Magnetic response of Sr$_2$RuO$_4$: quasi-local spin fluctuations due to Hund's coupling

TL;DR

This paper investigates the magnetic susceptibility of Sr$_2$RuO$_4$ using dynamical mean-field theory, revealing Hund's coupling driven quasi-local spin fluctuations consistent with recent neutron scattering data.

Contribution

It demonstrates that Hund's coupling induces quasi-local spin fluctuations in Sr$_2$RuO$_4$, providing a theoretical explanation aligned with experimental observations.

Findings

Identification of broad, quasi-local spin fluctuations centered around the $$ point.

Confirmation that Hund's coupling controls these fluctuations.

The planar xy orbital predominantly contributes to the $$ point response.

Abstract

We study the magnetic susceptibility in the normal state of Sr$_2$RuO$_4$ using dynamical mean-field theory including dynamical vertex corrections. Besides the well known incommensurate response, our calculations yield quasi-local spin fluctuations which are broad in momentum and centered around the $\Gamma$ point, in agreement with recent inelastic neutron scattering experiments [P. Steffens, et al., Phys. Rev. Lett. 122, 047004 (2019)]. We show that these quasi-local fluctuations are controlled by the Hund's coupling and account for the dominant contribution to the momentum-integrated response. While all orbitals contribute equally to the incommensurate response, the enhanced $\Gamma$ point response originates from the planar xy orbital.

Figures

Equations

$$\chi (\mathbf{Q})\equiv {\chi }_{S_z{S}_z}(\mathbf{Q})=\sum _{ab}{\chi }_{S_z^{(a)}{S}_z^{(b)}}(\mathbf{Q}),a,b\in \{xy,xz,yz\},$$

Full text

The importance of spin fluctuations for the physics of Sr2RuO4 has been emphasized long ago Sidis et al. (1999). This material is close to a spin-density-wave instability and small concentrations of impurities trigger ordering Minakata and Maeno (2001); Braden et al. (2002a). Inelastic neutron scattering experiments (INS) pioneered by Sidis *et al. *Sidis et al. (1999) and refined over the years Servant et al. (2002); Braden et al. (2002b, 2003, 2004); Iida et al. (2011, 2012); Steffens et al. (2019) have revealed that the magnetic response is essentially the sum of (i) a weakly momentum-dependent contribution centered at $\Gamma$ (in agreement with the Stoner enhancement factor of the uniform susceptibility by $\sim 7$ as compared to the band value Rice and Sigrist (1995); Mackenzie and Maeno (2003)) and (ii) a peak at an incommensurate wavevector $\mathbf{Q}_{SDW}\approx(0.3,0.3,0)$ 111In units of the reciprocal tetragonal lattice vectors $2\pi/a$ and $2\pi/c$. signaling the proximity to a spin-density-wave (SDW) instability Steffens et al. (2019). The peak at $\mathbf{Q}_{SDW}$ was predicted by Mazin and Singh Mazin and Singh (1999) using density functional theory (DFT) and the random phase approximation (RPA). However, DFT+RPA does not account for the broad structure at $\Gamma$, and it also predicts substantial anti-ferromagnetic fluctuations at the $X$ point, $\mathbf{Q}_{X}\!=\!(0.5,0.5,0)$, in contradiction to experiments Steffens et al. (2019).

More recently, however, it has been realized that the origin of the strong correlations in this material may not be associated with long-wavelength magnetic correlations, but with local correlations driven by the Hund’s coupling Mravlje et al. (2011); Georges et al. (2013). A successful description of an extensive set of physical properties of Sr2RuO4 has been obtained following this picture, supported by quantitative dynamical mean-field (DMFT) calculations. This includes the large mass enhancements of quasiparticles observed in de Haas-van Alphen experiments Mackenzie et al. (1996); *PhysRevLett.84.2662; *doi:10.1080/00018730310001621737 and angle resolved photo-emission spectroscopy Mackenzie et al. (1998); *PhysRevLett.85.5194; *PhysRevB.72.104514; *PhysRevB.72.205114; *PhysRevLett.105.226406; *PhysRevLett.109.066404; *Zabolotnyy:2013aa; *PhysRevLett.116.197003, as well as quasi-particle weights and lifetimes Mravlje et al. (2011), nuclear magnetic resonance Mravlje et al. (2011), optical conductivity Stricker et al. (2014); Deng et al. (2016), thermopower Mravlje and Georges (2016), Hall coefficient Zingl et al. (2019), quasiparticle dispersions Zhang et al. (2016); Kim et al. (2018); Tamai et al. (2018), and magnetic response Boehnke (2014); Boehnke et al. (2018); Gingras et al. (2018); Acharya et al. (2018).

