Asymmetric Ferromagnetic Criticality in Pyrochlore Ferromagnet Lu$_2$V$_2$O$_7$

TL;DR

This study investigates the unique asymmetric critical behavior of the ferromagnetic transition in Lu$_2$V$_2$O$_7$, revealing different universality classes on either side of the transition and linking it to microscopic anisotropic interactions.

Contribution

It provides the first detailed analysis of asymmetric critical exponents in a pyrochlore ferromagnet and connects the phenomena to Dzyaloshinskii-Moriya interactions within a Ginzburg-Landau framework.

Findings

Critical exponents are β=0.32(1), γ=1.41(1), δ=5.38.

Asymmetric critical behavior on T<Tc and T>Tc sides.

Consistent with 3D Ising and cubic universality classes.

Abstract

Critical phenomenon at the phase transition reveals the universal and long-distance properties of the criticality. We study the ferromagnetic criticality of the pyrochlore magnet Lu$_2$V$_2$O$_7$ at the ferromagnetic transition ${T_\text{c}\approx 70\, \text{K}}$ from the isotherms of magnetization $M(H)$ via an iteration process and the Kouvel-Fisher method. The critical exponents associated with the transition are determined as ${\beta = 0.32(1)}$, ${\gamma = 1.41(1)}$, and ${\delta = 5.38}$. The validity of these critical exponents is further verified by scaling all the $M(H)$ data in the vicinity of $T_\text{c}$ onto two universal curves in the plot of $M/|\varepsilon|^\beta$ versus $H/|\varepsilon|^{\beta+\gamma}$, where ${\varepsilon = T/T_\text{c} -1}$. The obtained $\beta$ and $\gamma$ values show asymmetric behaviors on the ${T < T_\text{c}}$ and the ${T > T_\text{c}}$ sides, and are consistent with the predicted values of 3D Ising and cubic universality classes, respectively. This makes Lu$_2$V$_2$O$_7$ a rare example in which the critical behaviors associated with a ferromagnetic transition belong to different universality classes. We describe the observed criticality from the Ginzburg-Landau theory with the quartic cubic anisotropy that microscopically originates from the anti-symmetric Dzyaloshinskii-Moriya interaction as revealed by recent magnon thermal Hall effect and theoretical investigations.

Figures

Tables

Table 1. Table 1: Critical exponents of the single-valent S = 1 / 2 𝑆 1 2 S=1/2 ferromagnetic oxides, Lu 2 V 2 O 7 with the pyrochlore structure and YTiO 3 with the perovskite structure, in comparison with the theoretical values from different models.

	$\beta$	$\gamma$	$\delta$	Ref.
Lu2V2O7	0.322	1.402	5.38	This work
YTiO3	0.328	1.441	5.39	Cheng et al. (2008)
3D Ising	0.325	1.241	4.82	Kaul (1985)
3D XY	0.345	1.316	4.81	Kaul (1985)
3D Heisenberg	0.365	1.386	4.8	Kaul (1985)
3D Cubic	0.364	1.390	4.82	Manuel Carmona et al. (2000)

Equations

$$M_{\text{s}}(T)$$

$${\chi }_0^{-1}(T)$$

$$M(H)$$

$$M_{\text{s}}(T)[d{M}_{\text{s}}/dT{]}^{-1}$$

$${\chi }_0^{-1}(T)[d{\chi }_0^{-1}/dT{]}^{-1}$$

$$M(H,\epsilon )=|\epsilon {|}^{\beta }{f}_{\pm }(H/|\epsilon {|}^{\beta +\gamma }),$$

$$\displaystyle\begin{split}\mathcal{L}=&\,a|\bm{M}|^{2}+b|\bm{M}|^{4}\\ &+\lambda_{\text{c}}\big{[}(M^{x})^{4}+(M^{y})^{4}+(M^{z})^{4}\big{]}+\cdots,\end{split}$$

