## Abstract

Efforts to find the magnetic monopole in real space have been made in cosmic rays and in particle accelerators, but there has not yet been any firm evidence for its existence because of its very heavy mass, ∼10^{16} giga–electron volts. We show that the magnetic monopole can appear in the crystal momentum space of solids in the accessible low-energy region (∼0.1 to 1 electron volts) in the context of the anomalous Hall effect. We report experimental results together with first-principles calculations on the ferromagnetic crystal SrRuO_{3} that provide evidence for the magnetic monopole in the crystal momentum space.

Dirac (*1*) postulated in 1931 the existence of a magnetic monopole (MM), searching for the symmetry between the electric and magnetic fields in the law of electromagnetism. A singularity in the vector potential is needed for this Dirac MM to exist. Theoretically, the MM was found (*2*, *3*) as the soliton solution to the equation of the non-Abelian gauge theory for grand unification. However, its energy is estimated to be extremely large, ∼10^{16} GeV, which makes its experimental observation difficult. In contrast to this MM in real space, one can consider its dual space, namely, the crystal momentum (**k**-) space of solids, and the Berry phase connection (*4*) of Bloch wave functions. This MM in momentum space (*5*) is closely related to the physical phenomenon of the anomalous Hall effect (AHE) observed in ferromagnetic metals.

The AHE is a phenomenon in which the transverse resistivity (ρ_{xy}) in ferromagnets contains a contribution from the magnetization (*M*) in addition to the usual Hall effect. The conventional expression for ρ_{xy} is (1) where *B* is the magnetic field, *R*_{0} is the usual Hall coefficient, and *R*_{s} is the anomalous Hall coefficient. This expression implicitly assumes that the additional contribution is proportional to *M*, and it is used as an experimental tool to measure *M* as a function of temperature. This analysis is extensively used in studies of ferromagnetic semiconductors with dilute magnetic impurities, which are the most promising materials for applications in spintronics (*6*). However, the mechanism of AHE has long been controversial (*7*–*11*). The key issues are whether the effect is intrinsic or extrinsic and how to treat the impurity and phonon scatterings. Karplus and Luttinger (*7*) were the first to propose the intrinsic mechanism of AHE, in which the matrix elements of the current operators are essential. Other theories (*8*, *11*) attribute the AHE to the impurity scattering modified by the spin-orbit interaction, namely, the skew scattering (*8*) and/or the side-jump mechanism (*11*). These extrinsic mechanisms are rather complicated and depend on the details of the impurities as well as on the band structure of the materials. Nevertheless, all these conventional theories (*7*–*11*) for the AHE derive Eq. 1, as they are based on the perturbative expansion of the spin-orbit coupling (SOC) λ and *M*; i.e., *R*_{s} ∝ λ.

Recently, the geometrical meaning of the intrinsic-origin AHE (*6*) has been recognized (*12*–*15*). The transverse conductivity (σ_{xy}) can be written as the integral of the Berry phase curvature (the gauge field) over the occupied electronic states in crystal momentum space (Eq. 5). The MM corresponds to the source or sink of the gauge field or curvature defined by this Berry phase connection. Therefore, the AHE can be a direct fingerprint of the MM in crystal momentum space. The presence of time-reversal symmetry results in σ_{xy} = ρ_{xy} = 0 in the dc limit from the generic argument, and the group theoretical condition for the nonzero σ_{xy} is equivalent to that of finite ferromagnetic moment (supporting online text). Therefore, ferromagnets are necessary to study σ_{xy}, even though the Berry phase connection is more universal and exists even in nonmagnetic materials.

We show by detailed first-principles band calculations combined with transport, optical, and magnetic measurements that the observed unconventional behavior of the AHE and Kerr rotation in the metallic ferromagnet SrRuO_{3} is of intrinsic origin and is determined by the existence of a MM in **k**-space. The conventional expression (Eq. 1) is not supported by our experimental data, which show a nonmonotonous temperature dependence that even includes a sign change.