In this letter, we bridge this gap between the spin fluctuations picture and the Hund’s metal picture of the normal state of Sr2RuO4 by analyzing the magnetic response function using DMFT. Our results reproduce the overall momentum dependence obtained in experiments Steffens et al. (2019), see Fig. 1, and reveal strong coupling effects which cannot be accounted for in RPA, such as a suppression of the antiferromagnetic response at $\mathbf{Q}_{X}$. We find that the response is dominated by quasi-local (weakly momentum-dependent) spin fluctuations, and show that these fluctuations are controlled by the strength of the Hund’s coupling. As discussed at the end of this letter, our findings have direct relevance for theories of the superconducting pairing mechanism, which is still an outstanding and much debated question Mackenzie et al. (2017).

We compute the magnetic susceptibility $\chi_{S_{z}S_{z}}(\mathbf{Q})$ using DMFT Georges et al. (1996), a DFT derived effective three-band $t_{2g}$ model without spin-orbit coupling 222The model is identical to the one used in Tamai et al. (2018) and derived from a density functional theory (DFT) calculation of Sr2RuO4 in the experimental structure Vogt and Buttrey (1995) (at $100\,$K) using Wien2k Blaha et al. (2018) with PBE Perdew et al. (1996) and $20^{3}$ k-points. Maximally-localized $t_{2g}$ Wannier functions were constructed using wien2wannier Kuneš et al. (2010) and Wannier90 Marzari and Vanderbilt (1997); Mostofi et al. (2008); Marzari et al. (2012) (using $10^{3}$ k-points)., and a local Kanamori interaction Kanamori (1963) with Hubbard $U=2.3\,\textrm{eV}$ and Hund’s $J=0.4\,\textrm{eV}$ Mravlje et al. (2011). The DMFT equations were solved using the hybridization expansion continuous time quantum Monte Carlo Werner et al. (2006); *Werner:2006qy; *Haule:2007ys; *Gull:2011lr implementation in TRIQS Parcollet et al. (2015); Seth et al. (2016). The DMFT particle-hole irreducible vertex was used to compute the static lattice susceptibility from the Bethe-Salpeter equation (BSE) Georges et al. (1996) as implemented in the TRIQS two-particle response function toolbox Strand (2019). Moreover, the static response at three specific momenta was computed down to much lower temperature, using self-consistent DMFT in applied magnetic fields by zero field extrapolation in supercells.

Fig. 1 displays the momentum dependent magnetic susceptibility from DMFT, with hot-spots at $\mathbf{Q}_{SDW}$ and ridges in $q_{x}$ and $q_{y}$ connecting these hot spots. This SDW component can be understood from the DFT electronic structure Oguchi (1995); *PhysRevB.52.1358; *Hase:1996aa; *McMullan:1996aa; *PhysRevB.59.9894; *PhysRevB.74.035115 of this material, which has three Ru(4$d$)-$t_{2g}$ bands crossing the Fermi level, filled with four electrons. The quasi-two-dimensional $\gamma$ band, with dominant xy orbital content, has a larger bandwidth (by a factor of $\sim\!2$) and slightly lower energy than the quasi-one-dimensional $\alpha$ and $\beta$ bands, originating mainly from the xz and yz orbitals. The peak at $\mathbf{Q}_{SDW}$ is generated by nesting in the $\alpha$ and $\beta$ (xz, yz) Fermi surface sheets, yielding ridges at $(0.3,q_{y},0)$ and $(q_{x},0.3,0)$ that cross and produce the peak at $\mathbf{Q}_{SDW}$ Mazin and Singh (1999).

