Unconventional orders in the maple-leaf ferro-antiferromagnetic Heisenberg model

TL;DR

This work analyzes the spin-$\tfrac{1}{2}$ Heisenberg model on the maple-leaf lattice with competing ferromagnetic and antiferromagnetic couplings to reveal a rich landscape of unconventional quantum phases. By combining unconstrained Luttinger-Tisza theory, cluster mean-field theory, and pseudo-fermion functional renormalization group methods, the authors map a phase diagram featuring conventional Néel, FM, and $c120^\circ$ orders embedded in a broad paramagnetic region. In this PM regime, they identify an extended hexagonal-singlet phase, a smaller dimerized hexagonal-singlet phase near the $c120^\circ$ boundary, and additional nonmagnetic correlations with potential quantum spin liquid behavior, plus a spin-nematic tendency near the FM boundary. The results highlight the potential for nontrivial quantum states in two-dimensional frustrated magnets and guide future variational studies and experimental explorations in maple-leaf materials.

Abstract

Motivated by the search for unconventional orders in frustrated quantum magnets, we present a multi-method investigation into the nature of the quantum phase diagram of the spin-$1/2$ Heisenberg model on the maple-leaf lattice with three symmetry-inequivalent nearest-neighbor interactions. It has been argued that the parameter regime with antiferromagnetic couplings on hexagons $J_h$ and ferromagnetic couplings on triangles $J_t$ and dimer $J_d$ bonds, is potentially host to a cornucopia of emergent phases with unconventional orders. Our analysis indeed identifies an extended region where any conventional dipolar magnetic order is absent. A hexagonal singlet state is found in the region around $J_{d}=J_{t}=0$, while a dimerized hexagonal singlet order of a lattice nematic character appears proximate to the phase boundary with the c$120^\circ$ antiferromagnetic order. Interestingly, upon traversing the bulk of the paramagnetic (PM) region, we find a variety of distinct correlation profiles, which are qualitatively different from those of the hexagonal singlet and dimerized hexagonal singlet orders but feature no appreciable spin-nematic response, while the boundary with the ferromagnetic phase shows evidence of spin-nematic order. This PM region is thus likely host to an ensemble of nonmagnetic phases which could putatively include quantum spin liquids. Our phase diagram is built from a complementary application of state-of-the-art implementations of the cluster mean-field and pseudo-fermion functional renormalization group approaches, together with an unconstrained Luttinger-Tisza treatment of the model providing insights from the semi-classical limit.

Figures (10)

