Semi-Dirac spin liquids and frustrated quantum magnetism on the trellis lattice

Abstract

Geometrical frustration in quantum magnets provides a fertile setting for unconventional phases of matter, including quantum spin liquids (QSLs). The trellis lattice, with its complex site arrangements and edge-sharing triangular motifs, presents a promising platform for such physics. In this work, we undertake a comprehensive classification of all fully symmetric QSLs on the trellis lattice using the projective symmetry group approach within the Abrikosov-fermion representation. We find 7 U(1) and 25 $Z_2$ short-ranged Ansätze and analyze the phase diagram in the mean-field parameter space, uncovering both gapped and Dirac QSLs as well as a semi-Dirac spin liquid that emerges at the level of projective symmetry group classification and mean-field band structure, in which the spinon dispersion is linear along one momentum direction but quadratic along the orthogonal one. We demonstrate that such dispersions can occur only at high-symmetry points in the Brillouin zone with $C_{2v}$ little groups and analyze their characteristic correlation signatures. Moreover, by optimizing over all symmetry-allowed mean-field states, we map out a phase diagram -- featuring six distinct phases -- of the nearest-neighbor Heisenberg Hamiltonian on the trellis lattice. Among these, we find four quasi-one-dimensional QSL phases, one dimer phase, and one Dirac QSL phase. Going beyond mean field, we also assess equal-time and dynamical spin structure factors of these phases using density-matrix renormalization group and Keldysh pseudofermion functional renormalization group calculations and compare qualitative momentum-space features of these spectra with those obtained at the mean-field level. Finally, we identify four cuprate and vanadate compounds as promising experimental realizations and provide spectroscopic predictions, based on first-principles Hamiltonians, as a guide for neutron-scattering studies.

Figures (16)

• Figure 1: (a) Trellis lattice with exchange couplings $J_v$ (green), $J_z$ (black), and $J_h$ (blue). The unit cell contains two sites labeled "1" and "2." The lattice parameters are denoted by $a$, $b$, and $c$. (b) Illustration of the lattice space-group symmetries. $C_2$ represents a twofold rotation about an axis perpendicular to the lattice plane. $\sigma_x$ and $\sigma_y$ denote reflections about the horizontal and vertical solid lines, respectively. $G_x$ is a glide symmetry consisting of a reflection about the horizontal dashed line followed by a translation $\frac{a}{2}\hat{\boldsymbol{X}} = \frac{1}{2}\boldsymbol{T}_1$. Similarly, $G_y$ combines a reflection along the vertical dashed line with a translation $(b+c)\hat{\boldsymbol{Y}} = -\frac{1}{2}\boldsymbol{T}_1 + \boldsymbol{T}_2$.
• Figure 2: (a), (b) Graphical illustration of the SU(2) flux operators on the trellis lattice with the base site labeled "1." Loop operators $P_h$, $P_s$, and $P_t$ correspond to hexagonal, square, and triangular plaquettes, respectively. (c) Definitions of reference bonds within the unit cell at $(x,y)=(0,0)$. (d)--(f) The white (gray) hexagons denote the first (extended) Brillouin zones for lattice parameters $b=1$, $c=1/2$, and $a=3/2$, with the extended zone obtained by a scaling factor of 3. Panels (d)--(f) indicate the paths along which dispersions are plotted for the U(1) Ansätze: (d) U1 and U2, (e) U3 and U4, where the light gray region shows the reduced Brillouin zone for doubling along $\boldsymbol{T}_1$, and (f) U5 and U6, with the reduced Brillouin zone for doubling along $\boldsymbol{T}_2$. $\boldsymbol{T}_1$ and $\boldsymbol{T}_2$ are defined in Fig. \ref{['fig:lattice']}.
• Figure 3: Schematic representation of the class A, B, and C Ansätze, described in Secs. \ref{['sec:u1_class_a']}, \ref{['sec:u1_class_b']}, and \ref{['sec:u1_class_c']}, respectively. Solid (dashed) lines indicate hoppings with positive (negative) signs. The green, blue, and black lines correspond to $\chi_v\tau^z$, $\chi_h\tau^z$, and $\chi_z\tau^z$, respectively. Black (gray) points represent positive (negative) onsite hoppings. The associated PSGs of these Ansätze are listed in Table \ref{['tab:theta_rho_PSG']}.
• Figure 4: Phase diagrams of the six U(1) Ansätze (a) U1, (b) U2, (c) U3, (g) U4, (h) U5, and (i) U6 in the $(\chi_v, \chi_z)$ parameter space, with $\chi_h$$=$$1$ fixed. The corresponding band structures for representative points, indicated by colored circles, are shown in panels (d)–(f) and (j)–(l), respectively. (d), (e) The energy dispersions are plotted along a high-symmetry path $K_3 \rightarrow \Gamma \rightarrow K_1 \rightarrow K_2 \rightarrow K_3$ in the Brillouin zone, where $K_3 = (0,2\pi/3)$, $\Gamma = (0, 0)$, $K_1 = (2\pi/3, 0)$, and $K_2 = (2\pi/3, 2\pi/3)$. (f), (j) The band dispersions are plotted along a path that connects the momenta $K_4^\prime=(\pi/3, \pi/6)$, $K_1^\prime=(0, \pi/6)$, $K_2^\prime=(0,\pi/2)$, $K_3^\prime =( \pi/3, \pi/2)$, and back to $K_4^\prime$. (k), (l) The spectra are displayed along the path $M_1 \rightarrow \Gamma \rightarrow M_2 \rightarrow M_3 \rightarrow M_1$, with $M_1 = (2\pi/3, 0)$, $\Gamma = (0, 0)$, $M_2 = (0, \pi/3)$, and $M_3 = (2\pi/3,\pi/3)$. In all cases, dashed lines indicate the Fermi energy.
• Figure 5: Evolution of the Dirac points as the system approaches the s-DSL phase from the DSL phase for the Ansätze U1, U3, U4, U5, and U6 (see Fig. \ref{['fig:u1_ansatz']}). The color intensity increases with progression toward the s-DSL, where two Dirac points merge to form the s-DSL. Each pair of dots with identical color corresponds to a particular value of the hopping parameter close to the s-DSL transition (see Fig. \ref{['fig:dispersion_all']}). The U4 s-DSL state is separated from the others by two intermediate gapped phases. Hollow circles mark the momentum-space path along which the dispersions are displayed in Fig. \ref{['fig:dispersion_all']}.