1 Introduction

Heterostructures at various interfaces have been fascinating hunting grounds for emergent phenomena that are absent in the bulk constituents [1–3]. One of the most well-known emergent phenomena that were discovered on the heterostructure interfaces were the quantum Hall effects on the GaAs/AlGaAs semiconductor heterostructure [4] and the coexisting superconductivity and ferromagnetism (FM) on the \(\mathrm{LaAlO} _{3}\)/\(\mathrm{SrTiO} _{3}\) oxide interface [5]. More recently, tremendous efforts have been devoted into investigation on the oxide interface formed by two structurally and chemically distinct transition-metal compounds [6, 7]. The interface is associated with the breaking of the spatial inversion symmetry by design. The lower dimensionality reduces the bandwidth and leads to an enhancement of electron correlations, thus, giving rise to interesting many-body phenomena including magnetism, metal-insulator transition, unconventional superconductivity [8]. From the experimental point of view, the heterostructure geometry enables otherwise unattainable manipulation with the external fields and probing measurements. For instance, the detection of the atomic-scale magnetic skyrmion at Fe ultra-thin film [9] on the Ir(111) surface by scanning tunnelling microscopy [10] and the electric-field-driven switching of individual magnetic skyrmions [11]. Recent technical advances in the atomic-scale fabrication of oxide heterostructures and experimental probe of electronic and magnetic orders in ultra-thin films [10–14] have rendered the interfaces a promising platform for the study of various electron correlation phenomena.

One prominent example is the heterostructure interface of rare-earth nickelate oxides [3, 13] that is reported to be a quantum critical metal owing to its proximity to an antiferromagnetic (AFM) quantum critical point (QCP) [14–16]. The critical fluctuation of the magnetic order parameter around the QCP, i.e. the antiferromagnetic-type fermion criticality [17–20], is known as the critical boson. The electrons near the Fermi surface, when coupled to such critical bosons, are strongly scattered and enter the respective non-Fermi liquid phases or regimes [21–24]. The non-Fermi liquid behaviors are observed by the transport measurements at low temperatures [15, 25, 26] that do not exist in the bulk counterparts [14]. The resistivity shows a peculiar temperature dependence \({\Delta \rho (T) \sim T^{\alpha}}\), and provides an evidence for the emergence of unusual metallic non-Fermi liquid phases with \({\alpha \ne 2}\). More exotically, the scaling exponent is tunable within the interval \({1\le \alpha \le 2}\) upon applying pressure by strain [15, 26, 27]. The burst of the experimental evidence on correlated phenomena in oxide heterostructures urge for theoretical frameworks and understanding of the underlying mechanism.

We consider the heterostructure interfaces with both itinerant electrons and magnetism. The itinerant electrons can arise from the interfacial reconstruction and the charge transfer from the bulk for polar-non-polar interfaces [6], or simply from the metallic side of the heterostructure. The magnetism comes from the Mott insulating inside of the heterostructure or arise from the enhanced correlation at the interfaces. One of the key ingredients at many interfaces, as mentioned, is the breaking of the spatial inversion symmetry. While this renders a Rashba spin-orbit coupling for the itinerant electrons, the Fermi surface is most likely still present and the itinerant electrons remain in the metallic phase. For magnetism, however, one immediate effect is the introduction of the Dzyaloshinskii-Moriya(DM) interaction between the magnetic degrees of freedom [28–30]. With the DM interaction, the interfacial magnetism can display a distinct behavior around the quantum phase transition. In contrast to the conventional AFM quantum criticality, the magnetic order parameter fluctuation is dominated by the critical boson modes on a continuously degenerated minima [31], for instance, the critical boson circle is illustrated in Fig. 1. This continuously degenerated bosonic minima induced quantum critical phenomena have aroused extensive attention from the theoretical aspects [32–35]. In this paper, we consider a quantum critical metal with itinerant magnetism due to strong electron correlation at the two-dimensional (2D) oxide interface. The itinerant electrons are strongly coupled with the critical bosonic contour in 2D. It is shown that, when the radius boson contour is much smaller than the Fermi momentum, there exists an interesting crossover regime where the dynamical critical exponent \({z=3}\) instead of \({z=2}\). Due to the large effective dimensions \({d+z}\) in this crossover regime, we adopt a self-consistent renormalization method to tame the critical fluctuation of the magnetic moments and study the physical property of the electrons. We show that the electronic state enters an unconventional non-Fermi liquid by evaluating the temperature dependence of the resistivity at low energy. The interfacial magnetism can be deciphered by the neutron scattering experiment whose spectral weight is proposed to be divergent around the critical boson contour. Finally, we analyze the effect of the external magnetic field and the property of the field-tuned QCP.

Figure 1
figure 1

(Color online.) The comparison of the representative low-energy boson dispersions with and without the DM interaction, where the left contains the degenerate contour minima

Full size image

The structure of paper is organized as follows. In Sect. 2, we introduce the field-theoretical model for the coupled bosons and fermions, and demonstrate the presence of the critical boson contour and its impact on the low-energy dynamics of bosons. In Sect. 3, we consider the fluctuations in the vicinity of the critical bosonic contour by means of scaling analysis at tree level and self-consistent renormalization at one-loop level. In Sect. 4, we analyze the non-Fermi liquid behavior in transport by evaluating the quasiparticle lifetime. Finally, in Sect. 5, we discuss on the effect of external magnetic field and the case where electronic dispersion shrinks to discrete band touchings.

2 Critical boson contour

The interfacial system that we are considering includes both itinerant electrons and magnetic degrees of freedom. Thus, the model for the system naturally contains three parts. The first part describes the kinetic energy of the itinerant electrons, the second part describes the Kondo-like coupling between the itinerant electron and the magnetic moment, and the third part describes the magnetic moments. Since we assume a Fermi liquid metal for the itinerant electron sector before incorporating the coupling with the magnetic sector and the electron interactions would merely renormalize the kinetic part, then we can safely start with a renormalized electron kinetic energy. This fermion-boson coupled model takes the following form,

$$\begin{aligned} {\mathcal{L}}\bigl[f^{\dagger},f;\phi \bigr] = & \sum _{i,\alpha} f^{\dagger}_{i \alpha}(\partial _{\tau }- \mu ) f_{i\alpha} - \sum_{ij,\alpha} t_{ij} f^{\dagger}_{i\alpha} f_{j\alpha} \\ &{}+ g \sum_{i,\alpha \beta} f^{\dagger}_{i\alpha} \vec{\sigma}_{ \alpha \beta} f_{i\beta} \cdot \vec{\phi}_{i} + {\mathcal{L}}_{ \mathrm{B}} [\vec{\phi}], \end{aligned}$$
(1)

the first line dictates a tight-binding model for the fermion hopping on a 2D lattice with \(f^{\dagger}(f)\) the fermion creation (annihilation) operator of the itinerant electron. The \(\vec{\sigma}_{\alpha \beta}\) is the Pauli matrix vector with \({\alpha , \beta =1,2}\) being the indices for spin-\(1/2\). The fermionic spin couples to the magnetic order parameter \(\vec{\phi}_{i}\) via a Kondo-like Yukawa coupling with a magnitude g. We adopt a coarse-grained Landau-Ginzburg expansion for the magnetic fluctuations,

$$ \begin{aligned} {\mathcal{L}}_{\mathrm{B}}[\vec{\phi}] & = \int d^{2}{\boldsymbol {x}} \frac{1}{2} \biggl\{ \vec{\phi} \cdot \bigl(r-J\nabla ^{2}\bigr) \vec{\phi} + \frac{u}{2} \bigl( \vec{\phi}^{2}\bigr)^{2} \\ &\quad {} + D \bigl[ {\hat{y}\cdot (\vec{\phi}\times \partial _{x} \vec{ \phi}) } -\hat{x}\cdot (\vec{\phi}\times \partial _{y} \vec{\phi}) \bigr] \biggr\} , \end{aligned} $$
(2)