The response in Fig. 1 also shows a large component, broad in momentum, with enhanced intensity centered at $\Gamma$ in comparison to the cold-spots at $M$ and $X$. This is the signature of the important quasi-local spin fluctuations. Antiferromagnetic fluctuations are suppressed, with the $X$ point being the global minimum of the response. This is qualitatively different from the results of weak-coupling approaches such as RPA Mazin and Singh (1999); Morr et al. (2001); Ng and Sigrist (2000); Eremin et al. (2002a, b) – even when basing RPA on the dressed DMFT Lindhard function Gingras et al. (2018) – or the fluctuation-exchange approximation Arakawa (2014, 2015). In contrast, these approximations yield an enhanced response at the $X$ point and fail to account for the quasi-local response. The latter was not discussed in previous DMFT work Boehnke et al. (2018), but noted in a recent DMFT+GW calculation Acharya et al. (2018). Both the quasi-local response and the suppression of the $X$-point fluctuations are in qualitative agreement with the recent INS experiments Steffens et al. (2019).

Studying the susceptibility along the high symmetry path $\Gamma$–$X$–$M$–$\Gamma$–$Z$ gives a quantitative picture of the response, see Fig. 2a. The incommensurate response at $\mathbf{Q}_{SDW}$ yields a peak on $\Gamma$–$X$ and the nesting ridges become local maxima on $X$–$M$ and $M$–$\Gamma$. The response at $\mathbf{Q}_{\Gamma}=(0,0,0)$ is enhanced relative to the cold-spots at $\mathbf{Q}_{X}=(0.5,0.5,0)$ and $\mathbf{Q}_{M}=(0.5,0,0)$ (green-shaded area) with $\mathbf{Q}_{X}$ being the global minimum. We note in passing that the negligible dispersion on $\Gamma$–$Z$ shows that the response is quasi two-dimensional. The quasi-local response (red- and green-shaded area) is the dominant part of the susceptibility, accounting for more than half of the momentum averaged response (yellow stars).

We also perform complementary calculations of the susceptibility down to much lower temperature through self-consistent DMFT in applied fields at $\mathbf{Q}_{\Gamma}$, $\mathbf{Q}_{X}$, and in the vicinity of the incommensurate wave vector $\mathbf{Q}_{SDW}$ at $\tilde{\mathbf{Q}}_{SDW}=(1/3,1/3,0)$ (using a $\sqrt{2}\times\sqrt{5}$ three site supercell), see Fig. 2b and 2c. The result is in quantitative agreement with the DMFT response obtained from the BSE after extrapolating to infinite fermionic frequency cutoff Luitz (2013), see markers in Fig. 2a. This serves as a non-trivial consistency check of our calculations and is, to the best of our knowledge, the first demonstration of thermodynamical consistency in DMFT at the two-particle level in a multiorbital model Georges et al. (1996); Potthoff (2006); Hafermann et al. (2014).

When lowering temperature the spin susceptibility is enhanced, see Fig. 2b. In particular, both $\chi(\mathbf{Q}_{\Gamma})$ and $\chi(\mathbf{Q}_{X})$ grow with decreasing temperature, where $\chi(\mathbf{Q}_{X})$ can be taken as a direct measure of the background response (red shaded area). However, the relative $\Gamma$ point enhancement (green shaded area) is robust and roughly constant, $\chi(\mathbf{Q}_{\Gamma})/\chi(\mathbf{Q}_{X})\approx 4/3$, in the studied temperature range. The precise value of this ratio, however, strongly depends on $J$ (see below). The DMFT local impurity susceptibility $\chi_{loc}$ shows a similar temperature dependence, and is approximately equal to the local susceptibility $\chi(\mathbf{r}=\mathbf{0})\equiv\frac{1}{V}\sum_{\mathbf{Q}}\chi(\mathbf{Q})$ at $T=464\,$K. This rough agreement strengthens the use of $\chi_{loc}$ as a proxy for the momentum average, $\chi_{loc}\sim\chi(\mathbf{r}=\mathbf{0})$ Mravlje et al. (2011); Mravlje and Georges (2016).