Full text

As a representative subject of quantum magnetism, pyrochlore antiferromagnets have attracted a significant attention in recent years Gardner et al. (2010). Many interesting phenomena including classical spin ice Castelnovo1 et al. (2008); Bramwell and Gingras (2001), quantum spin ice Molavian et al. (2007); Gingras and McClarty (2014); Ross et al. (2011), pyrochlore ice U(1) spin liquid Hermele et al. (2004); Savary and Balents (2012); Gingras and McClarty (2014); Lee et al. (2012); Huang et al. (2014), quantum order by disorder Savary et al. (2012); Hallas et al. (2018), spin nematics Taillefumier et al. (2017); Shannon et al. (2010), symmetry enriched topological orders Chen (2017a, b); Huang et al. (2014), topological magnetic excitations Li et al. (2016); Li and Chen (2018); Jian and Nie (2018) have been proposed and/or discovered for various compounds in the pyrochlore antiferromagnet families. While most efforts of this field have been devoted to the antiferromagnets, the pyrochlore ferromagnet Lu2V2O7 may stand out in the field of pyrochlore magnets by providing some rather unique and robust phenomena Bazuev et al. (1977); Shamoto et al. (2002); Ichikawa et al. (2005); Kiyama et al. (2006); Onose et al. (2010); Ideue et al. (2012). Lu2V2O7 is a vanadium based pyrochlore Mott insulator that orders ferromagnetically below about 70 K. Although it is a conventional ferromagnet, this material shows a remarkable magnon thermal Hall transport under magnetic field. Microscopically, the large thermal Hall effect Onose et al. (2010); Ideue et al. (2012) in this material is attributed to the non-trivial Berry curvature of the thermally populated magnon bands that originate from the antisymmetric Dzyaloshinskii-Moriya (DM) interaction Onose et al. (2010); Ideue et al. (2012); Mena et al. (2014); Xiang et al. (2011) between the V4+ spin-1/2 localized moments on the pyrochlore lattice. Since this discovery, Lu2V2O7 has become one of the stereotypes for the thermal Hall transports in Mott insulating materials.

To motivate our work, we here provide a general discussion about the magnetic properties of the pyrochlore ferromagnet Lu2V2O7 or any conventionally ordered matter. For the conventional ordered states, there are several basic and phenomenological aspects. The first one would be the order parameter and the ordering structure. Lu2V2O7 is a conventional ferromagnet. Being a ferromagnet is a global static property, the system should have a non-collinear spin structure within the four-sublattice unit cell. This is a direct consequence of the DM interaction. The second one would be the dynamic property or fluctuation with respect to the magnetic order. The thermal Hall transport is an overall effect caused by the magnetic excitations and reflects the weighted average of the Berry curvatures over the magnon bands Onose et al. (2010); Ideue et al. (2012). More detailed information about the dynamics would come from the magnon dispersion directly. This could be obtained through inelastic neutron scattering measurements. Theoretically, it has been suggested that the Weyl magnons Li et al. (2016) may be present from studying a minimal model for Lu2V2O7 and can be one origin for the thermal Hall effect in this system Su et al. (2017); Mook et al. (2016). The third aspect is the critical property due to the flucutations of order parameters associated with this ferromagnetic transition in Lu2V2O7. This aspect has not been carefully considered. Given the growing interests in this simple ferromagnetic pyrochlore oxide, we are motivated to explore its critical behaviors around $T_{\text{c}}$ via analyzing the isotherms of magnetization $M(H)$ with an iteration process and the Kouvel-Fisher method. Remarkably, we find that the critical exponents on both sides of $T_{\text{c}}$ are not equal and show 3D Ising-like and 3D cubic-like universality classes on each side. This differs from the conventional wisdom about the criticality that the asymmetry usually occurs in the non-universal prefactors of the scaling law rather than the scaling exponents. We discuss this asymmetric ferromagnetic criticality within Ginzburg-Landau theory and provide our view on the microscopic properties of the V-based pyrochlore magnet Lu2V2O7. We further give a general comment about the spin-orbit coupling and the Kitaev physics in the V-based magnets.