where the first line represents a standard \(\phi ^{4}\) theory. The second is the symmetry-allowed interfacial DM interaction. Due to the symmetry constraint, the DM vector \({\boldsymbol {D}}_{ij}\) is aligned along the normal n̂-direction of the interface. In the presence of this DM interaction, the global rotational symmetry is broken down to the U(1) rotation symmetry with respect to the n̂-direction. In reality, there exist other weak anisotropic interactions that would break the U(1) symmetry down to more discrete symmetries, and set new temperature and energy scales and drive different orders in the phase diagram of Eq. (2). We have assumed a dominant ferromagnetic exchange that favors an FM in the absence of DM interaction. Microscopically, the DM interaction could come from either the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [36] via the itinerant electrons or the superexchange interaction between the local moments themselves. In the former case, which is demonstrated in Fig. 2, the RKKY interaction could transfer the Rashba spin-orbit coupling of the interfacial itinerant electrons to the local moments and generates the DM interaction. By aligning the normal vector n̂ along the ẑ-axis, the DM interaction on the lattices can be transformed into the continuum as dictated in Eq. (2). In the latter case, the superexchange interaction contains the DM interaction as the consequence of the atomic spin-orbit coupling of the local moments. Generically, the DM interaction favors the canting of adjacent moments, when strong enough, leads to non-collinear and twisted magnetic configurations [10, 12]. Experimental evidence for the interfacial DM interaction have been accumulating from the observation of magnetic ground state configuration and corresponding excitation spectrum by means of various spectroscopy [10–12, 28]. The interfacial DM interaction has also been verified by the first principal-calculations for the Fe/Ir(111) interface [37] and so on [38].

Figure 2
figure 2

(Color online.) Schematic illustration for generating mechanism of the interfacial DM interaction. The magnetic moments in the upper(grey) layer is coupled to electrons from the lower(cyan) one. Interfaces between magnetic materials and materials with large spin-orbit interactions gives rise to interfacial DM interactions. The interfacial DM interaction Hamiltonian takes a form \(H_{\mathrm{DM}} = -{\boldsymbol {D}}_{ij}\cdot ({\boldsymbol {S}}_{i} \times {\mathbf{S}}_{j})\). \({\boldsymbol {D}}_{ij}\) is the DM vector originating from the RKKY process involving the electrons with large SOC from the lower layer. n̂ and \(\hat{r}_{ij}\) are the directional vectors along the normal direction of the interface and connecting adjacent spins \({\boldsymbol {S}}_{i}\), \({\boldsymbol {S}}_{j}\), respectively. By fixing the interfacial norm as \({\hat{n}}={\hat{z}}\), \({\boldsymbol {D}}_{ij} =( \hat{n}\times {\hat{r}}_{ij} )D\) lies within the interfacial plane. The DM interaction is formulated as \(\sim (\hat{z} \times {\hat{x}}) \cdot ({\mathbf{S}}\times \partial _{x} { \mathbf{S}}) + (\hat{z} \times {\hat{y}}) \cdot ({\mathbf{S}}\times \partial _{y} {\mathbf{S}}) \)

Full size image

2.1 DM interaction and critical boson contour

The presence of this 2D DM interaction fundamentally modifies the physics of the bosonic sector. Without the DM interaction, the bosons would condense at the momentum \({{\boldsymbol {q}} = {\boldsymbol {0}}}\) with the FM order for \({r=0}\). In the presence of the DM interaction, the system reaches a quantum critical point (QCP) locating at \({r=r_{c}}\), and the minima of the boson dispersion constitutes a degenerate contour in the 2D reciprocal space with a radius \(q_{0}\) (see Fig. 1). The eigenmodes on the contour corresponding to non-collinear and twisted magnetic structures. To explore the dynamic property of the bosonic sector, we follow the established Hertz-Millis scheme [21, 22] and firstly integrate out the fermionic degree of freedom as shown in Fig. 5(a). This process generates a Landau damping term for the bosons that arises from the particle-hole excitations around the Fermi level. The effective boson action takes a form

$$\begin{aligned}& {\mathcal{{S}}}_{\mathrm{B}} = {{{\mathcal{S}}_{\mathrm{B}}}^{(2)} } + \frac{u}{4} \int d \tau \int d^{2}{\boldsymbol {x}} \bigl[\phi ^{2}({ \boldsymbol {x}},\tau ) \bigr]^{2} , \end{aligned}$$
(3)
$$\begin{aligned}& {{{\mathcal{S}}_{\mathrm{B}}}^{(2)} } = \frac{1}{2}\sum _{{\boldsymbol {q}}, i\omega _{l}} \Pi _{\mu \nu}({\boldsymbol {q}},i\omega _{l}) \phi _{\mu}({\boldsymbol {q}}, i \omega _{l}) \phi _{\nu}(-{\boldsymbol {q}},-i\omega _{l}) , \end{aligned}$$
(4)

where \({\mu ,\nu =x,y,z}\) label the vector components of ϕ⃗, and q and \(\omega _{l}=2\pi l/\beta \) (\(l\in {\mathbb{Z}}\)) are the momentum and bosonic Matsubara frequency, respectively. The renormalized quadratic action, i.e. the polarization bubble in Fig. 5(a), takes the following form

$$ \Pi ({\boldsymbol {q}},i\omega _{l})= \begin{bmatrix} f({\boldsymbol {q}},i\omega _{l}) & 0 & iDq_{x} \\ 0 & f({\boldsymbol {q}},i\omega _{l}) & -iDq_{y} \\ -iDq_{x} & iDq_{y} & f({\boldsymbol {q}},i\omega _{l}) \end{bmatrix} . $$
(5)

The function on the diagonal entry is given by

$$ f(q,i\omega _{l}) = r+ J {q}^{2} + \delta \Pi (q,i\omega _{l}), $$
(6)

where \(\delta \Pi (q,i\omega _{l})\) represents the boson self-energy correction as derived in Sect. 2.3. Generically, we consider the characteristic form \(\delta \Pi (q,i\omega _{l}) = |\omega _{l}| /{{\Gamma}}_{q}\) which is known as the Landau damping term. We diagonalize the bare part of the quadratic boson action and obtain three branches of the eigenmodes,

$$\begin{aligned} & {\mathcal{E}}_{n}({\boldsymbol {q}},i\omega _{l}) = f({ \boldsymbol {q}},i\omega _{l}) +nDq, \quad n=0,\pm 1. \end{aligned}$$
(7)

The expression of the eigenmodes, particularly the lowest branch with \({n=-1}\), determines the nature of the bosonic sector at the QCP and the property of the magnetic phase transition.

We propose that the low-energy physics in the quantum critical regime is crucially affected even by a weak DM interaction (\({D\ll J}\)). As dictated in Eq. (4) and Eq. (5), the DM interaction term 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 where the critical boson mode reaches its minima only at discrete momenta such as the FM and AFM criticalities. In the presence of the interfacial DM interaction, there exist infinite critical bosons on continuously degenerate minima. Let us focus on the lowest eigenmode, i.e. the \({n=-1}\) branch of Eq. (7), in the static limit \({\omega _{l}=0}\). The dispersion of the lowest \({n=-1}\) branch is now abbreviated as \({\mathcal{E}}_{q}\) and reads

$$ {\mathcal{E}}_{q}= \delta + J(q-q_{0})^{2}, $$
(8)

which reaches its minimum on a contour in the momentum space described by \({q=q_{0} \equiv D/(2J)}\). The modes on the contour become simultaneously gapless when we approach the critical point locating at \({r=r_{c}= D^{2}/4J}\). The representative phase diagram is depicted in Fig. 3. On the right corner, the local moment (or bosonic) sector of the system realizes a paramagnetic phase with the gapped magnetic excitations. On the left corner, the local moment sector develops a helical magnetic order that spontaneously breaks the U(1) symmetry of the model in Eq. (2). At finite temperatures, there is no magnetic order in 2D, and there will be a Kosterlitz-Thouless-like transition due to the U(1) symmetry [39]. This U(1) symmetry is present due to the simplification of the chosen model. In reality, it can be broken by other weak anisotropies that will be mentioned in Sect. 5, and then there can be magnetic orders at finite temperatures. At the quantum phase transition, all the bosonic modes on the degenerate contour become critical at the same time, and thus we dub the degenerate contour “critical boson contour” (CBC).

