Introduction

Landau Fermi liquid theory is the major milestone of modern condensed matter physics, and illustrates the triumph of physical intuition1. The short-range repulsive interaction between the fermions was argued to be irrelevant as one approaches the low energy towards the Fermi surface. The singular long-range interactions, however, are not well coped in the framework of Fermi liquid theory and signifies the possibility of non-Fermi liquid (NFL) metals2,3. These singular interactions can come from (partially screened) long-range Coulomb interaction, the fluctuations of the gapless bosonic modes at the criticality, the Goldstone boson from the continuous symmetry breaking4, and the U(1) gauge boson5,6. The established theories describing NFL metals, particularly the experimentally relevant ones, are known as Hertz–Millis–Moriya theory3,7,8,9,10. This theory involves the coupling between gapless fermions near the Fermi surface and the critical bosons. If the number of the gapless fermions is finite such as Dirac fermion, Weyl fermion and the quadratic band touching, a controlled calculation with the perturbative renormalization group can be performed. In contrast, when the fermion sector is a Fermi surface, the physics become complex and this topic is under an active investigation in recent years.

So far there are two types of Fermi surface criticality associated NFLs5. The first involves an ordering at a finite wavevector, e.g., an antiferromagnetic (AFM) order11,12,13,14,15 or spin density wave16. The ordering wavevector connects a few “hotspots” on the Fermi surface, and the theoretical analysis focuses on the coupling between the hotspot fermions and the critical bosons. The second involves the critical bosons at the zero wavevector. This captures, for instance, the Ising-nematic criticality17,18,19,20,21, spinon-gauge coupling in the spinon Fermi surface U(1) spin liquid6, ferromagnetic (FM) criticality22,23. It was observed that the bosons that are tangential to the Fermi surface scatter the fermions strongly at low energies. Thus, the theoretical analysis of this Ising-nematic criticality is further reduced to the so-called patch theory where the tangential critical boson scatters the fermions from one patch or two patches. Various analytic techniques were developed. The early random phase approximation type of large-N expansion was questioned as this scheme of taming quantum fluctuations and organizing the Feynman diagrams misses the contribution from the processes involving the fermions on the Fermi surface6,20,24,25,26. The remedy was made by the double expansion that combines the large-N expansion and the ϵ-expansion27,28. Another remedy introduces the dimensional and co-dimensional regularization to the Fermi surface, and develops a systematic framework to regulate the quantum fluctuations14,21,29,30,31. It is hoped that the physical cases are located in the regimes where these development can be applied. Inspired by these developments, we turn our attention to another type of Fermi surface criticality and NFL. Compared to the efforts in the literature, here we are more inclined to exploring the mechanism and phenomenology of the NFLs.

In this paper, we study the system with the Fermi surface coupled to the critical bosons on a continuously degenerate manifold, i.e., a boson surface in three-dimensional space(3D) as shown in Fig. 1. The critical phase of bosons in one dimension is the Luttinger liquid32. Focusing on the stability of such highly degenerate critical phase in higher dimensions has been investigated in the context of Bose metal33,34,35,36,37,38, and recent studies declared that weakly interacting dilute bosonic systems with continuously degenerate minima in the low-energy bosonic excitations are stable in d = 2, 3 and thereby pointed to the concept of Bose Luttinger liquid39,40. Experimentally, the relevance of critical boson surface has been implied by the neutron scattering in MnSi41,42 where a nearly uniform intensity is measured on a sphere. Motivated by these developments, the investigation on the critical boson surface has received some attentions43,44,45. Here, we are neither dealing with nor relying on the stable phase of critical boson surfaces, instead we aim to improve our understanding of the critical boson surface-induced quantum criticality and its impact when it is coupled with gapless fermions.

Fig. 1: The coupling between Fermi surface and critical boson surface.
figure 1

a The blue sphere is the Fermi surface of the conduction electrons. The orange surface represents the critical boson surface. kF is the Fermi momentum, and q0 is the radius of the boson surface. The Yukawa coupling connects fermions with a ring of fermion modes on the Fermi surface. b The critical boson sphere.

