Disorder by disorder and flat bands in the kagome transverse field Ising model

We study the transverse field Ising model on a kagome and a triangular lattice using high-order series expansions about the high-field limit. For the triangular lattice our results confirm a second-order quantum phase transition in the $3d$ $XY$ universality class. Our findings for the kagome lattice indicate a notable instance of a disorder by disorder scenario in two dimensions. The latter follows from a combined analysis of the elementary gap in the high- and low-field limit which is shown to stay finite for all fields $h$. Furthermore, the lowest one-particle dispersion for the kagome lattice is extremely flat acquiring a dispersion only from order 8 in the $1/h$ limit. This behavior can be traced back to the existence of local modes and their breakdown which is understood intuitively via the linked cluster expansion.

Figure 1

(a) Countour plot of the one-particle dispersion ${\omega }^{\mathrm{tr}}\left(\vec{k}\right)$ as a function of the momentum $\vec{k}=(k_1,k_2)$ on the triangular lattice for $x=0.1$. The red (solid) line corresponds to the first Brillouin zone. (b) Illustration of the complex phase of the eigenstate corresponding to the one-particle gap at ${\vec{k}}_{\min }=(\frac{2}{3}\pi ,-\frac{2}{3}\pi )$. The state exhibits a $\sqrt{3}\times \sqrt{3}$ structure. Note that the structure for the second gap momentum ${\vec{k}}_{\min }=-(\frac{2}{3}\pi ,-\frac{2}{3}\pi )$ is also a $\sqrt{3}\times \sqrt{3}$ structure with a different orientation. We stress that the displayed arrows do not represent spins.

Figure 2

(a) Bare series and $d$log Padé extrapolants of the one-particle gap ${\Delta }^{\mathrm{tr}}$ as a function of $x$ for the triangular TFIM. Inset represents a zoom close to the quantum critical point. (b) Illustration of the convergence behavior of the critical values $x_{\mathrm{c}}$ resulting from the $d$log Padé extrapolants $[L/M]$ in dependence of the considered order $r=L+M+1$. Connected points of the same color correspond to extrapolants with constant $L-M$. The dashed line represents the quantum Monte Carlo result $x_{\mathrm{c}}^{\left(\mathrm{qmc}\right)}=0.303$ (Ref. 9). The horizontal dotted lines indicate the corresponding margin of error $\pm 0.009$. (c) Critical exponent $\nu$ extracted from $d$log Padé extrapolants shown in (b) as a function of $r$. Dashed horizontal line indicates the critical exponent ${\nu }^{3dXY}\approx 0.67$ of the $3d$ $XY$ universality class.

Figure 3

Lowest one-particle energy band ${\omega }_{\mathrm{low}}^{\left(13\right)}\left(\vec{k}\right)$ as a function of $\vec{k}=(k_1,k_2)$ for the kagome TFIM at $x=0.1$. Red solid line indicates the first Brillouin zone. Note the extreme smallness of the bandwidth.

Figure 4

(a) The kagome lattice has a three-site unit cell. We define unit vectors ${\vec{e}}_1$ and ${\vec{e}}_2$ as shown on the right-hand side of the upper panel. The position of a unit cell is then given by $(n_1,n_2)$ with $n_i\in \mathcal{Z}$. The three sites of the unit cell are denoted by $\{1,2,3\}$. (b) Illustration of the eigenstate corresponding to the one-particle gap. Arrows denote the complex phase of this eigenvector. The pattern realizes a $\sqrt{3}\times \sqrt{3}$ structure. Note that the structure for the second gap momentum ${\vec{k}}_{\min }=-(\frac{2}{3}\pi ,-\frac{2}{3}\pi )$ is also a $\sqrt{3}\times \sqrt{3}$ structure with a different orientation. We stress that the displayed arrows do not represent spins.

Figure 5

The finite-size samples of the kagome lattice with periodic boundary conditions which are used for the first-order degenerate perturbation theory in the low-field limit. Here $N$ denotes the total number of spins. The finite-size samples are built by the thick black triangles labeled by an integer number. Thin triangles are displayed to visualize periodic boundary conditions.

Figure 6

One-particle gap $\Delta /2h$ as a function of $u$. Lines correspond to $d$log Padé extrapolants in $u=\frac{x}{x+1}$ while symbols represent the first-order degenerate perturbation theory results of the low-field limit for the different finite-size samples. Left inset: Zoom around $u=1$. Right inset: First-order degenerate perturbation theory results of the one-particle gap against $1/N$. Solid black line represents a linear fit of the data.

Figure 7

Biased $d$log Padé extrapolations of the one-particle gap as a function of $u$. The extrapolants are biased to $\Delta (u=1)/2h=0.18$ obtained from the $1/N$ scaling in the low-field limit.

Figure 8

(a) Localized mode $|{\psi }_{\mathrm{loc}}\rangle$ on a hexagon of the kagome lattice. The notion $\pm$ denotes the sign in front of the one-particle states contained in $|{\psi }_{\mathrm{loc}}\rangle$. (b) Example of a destructive interference between the two virtual hopping processes $\left(1\right)$ (thick dashed bonds) and $\left(2\right)$ (thick solid bonds) exhibiting the same absolute value $t_{ij}^{\left(1\right)}=t_{i^{\prime }j}^{\left(2\right)}$ of the hopping amplitude. The initial states are given by the one-particle states $|i\rangle$ and $|i^{\prime }\rangle$ while the final state of both processes is given by $|j\rangle$. Due to the opposite prefactor of $|i\rangle$ and $|i^{\prime }\rangle$ in $|{\psi }_{\mathrm{loc}}\rangle$, both processes interfere destructively. In both figures filled dark circles correspond to the six sites $p^{*}$ surrounding the central hexagon.

Figure 9

(a) Left figure illustrates a hopping process $t_{ij}^{C_{\mathrm{sub}}}$ on a subcluster $C_{\mathrm{sub}}$ visualized as thick solid bonds. The right part represents the counterprocess $t_{i^{\prime }j}^{{\tilde{C}}_{\mathrm{sub}}}$ on ${\tilde{C}}_{\mathrm{sub}}$. The dashed vertical line is the symmetry axis through site $p^{*}$ when acting on $|{\psi }_{\mathrm{loc}}\rangle$. (b) Example of a hopping process that is not annihilated by a counterprocess. Any countercluster has a different topology and therefore a different contribution. This leads to a finite hopping amplitude of a particle leaving the hexagon. In all figures filled dark circles correspond to the six sites $p^{*}$ surrounding the central hexagon.

Figure 10

(a) Illustration of the only looped process in order $n=6$ as thick bonds. The particle in the initial and final state is located at the same site ($|i\rangle =|j\rangle$). Hence, this process does not contribute to the dispersion. (b) Hopping process contributing for orders $n\ge 7$. The corresponding counterprocess is represented by thick blue dashed bonds. (c) Hopping process contributing for orders $n\ge 8$. For this case any countercluster has a different topology leading to the breakdown of the local mode $|{\psi }_{\mathrm{loc}}\rangle$. In all figures filled dark circles correspond to the six sites $p^{*}$ surrounding the central hexagon.

Figure 11

Localized mode $|{\psi }_{\mathrm{loc}}^{\mathrm{sq}}\rangle =\frac{1}{2}\left(\right|1\rangle -|2\rangle +|3\rangle -|4\rangle )$ on the square plaquette of the checkerboard lattice. The displayed hopping process from site 1 to site $j$ involving five thick solid bonds leads to a dispersion as it closes a loop between the two different sites $p_{41}^{*}$ and $p_{12}^{*}$.