Figure 3
figure 3

(Color online.) The schematic phase diagram. At zero temperature, there exists a QCP at \({r=r_{c}}\) that separates the zero-temperature magnetic order and the paramagnet (PM). The PM extends to the finite-temperature regime where the fermions are in the Fermi liquid (FL) metal phase. In contrast, the magnetic order gives way to the quasi-long-range order at finite temperatures that further experiences a Kosterlitz-Thouless-like transition. The QCP mediates a quantum critical regime at finite temperatures where the infinite critical boson induces a novel type of non-Fermi liquid (NFL). The crossover to the conventional FM criticality-induced NFL is experienced at high temperatures. The dashed lines mark the crossover from infinite critical boson NFL to other quantum critical regimes

Full size image

2.2 Fermion-boson coupling

Here we consider the coupling between the fermions on the Fermi surface and the bosonic modes. In the paramagnetic phase, the bosonic modes are fully gapped and do not appear at low energies, thus the bosonic modes do not have qualitative impacts on the fermions on the Fermi surface. The fermion sector is a conventional Fermi liquid. We focus on the critical region where the CBC is relevant. Due to the CBC, each fermion mode (\({ \boldsymbol {k}}_{1}\) in Fig. 4) on the Fermi surface is scattered to two other fermion modes (\({\boldsymbol {k}}_{2}\) and \({\boldsymbol {k}}_{3}\) in Fig. 4) on the Fermi surface. As the fermion mode at \({\boldsymbol {k}}_{2}\) is further coupled to the fermion mode at \({\boldsymbol {k}}_{4}\), eventually the whole Fermi surface would be strongly coupled together except when the radius of the boson contour is commensurate with the Fermi momentum. The commensurability condition is established when multiple scatterings by the CBC bring the fermion back to the starting spot on the Fermi surface, i.e.

$$ \sin ^{-1}\biggl(\frac{q_{0}}{2k_{F}}\biggr) = \frac{m\pi}{n} , $$
(9)

where m and n are both integers. This means, n-times scattering is able to bring the fermion to encircle the Fermi surface m times and return to the starting point. For the commensurate case, the physics may look like the ‘hotspot theory’ for the AFM quantum criticality where the hotspots of the Fermi surface connected by the AFM ordering wavevectors become critical, and here each fermion on the Fermi surface is coupled to the a finite number of fermion modes connected by the relevant critical boson modes. But there is a key difference. In the current case, all the fermions on the Fermi surface are critical, while there exist a finite number of critical fermions for the AFM quantum criticality.

Figure 4
figure 4

(Color online.) (a) The coupling between the (dashed blue) CBC and the (solid gray) Fermi surface. \(k_{F}\) is the Fermi momentum, and \(q_{0}\) is the radius of the boson contour. (b) An example of commensurate coupling between the Fermi surface and the CBC. Here \({m=1,n=10}\) according to Eq. (9)

Full size image

For more generic cases, such a commensurability condition in Eq. (9) can not be satisfied. All the modes on the Fermi surface are eventually coupled together by the CBC which amounts to a ‘hot Fermi surface’. The presence of infinite many critical bosons on the CBC introduce considerable complexity for the fermion-boson coupled system. On one hand, the infinite many critical boson modes on the CBC resemble the FM quantum criticality; On the other hand, the magnitude of the critical boson modes is finite and identical which resembles the AFM quantum criticality. In this sense, we expect that the properties of both FM and AFM quantum criticalities are present in the fermion-boson coupled system with the CBC in distinct quantum critical regimes. It is extremely challenging to develop a unified theoretical framework to incorporate both FM and AFM quantum criticalities. Starting from the next section, we make an attempt to decipher the complicated problem by considering the FM or AFM quantum criticality in respective limits and relate the universal properties at the critical point to the transport phenomenon. In a previous study on 3D counterpart of the fermion-boson coupled problem [32], we are interested in the quantum critical regime mediated by the AFM quantum criticality; for the present study, we explore the quantum critical regime with FM quantum criticality. We hope these complementary studies unfold the complexity encoded in the quantum criticality associated with infinite critical bosons and drawn more attention to this intriguing problem.

2.3 Boson polarization function

For the 2D fermion-boson coupled system in the presence of the CBC, distinct low-energy dynamics can emerge for the critical bosons provided the ratio of the CBC and Fermi surface radius is in the respective limits. We demonstrate this point by evaluating the boson polarization function due to particle-hole excitation around the Fermi surface. We show that in the small boson radius limit \(q_{0}/k_{F} \ll 1\), the system mimics a FM quantum criticality with the dynamical critical exponent \({z=3}\). In a previous study [32], we explore the consequence of fermions interacting with a commensurate boson radius which amounts to the \(z=2\) dynamical critical exponent. In this study, we consider the \(z=3\) quantum criticality and explain the crossover region between the two different types of quantum critical regimes.

The boson polarization function is the renormalized boson self-energy correction at the one-loop level as illustrated by the Feynman diagram in Fig. 5(a), which reads

$$ \begin{aligned}[b] \delta \Pi ({\boldsymbol {q}},i\omega _{l}) = g^{2} \int \frac{d\epsilon _{n} d^{2}{\boldsymbol {k}}}{(2\pi )^{3}} \frac{1}{i\epsilon _{n} -\xi _{\boldsymbol {k}}} \frac{1}{i(\epsilon _{n}+\omega _{l}) -\xi _{{\boldsymbol {k}}+{\boldsymbol {q}}}}. \end{aligned} $$
(10)

\({\xi _{\boldsymbol {k}} = {{\boldsymbol {k}}^{2}}/{(2m)}-\epsilon _{F}}\) is the parabolic dispersion around the Fermi level \(\epsilon _{F}\). The kinematics of the Fermi surface and the CBC and their scattering configuration plays an important role in low-energy dynamics. We adopt the so-called global coordinate [40] for the fermion momentums where the Fermi momentums are parametrized by the surface radius \(k_{F}\) and a unit angular vector \(\hat{k}_{F}\). A given point near the Fermi surface is uniquely labeled by the nearest Fermi point and the radial small momentum variation

$$ {\boldsymbol {k}}= \hat{k}_{F}(k_{F} + \delta k). $$
(11)

The cubic form of the Yukawa interaction indicates that the boson momentum is the fermions momentum difference during the scattering. The boson momentum near the CBC can be written as (see Fig. 4)

$$ {\boldsymbol {q}} = {\boldsymbol {q}}_{0} + \delta {\boldsymbol {q}}, \quad \delta {\boldsymbol {q}} = \hat{q}_{0} \delta q_{\parallel }+ \delta {\boldsymbol {q}}_{\perp}. $$
(12)
Figure 5
figure 5

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

Full size image

In a small boson radius limit

$$ q_{0}/k_{F} \ll 1, $$
(13)

the fermion on the Fermi surface is only scattered to nearby regions and the Fermi momentum is approximately perpendicular to the bosonic one, i.e. \(|{\hat{k}}_{F} \cdot {\hat{q}}_{0}| \simeq \mathcal{O} ( q_{0}/k_{F} )\). The fermion dispersions can linearize approximately as

$$\begin{aligned}& \xi _{{\boldsymbol {k}}} \simeq \frac{{\boldsymbol {k}}_{F} \cdot \delta {\boldsymbol {k}}}{m}, \end{aligned}$$
(14)
$$\begin{aligned}& \xi _{{\boldsymbol {k}}+{\boldsymbol {q}}} \simeq \frac{ {\boldsymbol {k}}_{F} \cdot [ \delta {\boldsymbol {k}}+ ({\boldsymbol {q}}_{0} +\delta {\boldsymbol {q}}) ]}{m} \end{aligned}$$
(15)

