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Vol. 127 (2015) ACTA PHYSICA POLONICA A No. 2

Proceedings of the European Conference Physics of Magnetism, Pozna« 2014 The Higgs Amplitude Mode in Weak Ferromagnetic Metals Yi Zhanga,*, P.F. Farinasb and K.S. Bedella aDepartment of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA bDepartamento de Física, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil Using ferromagnetic Fermi theory, Bedell and Blagoev derived the collective low- excitations of a weak ferromagnet. They obtained the well-known magnon (NambuGoldstone) mode and found a new gapped mode that was never studied in weak ferromagnetic metals. In this article we have identied this mode as the Higgs (amplitude mode) of a ferromagnetic metal. This is identied as the Higgs since it can be shown that it corresponds to a uctuation of the amplitude of the order parameter. We use this model to describe the itinerant- ferromagnetic material MnSi. By tting the model with the existing experimental results, we calculate the dynamical structure function and see well-dened peaks contributed from the magnon and the Higgs. Our estimates of the relative intensity of the Higgs amplitude mode suggest that it can be seen in scattering experiments on MnSi. DOI: 10.12693/APhysPolA.127.153 PACS: 75.10.b, 71.10.Ay, 75.50.Cc

1. Introduction rst predicted by Bloch [14] and Slater [15], and observed The emergence of low-energy excitations in systems in iron in a experiment [16]. Abrikosov with spontaneously broken symmetry plays a very im- and Dzyaloshinskii [17] rst predicted in fer- portant role in our fundamental understanding of na- romagnetic Fermi using a ferromagnetic Fermi ture. There are two types of fundamental excitations liquid theory (FFLT) for itinerant ferromagnets. This () that may be present in the eld theory descrip- approach was put on a more microscopic foundation in tions of these spontaneously broken symmetries: massless the work of Dzyaloshinskii and Kondratenko [18]. Moriya Nambu-Goldstone (magnons, or modes) and Kawabata also developed a band theory (MK theory) and massive Higgs bosons (amplitude modes). Let the to study the long wavelength spin uctuation in itinerant order parameter that describes a state with a sponta- ferromagnetic systems [19]. Bedell and Blagoev [20] gen- eralized the FFLT and they discovered a new collective neously broken symmetry be φ(x) = ρ(x)e i θ(x). Then, mode of the system; they found a mode with a gap in the in the we have hφ(x)i = ρ e i θ0 , where - 0 excitation spectrum. We have identied this mode as the nite values for θ0 and ρ0 correspond to a broken symme- try. The two fundamental excitations are the Nambu Higgs amplitude mode of a ferromagnetic metal. , which corresponds to uctuations of 2. Model the phase of the order parameter with xed amplitude, We begin by quickly describing the FFLT approach ρ , and the , with uctuations of ρ(x) with 0 used by Bedell and Blagoev [20], in the small magnetic the phase xed at θ0. The understanding of the amplitude mode is partic- moment limit, to study the collective excitations in a fer- ularly important for the of elementary romagnetic metal. We then argue that the massive mode physics [1]. Recent experiments from LHC [2, 3] is indeed an amplitude mode. To estimate the collective which detected Higgs-like particles has drawn much at- modes' frequencies we use a simple one-band description tention to this subject. This Higgs-like mode was pre- with a spherical Fermi surface for the itinerant-electron dicted theoretically and found experimentally in many ferromagnet MnSi. By tting existing experimental re- condensed systems. This includes incommen- sults we can pin down some of the Landau parameters surate charge-density- (CDW) states [4], supercon- and use them to calculate dynamical structure function, ducting systems [58], and lattice Boson systems with S(q, ω). As we will see there are two sharp peaks in superuid and Mott transition [913], which S(q, ω) corresponding to the two collective modes, the can be realized by ultracold bosonic on an opti- well-known NambuGoldstone mode (sometimes referred cal lattice. to as the transverse ) and a gapped Higgs am- In this article we have identied the amplitude mode plitude mode. From our estimates of the spectral weight in another well-known system with spontaneously bro- of the amplitude mode it should be observable in neutron ken symmetry, a ferromagnetic metal. The Nambu scattering experiments. Goldstone magnon mode in ferromagnetic systems was We consider a three-dimensional weak ferromagnetic material below its Curie temperature. According to FFLT, in the weak moment limit, the system can be described by the quasi-particle distribution function

