Lattice effects on Laughlin wave functions and parent Hamiltonians

We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\nu =1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\mu$ particles per lattice site, where $\mu$ may be different than $\nu$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of $q$ and for different fillings $\mu$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\le 4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\nu =1/q$, while for larger $q$ there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor $1/q$. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\mu \le 1/q$ or $\mu \ge 1-1/q$ and when $q=4$ also at half filling.

Figure 1

Diagram of the ${\psi }_q^{\eta }$ states for the values of $q$ and $\eta$ considered in this work. $q$ is integer while $\eta$ is a rational number in the interval $[0,q]$, since the states for $\eta >q$ (gray region) are not defined. When $\eta \to 0$ or $\eta \to q$ (red lines), the state is a ferromagnet if $N$ is fixed, or a Laughlin state in the continuum if $N$ goes to infinity and $N\propto 1/\eta$ (if $\eta \to 0$) or $N\propto 1/(q-\eta )$ (if $\eta \to q$). The solid blue disks on the line $\eta =1$ correspond to lattice Laughlin states that satisfy $\mu =\nu =1/q$. The green disk is an integer quantum Hall state. On the orange line ($q$ not necessarily integer), the state in 1D is critical, while on the dotted orange line it has long-range order.

Figure 2

The mapping from a lattice on the complex plane to a cylinder. For the square lattice on the cylinder, the coordinates are $z_j=exp\left\{2\pi [(x_j-L_x/2+1/2)/L_y+(y_j-L_y/2+1/2)i/L_y]\right\},x_j\in \{0,...,L_x-1\},y_j\in \{0,...,L_y-1\}$. To compute the entanglement entropy of the state, the cylinder is cut into two halves and the Renyi entropy of the first half is computed using a Metropolis-Hastings algorithm. The topological entanglement entropy is extracted by varying the size $L_y$ of the cylinder.

Figure 3

Renyi entropy $S_A^{\left(2\right)}$ of the first half of the cylinder for the states ${\psi }_4^{\eta }$ and ${\psi }_4^{4-\eta }$ on a square $12\times 12$ lattice. The symmetry axis is shown as a red dashed line. The states at $\eta =1$ and $\eta =3$ have lattice filling factors $1/4$ and $3/4$ and correspond to the lattice versions of the $\nu =1/4$ and $\nu =3/4$ Laughlin states. The state at $\eta =2$ is particle-hole symmetric on a half-filled lattice. The errors estimated from the variation to the mean from Monte Carlo simulations with different initial conditions are below 0.01 for all points.

Figure 4

Characteristic length $d_q^{\eta }$ [Eq. (12)] estimating the correlation length of the ${\psi }_q^{\eta }$ states. The characteristic length is computed for $q$ integer, $\eta$ integer, and half integer using a Metropolis-Hastings algorithm on a square $12\times 12$ (or $10\times 12$ for $q=5$) lattice on the cylinder. The decay is found to be exponential everywhere, except when $\eta =q/2,q\ge 5$.

Figure 5

(a) Correlation function $C_{ij}$ between a fixed lattice site $i$ in the middle of the lattice and all other lattice sites $j$ in the state ${\psi }_q^{q/2}$ on a square $8\times 8$ lattice on the cylinder. (b) Absolute value of the correlation function $|C_{i(i+\mathrm{\Delta }y)}|$ between two lattice sites separated by a distance $\mathrm{\Delta }y$ in the periodic direction in the state ${\psi }_q^{q/2}$ on a square $16\times 16$ lattice on the cylinder. There is long-range antiferromagnetic order for $q\ge 6$, while the correlations decay exponentially for $q\le 4$.

Figure 6

(a) Scaling of the Renyi entropy $S_{L_y}^{\left(2\right)}$ of the state ${\psi }_4^{\eta }$ on an $L_x\times L_y$ square lattice on the cylinder (Fig. 2). The topological entanglement entropy of the Laughlin state at filling $1/4 [{\gamma }_0\left(4\right)=ln\left(2\right)\approx 0.693]$ is indicated with a red arrow. The size $L_x$ is taken to be 12, unless $\eta =2$ in which case $L_x$ is 20. Larger sizes are taken when $\eta =q/2$, to account for the longer correlation length and to get rid of finite-size effects. The black lines are linear fits and the values found for the topological entanglement entropy when $\eta$ equals to $0.5,1,1.5$, and 2 are respectively $0.698,0.734,0.643$, and 0.718. (b) Scaling of the Renyi entropy $S_{L_y}^{\left(2\right)}$ for the states at half filling ${\psi }_2^1$ and ${\psi }_3^{3/2}$ on the square lattice on the cylinder. Here $L_x$ is 12 for $q=2$ and 20 for $q=3$. The topological entanglement entropy of the Laughlin state at filling $1/2,{\gamma }_0\left(2\right)\approx 0.346$ [resp. at filling $1/3,{\gamma }_0\left(3\right)\approx 0.549]$ is indicated with a green (resp. blue) arrow. The values found for the topological entanglement entropy are respectively 0.375 and 0.536.

