Effect of Hilbert space truncation on Anderson localization

The 1D Anderson model possesses a completely localized spectrum of eigenstates for all values of the disorder. We consider the effect of projecting the Hamiltonian to a truncated Hilbert space, destroying time-reversal symmetry. We analyze the ensuing eigenstates using different measures such as inverse participation ratio and sample-averaged moments of the position operator. In addition, we examine amplitude fluctuations in detail to detect the possibility of multifractal behavior (characteristic of mobility edges) that may arise as a result of the truncation procedure.

Figure 1

Schematic of the energy-momentum dispersion relation of the zero-disorder 1D nearest-neighbor hopping problem. The solid black line indicates the tight-binding band $E\left(k\right)=-2cos\left(k\right)$. The light pink shaded region represents the part of the Hamiltonian projected out in the case of complete TRS breaking $\left(F=1\right)$. Likewise, the dark pink region represents the portion of the Hamiltonian projected out in the partial TRS breaking case $\left(F=1/4\right)$. We also depict the color code used to represent the three characteristic energies $(E=0,E=E_c=2cos\left(\frac{3\pi }{8}\right)\approx 0.77$ and $E=1.5)$ for which we calculate various ensemble-averaged quantities in the paper.

Figure 2

Log-log plot of the IPR localization length ${\xi }_{\mathrm{IPR}}$ (a) and the second moment based localization length ${\xi }_{M_2}$ (b) as a function of system size $N$ in the Anderson model in the small disorder regime $\left(W=1\right)$. We have chosen to plot the localization lengths for three representative energy values in the middle of the spectrum, namely $E=0,0.77\text{,}\text{and}1.5$. The error bars are not visible on this scale.

Figure 3

Log-log plot of the IPR localization length ${\xi }_{\mathrm{IPR}}$ (a) and the second moment based localization length ${\xi }_{M_2}$ (b) as a function of system size $N$ in the Anderson model with complete TRS breaking $\left(F=1\right)$. All $k$ states in $[-\pi ,0]$ are projected out. The disorder strength and energy windows are the same as in Fig. 2. It is clear that the localization lengths are independent of energy. The dotted black line with slope 1 is drawn as a guide to the eye, and is the same in both panels. The error bars are not visible on this scale.

Figure 4

A typical wave function at the center of the band $\left(E=0\right)$ of the Anderson model with complete TRS breaking $\left(F=1\right)$ at small disorder $\left(W=1\right)$ for a system with $N=8192$ sites. Wave functions for other values of $E$ (except in the tail of the density of states) are very similar.

Figure 5

Log-log plot of the IPR localization length ${\xi }_{\mathrm{IPR}}$ (a) and the second moment based localization length ${\xi }_{M_2}$ (b) as a function of system size $N$ for the case of partially broken TRS $(F=1/4$, with $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$ removed) at small disorder $\left(W=1\right)$. The three chosen energies plotted correspond to different regimes of behavior, as described in the text. The dotted black line with slope 1 is drawn as a guide to the eye, and is the same in both panels.

Figure 6

Typical wave functions at three characteristic energies in the Anderson model with partially broken TRS $(F=1/4$, with $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$ removed) at small disorder $\left(W=1\right)$ for a system with $N=8192$ sites. States at the center of the band (top) are clearly extended, with all sites having nearly equal amplitude. States at $E=1.50$ (bottom) seem to have a single localizedlike peak with a sinusoidal background (see text for discussion). States at $E_c=0.77$ (middle) seem to have fluctuations over several orders of magnitude, suggestive of critical states.

Figure 7

An Anderson localized wave function at the same disorder strength $\left(W=1\right)$ as Fig. 6 at $N=8192$ sites. The wave function has one central peak and is exponentially decaying away from it. The probability density $|{\psi }_m{|}^2$ drops by approximately 18 orders of magnitude over a span of nearly 1600 sites. An envelope of the functional form $e^{-2x/\xi }$, with this decay rate has $\xi \approx 75$, consistent with the values tabulated in Table 3.

Figure 8

Energy-resolved value of the ensemble averaged overlap $\langle v\rangle$ (see text for definition) between the truncated eigenstates and the Anderson localized eigenstates for the case $F=1$ at $W=1$ (small disorder) for four different system sizes. Here TRS is completely broken (all $k\in [-\pi ,0]$ are projected out). Red, blue, and green dots identify the energies $E=0$, 0.77, and $1.5$, respectively. In the inset, we show a log-log plot of how $\langle v\rangle$ changes as a function of system size $N$. Black dashed lines indicate fits proportional to $lnN/N$.

Figure 9

Energy-resolved value of the ensemble averaged overlap $\langle v\rangle$ between the truncated eigenstates and the Anderson localized eigenstates for the case $F=1/4$ at $W=1$ (small disorder) for four different system sizes. Here, TRS is partially broken by projecting out all $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$. Red, blue, and green dots identify the energies $E=0$, 0.77, and 1.5, respectively. In the inset, we show a log-log plot of how $\langle v\rangle$ changes as a function of system size $N$. At $E=0$ (red line), $\langle v\rangle$ is best fit by a function proportional to $lnN/N$. At the cutoff energy $E=E_c=0.77$ (blue line), we use a power law fit $\langle v\rangle \sim N^{-0.18}$.

Figure 10

(a) A typical quantum Hall critical state. The state is generated by exactly diagonalizing the Hamiltonian of the lowest Landau level with Gaussian white noise disorder on a square torus with $N_{\varphi }=4096$ flux quanta. Each side of the torus is $160.4l_B$, where $l_B$ is the magnetic length. The vertical axis is on a linear scale. (b) The wave function of the middle panel of Fig. 6, plotted on a linear scale for comparison.

