One-dimensional physics of the frustrated quantum magnet PHCC

Abstract

We report a comprehensive microscopic study of the frustrated quantum magnet PHCC, (C$_4$H$_{12}$N$_2$)Cu$_2$Cl$_6$, using density-functional band-structure calculations combined with numerical quantum many-body simulations of the underlying spin Hamiltonian. We show that the magnetism of PHCC is captured by a one-dimensional model of the frustrated spin chain with alternating nearest-neighbor couplings ($J_1=23.1$ K, $J_1'=7.0$ K) and uniform next-nearest-neighbor couplings ($J_2=13.9$ K). This model, which can also be thought of as a zigzag ladder, provides a quantitative description of the magnetic susceptibility and the magnetization process, and accounts for the observed dispersion of the single-triplet band and its merging into a continuum near the Brillouin zone center. We also make predictions for the existence of sharp bound (anti-bound) states of two triplets, below (above) the bottom (upper) edge of the two-particle scattering continuum.

Figures (9)

• Figure 1: (a) Crystal structure and (b) magnetic model of PHCC. Piperazinium ions are omitted for clarity. The dotted lines in the left panel show the long Cu--Cl bonds of 2.606 Å that are perpendicular to the CuCl$_4$ plaquettes. (c) The $J_1$--$J_1'$--$J_2$ frustrated zigzag chain with the relevant orbitals visualized using Wannier functions composed of the Cu $d_{x^2-y^2}$ and Cl $3p$ states. The arrows indicate the overlap of the adjacent Cl $3p$ orbitals resulting in the extended Cu--Cl--Cl--Cu superexchange pathways.
• Figure 2: Top: PBE density of states for PHCC, pip stands for the contribution of the atoms forming the piperazinium ion. The Fermi level is at zero energy. Bottom: Dispersion of the Cu $d_{x^2-y^2}$ bands calculated in PBE and its fit with the tight-binding (TB) model. The $k$-path is defined as follows: $\Gamma(0,0,0)$, $X(\frac{1}{2},0,0)$, $M(\frac{1}{2},\frac{1}{2},0)$, $Y(0,\frac{1}{2},0)$, $Z(0,0,\frac{1}{2})$, $T(\frac{1}{2},0,\frac{1}{2})$, $R(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, $A(0,\frac{1}{2},\frac{1}{2})$.
• Figure 3: Left: Fit of the experimental $\chi(T)$ using Eq. \ref{['eq:fit']} with the ED/TMRG-simulated magnetic susceptibility of the $J_1$--$J_1'$--$J_2$ model with $J_1\!=\!23.1$ K, $J_1'\!=\!7.0$ K, $J_2\!=\!13.9$ K, $g\!=\!2.023$, and $\chi_0\!=\!-1.6\times10^{-4}$ emu (mol Cu)$^{-1}$. Right: Fit of the experimental magnetization $M(H\!\parallel\!b)$ from Ref. stone2007 using Eq. (\ref{['eq:mh']}) with the DMRG-simulated magnetization, $g_b\!=\!2.1$, and the exchange parameters from the $\chi(T)$ fit.
• Figure 4: Excitation spectrum of the $J_1$--$J_1'$--$J_2$ model with $J_1:J_1':J_2=1:0.3:0.6$ calculated for the $24$-site chain with periodic boundary conditions. We show 30(1) lowest-lying excitations for $S^z=0,1$ ($S^z=2,3$) sectors. The solid curve is the excitation energy $\omega_{1t}^{(3)}(k)$ of the single-triplet band according to third-order perturbation theory Uhrig1997. The shaded (green) region is the corresponding (non-interacting) two-particle continuum delimited by the boundaries $\omega^{(3)}_{\text{2t,min}}(k) = \min_{q} \{ \omega^{(3)}_{1t}(q) + \omega^{(3)}_{1t}(k-q)\}$ and $\omega^{(3)}_{\text{2t,max}}(k) = \max_{q} \{ \omega^{(3)}_{1t}(q) + \omega^{(3)}_{1t}(k-q) \}$
• Figure 5: (a) Excitation energies of two-particle states of the $J_1$--$J_1'$--$J_2$ model on a system with 24 rungs and parameters $J_1=1$, $J_1'=0.3$ and $J_2=0.6$, as obtained from a numerical diagonalization of the matrix representation of $\mathcal{H}$ inside the two-particle sector [see Eq. (\ref{['eq:Vprojected']})]. The two-particle states with $S=0$, $1$ and $2$ are shown by blue, red and dark yellow symbols, respectively. The solid black lines show the first-order single-triplon energy $\omega_{1t}^{(1)}(k)$ of Eq. (\ref{['eq:w1ofk']}) and the corresponding predictions for the boundaries $\omega^{(1)}_{2\text{t,min}}$ and $\omega^{(1)}_{2\text{t,max}}$, of the non-interacting two-particle continuum. (b) Extrapolated values of the inverse of the coherence length $\langle\hat{d}\rangle$ of the lowest $S\!=\!0$, the lowest $S\!=\!1$ and highest $S\!=\!2$ bands, as obtained from diagonalizations of the Hamiltonian matrix inside the two-particle sector (see App. \ref{['app:PT']}), and using finite-size extrapolations of $1/\langle\hat{d}\rangle$ with system sizes with $N_r\!=\!48, 56, \cdots,160$ rungs.