Figure 7

Difference in the density ${\langle n_i\rangle }_{qh}-\langle n_i\rangle$ between the state ${\psi }_q^{\eta }$ with one quasihole of charge $1/q$ and one quasihole of charge $(q-1)/q$ and the state ${\psi }_q^{\eta }$ without quasiholes on a square $20\times 20$ (or $12\times 12$ for $q=10,\eta =5$) lattice on the cylinder. The coordinates $w_j$ of the quasiholes are placed in the center between 4 lattice sites $[x_j=10.5,y_j=5.5$ for the quasihole with charge $(q-1)/q$ and $x_j=10.5,y_j=15.5$ for the quasihole with charge $1/q]$ and are visible in blue as a lack of density on the neighboring sites. At $q=4$, it is found for all values of $\eta$ that the quasiholes are localized. For $q=10$ however, the quasiholes are localized when $\eta =1$, but at half filling ($\eta =5$) we observe that there is no splitting of the charge between a quasihole of charge $1/q$ and a quasihole of charge $(q-1)/q$ and thus no screening of the quasiholes.

Figure 8

Overlap, defined as $O_q\equiv |\langle {\psi }_{\text{Néel}}^1|{\psi }_q^{q/2}\rangle {|}^2+{\left|\langle {\psi }_{\text{Néel}}^2|{\psi }_q^{q/2}\rangle \right|}^2$, between the state at half filling ${\psi }_q^{q/2}$ and the two Néel states, computed using a Metropolis-Hastings algorithm on a square lattice of size $N_x\times N_x$ on the cylinder. The errors from the Monte Carlo simulations are below 0.02 for all points.

Figure 9

(a) Correlation function $C_{ij}$ between a fixed lattice site $i$ in the middle of the lattice and all other lattice sites $j$ in the state ${\psi }_q^{q/2}$ on a triangular and Kagome lattice on the cylinder. (b) Absolute value of the correlation function $|C_{i(i+\mathrm{\Delta }y)}|$ between two lattice sites separated by a distance $\mathrm{\Delta }y$ in the periodic direction in the state ${\psi }_q^{q/2}$ on a triangular $16\times 16$ lattice on the cylinder. Unlike on the square lattice for the same parameters, there is no long-range order.

Figure 10

Positions of the coordinates of the lattice sites along the interpolation between a square and a triangular lattice. The coordinates along the periodic direction $y_j$ are kept fixed to keep the periodicity, while the coordinates along the other direction $x_j$ are linearly interpolated between the square ($\tau =0$) and the triangular ($\tau =1$) lattice. The coordinates on the plane are then $z_j=e^{\frac{2\pi }{L_y}(x_j+i{y}_j)}$, as in Fig. 2.

Figure 11

Absolute value of the correlation function $|C_{i(i+\mathrm{\Delta }y)}|$ between two lattice sites separated by a distance $\mathrm{\Delta }y$ in the periodic direction in the state ${\psi }_6^3$ on a $16\times 16$ lattice interpolating between the square ($\tau =0$) and the triangular ($\tau =1$) lattice on the cylinder.

Figure 12

Scaling of the Renyi entropy $S_{L_y}^{\left(2\right)}$ for the states at half filling ${\psi }_2^1,{\psi }_4^2$, and ${\psi }_6^2$ on a triangular lattice on the cylinder. Here $L_x$ is taken to be 16. The topological entanglement entropy of the continuum Laughlin state at $q=2$ (resp. $q=4,q=6$) is indicated with a blue (resp. red, green) arrow. The values found for the topological entanglement entropy are respectively 0.347, 0.723, and 0.907.

Figure 13

(a) Square $12\times 12$ lattice on the cylinder with a charge (here $p=-6$) at infinity. The colors show the density of the state ${\psi }_2^1{\left[p\right]}_{\infty }$. The state with zero charge at infinity ${\psi }_2^1$ has uniform density $1/2$ but here the density is modified at the edge to account for the change of total particle number. (b) Difference of density between the ${\psi }_2^1{\left[p\right]}_{\infty }$ states and the ${\psi }_2^1$ state with respect to the distance to the edge $\mathrm{\Delta }x$. Values below $2\times {10}^{-4}$ are not converged. The change of density is exponentially localized at the edge.

Figure 14

Absolute value of the correlation function $|C_{i(i+\mathrm{\Delta }y)}|$ between two lattice sites in the bulk (middle of the cylinder) separated by a distance $\mathrm{\Delta }y$ in the periodic direction on a square $12\times 12$ lattice on the cylinder. The states considered here are the ${\psi }_q^1{\left[p\right]}_{\infty }$ state for different values of $p$. When $p=0$, the state is simply the ${\psi }_2^1$ state, while it is found that the other states have the same correlations.

Figure 15

Diagram of the ${\psi }_q^{\eta }$ states. The blue lines and dots represent values of the parameters for which exact parent Hamiltonians are derived in this section. The light blue line represents values of the parameters for which the parent Hamiltonians derived have a degenerate ground state on the plane.