(q_x\ell \right)}^3$ for different bond dimensions. Dashed curves in (c)–(e) show theoretical prediction with Luttinger parameter $K_{-}=1.125$. In (e), We estimate a gap ${\omega }_g$ equal to $0.01e^2/\ell$ using SMA.

Figure 11

Additional figures for CFL state with a Fermi sea composed of three wires, $L_y=9.1\ell$ in the presence of Coulomb interaction. (a) The connected correlation function, (b) the excess bipartite von Neumann entanglement entropy, (c) The error $\mathcal{E}\left(t\right)$ vs time. The blue and green curves correspond to the excited states obtained by applying the occupation number operator on the two sites inside the unit cell. (d) The static structure factor obtained from the ground state vs. the integral of the dynamical structure factor over $\omega >0$. (e) First moment of the dynamical structure factor $\langle \omega \rangle$ vs SMA.