The critical behaviors of Lu2V2O7 around the ferromagnetic transition can be described by a series of critical exponents $\beta$, $\gamma$, and $\delta$ that reflect the effective magnetic interactions at play Stanley (1971). Different critical exponents have been derived theoretically for different models, e.g. ${\beta=0.365}$ and ${\gamma=1.386}$ for 3D Heisenberg model, $\beta=0.345$ and $\gamma=1.316$ for 3D XY model, and $\beta=0.325$ and $\gamma=1.24$ for 3D Ising model, respectively Kaul (1985). These exponents are obtained by analyzing the isothermal magnetizations $M(H)$ near $T_{\text{c}}$, viz.

[TABLE]

where $M_{\text{s}}$ is the spontaneous magnetization and $\chi_{0}^{-1}$ is the inverse initial magnetic susceptibility, respectively. In a previous work, Zhou et al. Zhou et al. (2008) have obtained the critical exponents ${\beta=0.42}$ and ${\gamma=1.85}$ for Lu2V2O7 and ascribed them to a 3D Heisenberg model. However, these values show a large deviation from those predicted theoretically. In addition, the straight lines in the modified Arrott plot, $M^{1/\beta}$ versus $(H/M)^{1/\gamma}$, are not parallel at all. Here, we reinvestigate its critical behaviors around $T_{\text{c}}$ via analyzing the isotherms of magnetization $M(H)$ with an iteration process and the Kouvel-Fisher method. Interestingly, we found that the obtained ${\beta=0.322}$ at ${T<T_{\text{c}}}$ and ${\gamma=1.41}$ at $T>T_{\text{c}}$ are close to the predicted values of 3D Ising and 3D Heisenberg or cubic universality classes, respectively. Lu2V2O7 single crystals used in the present study were grown with the traveling-solvent floating-zone technique. Details about the crystal growth and sample characterizations have been published elsewhere Zhou et al. (2008). The magnetic properties of Lu2V2O7 were measured with a commercial Magnetic Property Measurement System (MPMS-III, Quantum Design).

In Fig. 1(a), we plot the temperature dependence of dc magnetic susceptibility $\chi(T)$ and its inverse $\chi^{-1}(T)$ measured under ${\mu_{0}H=0.1}$ T in both zero-field-cooled (ZFC) and field-cooled (FC) modes. As can be seen, the ZFC and FC $\chi(T)$ curves are overlapped with each other and the ferromagnetic transition around ${T_{\text{c}}\approx 70}$ K can be clearly visible from the sharp rise of $\chi(T)$. In the paramagnetic region above $T_{\text{c}}$, we have applied the Curie-Weiss (CW) fitting to $\chi^{-1}(T)$ in the temperature range 150-300 K and extracted the effective moment of ${\mu_{\text{eff}}=1.89}\,{\mu_{\text{B}}/\text{V}^{4+}}$ and the Weiss temperature ${\theta_{\text{CW}}=88.6}$ K. The obtained $\mu_{\text{eff}}$ is closed to expected value of $1.73\,\mu_{\text{B}}$ for ${S=1/2}$ of V4+, and the deviation from this ideal value originates from the spin-orbit coupling of the V4+ ion. The positive $\theta_{\text{CW}}$ signals the dominant ferromagnetic exchange interactions in this system. Fig. 1(b) displays the $M(H)$ curve at 2 K, which exhibits a typical ferromagnetic behavior and reaches a saturation moment of ${\sim 1.0\,\mu_{\text{B}}}$ as expected. All these results are consistent with those reported previously and confirm the high quality of the studied crystal Zhou et al. (2008).