While it is known that pristine Sr2RuO4 does not order magnetically Mackenzie and Maeno (2003), the question of whether DMFT yields SDW order at low temperature (like DFT Kim et al. (2017)) has not been addressed previously. To answer this question we make a linear extrapolation of $\chi^{-1}(\tilde{\mathbf{Q}}_{SDW})$ in temperature, see Fig. 2c. For the established values of $U$ and $J$ Mravlje et al. (2011), and in the absence of spin-orbit coupling, we find that DMFT yields SDW order at $T_{SDW}\approx 123$ K, much lower than RPA 333This is a drastic suppression compared to RPA with bare interactions on the DMFT bubble with $T_{SDW}\sim 2500\,$K and the Hartree-Fock bubble with $T_{SDW}\sim 6850\,$K.. However, the transition temperature is very sensitive to the precise value of the microscopic parameters, in particular the Hund’s coupling $J$ (not shown).

The sensitivity in $J$ raises the question how the relatively small spin-orbit coupling (SOC) $\lambda_{DFT}\approx 0.1\,\textrm{eV}$ Zhang et al. (2016) affects the ordering temperature. Full DMFT calculations with SOC, in the relevant temperature range, are out of reach with currently available algorithms. Instead we resort to an approximate treatment – following Ref. Kim et al., 2018 – and add a static self-energy correction to the DMFT bubble in the BSE, with a correlation-enhanced SOC coupling $\lambda=2\lambda_{DFT}$, see also Ref. Liu et al., 2008. This accounts for the first order SOC contributions to the DMFT bubble $\chi^{(0)}$ but neglects the effect of SOC on the vertex. In momentum space the magnetic susceptibility with SOC corrections exhibits an overall suppression of the incommensurate and ridge response while the $\Gamma$, $X$, and $M$ points are only weakly affected, see Fig. 2a. The reduced incommensurate response yields a higher inverse susceptibility, see Fig. 2c, shifting the transition to lower temperature. Using the linear slope of the non-SOC case gives $T^{SOC}_{SDW}\lesssim 0$. Our tentative conclusion is thus that a full DMFT+SOC calculation down to low temperature is likely not to yield SDW ordering. This obviously deserves further studies. Since the inclusion of SOC primarily affects the SDW response, which is not the main focus of our study, we will neglect it in the following.

To disentangle the microscopic mechanisms driving the different components of the magnetic response, we study their dependence on the Hund’s coupling $J$, see Fig. 3. While the incommensurate spin-density-wave response $\chi(\tilde{\mathbf{Q}}_{SDW})$ displays a non-monotonic behavior in $J$, increasing $J$ suppresses $\chi(\mathbf{Q}_{X})$ and drastically increases $\chi(\mathbf{Q}_{\Gamma})$ and $\chi_{loc}$. Hence, the Hund’s coupling drives the observed $\Gamma$ point enhancement (green shaded area in Figs. 2a and 2b), as well as the enhancement of the local susceptibility. Since the response around the $\Gamma$ point is very broad in momentum space (see Fig. 1) this in turn suggests that the Hund’s coupling is responsible for the overall quasi-local magnetic response. We note in passing that the opposite trends of $\chi(\mathbf{Q}_{\Gamma})$ and $\chi(\mathbf{Q}_{X})$ as a function of $J$ produces a qualitative change of the magnetic response at $J\sim 0.32\,\textrm{eV}$ where the two terms cross. We conclude that the Hund’s coupling is responsible for the enhanced quasi-local fluctuations and plays a key role in the overall momentum space structure of the magnetic response.

We finally investigate how the magnetic response is distributed over the planar $xy$, and out-of-plane $xz$ and $yz$ orbitals, by studying the decomposition

[TABLE]

shown in Fig. 4a. We find that the orbital-off-diagonal response ($a\neq b$) is roughly 50% of the total magnetic response and confirm Boehnke et al. (2018) that $xy$, $xz$, and $yz$ contribute approximately equally to the $\mathbf{Q}_{SDW}$ response, see markers in Fig. 4. However, $\chi_{S_{z}^{(xy)}S_{z}^{(xy)}}(\mathbf{Q})$ is markedly higher than $\chi_{S_{z}^{(xz)}S_{z}^{(xz)}}(\mathbf{Q})$ around $\Gamma$ and along $M\!-\!\Gamma$. It is this part of the $\chi_{S_{z}^{(xy)}S_{z}^{(xy)}}(\mathbf{Q})$ response, shown in Fig. 4b, that is the origin of the broad plateau around $\Gamma$ and cold spots at $X$ and $M$ in Fig. 1, and the $\mathbf{Q}_{\Gamma}$ enhancement (green area) in Fig. 2a. While this only gives a weak momentum dependence to the large quasi-local magnetic response (red and green areas in Fig. 2a), the momentum space variations are extremely sensitive to the Hund’s coupling, as seen in Fig. 3.