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We consider the 3D itinerant magnets that comprise two distinct types of degrees of freedom: (i) conduction electrons, (ii) local magnetic moments. In the absence of inversion symmetry, there exists a Dzyaloshinskii–Moriya interaction between the local moments that is responsible for the generation of infinite critical bosons on a spherical surface at the phase transition. The critical boson surface is coupled to the fermions on the Fermi surface at low energies through a Yukawa-type interaction. Microscopically, this Yukawa coupling arises from the Hund’s or Kondo-like coupling between the conduction electron and the local moments. As illustrated in Fig. 1a, each fermion is coupled to a ring of fermions on the Fermi surface, and the fermions on this ring are further coupled to many other rings of fermions. Thus, infinite number of gapless fermions are scattered by the infinite number of critical bosons, which makes the whole Fermi surface critical. This fermion–boson-coupled model is fundamentally different from the AFM criticality or the Ising-nematic criticality where only a finite number of critical bosons are involved. Thus, neither the hotspot treatment for the AFM criticality nor the conventional patch theory is applicable. We establish the basic properties and the global phase diagram of the system, and adopt the self-consistent renormalization theory10 to address the properties near the criticality. We show that, due to the novel type of fermion–boson coupling, the system becomes a NFL metal with distinct power law behaviors in the vicinity of the transition. We analyze the fermion and the boson properties as well as the related crossovers at the criticality and in the ordered regime.

Results

Model

The full model for the itinerant magnets contains three parts, the conducting electrons, the local moments, and the coupling between them, which is then described by a fermion–boson-coupled Lagrangian

$$\begin{array}{ll}{{{\mathcal{L}}}}[{f}^{{\dagger} },f;\overrightarrow{\phi }]=\mathop{\sum}\limits_{l,\alpha }{f}_{l\alpha }^{{\dagger} }({\partial }_{\tau }-\mu ){f}_{l\alpha }-\mathop{\sum}\limits_{l{l}^{{\prime} },\alpha }{t}_{l{l}^{{\prime} }}{f}_{l\alpha }^{{\dagger} }{f}_{{l}^{{\prime} }\alpha }\\ \qquad\qquad\qquad+\,g\mathop{\sum}\limits_{l,\alpha \beta }{f}_{l\alpha }^{{\dagger} }{\overrightarrow{\sigma }}_{\alpha \beta }{f}_{l\beta }\cdot {\overrightarrow{\phi }}_{l}+{{{{\mathcal{L}}}}}_{{{{\rm{B}}}}}[\overrightarrow{\phi }].\end{array}$$
(1)

The first line of Eq. (1) dictates an electron model where f†(f) is the fermion creation (annihilation) operator. The itinerant electrons hop on a 3D lattice denoted by the site index l. The electron spin couples to a magnetic moment field \({\overrightarrow{\phi }}_{l}\) with a Kondo-like Yukawa coupling. Here, the bosonic field \({\overrightarrow{\phi }}_{l}\) is a three-component vector defined on site \(l,{\overrightarrow{\sigma }}_{\alpha \beta }\) is the Pauli matrix vector with α, β being the spin indices. The magnetic fluctuation near the phase transition admits a standard Ginzburg–Landau expansion up to \({{{\mathcal{O}}}}({\phi }^{4})\) with

$${{{{\mathcal{L}}}}}_{{{{\rm{B}}}}}[\overrightarrow{\phi }]=\mathop{\sum}\limits_{l}\frac{1}{2}{\overrightarrow{\phi }}_{l}\cdot (r-J{\nabla }^{2}){\overrightarrow{\phi }}_{l}+\frac{u}{4}{\left({\overrightarrow{\phi }}_{l}^{2}\right)}^{2}+\frac{D}{2}{\overrightarrow{\phi }}_{l}\cdot (\nabla \times {\overrightarrow{\phi }}_{l}),$$
(2)

The first two terms of \({{{{\mathcal{L}}}}}_{{{{\rm{B}}}}}[\overrightarrow{\phi }]\) represent a standard ϕ4-theory with an order parameter \(\overrightarrow{\phi }\) for the magnetic moment and r is the boson mass. The last term is an anti-symmetric Dzyaloshinskii–Moriya (DM) interaction, which fundamentally alters the critical phenomenon and leads to a rich phase diagram in Fig. 2.

