Quantum disorder and local modes of the fully-frustrated transverse field Ising model on a diamond chain

We investigate the transverse field Ising model on a diamond chain using series expansions about the high-field limit and exact diagonalizations. For the unfrustrated case we accurately determine the quantum critical point and its expected 2d Ising universality separating the polarized and the ${\mathcal{Z}}_2$ symmetry broken phase. In contrast, we find strong evidence for a disorder by disorder scenario for the fully-frustrated transverse field Ising model, i.e., except for the pure Ising model, having an extensive number of ground states, the system is always in a quantum disordered polarized phase. The low-energy excitations in this polarized phase are understood in terms of exact local modes of the model. Furthermore, an effective low-energy description for an infinitesimal transverse field allows us to pinpoint the quantum disordered nature of the ground state via mapping to an effective transverse field Ising chain and to determine the induced gap to the elementary effective domain wall excitation very accurately.

Figure 1

Illustration of the TFIM on a diamond chain. If all Ising couplings $J_{ij}=-J$ are equal, the system is unfrustrated (upper panel). The system is highly frustrated if the Ising coupling on the thick red bonds for each plaquette $p$ has a different sign compared to the other three bonds in the same plaquette (lower panel). In both cases one can use the three-site unit cell with sites 1, 2, and 3 illustrated as a black box.

Figure 2

Illustration of the local modes of the (a) unfrustrated and (b), (c) frustrated diamond chain. Thick red (thin gray) bonds denote an Ising interaction $+J$ ($-J$). The number in parentheses corresponds for each site $(\nu ,\alpha )$ to the amplitude of the single-particle state $|\nu ,\alpha \rangle$ in Eqs. (8) and (10) omitting the normalization constant.

Figure 3

One-particle gap ${\Delta }_{\mathrm{uf}}/2h$ as a function of $x$ for the unfrustrated TFIM. Lines correspond to DlogPadé extrapolants while symbols represent exact diagonalization data of different finite-size samples with $N$ sites. Inset: Zoom on the critical region.

Figure 4

One-particle gap ${\Delta }_{\mathrm{ff}}/2h$ as a function of $u$. Lines correspond to DlogPadé extrapolants in $u=x/(x+1)$, while symbols represent exact diagonalization data for different finite-size samples. Inset: Limiting value $\Delta (u=1)/2h$ from ED as a function of $1/N$.

Figure 5

Illustration of the low-energy description in the low-field limit: (a) Frustrated bonds live on the squares which build the effective dual lattice of bonds. The black rectangle represents the unit cell containing the four squares denoted by 1, 2, 3, and 4. (b) Processes of the Hamiltonian $H_{◊}$ which take place inside a unit cell. Black squares (white squares) correspond to the presence (absence) of a frustrated bond. (c) Local spectrum of $H_{◊}$ with eigenstates ${|s\rangle }_{\pm }$ and ${|a\rangle }_{\pm }$. (d) Processes of the Hamiltonian $H_{\times }$ acting on nearest-neighbor unit cells such that the number of frustrated bonds equals one in each unit cell.