• Figure 1: Maple-leaf lattice in real- and momentum-space (a) The maple-leaf lattice, with the three symmetry-inequivalent nearest-neighbor couplings $J_t$, $J_d$, and $J_h$ highlighted in different colors. In this work, we consider $J_h > 0$ and $J_d, J_t \leq 0$. (b) The dashed line indicate the Brillouin zone of the maple-leaf lattice, while the solid line represents the extended Brillouin zone corresponding to the first Brillouin zone of the triangular lattice, which becomes the maple-leaf lattice upon 1/7 depletion. The extended Brillouin zone can also be obtained by scaling the original Brillouin zone by a factor of $\sqrt{7}$ and rotating it by an angle $\phi=\arccos{\frac{5}{2\sqrt{7}}}$. Dots mark the allowed momenta on the eighteen-site clusters (shown in Fig. \ref{['fig:cmft-phasediagrams']}) we consider in our CMFT analysis. Among those, the five symmetry inequivalent momenta are additionally labeled.
• Figure 2: Schematic phase diagram and representative states summarizing the results from our Luttinger-Tisza, CMFT and pf-FRG analyses. We consider antiferromagnetic $J_h > 0$ and ferromagnetic $J_d, J_t \leq 0$. The phase boundaries (solid lines) are obtained from pf-FRG. For large negative $J_d$ and/or $J_t$, conventionally ordered Néel, FM and c$120^\circ$ phases appear in all methods. The corresponding real-space spin configurations are depicted in panels (b)-(d), where the dashed blue lines indicate possible choices for the six (Néel and FM) and eighteen (c$120^\circ$) site magnetic unit cells. Between these ordered phases lies a broad paramagnetic (PM) regime without conventional dipolar order, whose correlation profile suggests the presence of several different unconventional phases within. Both CMFT and pf-FRG indicate an extended region around $J_d = J_t = 0$ that realizes the hexagonal singlet (HS) state. Additionally, our results support the presence of a smaller phase with dimerized hexagonal singlet (d-HS) correlations near the boundary of the c$120^\circ$ phase, though its precise extent remains unclear. Panels (e) and (f) show the nearest-neighbor correlation of the singlet states, where red implies antiferromagnetic and blue ferromagnetic correlations. At the boundary between the PM regime and the FM phase, pf-FRG shows a strong tendency towards spin-nematic order. Furthermore, within the PM regime, we identify a region (marked in green) showing qualitatively different correlations from the HS and d-HS state and no strong nematic response. This may indicate the presence of an additional, non-magnetic, putative quantum spin liquid (QSL) phase. Note that we are not able to determine precise phase boundaries in the PM regime and have thus indicated the approximate location of different phases by color gradients.
• Figure 3: Classical phase diagram from Luttinger-Tisza. (a) Magnitude $q = |\mathbf{q}^\mathrm{min}|$ of the q-vectors with minimal Luttinger-Tisza eigenvalue as a function of the ferromagnetic couplings $J_t$ and $J_d$ (with $J_h > 0)$. (b) Corresponding $\mathbf{q}^\mathrm{min}$-vectors in the first Brillouin zone for the different phases. In the Néel and the FM phase the minimal momenta is located at $\mathbf{q}^\mathrm{min} = \boldsymbol{\Gamma}$, while in the c$120^\circ$ phase it coincides with $\mathbf{q}^\mathrm{min} = \mathbf{K}$. Between these phases, the $\mathbf{q}^\mathrm{min}$-vectors lie at incommensurate (ICS) momenta, continuously interpolating between the $\boldsymbol{\Gamma}$ and $\mathbf{K}$ points. In all phases except the ICS phase, the hard spin-length constraint is satisfied, allowing real-space spin configurations to be directly constructed from the Luttinger-Tisza eigenvectors. These configurations are depicted in Figs. \ref{['fig:approximate-phasediagram']}(b)-(d). In the ICS phase, only the soft spin constraint is satisfied.
• Figure 4: Quantum phase diagram from CMFT for three different clusters. The top row shows the three eighteen-site clusters used in the analysis. Solid lines represent interactions treated exactly, while dashed lines correspond to interactions approximated via a mean-field decoupling under periodic boundary conditions. The bottom row displays the corresponding phase diagram as function of the ferromagnetic couplings $J_t$ and $J_d$ (with $J_h > 0)$. Dashed lines indicate qualitative changes in the ground state, which may correspond to either crossovers or true phase transitions. The color scale encodes the average local magnetization $|\langle \mathbf{S}_i \rangle|$. Black regions indicate non-magnetic phases where all $|\langle \mathbf{S}_i \rangle| = 0$. We find that the hexagonal singlet (HS) state is realized as an extended phase, for all clusters. Only cluster (b) respects the symmetries required for the dimerized hexagonal singlet (d-HS) state, which indeed appears as a small but distinct region in its corresponding phase diagram. Additionally, we find a paramagnetic (PM) region for cluster (a) that neither resembles the HS nor the d-HS state. Colored regions correspond to magnetic phases, for which we label regions where neither Néel, FM or c$120^\circ$ order is fully realized by the momentum $k=\mathbf{k}^\mathrm{max}$ at which the corresponding structure factor is maximal.
• Figure 5: Observables characterizing the magnetic phases in CMFT. Columns (a)-(c) correspond to clusters shown in Fig. \ref{['fig:cmft-phasediagrams']}. The first three rows show the order-parameters that characterize the FM, Néel and c$120^\circ$ magnetic phases. In regions where none, or multiple of those order-parameters are finite, we characterize the phase by the momentum $k^\mathrm{max}$ at which the structure factor is maximal, shown in the last row. The finite allowed momenta for the eighteen-site clusters and their labels are shown in Fig. \ref{['fig:lattice']}.