Fig. 2: The global phase diagram.
figure 2

The left corner is a helimagnet and a Goldstone-NFL where the NFL is induced by the gapless Goldstone boson. The right corner is a paramagnet and Fermi liquid metal. The central region is the quantum critical regime where a distinct NFL with infinite critical bosons on the boson surface is realized. As the temperature rises to the point where the thermal fluctuation submerges the boson surface, the system experiences a crossover to a FM criticality-like NFL behavior. The solid (dashed) line refers to the phase transition (thermal crossover).

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To tackle with the bosonic fluctuation, a saddle point solution of \(\overrightarrow{\phi }\) is required, on top of which the lowest order expansion counts for the fluctuations. As shown in Fig. 3, we follow the Hertz approach by integrating out the gapless fermions. This gives rise to the Landau damping that dominates the low-energy boson dynamics. The effective action for the bosonic sector is written as,

$$\begin{array}{ll}{{{{\mathcal{S}}}}}_{{{{\rm{B}}}}}=\frac{1}{2}\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}{\Pi }_{\mu \nu }({{{\bf{q}}}},i{\omega }_{l}){\phi }_{\mu }({{{\bf{q}}}},i{\omega }_{l}){\phi }_{\nu }(-{{{\bf{q}}}},-i{\omega }_{l})\\ \qquad\quad\,\,+\,\frac{u}{4}\int\,d\tau \int\,{d}^{3}{{{\bf{r}}}}{[{\overrightarrow{\phi }}^{2}({{{\bf{r}}}},\tau )]}^{2},\end{array}$$
(3)

where μ, ν, λ label the vector components of \(\overrightarrow{\phi }\), and we have converted the 3D lattice index to a continuous, real space coordinate r. The fermionic bubble is illustrated in Fig. 3 and the renormalized boson polarization takes the form

$$\begin{array}{rcl}{\Pi }_{\mu \nu }({{{\bf{q}}}},i{\omega }_{l})&\!\!=\!\!&f(q,i{\omega }_{l}){\delta }_{\mu \nu }-iD{\epsilon }_{\mu \nu \lambda }\,{q}_{\lambda },\\ f(q,i{\omega }_{l})&\!\!=&r+J{q}^{2}+\frac{| {\omega }_{l}| }{{\Gamma }_{q}},\end{array}$$
(4)

where \({\omega }_{l}=2\pi l{\beta }^{-1}(l\in {\mathbb{Z}})\) is the Matsubara frequency for the bosons, and the ∣ωl∣/Γq is the Landau damping term. In general, the function Γq takes a form Γq = Γq with q = ∣q∣.

Fig. 3: Renormalized bubble diagrams.
figure 3

a The fermion bubble induced boson dynamics. b The renormalized fermion propagator from the renormalized boson correlator. The light and bold curly line represent the bare and renormalized boson correlators, respectively.

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Critical boson surface

The DM interaction complicates the low-energy theories by introducing the vector index into the bosonic sector. The dispersion of the bosonic modes is modified compared to the D = 0 case. Diagonalizing the bare quadratic bosonic part at ωl = 0, we obtain three branches of bosonic modes with dispersions given by

$${E}_{n}(q)=r+J{q}^{2}+nDq,\quad n=-1,0,+1.$$
(5)

The lowest branch E−1(q) is of particular interest, which reaches its minima on a spherical surface in the momentum space q = q0 ≡ D/(2J). Approaching the criticality at rc = D2/(4J), the lowest mode E−1(q) becomes gapless at the surface q = q0. Thus, the boson modes on the entire sphere q = q0 become critical simultaneously, which is then dubbed “critical boson surface”. The finite radius of the sphere is guaranteed by the DM interaction. The function Γq ≈ Γq0, and Γ is a constant due to the finite density of states on the Fermi surface.