Figure 11

Energy-resolved value of the anomalous multifractal exponent ${\mathrm{\Delta }}_{\frac{1}{2}}$ for the three Hamiltonians considered in this paper at small disorder $\left(W=1\right)$. For reference, the known values of ${\mathrm{\Delta }}_{\frac{1}{2}}$ for an extended metallic wave function, localized insulating wave function, Quantum Hall critical state (purple), and a critical state at the 3D Anderson metal-insulator transition (brown) from Table 4 are also shown.

Figure 12

Mean current as a function of energy for the three different kinds of Hamiltonians at $N=8192$ sites in the small disorder regime $\left(W=1\right)$. The black dashed curve shows the current for the zero-disorder case $J\left(E\right)=\sqrt{1-\frac{E^2}{4}}$.

Figure 13

Mean eigenvalue spacing ratio $\langle r\rangle$ as a function of energy for the three different kinds of Hamiltonians at $N=8192$ sites at small disorder $\left(W=1\right)$. In the upper panels, we plot the distributions $P\left(s\right)$ of scaled eigenvalue spacings.

Figure 14

Log-log plot of the IPR localization length ${\xi }_{\mathrm{IPR}}$ (a) and the second moment based localization length ${\xi }_{M_2}$ (b) as a function of system size $N$ in the Anderson model with partial TRS breaking $\left(F=1/4\right)$ at large disorder $\left(W=16\right)$. All $k$ states in $[-\frac{5\pi }{8},-\frac{3\pi }{8}]$ are projected out. The dotted black line is the same in both panels and has slope 1. The error bars are not visible on this scale.

Figure 15

Energy resolved value of the anomalous multifractal exponent ${\mathrm{\Delta }}_{\frac{1}{2}}$ for the three Hamiltonians considered in this paper at large disorder $\left(W=16\right)$. For reference, the known values of ${\mathrm{\Delta }}_{\frac{1}{2}}$ for an extended metallic wave function, localized insulating wave function, Quantum Hall critical state (purple), and a critical state at the 3D Anderson metal-insulator transition (brown) from Table 4 are also shown.

Figure 16

Mean eigenvalue spacing ratio $\langle r\rangle$ as a function of energy for the three different kinds of Hamiltonians at $N=8192$ sites at large disorder $\left(W=16\right)$. In the upper panels, we plot the distributions $P\left(s\right)$ of scaled eigenvalue spacings.

Figure 17

Multifractal spectrum $f\left(\alpha \right)$ for wave functions in the Anderson model at small disorder $\left(W=1\right)$ at fixed energy $E=0.77$. The strong finite-size effects and lack of universality are characteristic of exponentially localized wave functions as discussed in the text.

Figure 18

Multifractal spectrum $f\left(\alpha \right)$ for wave functions in the case of complete TRS breaking $(F=1$, with $k\in [-\pi ,0]$ removed) at small disorder $\left(W=1\right)$. The energy is fixed at $E=0.77$. Note the scale is different than that for Fig. 17.

Figure 19

Multifractal spectrum $f\left(\alpha \right)$ for the case of partially broken TRS $(F=1/4$, with $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$ removed) at small disorder $\left(W=1\right)$. We show four different system sizes at each of three representative energies: $E=0$ (top), $E=E_c=0.77$ (middle), and $E=1.5$ (bottom). The scales for each of these figures are the same. On this scale, $f\left(\alpha \right)$ for states in the center of the band reduces to a single point (top). In the inset of the top panel, we show a magnified version of $f\left(\alpha \right)$, showing it is tightly concentrated around (1,1).

Figure 20

(a) Multifractal spectrum $f\left(\alpha \right)$ for the case of partially broken TRS $(F=1/4$, with $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$ removed) at large disorder $\left(W=16\right)$ at fixed energy $E=0.77$. (b) A typical wave function at $E=0.77$. The inset shows the same wave function on a log scale. Similar behavior is seen at other energies as well as other values of $F$.

Figure 21

Probability distribution by decade of the site probability $P\left({log}_{10}\right|\psi {|}^2)$ for the case of partially broken TRS $(F=1/4$, with $k\in [-\frac{5\pi }{8},-\frac{3\pi }{8}]$ removed) at small disorder $\left(W=1\right)$. We show two different system sizes at each of three representative energies: $E=0,E=E_c=0.77$ and $E=1.5$. The inset shows the same probability distribution, plotted as a function of the system-size independent variable $\alpha$.

Figure 22

Probability distribution by decade of the site probability $P\left({log}_{10}\right|\psi {|}^2)$ for the case of noninteracting electrons at the center of the band in the lowest Landau level with Gaussian white noise disorder. We obtain eigenstates closest to $E=0$ on a square torus with three different values of $N_{\varphi }$ (number of flux quanta). To obtain this plot, we then divide the torus into plaquettes of $\frac{l_B}{3}\times \frac{l_B}{3}$ and bin the values of ${\left|\psi \left(\mathbf{r}\right)\right|}^2$ integrated over each plaquette over an ensemble of several disorder realizations. The inset shows the scaling collapse of the the three distributions when plotted against the variable $\alpha$ [Eq. (A6)].

Figure 23

Comparison of ensemble averaged absolute values of Hamiltonian matrix elements. At small disorder $\left(W=1\right)$, the hopping term (blue) dominates the disorder term (green). At large disorder, the disorder term (yellow) dominates. The black lines are the approximations (B6) and (B7). Data are cut off at half the chain length due to periodic boundary conditions.