The boson polarization function is evaluated in the static limit \(|\omega _{l}|\ll v_{F} \delta q\), which yields

$$\begin{aligned} \delta \Pi (\delta{\boldsymbol {q}},i\omega _{l}) =& g^{2} \int \frac{d\epsilon _{n} \,d(\delta k) k_{F} \,d\hat{k}_{F}}{(2\pi )^{4}} \frac{1}{i\epsilon _{n} -v_{F} \delta k} \\ &{}\times \frac{1}{i(\epsilon _{n}+\omega _{l}) -v_{F}(\delta k + {\hat{k}}_{F} \cdot \delta {\boldsymbol {q}} + q_{0}^{2}/k_{F}) } \\ = & i \frac{g^{2} k_{F}}{v_{F}} \int \frac{d\epsilon _{n} \,d\hat{k}_{F}}{(2\pi )^{3}} \frac{{\mathrm{sgn}}(\epsilon _{n}) - {\mathrm{sgn}}(\epsilon _{n}+\omega _{l})}{v_{F} {\hat{k}}_{F} \cdot \delta {\boldsymbol {q}} + q_{0}^{2}/m - i \omega _{l} } \\ \simeq & \frac{g^{2} k_{F}}{4\pi ^{2} v_{F}} \frac{\omega _{l} {\mathrm{sgn}}(\omega _{l})}{ \sqrt{ (v_{F} \delta q)^{2}- (q_{0}^{2}/m)^{2} }}. \end{aligned}$$
(16)

We consider an intermediate momentum regime

$$ q_{0}(q_{0}/k_{F}) \ll \delta q < q_{0}. $$
(17)

This is a feasible regime because \(q_{0}/k_{F}\ll 1\). This regime recovers the Landau damping term that amounts to the dynamic exponent \(z=3\)

$$\begin{aligned}& \delta \Pi (\delta {\boldsymbol {q}},i\omega _{l}) = \frac{ \vert \omega _{l} \vert }{{{\Gamma}}_{q}}, \end{aligned}$$
(18)
$$\begin{aligned}& {{\Gamma}}_{q} \equiv {{\Gamma}} (\delta q), \ {{\Gamma}}= \frac{4\pi ^{2}v_{F}^{2}}{g^{2} k_{F}}. \end{aligned}$$
(19)

In a small momentum regime \(\delta q < q_{0}(q_{0}/k_{F})\), we obtain the Gilbert damping term \(\sim |\omega _{l}|/{{\Gamma}}_{q_{0}}\) with \({{\Gamma}}_{q_{0}} = \frac{g^{2} m^{2}}{4\pi ^{2} q_{0}^{2}}\).

In addition, we check that when the small boson radius condition[Eq. (13)] is violated, the Gilbert damping with \(z=2\) is obtained irrespective of the momentum regimes. This is due to different ways of linearizing the fermionic dispersion around Fermi surface. For a finite boson radius \({q_{0} \lesssim k_{F}}\), a given Fermi point \({\boldsymbol {k}}_{F}\) in the vicinity of the Fermi surface is clearly scattered to a distinct point \({\boldsymbol {k}}_{F}^{\prime }= {\boldsymbol {k}}_{F} + {\boldsymbol {q}}_{0}\) which yields

$$\begin{aligned}& \xi _{{\boldsymbol {k}}} \simeq {\boldsymbol {k}}_{F} \cdot \delta { \boldsymbol {k}}/m, \end{aligned}$$
(20)
$$\begin{aligned}& \xi _{{\boldsymbol {k}}+{\boldsymbol {q}}} \simeq {\boldsymbol {k}}_{F}^{\prime } \cdot ( \delta { \boldsymbol {k}} + \delta {\boldsymbol {q}})/m. \end{aligned}$$
(21)

The 2D momentum integral can be carried out in terms of two new variables: \(k_{1} = {\boldsymbol {k}}_{F}/m \cdot \delta {\boldsymbol {k}}\), \(k_{2} = {\boldsymbol {k}}_{F}^{ \prime}/m \cdot \delta {\boldsymbol {k}}\). The boson polarization function is calculated as,

$$\begin{aligned}& \delta \Pi (\delta {\boldsymbol {q}},i\omega _{l}) \\& \quad = \frac{g^{2} }{ \vert {\boldsymbol {k}}_{F} \times {\boldsymbol {k}}_{F}^{\prime} \vert /m^{2}} \int \frac{d\epsilon _{n} \,dk_{1} \,dk_{2} }{(2\pi )^{3}} \frac{1}{i\epsilon _{n} - k_{1}} \frac{1}{i(\epsilon _{n}+\omega _{l}) -k_{2} } \\& \quad = \frac{g^{2} }{ \vert {\boldsymbol {k}}_{F} \times {\boldsymbol {q}}_{0} \vert /m^{2}} \int \frac{dk_{1} \,dk_{2}}{(2\pi )^{2}} \frac{{\mathrm{sgn}}(k_{2})-{\mathrm{sgn}}(k_{1})}{k_{1}-k_{2} - i\omega _{l}} \\& \quad \simeq -\frac{g^{2}}{4\pi k_{F} q_{0}/m^{2} } \vert \omega _{l} \vert . \end{aligned}$$
(22)

The two different damping terms affect the low-energy physics of the fermion-boson coupled system in the distinct fashions. We have explored the NFL behaviors induced by the infinite many critical boson modes with \({z=2}\) in a previous study for the three-dimensional case [32]. In the present study, we make use of the Landau damping term obtained in the static limit and study the NFL on the heterostructure interfaces induced by the magnetic fluctuations that are described by an infinite number of critical boson modes with the university class \({z=3, d=2}\). We note that developing a unified theoretical framework to incorporate both regimes is quite difficult. We hope that the attempt here would raise the interest in investigating the quantum criticality induced by infinitely many critical bosons and its associated non-Fermi liquid behaviors when coupled to the fermionic degree of freedom.

3 Magnetic fluctuations with critical boson contour

In this section, we address the magnetic fluctuations due to the critical boson contour. The Ginzburg-Landau theory describes a continuous magnetic transition at zero temperature upon the variation of the tuning parameter r. Near the quantum critical point, the wild quantum fluctuation induces a non-Fermi liquid behavior at low yet finite temperatures. J. Hertz pointed out that the dynamical and static properties are intertwined and thus should be treated on equal footing [21]. A distinct universality class can be identified according to the dynamic exponent that is determined by the coupling between the magnetic order parameter and the fermionic quasi-particles. The bare term receives a self-energy correction taking the form of the Landau damping term as derived in Eq. (19). The Landau damping term overwrites the original dynamic term in the low-energy limit and the tangential component is introduced which turns out to be important for the low-energy and low-temperature fluctuations.

In order to analyze the phenomena related to the novel boson fluctuations, we first carry out a scaling analysis for physical parameters and field operators on the tree level. The validity of the Hertz-Millis method is examined for the university class \(d=2\), \(z=3\) with the presence of a critical boson contour. Then, we implement a self-consistent renormalization (SCR) developed by T. Moriya to incorporate the effect of the interaction on the one-loop level. The SCR scheme is built on the Hertz-Millis model and is used to tame the wild fluctuation around QCP.