In Fig. 2(a), we plot the isothermal $M(H)$ curves of Lu2V2O7 in the temperature range of 60-80 K, which covers the ferromagnetic transition. The demagnetization effect has been corrected. These $M(H)$ data are replotted in the Arrott plot $M^{2}$ vs $H/M$ in Fig. 2(b), and in the modified Arrott plots $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ with the critical exponents of 3D Heisenberg and 3D Ising models in Fig. 2(c) and (d), respectively. The curved Arrott plot in Fig. 2(b) rules out the possibility of mean-field model, but the positive slope of the $M^{2}$ vs $H/M$ confirms that the paramagnet-ferromagnet transition is a continuous transition. On the other hand, the modified Arrott plots in Fig. 2(c) and (d) gave roughly parallel straight lines, and it is hard to distinguish visually which model could better describe the ferromagnetism of Lu2V2O7.

In order to determine precisely the critical exponents, we employed an iteration process in analyzing the isothermal $M(H)$ data near $T_{\text{c}}$ based on the general formula (1) to (3) given above Yang et al. (2001); Yanagihara et al. (2002). Starting from the Arrott plot shown in Fig. 2(b), we obtain the first set of $M_{\text{s}}(T)$ and $\chi_{0}^{-1}(T)$ by extrapolating the corresponding $M^{2}$ vs $H/M$ curves to the vertical and horizontal axes, respectively. As shown in Fig. 3(a), the obtained $M_{\text{s}}(T)$ and $\chi_{0}^{-1}(T)$ are fitted with the Eqs. (1) and (2), respectively, to extract the first set of critical exponents and critical temperatures, i.e. ${\beta=0.414(6)}$, ${T_{\text{c}}^{-}=71.38(3)}$ K, and ${\gamma=1.34(5)}$, ${T_{\text{c}}^{+}=69.2(3)}$ K as listed in the figure. By using the obtained $\beta$ and $\gamma$ values, we then construct a modified Arrott plot $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ and repeat the above process to obtain the second set of $M_{\text{s}}(T)$ and $\chi_{0}^{-1}(T)$ and the corresponding critical exponents and critical temperatures. As illustrated in Fig. 3(a), after three iterations the fitting parameters are converged to ${\beta=0.322(1)}$, ${T_{\text{c}}^{-}=67.91(1)}$ K, and ${\gamma=1.402(3)}$, ${T_{\text{c}}^{+}=67.7(1)}$ K. These critical exponents are different from those reported previously Zhou et al. (2008).

We further determine the critical exponents by employing the Kouvel-Fisher relation Kouvel and Fisher (1964), viz.

[TABLE]

in which the $M_{\text{s}}(T)$ and $\chi_{0}^{-1}(T)$ were obtained from the modified Arrott plot with the final critical exponents obtained above. As shown in Fig. 3(b), linear fittings to the plots of $M_{\text{s}}[dM_{\text{s}}/dT]^{-1}$ and $\chi_{0}^{-1}[d\chi_{0}^{-1}/dT]^{-1}$ versus $T$ yield ${\beta=0.322(6)}$, ${T_{\text{c}}^{-}=67.8(1)}$ K, and ${\gamma=1.417(4)}$, ${T_{\text{c}}^{+}=67.7(2)}$ K. Both values of $\beta$ and $\gamma$ obtained by the Kouvel-Fisher relation agree well with results from the iterations of modified Arrott plot, confirming the validity of these above analysis. For the completeness, we also estimate the critical exponent $\delta$. In Fig. 3(c), we display the double logarithmic plot of $M$ vs $H$ at 68 K, which is very close to the critical temperature ${T_{\text{c}}=67.8(1)}$ K determined above. As can be seen, the data falls nearly on a straight line with a slope of $1/\delta$ according to Eq. (3), and a linear fitting to the data at high-field region gives ${\delta=5.38}$, which fulfills the Widom scaling relation perfectly Widom (1964), i.e. ${\delta=1+\gamma/\beta}$ by using ${\gamma=1.417}$ and ${\beta=0.322}$ obtained above.