*corresponding author; e-mail: [email protected] npαα0 (r, t) = np(r, t)δαα0 + mp(r, t) · σαα0 , and the quasi-particle energy function pαα0 (r, t) = p(r, t)δαα0 +

(153) 154 Yi Zhang, P.F. Farinas, K.S. Bedell

hp(r, t) · σαα0 [17, 21], with mp the quasi-particle mag- Fermi surface. The ± signs correspond simply to the dif- netization and P a 0 the eective ferent precessional directions so that we eectively have hp = −B + 2 p0 fpp0 m p magnetic eld which includes the external magnetic eld two modes. B and the internal magnetic eld generated by quasi- Before we go to the full calculation of S(q, ω), we can particle interactions a . Here and throughout the pa- fpp0 extract the basic physics from these results. The stan- per we set γ~ and . The quasi- dard model of a ferromagnetic metal is often referred to µmag = 2 = 1 ~ = 1 particle interaction can be expanded in Legendre polyno- as the Stoner model and it can best be characterized by mials, and in this weak moment limit, it can be treated its elementary excitations. To begin with there are the as spin rotation invariant [22], so that it can be sepa- spin 1/2 particle-like excitations (sometimes referred to rated into the spin symmetric and spin antisymmetric as quasi-particles), consisting of up spins (σ =↑), major- parts σσ0 P s a ˆ0 Here, ity spins, and down spins ( ), minority spins. In the N(0)fpp0 = l(Fl + Fl σ · σ0)Pl(pˆ · p ). σ =↓ N(0) is the average over the two Fermi presence of spontaneous long range ferromagnetic order surfaces and in the ferromagnetic phase, a . there are also gapless transverse spin waves (the Nambu F0 < −1 This state can be said to be protected by the general- Goldstone mode). This mode has a total spin of ±1 as- ized Pomeranchuck stability condition [23, 24], and its sociated with the two precessional directions of the equi- ground state is described by the distribution function librium magnetization. For simplicity, in what follows we 0 0 0 where is the equilibrium mag- will consider only the spin +1 excitations. If we think in mp = −m0∂np/∂εp, m0 netization divided by N(0). Fluctuations about equi- terms of the order parameter for the ferromagnetic state this would be a uctuation of the phase (rotation about librium, δmp, are described by a linearized spin kinetic equation [21]: the z axis in space) with the magnitude of the order pa- rameter xed at its equilibrium value, m . In addition ∂δm (r, t) ∂n0 0 p p to these spin waves, there are other spin +1 excitations + vp · ∇(δmp(r, t) − 0 δhp(r, t)) = ∂t ∂εp in the Stoner model that correspond to uctuations in 0 −2(mp(r, t) × δhp(r, t) + δmp(r, t) the magnitude of the order parameter at a xed phase. At q = 0 these excitations have a gap in their spec- 0 (1) ×hp(r, t)) + I[mp], trum usually referred to as the Stoner gap. For q 6= 0, where 0 P a 0 and these spin excitations have a range of frequencies hp = −B + 2 p0 fpp0 m p δhp = −δB + +1 P a 0 (shaded region shown in Fig. 1) and these are the inco- 2 0 f 0 δm p are the eective equilibrium eld and its p pp herent particlehole excitations, ± a . uctuation, respectively. Here, a small magnetic eld ωp−h = ∓2m0F0 +q·vp δB is set transverse to the equilibrium magnetization. These excitations are not collective, thus, there is no A direct way to obtain the dispersions is to take the Higgs amplitude mode in the Stoner model: When the limit of free oscillations of Eq. (1) by setting B = 0 momentum-transfer, q, is large enough the Goldstone and δB = 0, and looking at the low temperature limit mode decays into these incoherent particlehole excita- tions (Landau damping), while the gapped excitations for which the collision integral I[mp] can be ignored. This is the quantum spin hydrodynamic regime whose are Landau damped for all values of q. details of the derivation are more explicit in Ref. [20]. The FFLT description of Bedell and Blagoev [20] for In addition to the free oscillation limit, it is also pos- small momentum transfers is qualitatively the same as sible to use Eq. (1) to calculate the response function the Stoner model if we set all Landau parameters, a Fl = that relates with its induced transverse re- , for all , and keep only a. If we keep only χ(q, ω) δB 0 l > 0 F0 sponse P ( ) and from it the the and moments of the spin density dis- δm = p δmp δm = χ(q, ω)δB l = 0 l = 1 dynamical structure function S(q, ω) = −Im[χ(q, ω)]/π. tribution function (ν0 and ν1), we get the two modes, The calculations are done by projecting out the l compo- Eqs. (2) and (3). The rst mode is just the Nambu nents of both the kinetic equation and the spin density Goldstone mode and the second mode has a gap in its distribution function 0 0 P . spectrum, where at , it is just the Stoner gap, δmp = −(∂np/∂εp) l νlPl(pˆ · zˆ) q = 0 + a . This excitation causes a change in the We will return to the full calculation of S(q, ω) a lit- ω2 = 2 |F0 | m0 tle later. magnitude (amplitude) of the order parameter since it is a spin ip process. In the spin ip process we take We can use the kinetic equation to derive the conti- a down spin and ip it to an up spin causing an ampli- nuity equation for the magnetization and the equation of tude uctuation since we decreased the number of down the spin current dened as σ P ji (r, t) = p vpimp(r, t)(1+ spins while increasing the number of up spins during this a . The dispersions found using the free-oscillation F1 /3) uctuation. These uctuations of the amplitude of the limit are order parameter could have been the Higgs amplitude c2 mode, however, in the Stoner model this mode sits in the ω±(q) = s q2, (2) 1 ω± particlehole continuum and it is Landau damped; it is c2 not a collective mode. ω±(q) = ω± − s q2, (3) 2 ω± The Fermi liquid description of the collective modes of where ± a a , 2 a a ferromagnetic metal [20] goes beyond the Stoner model ω = ±2m0 |F0 − F1 /3| cs = |1 + F0 | (1 + a 2 , and is the Fermi velocity on the average described above in a simple but most important way. F1 /3)vF/3 vF The Higgs Amplitude Mode in Weak Ferromagnetic Metals 155