3.1 Magnetic correlation

We have shown that the low-energy physics of the interfacial magnetism is dominated by the CBC, whose finite radius is guaranteed by the DM interaction. The stability of the metallic Bose state was proposed theoretically a long time ago [41–44], which remains elusive until recent experimental observation in high-\({T_{c}}\) superconductor films [45, 46] Recently, a fascinating Bose metallic phase with the CBC is proposed by the name “Bose Luttinger liquid” [31, 33]. The CBC plays a similar role as the Fermi surface in a Fermi liquid. The stability of this unconventional critical phase is elucidated in terms of a patch theory [31]. The momentum-space annulus around the CBC is decomposed into patches, and the inter-patch interaction is shown to be negligible in a certain limit. The low-energy dispersion on each patch depends solely on the parallel component \(\delta q_{\parallel}\). This quasi-one-dimensional nature of the critical fluctuation is captured by a multi-dimensional bosonization scheme, or equivalently the coupled wire construction [47–51]. These methods are known to be effective in studying competing orders above one dimension, yielding a sliding Luttinger liquid description. In this study, we are not concerned about the stability of the critical phase of bosonic matter which has been addressed extensively. Rather, we assume the interfacial magnetic phase transition stabilizes the CBC and investigate the physical consequences in the quantum critical regime for the magnetic sector as well as the electronic sector.

The structure in the boson correlation function sheds light on the existence of the CBC. We evaluate the correlation function in the momentum and frequency space at the quadratic level. The correlation function is given by the susceptibility matrix \(M_{\mu \nu} ({{\boldsymbol {q}}, i\omega _{l}})\)

$$\begin{aligned} & \bigl\langle \phi _{\mu} ({\boldsymbol {q}}, i \omega _{l}) \phi _{\nu} \bigl({ \boldsymbol {q}}', i \omega _{l'}\bigr) \bigr\rangle _{{\mathcal{S}}^{(2)}_{\mathrm{B}}} \\ & \quad = M_{\mu \nu} ({{\boldsymbol {q}}, i \omega _{l}}) \delta _{{\boldsymbol {q}} ,-{\boldsymbol {q}}'} \delta _{i \omega _{l} ,-i \omega _{l'}} , \end{aligned}$$
(23)

where the subindex \({{{\mathcal{S}}_{\mathrm{B}}}^{(2)}}\) refers to the statistical average over the quadratic action \({{{\mathcal{S}}_{\mathrm{B}}}^{(2)}}\). The susceptibility matrix is an inverse of the polarization function that reads

$$\begin{aligned} \begin{aligned}[b] & M({\boldsymbol {q}}, i\omega _{l}) \\ & \quad = \frac{1}{{\mathrm{Det}} [\Pi ]} \begin{bmatrix} f^{2}-D^{2} q_{y}^{2} & -D^{2}q_{x}q_{y} & -i f Dq_{x} \\ -D^{2}q_{x}q_{y} & f^{2}-D^{2} q_{x}^{2} & i f Dq_{y} \\ i f Dq_{x} & -i f Dq_{y} & f^{2} \end{bmatrix}, \end{aligned} \end{aligned}$$
(24)

where the trace of the polarization function reads \({\mathrm{Det}}[\Pi ] = 3f^{2}-D^{2}q^{2}\). The trace of Eq. (24) corresponds to the angular averaged spectrum at the criticality,

$$\begin{aligned} {\mathcal{F}}({\boldsymbol {q}},\omega ) \sim &{\mathrm{Tr}} \bigl[ M({ \boldsymbol {q}},\omega ) \bigr] \\ =& \frac{1}{f-Dq} +\frac{1}{ f}+ \frac{1}{ f+Dq}. \end{aligned}$$
(25)

We notice that the first term conceives a divergent contribution from the CBC. Therefore, we single out the divergent contribution at the criticality and the spectrum displays a divergent behavior around the CBC,

$$ {\mathcal{F}}({\boldsymbol {q}},\omega ) \sim \frac{(\omega /{{\Gamma}}_{q})}{J^{2}(q-q_{0})^{4}+(\omega /{{\Gamma}}_{q})^{2}}. $$
(26)

In the static limit \({\omega /{{\Gamma}}_{q}\rightarrow 0}\), the expression of the spectrum in Eq. (26) corresponds to the Dirac delta function divergence in the momentum space for the approximation of the quadratic action.

3.2 Scaling analysis

At the mean-field level, the magnetic correlation function gives a clear evidence for the existence of the CBC which can be observed by spectroscopic detection. The thermodynamic and transport measurement involves finer details that go beyond the mean-field method, which requires properly taming the wild fluctuation. We employ the SCR method to include the boson interactions in a perturbative manner. The validity of SCR is built on the foundation of the Hertz-Millis method: the bosonic sector has a Gaussian fixed point and the critical phenomenon is described by a Ginzburg-Landau expansion [see Eq. (2)]. Moreover, the Gaussian fixed point is immune to introducing fermion-boson coupling.

We justify the applicability of the Hertz-Millis method by carrying out the scaling analysis at the tree-level renormalization. The scaling dimensions of the fermion and boson momentum variations are determined by the low-energy scattering configuration illustrated in Fig. 6. We adopted the global coordinate to uniquely label the momentum variation around the Fermi surface, namely only the radial component has a finite scaling dimension \({\mathrm{dim}}[\delta k]=1\) while the tangential component is treated as an angular variable with zero scaling dimension. For a given Fermi momentum point, the boson momentum can be decomposed into components as in Eq. (12). The radial and tangential components of boson momentums have scaling dimensions: \({{\mathrm{dim}}[\delta q_{\parallel}]=1, \ {\mathrm{dim}}[\delta q_{\perp}]= \gamma}\) where the value of \(\gamma \in [\frac{1}{2},1]\) reflects the nature of curved boson contour [52]. We further take the convention that the fermionic and bosonic sectors have the same dynamic exponent \({\mathrm{dim}}[\epsilon _{n}]={\mathrm{dim}}[\omega _{l}] = z\).

Figure 6
figure 6

(Color online.) The scattering process of the fermions on the Fermi surface by the bosons near the CBC. The (gray) curve stands for the Fermi surface that is coupled to the (dashed) CBC of the bosonic sector. The (red) vectors are the small momentum variation from the CBC. The boson momenta \({\boldsymbol {q}}_{0}\) and δq are exaggerated for clarity. The boson momentum deviation δq is decomposed into the parallel (perpendicular) component \(\delta {\boldsymbol {q}}_{\parallel}\) (\(\delta { \boldsymbol {q}}_{\perp}\)). The parallel direction is aligned to the vector \({\hat{q}}_{0}\)

Full size image

By requiring that the Gaussian parts of the bosonic and fermionic theories are invariant under renormalization, we obtain scaling dimensions for various field operators and interactions [see detailed derivation in Appendix B]. The four-fermion interaction is irrelevant ensuring the Fermi liquid fixed point. The quartic boson interaction u, with a scaling dimension \({\mathrm{dim}}[u]=3-z-(d-1)\gamma \), as well as all higher-order interactions are all irrelevant for the university class \(d=2\), \(z=3\). In addition, the Yukawa coupling constant is also irrelevant \({\mathrm{dim}}[g]=- [z-1+(d-3)\gamma ]/2\). These results are independent of the choice of γ brought by the boson curvature effect. In summary, the Ginzburg-Landau action [in Eq. (4)] is a faithful expansion for the critical bosons which amounts to a Gaussian fixed point. The Yukawa coupling can be therefore treated perturbatively. and the Gaussian fixed point is stable against the fermion-boson coupling.

3.3 Self-consistent renormalization

In the vicinity of the FM or AFM quantum critical points, the fluctuation is governed by the ordering wavevector. In contrast, in the presence of the CBC, the pattern of the fluctuation is completely altered. The low-temperature fluctuation would be governed by the CBC. To understand the novel features of the fluctuation in the critical properties, we adopt the self-consistent renormalization (SCR) [23, 53]. This approach was found to be quite effective in the study of ferromagnetic and antiferromagnetic criticalities, and gave consistent results about physical properties with the experimental measurements. The spirit of the SCR approach is essentially variational. It is to find the most appropriate variational free energy at the quadratic level to replace the original free energy that contains the quartic interaction. This approach amounts to decoupling the quartic interaction and incorporating the interaction with a self-consistent Born approximation. Technically, it corresponds to conducting a renormalized one-loop approximation for the scattering of the order parameters. The renormalization process endows the non-thermal parameter r with a temperature dependence \({r(T)=r_{c} +\delta (T)}\) after considering the effect of thermal fluctuation and correlation. Here, \(\delta (T)\) measures the relative distance towards the critical point and is associated with the inverse square of the correlation length which diverges from approaching the critical point. The temperature dependence of δ (or equivalently r) plays an important role in determining low-temperature thermodynamics.

