Emergent spin-1 trimerized valence bond crystal in the spin-$\frac{1}{2}$ Heisenberg model on the star lattice

We explore the frustrated spin-$\frac{1}{2}$ Heisenberg model on the star lattice with antiferromagnetic (AF) couplings inside each triangle and ferromagnetic (FM) intertriangle couplings ($J_e<0$), and calculate its magnetic and thermodynamic properties. We show that the FM couplings do not sabotage the magnetic disordering of the ground state due to the frustration from the AF interactions inside each triangle, but trigger a fully gapped inversion-symmetry-breaking trimerized valence bond crystal (TVBC) with emergent spin-1 degrees of freedom. We discover that with strengthening $J_e$, the system exhibits a universal scaling behavior either with or without a magnetic field $h$: the order parameter, the five critical fields that separate the $J_e\text{−}h$ ground-state phase diagram into six phases, and the excitation gap obtained by low-temperature specific heat, all depend exponentially on $J_e$. Our work implies that the spin-1 VBCs can be stabilized by introducing small FM couplings in the geometrically frustrated spin-$\frac{1}{2}$ systems.

Figure 1

The ground-state phase diagram of the star Heisenberg model. For $J_e>0$, previous studies show various possible VBCs and spin liquids, where one recent work found a $\sqrt{3}\times \sqrt{3}$ VBC and a $J_e$-bond VBC [32, 33]. The phase boundary $J_e^c$ has not been settled yet. For $J_e<0$, we show that the system is in a trimerized valence bond crystal (TVBC) phase, where a triplet appears at each $J_e$ bond and the inversion symmetry of up and down triangles (marked by blue and yellow, respectively) is broken.

Figure 2

(a) The ground-state energy $E_0$ of the star Heisenberg model for $J_e=-1$ and $J_t=1$ obtained by the plain and SU(2) simple update algorithm, where $E_0$ converges versus number of states ($D$) and number of multiplets ($D^{*}$, red solid dots), respectively, with clearly superior performance of the SU(2)-based calculations. For comparison only, we also translate the number of multiplets $D^{*}$ into the corresponding actual number of states $D$ (black lines and symbols). Taking a fixed $D^{*}$, $D$ of different virtual bonds may vary, according to the SU(2) fusion rules and the specific set of multiplets associated with each bond. We show the minimum, maximum, and average values of $D$ over the three virtual bond indices of a tensor. (b) $E_0$ versus $J_e$, obtained by SU(2) simple update and DMRG simulations, which show very good agreement with each other in the whole parameter range. The inset sketches the local tensors of the TN state.

Figure 3

The bond correlation $\langle S_i{S}_j\rangle$ versus distance $x$ from the pinned boundary (using $J_{\mathrm{pin}}=2$) on a YC4-36 cylinder for $J_e=-4$ calculated by DMRG keeping 2000 SU(2) multiplets. The thickness indicates the strength of the bond energies.

Figure 4

(a) Log-linear plot of the inversion-symmetry-breaking parameter $\delta$ as a function of the distance $x$ from the boundary for the YC4-36 cylinder with boundary pinning. (b) The $J_e$ dependence of $\delta$ obtained from SU(2) simple update simulations. As long as $J_e<0$, the system has a nonzero $\delta$. We find by fitting that for approximately $-J_e\gg 4$, $\delta$ fulfills the relation $\delta =\tilde{\delta }(1-e^{\mu J_e})$, where $\mu =0.28$ and $\tilde{\delta }=0.1$ that gives the value of $\delta$ for $J_e\to -\infty$. In contrast, for small $|J_e|$, we find $\delta =0.03J_e^2$ for $J_e\to 0$ (see inset).

Figure 5

The field dependence of (a) the magnetization $M_z$ and (b) the TVBC order parameter $\delta$. Five critical fields $h_{ci}$ ($i=1,2,...,5$) is determined by $M_z$ and $\delta$, which determine six phases in the $J_e\text{−}h$ diagram as shown in (c). For $0\le h<h_{c1}$, $M_z=0$ and $\delta$ is intact. For $h_{c1}<h<h_{c2}$, $M_z$ increases, and $\delta$ starts to diminish and vanishes at $h=h_{c2}$, where the inversion symmetric and symmetry-breaking phases are separated and one always has $M_z=1/30$. For $h_{c3}<h<h_{c4}$, the system is in a conventional $\frac{1}{3}$-plateau solid phase. Here, we use the simple update algorithm of PEPS with $D\sim 30$.

Figure 6

(a) The temperature dependence of specific heat $C$ for various $J_e$, where multipeak structures are observed. We use the NCD algorithm with $D=12$. The position of the low-temperature peak moves to the higher temperature as $-J_e$ increases. Inset: the curve of $lnC$ versus inverse temperature $1/T$. Below the low-temperature peak, one can see that the specific heat decays exponentially with $1/T$ as $lnC=-\mathrm{\Delta }/T+\mathrm{const}$, where $\mathrm{\Delta }$ is the ($J_e$-dependent) excitation gap. (b) By fitting specific heat, the excitation gap $\mathrm{\Delta }$ for different $J_e$ is obtained and shown to fulfill the relation $\mathrm{\Delta }=\tilde{\mathrm{\Delta }}(1-e^{\kappa J_e})$. The error bars are given by the linearity of $lnC$ at the low temperatures [50]. By fitting the $\mathrm{\Delta }-J_e$ curve, we have $\tilde{\mathrm{\Delta }}=0.175$ that gives the excitation gap in the $J_e\to -\infty$ limit and the constant $\kappa =0.5$.