For a pure classical boson system, the large phase space provided by the critical boson surface always results in a fluctuation-driven first-order transition46. The low-energy dynamics of the fermion-boson coupled system is not determined by the critical boson surface alone; rather the boson receives renormalization from the particle-hole excitation around Fermi surface. A previous work by Schmalian and Turlakov47 shed light on the nature of the quantum phase transition in the presence of the critical boson surface for a fermion-boson coupled system. The effective low-energy theory of the critical bosons are obtained by projecting onto the n = − 1 mode in Eq. (5), which turns out to be a ϕ4-theory with multiple quartic interaction constants. In certain parameter regimes, the transition can be a second-order transition in the university class d = 3, z = 2, and a mean-field theory of this second-order transition has been developed48. In the following, we regard that our system undergoes a second-order quantum transition and treat the critical fluctuation around the quantum critical point perturbatively. This physical scenario is realized within a crossover regime ξGi ≫ ξ ≫ ξDM47,49. Here \({\xi }_{{{{\rm{DM}}}}} \sim {q}_{0}^{-1}\) is the length scale of DM interaction. When the correlation length ξ ≫ ξDM, the fluctuation of the bosons is dominated by the critical boson surface. The opposite limit ξ ≪ ξDM dictates a high-temperature regime in the phase diagram of Fig. 2. In addition, we regard that the fluctuations are weakly interacting according to the Ginzburg criteria ξ ≫ ξGi, and will carry out the self-consistent renormalization study next.

Self-consistent renormalization theory

Although the patch and the hotspot theories are inadequate for our fermion–boson-coupled model, a phenomenological technique, dubbed the self-consistent renormalization (SCR) theory10,50, captures the key features of the fermion–boson coupling and provides the evidence for the behaviors near the criticality. The spirit of the SCR approach is to find the most appropriate quadratic action that encodes the effective renormalized non-linear interactions. This approach works well with d = 3, z = 2 and quantitatively produces the experimental results in many itinerant magnets of various dimensions2 and works for continuous and nearly continuous transitions8,9. We here make an attempt to implement the SCR calculation for our fermion–boson-coupled model and hope to gain some understanding about the physical properties of the model. Moreover, since the SCR approach has never been applied to itinerant magnets with degenerate low-energy critical modes, our attempt would add one physical example to the SCR theory.

To search for the best action in the SCR approach, one relies on Feynman’s variational method to optimize the free energy. Following the procedure by Moriya10, we consider a trial quadratic action with the following form

$$\begin{array}{lll}{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}[\tilde{r}]=\frac{1}{2}\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}\left[\left(\tilde{r}+J{q}^{2}+\frac{| {\omega }_{l}| }{{\Gamma }_{q}}\right){\delta }_{\mu \nu }-iD{\epsilon }_{\mu \nu \lambda }{q}_{\lambda }\right]\\ \qquad\qquad\times\, {\phi }_{\mu }({{{\bf{q}}}},i{\omega }_{l}){\phi }_{\nu }(-{{{\bf{q}}}},-i{\omega }_{l}),\end{array}$$
(6)

where we replace r with a variational parameter \(\tilde{r}\) that is to be determined. The boson correlator is given as a hermitian matrix Mμν (see Supplementary Note 6 for derivations),

$${\langle {\phi }_{\mu }({{{\bf{q}}}},i{\omega }_{l}){\phi }_{\nu }({{{{\bf{q}}}}}^{{\prime} },i{\omega }_{{l}^{{\prime} }})\rangle }_{{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}}\simeq {M}_{\mu \nu }({{{\bf{q}}}},i{\omega }_{l}){\delta }_{{{{\bf{q}}}},-{{{{\bf{q}}}}}^{{\prime} }}{\delta }_{i{\omega }_{l},-i{\omega }_{{l}^{{\prime} }}}$$
(7)

where \({\langle \cdots \rangle }_{{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}}\) refers to the statistical average against \({\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}\). This correlation conceives the information of the critical boson surface, and serves as a key ingredient for the mechanism of the proposed NFL that can be detected by the neutron scattering experiment. At the criticality, the powder-averaged neutron scattering spectrum is given as \(\,{{\mbox{Tr}}}\,M({{{\bf{q}}}},\omega ) \sim (\omega /{\Gamma }_{{q}_{0}})/[{J}^{2}{(q-{q}_{0})}^{4}+{(\omega /{\Gamma }_{{q}_{0}})}^{2}]\). Around the critical boson surface ∣q∣ = q0, the spectrum displays a divergent behavior. In the real space, the boson surface momentum q0 provides a characteristic length scale 1/q0, which endows the correlation function with a spatial modulation in all directions, and the correlator from the elastic neutron scattering is given by an envelop function on top of the usual power law decaying in the long-distance limit as \({\sum }_{\mu }{\langle {\phi }_{\mu }({{{\bf{r}}}}){\phi }_{\mu }(0)\rangle }_{{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}} \sim \sin ({q}_{0}| {{{\bf{r}}}}| )/| {{{\bf{r}}}}{| }^{2}\).

