Infinite density matrix renormalization group for multicomponent quantum Hall systems

While the simplest quantum Hall plateaus, such as the $\nu =1/3$ state in GaAs, can be conveniently analyzed by assuming only a single active Landau level participates, for many phases the spin, valley, bilayer, subband, or higher-Landau-level indices play an important role. These “multicomponent” problems are difficult to study using exact diagonalization because each component increases the difficulty exponentially. An important example is the plateau at $\nu =5/2$, where scattering into higher Landau levels chooses between the competing non-Abelian Pfaffian and anti-Pfaffian states. We address the methodological issues required to apply the infinite density matrix renormalization group to quantum Hall systems with multiple components and long-range Coulomb interactions, greatly extending accessible system sizes. As an initial application we study the problem of Landau-level mixing in the $\nu =5/2$ state. Within the approach to Landau-level mixing used here, we find that at the Coulomb point the anti-Pfaffian state is preferred over the Pfaffian state over a range of Landau-level mixing up to the experimentally relevant values.

Figure 1

An infinitely long cylinder of circumference $L$. The coordinate $x$ runs along the closed direction, and $y$ runs along the infinite direction. The dotted lines indicate the guiding centers for the orbitals in the Landau gauge, which we map to a 1D chain for DMRG (shown as yellow circles above the cylinder). In an $N$-component system, we have $N$ orbitals per guiding center and thus $N$ sites in the 1D chain per dotted line. (Here $N=2$ is illustrated, with components $\mu =\uparrow ,\downarrow$.)

Figure 2

Occupation of the $n=1$ Landau level as a function of the Landau-level splitting $\omega$, in units with $\hslash =e^2/\epsilon {\ell }_B=1$. The lowest-order perturbation theory in ${\omega }^{-1}$ predicts $\langle {\widehat{N}}_1\rangle \sim {\omega }^{-2}$, which we verify for $\omega \gg 2$, but modest higher-order effects appear near $\omega \sim 1$, the regime of physical interest. iDMRG at $L=17{\ell }_B$ (solid) is in good agreement with exact diagonalization (dashed lines).

Figure 3

Extrapolation of the energy per electron at $\nu =1/3$ with interaction given by Eq. (5) for $n=1$. The energy $E\left({\xi }_V\right)$ is plotted as a function of ${\xi }_V^{-4}{\ell }_B^4$, where ${\xi }_V$ is the “cutoff” length scale. Extrapolating the energy to ${\xi }_V\to \infty$ via the functional form (6), we find $E\left(\infty \right)=-0.410164\left(4\right)$ for the Coulomb potential, in good agreement with previous DMRG study of $E=-0.41016\left(2\right)$ [45].

Figure 4

(top) Plot of various quantities as a function of circumference $L$ for the one-third-filled spin-polarized state in the zeroth $\left(\nu =1/3\right)$ and first $\left(\nu =7/3\right)$ Landau levels, with no Landau-level mixing. The simulations were done at ${\xi }_V=5$ with MPS bond dimensions up to $\chi =15000$. The five plots are, from top to bottom, entanglement entropy $S$, topological entanglement entropy $\gamma =L\frac{dS}{dL}-S$, correlation length $\xi$, shift $\mathcal{S}$, and chiral central charge $c$, with the theoretical values given by the red dashed lines. The oscillations are far more pronounced in the $\nu =7/3$ state than in the $\nu =1/3$ state. (bottom) Orbital entanglement spectra for the two states at $L=25{\ell }_B$, organized by their momenta. Here only Schmidt states in the neutral charge sector (fixed number of particles on each side of the entanglement cut) are shown. The “low-energy” portion of the spectra is highlighted.

Figure 5

Landau-level mixing at $\nu =5/2,\kappa =1.38,L=20{\ell }_B$. We find that when two LLs $\left(n=0,1\right)$ are kept, as shown in (a), the ground state is the Pfaffian state, while when three LLs $\left(n=0,1,2\right)$ are kept, as shown in (b), the ground state is the anti-Pfaffian state. To verify this, in two separate runs we initialize the iDMRG simulation with model wave functions for the Pfaffian (red) and anti-Pfaffian (blue) states, for which exact matrix product state representations are known. The wave functions are then optimized using DMRG, and the energy is computed as the DMRG proceeds. (a) For two LLs, the MPS bond dimension is restricted to $\chi =2600$ during the initial sweeps and is then increased in three steps up to its final value of 6300. The anti-Pfaffian appears to be metastable, with an energy per flux of the $\delta E\sim 0.5\times {10}^{-3}$ higher than the Pfaffian. The entanglement spectrum, shown in the inset, initially has a left-moving chirality, a clear signature of the anti-Pfaffian in our convention. As $\chi$ is allowed to increase (leading to the steplike energy decreases), the entanglement spectrum becomes achiral, then collapses to the Pfaffian. The wave functions for the two runs are identical during the final sweeps of the DMRG. (b) For three LLs, the analysis is equivalent, but we find the anti-Pfaffian is preferred.

Figure 6

The convergence of the energy per flux during the DMRG simulation when initialized from the anti-Pfaffian (blue) and Pfaffian (red) states, like in Fig. 5. As in Fig. 5, we simulate $L_x=20{\ell }_B$ while keeping the lowest $N_{\mathrm{LL}}=3$ Landau levels. However, we now consider a range of Landau-level mixing $\kappa =1.38,1.38/1.5,\cdots$. For larger $\kappa$, the Pfaffian state tunnels into the anti-Pfaffian state. For smaller $\kappa$ the Pfaffian state remains metastable since the finite DMRG bond dimension discourages tunneling once the energetic splitting is too low. Regardless, there is an energetic splitting favoring the anti-Pfaffian state.

Figure 7

An estimate of the splitting per flux between the Pfaffian and anti-Pfaffian states as a function of $\kappa$. To estimate the splitting, we calculate the energy difference per flux ${\tilde{E}}_{\mathrm{Pf}}-{\tilde{E}}_{\mathrm{aPf}}$ between simulations initialized with the model Pfaffian and anti-Pfaffian states at the second sweep of the DMRG calculations, as shown in Fig. 6. The energy splitting is metastable during the initial sweeps of DMRG; while it is not the true energetic splitting, it appears to be close to the true splitting found in exact diagonalization of a smaller torus. As expected, we find a roughly linear dependence on $\kappa$.

Figure 8

The entanglement entropy $S_E\left(\kappa \right)$ as a function of the Landau-level mixing strength $\kappa$. We measure the entanglement entropy as the DMRG bond dimension is increased from $\chi =2400$ to 6300. The state is not fully converged at $\chi =6300$, so we extrapolate $S_E$ in $1/\chi$ to obtain an estimate of the converged result, shown as a dashed line. The resulting $S_E$ depends only weakly on $\kappa$, supporting a continuous dependence on $\kappa$ up to $\kappa =1.38$.

Figure 9

The splitting of the variational energies $E_{\mathrm{Pf}}-E_{\mathrm{aPf}}$ per flux when the iDMRG is restricted to $\chi =2000$. Again, the iDMRG is initialized with the Pf and aPf states, but for small $\chi$ the DMRG cannot tunnel between the two. The resulting energy splitting is measured while keeping the lowest $N_{\mathrm{LL}}=2,3,4,5$ LLs. In agreement with cases (i) and (ii), only the $N_{\mathrm{LL}}=2$ case prefers the Pf.