Nature of quantum criticality in the Ising ferromagnet TbV$_6$Sn$_6$

Abstract

TbV$_6$Sn$_6$ is a topological metal where ferromagnetic Tb ions with strong uniaxial magnetic anisotropy interact with V kagome layers. Inelastic neutron scattering (INS) measurements show that the Tb ions adopt an Ising doublet ground state. Here, we consider whether a transverse magnetic field can drive TbV$_6$Sn$_6$ toward a quantum critical point, providing a rare example of transverse-field Ising criticality in a metallic compound. High-field magnetization measurements reveal a first-order-like spin-reorientation transition at 25.6 T. Our INS-based magnetic model finds that this is caused by an avoided crossing of an excited-state singlet with the ground-state doublet. Surprisingly, our model predicts that quantum critical and tricritical points are accessible within the range of experimentally determined model parameters and may be reached by varying the direction of an applied magnetic field.

Figures (3)

• Figure 1: Ising anisotropy in TbV$_6$Sn$_6$. The $D_{6h}$ point-group symmetry environment of the Tb$^{3+}$ ion [panel (a)] and its oblate-shaped Tb-4$f$ charge density [panel (b)]. The INS data with incident neutron energy of (c) $E_{i} = 12$ meV and (d) $E_{i} = 3.32$ meV. Data near and below the yellow dashed line in panel (c) are corrupted by spurious instrumental background. (e) The energy dependence of the combined $E_{i} = 3.32$ and 12 meV INS spectra at various temperatures (dots). Intensities at each $E_i$ and $T$ are averaged over the same $Q$ range indicated by the dashed red boxes in panels (c) and (d) and have a fitted background removed. Data for different $E_i$ are placed on the same intensity scale using procedures described in the Supplemental Material SI. Fits to the INS spectra from CEF calculations are plotted as solid lines in panel (e). (f) Zero-field CEF energy levels versus $B_6^6$ while keeping other Case I CEF parameters fixed. CEF states are pure $\ket{\pm m}$ eigenstates when $B_6^6=0$.
• Figure 2: Transverse field quantum phase transition. (a) Magnetization data measured in a pulsed transverse field along (100) and (210) directions at $T=625$ mK. Low-field dd data measured at 1.8 K are shown for comparison. (b) Triangular Tb layer showing field directions. (c) Case I mean-field model calculations of the magnetization components parallel to the (100) and (210) applied field directions ($M_{\perp}$, solid lines) and along (001) ($M_z$, dashed lines). (d) Evolution of crystal-field levels with transverse field applied along (100). The Ising state doublet (red lines) and the state that evolves from the zero-field $\ket{m=0}$ state to the planar $\ket{m_{\perp}=6}$ state (blue line) are highlighted. (e), (f) Mean-field calculations for Case II parameters. Panels (g) and (h) show the evolution of the Case I and Case II CEF levels with transverse field along (210). Lines are colored according to projection of their wavefunction $\ket{\Gamma}$ onto the $\ket{6}$ state. Dashed lines show the critical field and the green bar is $\sim k_{\rm B} T_{\rm C}$.
• Figure 3: Quantum criticality in TbV$_6$Sn$_6$. (a)$-$(f) Mean-field phase diagrams in the $H_x$-$H_y$ plane [with $(100)$ and $(210)$ directions shown in panel (a)] for Case I CEF parameters with variable $B_6^6$ values; the $B_l^0$ and $\mathcal{J}_0$ are held fixed. The blue regions are ferromagnetic and the white regions are quantum paramagnetic. There are two critical $B_6^6$ values: a lower $B_6^6 = 4.6\times 10^{-6}$meV, below which all transitions are first order (black), and an upper $B_6^6= 33\times 10^{-6}$meV, above which all transitions are second order (gray). In-between, there are QTCPs at certain angles (red dots) that separate the first- and second-order lines. Case II behaves very similarly to $B_6^6 = 20\times 10^{-6}$meV. (g) Quantum critical wing structure in $H_x$-$H_z$ plane for Case I parameters. (h) Evolution of the angular dependence of the QCEPs with $B_6^6$, in the $H_x$-$H_y$ plane. The red dots indicate where the QCEPs evolve into QTCPs for larger $B_6^6$.