The variational free energy for the bosonic sector is

$$\begin{array}{ll}F(\tilde{r})\equiv \tilde{F}(\tilde{r})+\frac{1}{\beta }{\langle {{{{\mathcal{S}}}}}_{{{{\rm{B}}}}}-{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}\rangle }_{{\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}}\\ \qquad \,\,=\tilde{F}(\tilde{r})+\frac{1}{2\beta }(r-\tilde{r})\left[\,{{\mbox{Tr}}}\,\,\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}M({{{\bf{q}}}},i{\omega }_{l})\right]+\frac{u}{4{\beta }^{2}V}\\ \qquad\qquad\,\,\,\times\,\left\{{\left[{{\mbox{Tr}}}\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}M({{{\bf{q}}}},i{\omega }_{l})\right]}^{2}+2\,{{\mbox{Tr}}}\,{\left[\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}M({{{\bf{q}}}},i{\omega }_{l})\right]}^{2}\right\},\end{array}$$
(8)

where \(\tilde{F}(\tilde{r})\) is the free energy corresponding to the trial quadratic action \({\tilde{{{{\mathcal{S}}}}}}_{{{{\rm{B}}}}}\), and V is the system volume. The variational parameter \(\tilde{r}\) is determined from the saddle point equation \({\partial }_{\tilde{r}}F(\tilde{r})=0\). After the cumbersome calculation detailed in Supplementary Note 1, the optimization procedure results in the following self-consistent equation for the parameter \(\tilde{r}\),

$$\tilde{r}-r=\frac{uc}{\beta V}\mathop{\sum}\limits_{n=\pm 1,0}\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}\frac{1}{| {\omega }_{l}| /{\Gamma }_{q}+{{{{\mathcal{E}}}}}_{n}(q)},$$
(9)

where \({{{{\mathcal{E}}}}}_{n}(q)={E}_{n}(q){| }_{r\to \tilde{r}}\) and c is a constant prefactor. We further set \(\delta \equiv \tilde{r}-{r}_{{{{\rm{c}}}}}\) and δ0 ≡ r − rc. Here, δ measures the distance from the quantum critical point δc = 0 and is related to the correlation length ξ(T) that is expected to diverge at the criticality at low temperatures with δ(T) = ξ−2(T). From the above self-consistent equation, we find that the quantum fluctuation at finite temperatures is encoded in the function δ(T) that scales as (see Supplementary Note 2 for derivations)

$$\delta (T) \sim {T}^{\alpha },\quad \alpha =\frac{4}{5}.$$
(10)

It is illuminating to compare with the behaviors for the 3D FM (AFM) criticality where the dynamic exponent z = 3(z = 2) and the SCR calculation yields α = 4/3(α = 3/2). In fact, the same exponents were obtained from a simple scaling counting using Millis’s renormalization α = (d + z − 2)/z2. For both FM and AFM criticality, α > 1. In contrast, α < 1 in Eq. (10) indicates much stronger fluctuations due to the extensive phase space provided by the critical boson surface.

NFL behavior from critical boson surface

Unlike the FM or AFM criticalities where the low-energy fluctuations are at discrete momenta, the low-energy fluctuations near a finite boson surface strongly scatter the itinerant electrons and reduce the lifetime of electron quasiparticles. We use the renormalized boson correlator and the Feymann diagram in Fig. 3 to calculate the self-energy of the conduction electron,

$$\Sigma ({{{\bf{k}}}},i{\epsilon }_{n})\simeq \frac{{g}^{2}}{\beta V}\mathop{\sum}\limits_{{{{\bf{q}}}},i{\omega }_{l}}{G}_{0}({{{\bf{k}}}}+{{{\bf{q}}}},i{\epsilon }_{n}+i{\omega }_{l})\,{{\mbox{Tr}}}\,[M({{{\bf{q}}}},i{\omega }_{l})],$$
(11)