As can be seen from Eqs. (2) and (3) there are two modes, the rst one is the Goldstone mode. The second mode has a gap in its spectrum given by + a a . ω2 = 2m0 |F0 − F1 /3| The introduction of the higher order Fermi liquid pa- rameter a is responsible for the propagation of the F1 mode. This parameter couples the momentum of the quasi-particle to its spin and it is responsible for pushing the mode out of the particlehole continuum. Most of the spectral weight in this propagating mode comes from the incoherent particlehole continuum. As we noted earlier the particlehole continuum is made up from incoher- ent spin ip (spin +1) excitations and they correspond Fig. 1. Collective modes together with the ph con- tinuum in the case (a) a , a and to an amplitude uctuation. This mode can sit above F0 = −1.18 F1 = −0.84 (b) a , a . The green-dashed lines the Stoner gap for positive a and below the Stoner gap F0 = −1.16 F1 = 1.32 F1 represent the dispersion calculated using the hydrody- for negative a, with the lower bound, a [24]. F1 F1 > −3 namic approach, Eqs. (2) and (3) while the blue- It propagates and is built out of uctuations that change lines are the dispersions taken from the poles of the re- the amplitude of the order parameter, thus it is the fer- sponse function. Plots obtained by using parameters t- romagnetic metal example of the Higgs amplitude mode. ted to early experiments (see text), including the follow- 29 −1 −3 −1 ∗ Also since it couples to the spin uctuations it can be ing: N(0) = 6.4 × 10 eV m , kF = 1.23 Å , m = , eV, and 4 m s−1. seen in the transverse spin uctuation response function 39.3m e EF = 0.147 vF = 3.63 × 10 that could be measured in a neutron scattering experi- ment, as we see below. 3. Response function We calculate the dynamical spinspin response func- tion based on the kinetic equation, Eq. (1). Here we go beyond the hydrodynamic approach by keeping νl for arbitrary values of and the Landau parameters a up l Fl to l = 1. The response function obtained is given by a + 2m0F1 + χ(q, ω) −χ0 + qv (1+F a/3) χ1 F 1 (4) = F a , N(0) a + ω 1 + 1 − F0 χ0 − a χ1 qvF 1+F1 /3 where