SrRuO_{3} with perovskite structure is an itinerant (metallic) ferromagnet. Ru^{4+} has four *t*_{2g} electrons with low spin configurations. The 4*d* orbitals of SrRuO_{3} are relatively extended and the bandwidth is large compared with its Coulombic interaction. The relativistic SOC is also large in its 4*d* electrons, because of the heavy atomic mass (on the order of 0.3 eV for the Ru atom). High-quality single crystal is now available with a residual resistivity on the order of 10 μΩcm. These aspects make this system an ideal candidate in which to observe the AHE due to the **k**-space gauge field. Stoichiometric SrRuO_{3} (bulk and thin film) and Ca-doped (Sr_{0.8}Ca_{0.2}RuO_{3} thin film) single crystals were prepared (*16*). The *M* curve (Fig. 1) of SrRuO_{3} film is quite similar to that of bulk single crystal, except that the Curie temperature (*T*_{c}, ∼150 K) is slightly lower than that of bulk (∼160 K), because of the strain effects. However, the isovalent Ca-doping markedly suppresses *T*_{c} and *M*. All of the samples were used for transport measurements in the dc limit. As seen from Fig. 1C, the ρ_{xy} changes nonmonotonously with temperature and even includes a sign change. Such behavior is far beyond the expectation based on the conventional expression in Eq. 1. In addition to the transport measurement, the frequency (ω)–dependent conductivities (Fig. 2) were measured for SrRuO_{3} film by an optical method (*16*). In addition to the strong structures around 3.0 eV, which are mostly due to the charge transfer from O-2*p* to Ru-4*d*, sharp structures were also observed for both the real and the imaginary part of σ_{xy}(ω) below 0.5 eV. These low-energy sharp structures cannot be fitted by extended Drude analysis. The lower the energy is, the stronger the deviations from fitting are. We will show, by combination with first-principles calculations, that these unconventional behaviors actually originate from the singular behavior of MMs in momentum space.

We now turn to some details of the theoretical analysis. The Berry phase is the quantal phase acquired by the wave function that is associated with the adiabatic change of the Hamiltonian (*4*, *17*). Let |*n*(**α**)〉 be the *n*th eigenstate of the Hamiltonian *H*(**α**), with **α** = (α_{1},....,α_{m}) being the set of parameters. The Berry's connection is the overlap of the two wave functions infinitesimally separated in **α**-space, i.e. (2) where the vector potential **a**_{n}(**α**) is defined by **a**_{n}(**α**) = *i*〈*n*(**α**)|∇_{α}|*n*(**α**)〉. Although the concept of the Berry phase has broad applications in physics (*17*), its relevance to the band structure in solids has been recognized only recently and in limited situations, such as the quantum Hall effect under a strong magnetic field (*18*) and the calculation of electronic polarization in ferroelectrics (*19*, *20*). In this case, the parameter **α** is the crystal momentum **k**. For the Bloch wave function ψ_{n}** _{k}**(

*) =*

**r***e*

^{i}

**⚫**

^{k}

^{r}*u*

_{n}

**(**

_{k}*), where*

**r***n*denotes the band index and

*u*

_{n}

**is the periodic part, the vector potential for the Berry phase**

_{k}*a*

_{nμ}(

*) is (3) where μ is the chemical potential. With this vector potential, the gauge covariant position operator*

**k***x*

_{μ}for the wave packet made out of the band

*n*is given by

*x*

_{μ}=

*i*∂

_{kμ}–

*a*

_{nμ}(

**k**). Therefore, the commutation relation between

*x*

_{μ}and

*x*

_{ν}includes the gauge field

*F*

_{μν}= ∂

_{kμ}

*a*

_{nν}–

_{k∂}

*a*

_{nμ}, as (4) which leads to the additional (anomalous) velocity –

*i*[

*x*

_{μ},

*V*(

*x*)] = –

*F*

_{μν}∂

*V*(

*x*)/∂

*x*

_{ν}, being transverse to the electric field

*E*

_{ν}= –∂

*V*(

*x*)/∂

*x*

_{ν}. Therefore, the transverse conductivity σ

_{xy}is given by the sum of this anomalous velocity over the occupied states as (

*18*) (5) where

*b*

_{z}(

**k**) =

*F*

_{xy}(

**k**), ϵ is ϵ

_{n}(

**k**) is the eigen energy of ψ

_{n}

**(**

_{k}**r**), β is the inverse temperature, and

*n*

_{F}[ϵ] = 1/(

*e*

^{β(ϵ–μ)}+ 1) is the Fermi distribution function. Hence, the behavior of gauge field

*b*

_{z}(

**k**) in

**k**-space (

*21*) determines that of σ

_{xy}. One might imagine that it is a slowly varying function of

**k**, but that is not the case. Fig. 3B is the calculated result for

*b*

_{z}(

**k**) in the real system SrRuO

_{3}. It has a very sharp peak near the Γ-point and ridges along the diagonals. The origin for this sharp structure is the (near) degeneracy and/or the band crossing, which act as MMs. Consider the general case where the two-band Hamiltonian matrix