To find the appropriate action and determine the scaling law of \(\delta (T)\), one relies on Feynman’s variational approach to optimize the free energy. The proposed variational free energy for the bosonic sector is of the following form,

$$ F ({\tilde{r}}) \equiv \tilde{F} ({\tilde{r}}) + \frac{1}{\beta} \langle { \mathcal{S}}_{\mathrm{B}} - \tilde{\mathcal{S}}_{\mathrm{B}} \rangle _{ \tilde{\mathcal{S}}_{\mathrm{B}}}, $$
(27)

where the quadratic part of the variational free energy is

$$ \tilde{F} (\tilde{r}) \equiv -\beta ^{-1} \ln {\mathrm{Tr}} \bigl(e^{- \tilde{\mathcal{S}}_{\mathrm{B}}} \bigr). $$
(28)

The variational free energy \({\tilde{\mathcal{S}}_{\mathrm{B}}} (\tilde{r})\) is defined by introducing a variational parameter r̃, which remains undetermined, in exchange for the control parameter r in Eq. (4) and Eq. (5). We then establish the saddle point equation with respect to r̃. The SCR equation \({\partial _{\tilde{r}} F(\tilde{r}) =0 }\) has an expression

$$ \tilde{r} = r + \frac{cu}{\beta V} \sum_{{\boldsymbol {q}}, i\omega _{l}} { \mathrm{Tr}} \bigl[M({\boldsymbol {q}}, i\omega _{l}) \bigr], $$
(29)

where c is a non-universal number. We solve the self-consistent equation at low temperatures and provide the technical details in a step-by-step manner in the appendix. The solution in favor of \(\delta (T)\) exhibits a simple scaling law

$$ \delta (T) = T^{\alpha}, \quad \alpha =\frac{5}{9} . $$
(30)

As promised, the non-thermal parameter acquires a T-dependence \({r(T)= r_{c} + \delta (T)}\) as a result of the SCR procedure. The exponent is much smaller than the values for 2D FM (\({\alpha =4/3}\)) and AFM (\({ \alpha =3/2}\)) criticalities. Moreover, this value is different from the one obtained for the 3D itinerant quantum magnets in the presence of a critical boson sphere [32]. On one hand, the fluctuating modes near the CBC are dispersive along the radial direction which is illustrated in Fig. 6. Namely, the system is of quasi-one-dimensional nature regardless of the dimensionality of the critical boson manifold. On the other hand, the boson polarization in the long-distance limit takes the form of Landau damping term with \(z=3\) that contains the tangential components. The Landau damping term completely alters the low-energy dynamics of the fermion-boson coupled system and leads to a peculiar power-law scaling for \(\delta (T)\).

In the next section, we show that the structure of the CBC makes a huge difference when considering the scattering process with the fermions. The parameter \(\delta (T)\) contributes to the thermodynamic properties of itinerant electrons in an important way.

3.4 Thermal crossover to FM criticality

As we have previously emphasized, the interfacial DM interaction would convert the interfacial ferromagnet into a frustrated magnet where there exists a degenerate momentum contour for the system to select the magnetic order. For the continuum model that was discussed above, the fluctuation near the CBC dominates the low-temperature and the-energy physics. As the temperature further increases, the thermal fluctuation overcomes the curvature of the CBC such that the temperature is comparable to the energy difference between the Γ point and the CBC. This crossover temperature is estimated to be of the order of \(\mathcal{O} (D^{2}/J)\). Above this crossover temperature and in the critical regime, compared to the large thermal fluctuation, the CBC behaves very much like a point. Therefore, the system would experience the crossover to the ferromagnetic criticality (see Fig. 3). This crossover would also show up in the transport properties of the itinerant electrons.

4 Non-Fermi liquid behaviors

In the conventional non-Fermi liquid near the FM or AFM criticality, the low-energy theory of the bosonic sector comprises discrete momentum points. Here, we encounter an infinite number of bosonic modes on a contour that become critical simultaneously. The quasiparticles near the Fermi energy are scattered by the CBC, resulting in a novel type of non-Fermi liquid in 2D. This is depicted in the quantum critical regime (painted in blue) in the \((T,r)\)-phase diagram in Fig. 3. In this section, we first evaluate the frequency dependence of the fermionic self-energy correction at zero temperature, which indicates that the quasiparticles no longer exist. Then, we evaluate the temperature dependence of the fermionic self-energy with zero external frequency and predict the non-Ferm liquid behaviors in transport experimental observables.

4.1 Zero temperature

The existence of quasiparticles in fermionic sector has to be re-examined in the presence of scattering process introduced by the CBC. Since the fermion-boson cubic interaction flows to a weak coupling limit, we can estimate the fate of quasiparticles perturbatively in terms of Yukawa coupling g. We evaluate the real frequency dependence of the fermionic self-energy at zero temperature \(\Sigma ({\boldsymbol {k}}_{F},\omega ;T=0)\). Specifically, the fermion self-energy correction induced by the CBC is given by,

$$ \begin{aligned}[b] &\Sigma ({\boldsymbol {k}}_{F},i\epsilon _{n};T=0) \\ &\quad = g^{2} \int \frac{d^{2} {\boldsymbol {q}}}{(2\pi )^{2}}\frac{d\omega _{l}}{2\pi} \frac{1}{(q-q_{0})^{2}+ \vert \omega _{l} \vert /\Gamma _{q}} \\ &\qquad{}\times \frac{1}{i(\epsilon _{n}+\omega _{l})- \xi _{{\boldsymbol {k}}_{F}+{\boldsymbol {q}}}}. \end{aligned} $$
(31)

The momentums are decomposed in the same way as in Eq. (11), (12). The integrand involves all components of δq which is due to the special scattering configuration facilitated by the CBC. The integral can be carried out by using a polar coordinate \(\delta {\boldsymbol {q}}= \delta q(\cos \phi ,\sin \phi )\), which reads

$$ \begin{aligned}[b] &\Sigma ({\boldsymbol {k}}_{F},i\epsilon _{n};T=0) \\ &\quad = \frac{g^{2}}{4\pi ^{2}} \int \frac{d\omega _{l}}{2\pi} \int _{0}^{+\infty} d(\delta q) \delta q \int _{0}^{2\pi} d\phi \\ &\qquad{} \times \frac{1}{ (\delta q)^{2} \sin ^{2}\phi + \vert \omega _{l} \vert /(\Gamma \delta q)} \frac{1}{i(\epsilon _{n}+\omega _{l})- v_{F} \delta q \cos \phi}. \end{aligned} $$
(32)

The leading contribution in the low-energy limit is calculated in Appendix C which takes a form \(\Sigma ({\boldsymbol {k}}_{F},i\epsilon _{n};T=0) \sim i {\mathrm{sgn}}(\epsilon _{n}) |\epsilon _{n}|^{2/3}\). By performing an analytic continuation to the real frequency \(i\epsilon _{n} \rightarrow \omega + i 0^{+}\), we obtain

$$ {\mathrm{Im}}\Sigma ({\boldsymbol {k}}_{F},\omega ;0) \sim { \mathrm{sgn}}(\omega ) \vert \omega \vert ^{2/3}. $$
(33)

The broadening of the quasiparticle peaks is significantly enlarged compared with the Fermi liquids \(\sim \omega ^{2}\). We further use the Kramer-Kronig relation to obtain the real part of the self-energy which leads to the definition quantity for Fermi liquid: the quasiparticle weight \(Z({\boldsymbol {k}}_{F}, \omega ) = [ 1- \partial _{\omega }{\mathrm{Re}} \Sigma ({\boldsymbol {k}}_{F},\omega ;0) ]^{-1}\). In the low-energy limit \(\omega \rightarrow 0\), one can readily see that the quasiparticle weight vanishes. All the evidences indicate the destroy of quasiparticles and emergence of non-Fermi liquids.