+ 1 χ0 (q, ω) = −1 + 2qvF Fig. 2. Dynamic structure function for dierent val-  a  ω + 2m0F + qvF ues of of a. As a is switched on, the Higgs builds a 0 (5) F1 F1 ×[ω + 2m0(1 + F0 )] ln a , ω + 2m0F − qvF out of the particlehole continuum (the wider peak in 0 the middle) while the Goldstone mode remains approx- + 1 a imately unchanged. χ1 (q, ω)=− [ω+2m0(1+F0 )] qvF   a  1 a ω+2m0F0 +qvF (6) As we noted earlier, all of the spectral strength for the × 1− (ω+2m0F0 ) ln a . 2qvF ω+2m0F0 −qvF Higgs mode comes from the continuum, which makes the From the response function, we obtain the dynamic observation of the continuum itself dicult, as it is known structure function S(q, ω) = −Im[χ(q, ω)]/π, whose from neutron scattering experiments. poles will give, for small s, + and + + q ω1 (q) ω¯2 (q) = ω − The itinerant ferromagnetic material MnSi is a very c2 s 2 with 2 a , correspond- good candidate for the experimental search of the Higgs ( ω+ − α)q α = 2vF(1 + 3/F1 )/15m0 ing to the two collective modes derived in hydrodynamic mode. MnSi is metallic and it behaves ferromagnetically approach respectively. In [20] it was noted that ν2 is in a magnetic eld [2528]. Neutron scattering leads to 2 −2 of the same order of ν1 while νl  ν2 for all l ≥ 3. the magnon dispersion [29], ω(meV) = 0.13 + 52q ( ) In our calculation we look for the pole of the response and specic heat measurement gives [30, 31] (C/T )0 = −4 2 function, which includes all νls. A comparison to the 85 × 10 cal/(K mole). From the band structure cal- modes obtained in Ref. [20] with the hydrodynamic ap- culation [32], we know the density of the quasi-particles. proach is illustrated in Fig. 1. It shows, as expected, We t the experimental data to a single band descrip- agreement of the two results over a considerable range of tion with the quadratic dispersion E = ~2k2/2m∗, which small values for q. However, the full calculation of S(q, ω) keeps the volume of the Fermi surface unchanged. We ob- captures additional features, as, e.g., seen in Fig. 2 tain all the parameters we need and a relationship be- where the value of a is varied to illustrate the building tween two of the Landau parameters, a F1 F1 = −(375 + of the Higgs mode out of the particlehole continuum. a a . Since the system is weakly 321F0 )/(143 + 125F0 ) 156 Yi Zhang, P.F. Farinas, K.S. Bedell ferromagnetic, a should be close to and smaller than [5] R. Sooryakumar, M.V. Klein, Phys. Rev. Lett. 45, F0 , and a depends on a very sensitively in this re- 660 (1980). −1 F1 F0 gion. We do not have extra experimental results to pin [6] P.B. Littlewood, C.M. Varma, Phys. Rev. Lett. 47, down the sign of a, so we take the values of ( a, a) to 811 (1981). F1 F0 F1 be (−1.16, 1.32) and (−1.18, −0.84) as typical examples [7] P.B. Littlewood, C.M. Varma, Phys. Rev. B 26, to show the two collective modes in these two cases. 4883 (1982). [8] C.M. Varma, J. Low Temp. Phys. 126, 901 (2002). [9] D. Podolsky, A. Auerbach, D.P. Arovas, Phys. Rev. B 84, 174522 (2011). [10] E. Altman, A. Auerbach, Phys. Rev. Lett. 89, 250404 (2002). [11] L. Pollet, N. Prokof'ev, Phys. Rev. Lett. 109, 010401 (2012). [12] S.D. Huber, E. Altman, H.P. Büchler, G. Blatter, Phys. Rev. B 75, 085106 (2007). [13] M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schaub, C. Gross, E. Demler, S. Kuhr, I. Bloch, Nature 487, 454 (2012). [14] F. Bloch, Z. Phys. 61, 206 (1930). Fig. 3. Dynamical structure function showing both the [15] J.C. Slater, Phys. Rev. 35, 509 (1930). collective modes and the ph continuum in the cases of a a a a [16] R.D. Lowde, Proc. R. Soc. A 235, 305 (1956). F0 = −1.16, F1 = 1.32 (a) and F0 = −1.18, F1 = −0.84 (b). [17] A.A. Abrikosov, I.E. Dzyaloshinskii, Sov. Phys. JETP 35, 535 (1959). Figure 3 shows the dynamic structure function in [18] I.E. Dzyaloshinskii, P.S. Kondratenko, Zh. Eksp. the two typical cases for dierent momentum transfer. Teor. Fiz. 70, 1987 (1976). 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