*H*(

**k**) at

**k**can be written as

*H*(

**k**) = Σ

_{μ = 0,1,2,3}

*f*

_{μ}(

**k**)σ

_{μ}where σ

_{1,2,3}are the Pauli matrices and σ

_{0}is the unit matrix. When

**k**is mapped to the vector

**f**(

**k**) = [

*f*

_{1}(

**k**),

*f*

_{2}(

**k**),

*f*

_{3}(

**k**)], then the contribution to σ

_{xy}from the neighborhood of this degeneracy region is given by the solid angle

*d*Ω

**for the infinitesimal**

_{f}*dk*

_{x}

*dk*

_{y}integrated over

**k**. Therefore, the gauge flux near the MM, namely, the degeneracy point

**f**=

**0**, is singularly enhanced (Fig. 3) (supporting online text).

We studied the behavior of σ_{xy} by first-principles calculations (*16*). The calculated density of states is not so different between the cases with and without SOC (Fig. 4A), whereas the σ_{xy} should be very sensitive to the Bloch wave functions and depends on the Fermi-level position and the spin-splitting (magnetization) substantially, as predicted by the discussion above. We determined the behavior of σ_{xy} as a function of the Fermi-level position by using a small broadening parameter for the lifetime δ (70 meV) (Fig. 4B). When the Fermi level was shifted, not only the absolute value but also the sign of σ_{xy} was found to change. The sharp and spiky structures are the natural results of the singular behavior of the MM (Fig. 3). For the case without any shift of Fermi level, we obtained a value of σ_{xy} = –60 Ω^{–1} cm^{–1}, which has the same sign as and is comparable with the experimental value (about –100 Ω^{–1} cm^{–1}). Such a spiky behavior should also be reflected in the ω-dependent σ_{xy}, especially for the low-energy range with longer lifetime, whereas it should be suppressed at higher activation energies with shorter lifetime. As shown in Fig. 2 for the ω dependence of optical conductivity, the high-energy (>0.5 eV) part, which is dominated by the *p* – *d* charge transfer peak, is usual and can be well reproduced by our calculations, whereas the observed peak structure of σ_{xy}(ω) below 0.5 eV is a clear demonstration of the predicted spiky behavior. The spectra below 0.2 eV were not measured because of the technical difficulty, but structure there should be even sharper, because the dc limit *Re*(σ_{x}* _{y}*) ≈ –100 Ω

^{–1}cm

^{–1}has the opposite sign [the

*Im*(σ

_{xy}) at the dc limit should go back to zero]. Such low-energy behavior is well represented by our calculations, providing further evidence for the existence of MMs.

It is straightforward to understand the results of our transport measurement for σ_{xy}. We attribute the temperature (*T*) dependence of σ_{xy} to that of the magnetization *M*(*MT*). As the result of **k**-space integration over occupied states, the calculated σ_{xy} is nonmonotonous as a function of *M* (Fig. 1D). With the reduction of spin-splitting, the calculated σ_{xy}, after the initial increase, decreases sharply, then increases and changes sign (becoming positive), and finally decreases again, capturing the basic features of the experimental results. Even more surprisingly, when the measured ρ_{xy} versus *T* curves shown in Fig. 1C are converted into the σ_{xy} versus *M* curves shown in Fig. 1D, they now all follow the same trend and match with our calculations. The curves are measured for different samples (with different saturation moments), but all follow the same rule qualitatively and could be simply explained by the reduction of *M* (*22*) (Fig. 1A). However, the comparison between the experiments and the calculations should be semi-quantitative, because the results are sensitive to the lattice structures. The calculated σ_{xy} for the fictitious cubic structure shows a strong deviation from that obtained for orthorhombic structure, and it changes the sign to be positive at low temperature (at large *M*). Therefore, more accurate information on the structure is needed to obtain the quantitative result. However, such a sensitivity does not affect our main results, i.e., the nonmonotonous behavior of σ_{xy}. Even the calculations for cubic structure show such behavior and may be used as a guide of possible deviation.

The results and analysis presented here should stimulate and urge the reconsideration of the electronic states in magnetic materials from a very fundamental viewpoint. For example, the MM is accompanied by the singularity of the vector potential, i.e., the Dirac string (*1*). As shown by Wu and Yang (*23*) this means that more than two overlapping regions have to be introduced, in each of which the gauge of the wave function is defined smoothly. This means that one cannot define the phase of the Bloch wave functions in a single-gauge choice when the MM is present in the crystal momentum space. This leads to some nontrivial consequences, such as the vortex in the superconducting order parameter as a function of **k** (*24*), and many others are left for future studies.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/302/5642/92/DC1

Materials and Methods

SOM Text

References and Notes