We note that despite the presence of the CBC, the fermionic self-energy correction at zero temperature is exactly same as the Ising nematic QCP (with \(d=2\), \(z=3\)) induced NFLs [54]. The Hertz-Millis scheme relies on integrating out the gapless fermions first which generates low-energy dynamical term for bosons; and, the renormalized bosons are used to further evaluate the fermionic self-energy. With the fermionic self-energy correction, the self-consistency of the Hertz-Millis scheme has been checked in literatures [54, 55] by re-inserting the renormalized fermionic Green function into the bubble diagram in Fig. 5(a). The one-loop expressions in fact satisfy the Eliashberg-type equations and are therefore self-consistent solutions.

The frequency dependence can not be extended to finite temperatures by simply assuming the \(\omega /T\) scaling behavior. The T-dependence for the self-energy has multiple sources due to fluctuations around the CBC which reflects in the peculiar exponent derived by SCR method [see Eq. (30)]. In the next section, we carefully take all the contributions into account and predict non-Fermi liquid behaviors at finite temperatures.

4.2 Finite temperature

In the vicinity of the critical point, the non-Fermi liquid behavior manifests as a characteristic temperature scaling in the thermodynamic properties. In particular, we focus on the resistivity that describes the transport property. To this end, we evaluate the (renormalized) fermionic self-energy making use of the renormalized boson correlation function. The one-loop renormalization sketched in Fig. 5(b) is explicitly expressed as

$$ \Sigma ( {\boldsymbol {k}}, i\epsilon _{n} ) \simeq \frac{g^{2}}{\beta V} \sum_{{\boldsymbol {q}},i\omega _{l}} G_{0}({ \boldsymbol {k}}+{\boldsymbol {q}},i\epsilon _{n}+i \omega _{l}) \text{Tr} \bigl[ M({\boldsymbol {q}},i \omega _{l}) \bigr] , $$
(34)

where \({\epsilon _{n}\equiv (2n+1)\pi /\beta}\) (\(n \in {\mathbb{Z}}\)) is the Matsubara frequency for the fermions, and \({G_{0}({\boldsymbol {k}},i\epsilon _{n})= (i\epsilon _{n}-\xi _{\boldsymbol {k}})^{-1}}\) is the bare Green’s function of the fermions. We follow the argument in the paragraphs near Eq. (24) and single out the term that contains the CBC in the boson susceptibility. After transforming to the dispersive representation and performing an analytic continuation \({i\epsilon _{n} \rightarrow \omega +i 0^{+}}\), we express the imaginary part of the fermion self-energy as

$$ \begin{aligned}[b] {\mathrm{Im}} \Sigma ^{\mathrm{R}}({\boldsymbol {k}}, \omega ;T) & = {-\frac{g^{2}}{V} } \sum_{\boldsymbol {q}} \int _{-{{\Gamma}}_{q}}^{+{{\Gamma}}_{q}} d \epsilon \bigl[ f_{B}( \epsilon ) +f_{F}(\epsilon +\omega ) \bigr] \\ &\quad{}\times \delta (\omega +\epsilon -\xi _{\boldsymbol {k}+q}) \frac{\epsilon /{{\Gamma}}_{q}}{{\mathcal{E}}_{q}^{2}+(\epsilon /{{\Gamma}}_{q})^{2}}, \end{aligned} $$
(35)

where we have used the relation \({|\epsilon |<{{\Gamma}}_{q}}\). We are interested in the \(\omega =0\) static component at low temperatures \(|\epsilon |\ll T\). Together with the zero temperature limit \(|\epsilon |\ll T\), we are interested in the regime set by a sequence of energy scales

$$ \vert \epsilon \vert < {{\Gamma}}_{q} \ll T. $$
(36)

This condition imposes a temperature-dependent bound on the magnitude of the momentum variation: \(\delta q \le q_{c} = T/{{\Gamma}}\). The following calculations are performed under the condition in Eq. (36) which yields

$$ -{\mathrm{Im}} \Sigma ^{\mathrm{R}}({\boldsymbol {k}},0;T) = \frac{g^{2}}{V} \sum_{ \boldsymbol {q}} \frac{T/{{\Gamma}}_{q}}{{\mathcal{E}}_{q}^{2}+(\xi _{\boldsymbol {k}+q}/{{\Gamma}}_{q})^{2}} , $$
(37)

where the integration involving the delta function generates a finite result provided a reasonable condition \(v_{F}\le {{\Gamma}}^{-1}\) is satisfied with a small interacting strength g.

To properly extract the low-temperature behavior from the imaginary part of the fermion self-energy, we keep track of all the temperature dependence in Eq. (37) that are conceived in: i) the renormalized thermal parameter \(\delta (T)\); ii) the upper bound for the momentum integrals. The scattering processes that are relevant at the low temperatures are the elastic scattering process where the exchanged energy is nearly zero \({\xi _{\boldsymbol {k}+q}\simeq 0}\). In other words, the Fermi momenta on the Fermi surface \({|{\boldsymbol {k}}| = k_{F}}\) are scattered back to the vicinity of the Fermi surface with \({|{\boldsymbol {k}}+{\boldsymbol {q}}_{0}| = k_{F}}\). We schematically show this scattering configuration in Fig. 6. The deep blue dashed line is the CBC and the light blue region represents the low-energy shell around the CBC. Let us consider the low-energy scattering process associated with the intersecting region between the Fermi surface and the CBC. Here, we follow the same convention as the calculation of the boson polarization function in Sect. 2.3. The fermionic and bosonic momentums are expanded in the manner as explained in Eq. (11) and Eq. (12). And, we consider the reasonable limiting cases with: i) the static limit dictated in Eq. (17); ii) the small boson radius limit in Eq. (13). Under these conditions, the directional vectors \({\hat{k}}_{F}\), \({\hat{q}}_{0}\) form the local basis of the 2D coordinate. The bosonic momentum variation can be further decomposed into components that are parallel and perpendicular with respect to \(\hat{q}_{0}\), which reads

$$ \delta {\boldsymbol {q}} = \delta {\boldsymbol {q}}_{\perp }+ \delta { \boldsymbol {q}}_{\parallel}, \qquad \delta {\boldsymbol {q}}_{\parallel }= \delta q_{\parallel }\hat{q}_{0}, \qquad \delta {\boldsymbol {q}}_{\perp }= \delta q \hat{k}_{F}. $$
(38)

The fermion dispersion function is expanded with respect to small bosonic momentum fluctuations

$$ \xi _{\boldsymbol {k}+q} = \frac{({\boldsymbol {k}}+{\boldsymbol {q}}_{0} + \delta {\boldsymbol {q}})^{2}}{2m} - \epsilon _{F} \simeq v_{F} \delta q_{\perp}, $$
(39)

where the leading order term \({o(\delta q)}\) is retained. Observing the δq-integral in Eq. (35), we note that the dominant contribution comes from the region \(\delta +\frac{1}{2} (\delta q_{\parallel})^{2} \sim \xi _{{\boldsymbol {k}}+{ \boldsymbol {q}}}/\Gamma _{q} \sim (v_{F}/\Gamma ) (\delta q_{\perp}/\delta q) \). We can safely drop the term with \({\mathcal {O}}(\delta q_{\parallel})^{2}\) and carry out the integration in the polar coordinate \((\delta {\boldsymbol {q}}_{\parallel}, \delta {\boldsymbol {q}}_{\perp})= \delta q ( \cos \phi ,\sin \phi )\), which leads to

$$ \begin{aligned}[b] - {\mathrm{Im}} \Sigma ^{\mathrm{R}}({\boldsymbol {k}},0;T) & = g^{2} \int _{0}^{q_{c}} ( \delta q) \,d(\delta q) \frac{T}{\Gamma (\delta q)} \\ &\quad{}\times \int _{0}^{2\pi}d\phi \frac{1}{{\delta}^{2}+ [ (v_{F}/\Gamma ) \cos \phi ]^{2}}\\ & \sim \frac{T q_{c}}{\delta}. \end{aligned} $$
(40)

Substituting the expression of \(\delta (T)\) into Eq. (40), we end up with a temperature dependence for the imaginary part of the fermion self-energy in the low-temperature limit

$$\begin{aligned} - {\mathrm{Im}} \Sigma ^{\mathrm{R}}(k_{F},0;T) \sim & T^{2-\alpha}\sim T^{1.44}. \end{aligned}$$
(41)

We note the temperature dependence with \(\omega =0\) obtained here has a distinct power than the frequency dependence at \(T=0\) in Eq. (33). This is largely attributed to the anomalous exponent α of the boson mass \(\delta (T) \sim T^{\alpha}\) obtained by the SCR method in Sect. 3.3. The original non-thermal parameter δ acquires a peculiar temperature dependence to the wild fluctuation near the critical boson contour.