To check the reliability of our analysis on the critical behavior of Lu2V2O7, we test the obtained critical exponents according to the prediction of the scaling hypothesis Stanley (1971). In the critical asymptotic region, the magnetic equation of state could be expressed as

[TABLE]

where $f_{+}$ for ${T>T_{\text{c}}}$ and $f_{-}$ for ${T<T_{\text{c}}}$ are regular analytical functions, and ${\varepsilon=T/T_{\text{c}}-1}$ is the reduced temperature. Eq. (6) implies that for the right choice of $\beta$, $\gamma$, and $\delta$ values, $M/|\varepsilon|^{\beta}$ as a function of $H/|\varepsilon|^{\beta+\gamma}$ should produce two universal curves: one for ${T>T_{\text{c}}}$ and the other for ${T<T_{\text{c}}}$. By using the values of $\beta$ and $\gamma$ obtained by the Kouvel-Fisher method and ${T_{\text{c}}=67.8}$ K, we have obtained the scaling plot shown in Fig. 4. It is clearly seen that all the points indeed collapse into two separate branches. These well-scaled curves in Fig. 4 thus further confirm the reliability of the obtained critical exponents.

We have listed the critical exponents of Lu2V2O7 in Table 1 and compared with those of ferromagnetic perovskite YTiO3 Cheng et al. (2008) as well as the theoretical values from different models Kaul (1985); Manuel Carmona et al. (2000). Firstly, we found that the critical exponents for these two ferromagnetic oxides with ${S=1/2}$ are stunningly similar with each other. Secondly, it is interesting to note that their $\beta$ and $\gamma$ values are not consistent with those predicted by a single model. Instead, $\beta$ is very close to the predicted value of 3D Ising universality class, whereas $\gamma$ is close to the predicated value of 3D Heisenberg or cubic universality class. This makes Lu2V2O7 and YTiO3 rather rare cases in which the critical behaviors above and below $T_{\text{c}}$ are described by two different universality classes. This finding is intriguing because the critical behavior associated with a continuous phase transition should be described by one universality class with a single set of critical exponents, and the asymmetric behaviors usually occur in the non-universal prefactors of the scaling relations. The transition from the 3D Heisenberg$/$cubic to Ising universality class across $T_{\text{c}}$ implies the reduction of spin dimensionality or effective spin components upon the ferromagnetic ordering. As mentioned above, recent experimental and theoretical investigations on Lu2V2O7 have revealed the presence of significant anisotropic interactions in the form of DM interaction Onose et al. (2010); Ideue et al. (2012); Mena et al. (2014); Xiang et al. (2011), which should be responsible for such a transition.

The observation of similar critical behaviors in Lu2V2O7 and YTiO3 is not completely unexpected since both V4+ and Ti3+ ions have identical 3d1 electronic configuration with an active orbital degree of freedom in the octahedrally coordinated environments. The active orbitals are the lower $t_{2g}$ orbitals for both compounds. For Lu2V2O7, we know that the trigonal distortion would further split the $t_{2g}$ orbitals into $a_{1g}$ orbital and $e_{2g}^{\prime}$ orbitals. On the other hand, the atomic spin-orbit coupling is active for the $t_{2g}$ orbitals. The spin-1/2 local moment of the V4+ ion should be interpreted as the effective spin-1/2 of the ground state Kramers doublet for a local single-ion Hamiltonian with the spin-orbit coupling and the trigonal distortion. From this perspective, the nature of the V4+ local moment would be similar to the one for the Ir4+ ion, where the latter can be thought as a 5d1 hole Chen and Balents (2008) (instead of 3d1 electron for V4+). An interesting consequence of this correspondence is that, the rich physics due to the spin-orbit entanglement for iridates, such as Kitaev interaction Jackeli and Khaliullin (2009) and/or highly anisotropic exchange Chen and Balents (2008), may be found among the 3d transition metal oxides like vanadates.