where \({\epsilon }_{n}\equiv (2n+1)\pi /\beta (n\in {\mathbb{Z}})\) is the fermion Matsubara frequency, and \({G}_{0}({{{\bf{k}}}},i{\epsilon }_{n})={(i{\epsilon }_{n}-{\xi }_{{{{\bf{k}}}}})}^{-1}\) is a bare Green’s function of the electrons with a dispersion ξk ≡ k2/(2m) − ϵF. By performing an analytic continuation iϵn → ω + iη, we obtain the T-dependence for the retarded self-energy in the static limit ω = 0 with ∣k∣ = kF located on the Fermi surface (see Supplementary Note 4 for derivations),

$${{{\rm{Im}}}}{\Sigma }^{{{{\rm{R}}}}}({{{\bf{k}}}},\omega =0;T) \sim {T}^{3}{\delta }^{-2} \sim {T}^{3-2\alpha }.$$
(12)

We note the T-dependence here has multiple sources that might render a simple scaling ~ ω/T inappropriate. We can see this by evaluating the ω-dependence of the self-energy in the zero temperature limit is evaluated in Supplementary Note 5, which yields \({{{\rm{Im}}}}{\Sigma }^{{{{\rm{R}}}}}({{{\bf{k}}}},\omega ;T=0) \sim \sqrt{\omega }\). This result is consistent with literature47.

The imaginary part of the self-energy in Eq. (12) determines the scattering rate for the forward scattering process provided the exchanged boson momentum is small q0 ≪ kF. The electronic resistivity (or the inverse transport lifetime) is obtained from the scattering rate by multiplying an angular factor \((1-\cos \theta )\), where θ ≃ q0/kF ≪ 1 is the small forward scattering angle. The narrow scattering angle suppress the resistivity by \(\sim {({q}_{0}/{k}_{{{{\rm{F}}}}})}^{2}\), and more importantly, the T-dependence is inherited from Eq. (12) as,

$$\rho (T) \sim \frac{{q}_{0}^{2}}{2{k}_{{{{\rm{F}}}}}^{2}}{{{\rm{Im}}}}{\Sigma }^{{{{\rm{R}}}}}({{{\bf{k}}}},\omega =0;T) \sim {T}^{7/5}.$$
(13)

This peculiar power law T-dependence indicates a distinct NFL behavior owing to the scattering of the electrons by the extensive critical fluctuations on the boson surface. The accuracy of this study is limited by the phenomenological method we used, namely, we only considered small-angle forward scattering. Whereas the large-angle scattering processes enabled by multiple scatterings on the Fermi surface are neglected. Moreover, the SCR method represents a way to bypass the significant challenge that it is extremely difficult to cook up a low-energy effective theory.

Crossover to FM-NFL at high-T

We have discussed the NFL behavior arised from the strong fluctuations near the boson surface at low-T which is quoted as an infinite critical boson NFL. When the temperature is further increased to be larger than a characteristic energy associated with the boson sphere radius q0, i.e., \(T\gg {\Gamma }_{{q}_{0}}\), the fluctuation is no longer dominated by the boson modes near the critical surface. In fact, the structure of the critical boson surface is no longer discernible at high temperatures. The boson sphere can be regarded as a point object in the reciprocal space, resembling the case of the FM criticality. In this high-temperature regime above the criticality, the temperature dependence of the variational parameter crossovers to scale as δ(T) ~ T4/3, which coincides with the case of the NFL from the FM fluctuations in 3D. The calculation is shown in Supplementary Note 3. Thus, the system undergoes a crossover between two distinct types of NFLs (see Fig. 2), and the crossover temperature can be approximately set by the difference of the boson energies at the center and at the surface of the boson sphere,

$${T}_{c} \sim {E}_{-1}(q=0)-{E}_{-1}({q}_{0}) \sim {{{\mathcal{O}}}}\left(\frac{{D}^{2}}{4J}\right).$$
(14)