The imaginary part of the self-energy in Eq. (41) corresponds to the inverse lifetime function of the electron on the Fermi surface \({|{\boldsymbol {k}}|=k_{F}}\). According to the Drude formula, the electronic resistivity is proportional to the inverse mean lifetime, which exhibits a temperature dependence

$$ \begin{aligned} \rho (T)\sim \frac{1}{\tau}\sim T^{1.44 }. \end{aligned} $$
(42)

This peculiar temperature scaling exponent of the resistivity signatures a novel type of non-Fermi liquid behavior. The power-law \(\rho (T)\) can be compared with the FM and AFM criticalities in 2D, where the standard Hertz-Millis theory yields \({\rho (T) \sim T^{4/3}}\) and ∼T respectively [56]. We conclude by claiming that the existence of the CBC fundamentally changes the scattering process and generates a distinct power law for the resistivity at finite yet low temperatures.

5 Discussion and summary

The external magnetic field has an important impact on the physical properties near a quantum critical point. The magnetic field is often used to tune the properties of the quantum criticality. Here, an external magnetic field explicitly breaks the time-reversal symmetry and immediately modifies the universality class of the phase transition. The temporal quantum fluctuations of the order parameter (boson field) is crucial in determining the university class of the quantum phase transitions. Within the framework of spin fluctuation theory, the presence of a magnetic field introduces a new type of dynamics in addition to the Landau damping. The external magnetic field induces a precession for the spin order parameter, and the Lagrangian in Eq. (2) would be supplemented with a dynamic precession term \(\sim \int d^{2}{\boldsymbol {x}}\ {\boldsymbol {B}}\cdot i (\phi \times \partial _{ \tau }\phi )\) [57]. For conventional antiferromagnetic criticality, however, the Landau damping and precession belongs to the same university class with \({z=2}\). The interplay of the two dynamic processes in 3D merely leads to non-universal changes in various thermodynamic quantities depending on the relative magnitudes [57, 58]. In contrast, the precession of the spin order parameter changes the nature of the 2D interfacial magnetic transition qualitatively. In the strong out-of-plane field limit, the critical boson contour is preserved and the dynamical critical exponent is converted to \({z=1}\), which is in the same class as conventional insulating magnets. The result in this work no longer applies, and the impact on the itinerant electrons will be explored in future work. The in-plane magnetic field, however, would compete with the DM interaction, and the critical boson surface for the minima would not be satisfied.

In the formulation of our theory, we have assumed a global U(1) symmetry and the interfacial DM interaction respects this continuous U(1) symmetry. In reality, this symmetry can be weakly broken by various anisotropy in the system, and the continuous degenerate contour will be lifted. As long as the temperature or the energy scale appears above the anisotropic energy scale, the results of the non-Fermi liquid and critical boson contour would hold. We thus vision that our results can apply to a large temperature/energy window. In addition, the contour or line degeneracy appears constantly in frustrated magnetic systems in frustrated magnets like honeycomb lattice antiferromagnets [59–62], and can even be present in more exotic excitations like spinons or magnetic monopoles in spin liquids [63]. Therefore, in the itinerant frustrated materials, such non-Fermi liquid behaviors can potentially emerge. Finally, how the strong fluctuations of the quasi-long range ordered regime in Fig. 3 impact on the itinerant electrons is an interesting topic on its own. This problem differs qualitatively from the coupling between the gapless Goldstone mode and the itinerant electrons where there can lead to non-Fermi liquids [32, 64], and will be considered in future works.

It is worthwhile to comment on the case when the itinerant electrons on the heterostructure interface are Dirac fermions or involve discrete band touchings. This happens for example on the surface or the termination of a three-dimensional topological insulator where magnetism can occur due to the correlation. At the charge neutrality point, the Fermi surface is composed of discrete momentum points. Only if there exist boson modes on the CBC that connect the momentum points, the low-energy physics can be reduced to the coupling between these fermions and the relevant critical bosons. Thus, most often, the CBC scatters the fermions at the band touching points to the gapped region in the reciprocal space. Such processes are certainly not low-energy physics, and cannot provide a good starting point for the analysis of low-energy physics. In fact, the optimization of the kinetic energy for the fermions could possibly select certain magnetic ordering wavevectors from the boson contour in the ordered regime. This is quoted as “fermion order by disorder”.

We briefly clarify the relation and distinction between the physical circumstance considered in the present study and the periodic Anderson model for the Kondo lattice problem. The physical degrees of freedom at the heterostructure interface comprise the itinerant electrons with spins and the local magnetic moments that resemble the situation in the Kondo lattice system. In the case of the Kondo lattice problem, the local magnetic moments interact with the spins of the conduction electrons. The conduction electrons have two competing effects on the local moments [58, 65]: i) the local moment can be screened by the spins of the conduction electrons leading to the formation of Kondo singlets; ii) the RKKY interaction mediated by the conduction electrons favors the magnetic orders. With the Kondo singlet formation for the former case, the local moments dissolve into the Fermi sea of the conduction electrons, and the whole system becomes the heavy Fermi liquid with a large Fermi surface. For the latter case, the system develops the magnetic orders for the local moments. The magnetic order may be viewed as the instability of the Fermi surface of the heavy Fermi liquid, or Thus, by increasing the RKKY interaction and/or decreasing the Kondo temperature, the system undergoes a quantum phase transition to a magnetic ordered phase; during which the Kondo singlets can be either preserved or destroyed. Here, one may discuss whether the magnetic order is from the Fermi surface instability of the heavy Fermi liquid or from the interacting local moments that do not participate to the Fermi volume. The latter scenario was for example studied in Ref. [66] that extended the dynamic mean field theory and translate the problem into the Bose-Fermi Kondo model where a single spin is coupled to fermionic and bosonic baths. The itinerant and local degrees of freedom are solved consistently. In more exotic scenarios [58], the local moments may even form a (gapped) quantum spin liquid with a small Fermi surface from the conduction electrons, then the distinction from the Fermi surface instability of the heavy Fermi liquid is more visible. In our problem, we did not invoke the Kondo screening at all. At zero temperature, the local moments either develop the magnetic order or become gapped and then absent from the low-energy physics. The local moments did not participate to the Fermi volume throughout our phase diagram of this work. The Kondo screening effect would be another parameter axis that is perpendicular to our phase diagram.

In summary, we have studied the itinerant quantum magnets at the interface of the magnetic heterostructure. We propose a novel type of non-Fermi liquid that is induced by infinite many critical boson modes due to the existence of interfacial DM interaction. We uncover the physical consequences of the corresponding experimental features for the itinerant electrons and local spins in the spectroscopy and transport, respectively. Our study has broadened the physical context for the exotic bosonic critical phases of matter with continuously degenerate minima, which seem to become a focused, rapidly developing research field recently [32–35, 50].