The orbital nature of the V4+ local moment is consistent with the presence of significant (anti-symmetric) DM interaction. Moreover, for the same reason, we expect that the symmetric pseudo-dipolar interaction should be present and equally important as the DM interaction. This may not effect certain topological properties of the magnon bands too much as these properties are topologically robust. Since the local moment is effective spin-1/2, the single-ion anisotropy should not be present. As for the universal long-distance property in the vicinity of the transition, one does not need to worry too much about the detailed form of the interaction. Due to the orbital character of the local moment, the lattice symmetry directly acts on the effective spin components that effectively reduces the ferromagnetic order parameters from having an O(3) symmetry to the cubic symmetry. From the Ginzburg-Landau symmetry analysis, we propose the following action to describe the behavior of Lu2V2O7 near the transition,

[TABLE]

where $\bm{M}$ is the coarse-grained ferromagnetic order parameter, and “$\cdots$” represents the higher-order terms neglected here. The first two terms are isotropic with $a\approx a^{\prime}(T-T_{\text{c}})$, $a^{\prime}>0$ and $b>0$. They can describe the conventional ferromagnetic transition for $O(3)$-type order parameter, belong to the 3D Heisenberg universality class. We include the cubic anisotropy of the Lu2V2O7 system via $\lambda_{\text{c}}$ and require $\lambda_{\text{c}}<0$. This cubic anisotropy then favors the order parameter $\bm{M}$ to be aligned with the $\langle 001\rangle$ directions, which is consistent with the experimental observation Onose et al. (2010). For a small $\lambda_{\text{c}}$, the system should be found still in the Heisenberg universality class when $T$ is above but not too close to $T_{\text{c}}$. When $T$ is further tuned to $T_{\text{c}}$, one expects a crossover from the Heisenberg universality class to the cubic universality class Aharony (1973). We mention that the critical exponents belong to these two universality classes are very close, Table 1 thus difficult to be distinguished from each other in experiments. Recent results in literature give ${\gamma=1.3895}$ for the Heisenberg class Guida and Zinn-Justin (1998) and ${\gamma=1.390}$ for the cubic class Manuel Carmona et al. (2000), which are indeed close to the experimental result ${\gamma=1.402}$ obtained in this work. At ${T<T_{\text{c}}}$, the cubic symmetry is spontaneously broken. The system has chosen one of the ${\langle 001\rangle}$ directions and the order parameter becomes Ising-like, leading the critical exponent $\beta$ belong to the Ising universality class.

In summary, we have investigated the critical behavior of ferromagnetic pyrochlore Lu2V2O7 based on the measurements of isothermal magnetization $M(H)$ around $T_{\text{c}}$. We found that the ${\beta=0.322}$ at ${T<T_{\text{c}}}$ and ${\gamma=1.417}$ at ${T>T_{\text{c}}}$ are close to the predicted values of 3D Ising and cubic universality classes, respectively. This makes Lu2V2O7 a rare example in which the critical behaviors associated with a ferromagnetic transition belong to different universality classes. We have rationalized the observed criticality from the Ginzburg-Landau theory with the quartic cubic anisotropy that microscopically originates from the anti-symmetric DM interactions.

Acknowledgments.—This work is supported by the National Key R&D Program of China (Grant Nos. 2018YFA0305700, 2018YFA0305800, 2016YFA0301001, 2016YFA0300500), the National Natural Science Foundation of China (Grant Nos. 11574377, 11834016, 11874400), the Strategic Priority Research Program and Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (Grant Nos. XDB25000000, XDB07020100 and QYZDB-SSW-SLH013). J.P.S. and Y.Y.J. acknowledge support from the China Postdoctoral Science Foundation and the Postdoctoral Innovative Talent program.