NFL in the helimagnetic ordered phase

When r < rc, the system develops a magnetic order by spontaneously selecting the ordering wavevector from the degenerate boson surface [see Supplementary Note 7 for more discussions]. In the left corner of the phase diagram in Fig. 2, a helical order with an ordering wavevector at \({{{{\bf{q}}}}}_{1}={q}_{0}\hat{n}\) is picked up where \(\hat{n}\) is the propagating direction of the helimagnet. The original model in Eq. (1) is invariant under a combined rotation with respect to the real space and the internal space of the magnetic orders. The helimagnet spontaneously breaks this continuous symmetry and thus generates gapless Goldstone modes. In the helimagnetic phase, a small fluctuation above the helical order parameter couples to the itinerant electrons with

$$\sim g\int\,{d}^{3}{{{\bf{r}}}}\,{f}_{\alpha }^{{\dagger} }({{{\bf{r}}}}){\overrightarrow{\sigma }}_{\alpha \beta }{f}_{\beta }({{{\bf{r}}}})\cdot \delta \overrightarrow{\phi }({{{\bf{r}}}}),$$
(15)

where \(\delta \overrightarrow{\phi }({{{\bf{r}}}})={\phi }_{0}[-\varphi ({{{\bf{r}}}})\sin {q}_{0}z,\varphi ({{{\bf{r}}}})\cos {q}_{0}z,\theta ({{{\bf{r}}}})]\) with θ(r) and φ(r) describing the polar and the azimuthal phase fluctuations against the helical order \(\langle \overrightarrow{\phi }({{{\bf{r}}}})\rangle ={\phi }_{0}[\cos ({q}_{0}z)\hat{x}+\sin ({q}_{0}z)\hat{y}]\) with \(\hat{n}=\hat{z}\). These phase fluctuations give rise to gapless Goldstone modes51. The Yukawa coupling, g, remains finite in the low-energy limit. This is due to the fact that the generator for the continuous symmetry involves the orbital angular momentum and thus does not commute with the translation operator, i.e., the total momentum. Applying the general criteria in ref. 4, we see that the Goldstone mode converts the fermion sector into a NFL which is dubbed “Goldstone-NFL”. The NFL behavior in the helimagnetic phase, e.g., the electronic resistivity, has been discussed52,53.

Discussion

In summary, we have studied a distinct type of NFL in the 3D itinerant magnets that are not captured by the conventional patch and hotspot theories. The infinite critical boson modes on the boson sphere connect all momentum points on the Fermi surface for the itinerant electrons, leading to unconventional consequences for both local moments and itinerant electrons. The SCR approach adopted in our study is a phenomenological method that has been proven to be successful in explaining the FM and the AFM fermion criticalities10. For the continuously degenerate boson surface criticality, the SCR approach incorporates the large scattering phase space between fermions and bosons where the renormalized fermionic and bosonic correlations are taken into consideration sequentially.

On the more experimental side, several further investigations with material-based simulations can be expected. In reality, the boson sector would experience a cubic anisotropy in the interaction that would favor the wavevectors along either 001 or 111 directions and thus lift the degeneracy of the boson surface. This cubic anisotropy could set another crossover energy scale for the problem. The effect of external fields such as the pressure and magnetic field is anticipated due to its relevance to the experiments on MnSi41,42. An anomalous NFL transport behavior is observed in a widespread region of the phase diagram spanned by pressure and magnetic field. The experimental relevance of the infinite critical boson NFL to physical systems like MnSi may be addressed in a specific study with a more realistic consideration. In general, the role of a finite uniform magnetic field is two-folded. First, it leads to the procession of the spin order parameter, which competes with the Landau damping dynamics caused by the fermion. Second, this external field introduces anisotropy for the quantum critical points, which may effectively reduce the fluctuating dimension of the boson sector. Intuitively, the 2D counterpart of the present problem can be readily considered where a Fermi circle is coupled to a 1D critical boson contour. This critical boson contour can appear, for instance, on the interface of magnetic heterostructure54 or in 2D-frustrated magnets55. Particular interest lies in the situation where the radius of the boson contour can be commensurate or incommensurate to the Fermi circle. For the commensurate cases, only finite Fermi points are connected by the boson contour. The crossover/transition to incommensurate cases bridges the infinite critical boson NFL with the conventional one described by the hotspot theory. Moreover, one can consider the fermion–boson-coupled system in mixed dimensions, namely, the dimension of the Fermi surface is incompatible with critical boson modes. One intuitive example can be found in the 3D fractional quantum Hall system56. Soft gauge bosons in 3D bulk can couple with the chiral Fermi level of the partons on the 2D surface56, which may trigger a NFL instability on the surface. Beyond the scope of condensed matter physics, this general framework may apply to the meson–neutron coupling in neutron stars where the meson condensation forms a degenerate boson surface57.