We now compare our results to simpler approximations and show that the DMFT results are the only one qualitatively compatible with experiments and that the full frequency-dependent vertex is a crucial part of the calculation which can not be neglected 444Note that a similar observation has recently been made for the charge response of the cuprates Nourafkan et al. (2019).. Indeed, in Fig. 5a the DMFT result is compared to the bare DFT and DMFT bubbles ($\chi^{(0)}_{\text{DFT}}$, $\chi^{(0)}_{\text{DMFT}}$) and the screened RPA result $\chi_{\text{RPA}}$. The RPA calculation uses – in spirit of Ref. Gingras et al., 2018 – the DMFT bubble $\chi^{(0)}$ and screened effective interaction parameters $\tilde{U}=1.37\,\textrm{eV}$ and $\tilde{J}/\tilde{U}=0.4/2.3$, where $\tilde{U}$ has been taken to reproduce the local susceptibility $\chi(\mathbf{r}=\mathbf{0})\approx 7.3\,\mu_{B}^{2}/\textrm{eV}$ of DMFT. The frequency dependent particle-hole vertex is clearly essential in the DMFT calculation, as $\chi^{(0)}_{\text{DMFT}}$ is much smaller than the DMFT result $\chi_{\text{DMFT}}$. $\chi^{(0)}_{\text{DFT}}$ is also strongly suppressed compared to $\chi_{\text{DMFT}}$, and the $X$-point response is higher than both the $\Gamma$ and $M$ points (see Fig. 5b). Finally, the screened RPA using the DMFT bubble $\chi_{\text{RPA}}$ severely overestimates the strength of the nesting peaks, underestimates the constant background response, and fails both to enhance $\chi(\mathbf{Q}_{\Gamma})$ and to suppress $\chi(\mathbf{Q}_{X})$, see Fig. 5c.

In conclusion, we have analyzed the momentum-dependent magnetic response of Sr2RuO4 using dynamical mean-field theory, taking full account of vertex corrections. The latter are found to play a crucial role, leading to key effects absent at the RPA level such as the suppression of the antiferromagnetic response at $\mathbf{Q}_{X}$. In agreement with neutron scattering experiments Steffens et al. (2019), the magnetic response has two main components: an SDW incommensurate response at $\mathbf{Q}_{SDW}$ and a quasi-local weakly momentum-dependent component, which provides the main contribution to the overall momentum integrated response. Our main result, on a qualitative level, is the demonstration that the physical origin of the quasi-local magnetic response is the Hund’s coupling, hence reconciling the experimental emphasis put on spin fluctuations in this material with the theoretical picture of Sr2RuO4 as a ‘Hund’s metal’.

This has far-reaching consequences: both our theoretical calculations and neutron scattering experiments indicate that there is no dispersing ‘quasi-ferromagnetic’ spin fluctuation mode in Sr2RuO4. Hence, pairing mechanisms based on a mediating bosonic mode (‘glue’) associated with ferromagnetic spin fluctuations Rice and Sigrist (1995); Mazin and Singh (1997, 1999); Mackenzie and Maeno (2003) have to be seriously reconsidered. The observed suppression of the magnetic response at the $X$ point also invalidates an antiferromagnetic ‘glue’. Instead, pairing mechanisms based on a quasi-local mode associated with Hund’s coupling offer a promising route. Recent work has appeared in this direction for model Hamiltonians Hoshino and Werner (2015, 2016); Steiner et al. (2016) and for iron-based superconductors Lee et al. (2018). However, these mechanisms were proposed in the regime of slow spin fluctuations above the Fermi liquid temperature, and need to be extended to be applicable to Sr2RuO4. This is a key agenda for future work aiming at solving the 25-years old puzzle of superconductivity in this material Mackenzie et al. (2017).