Robust surface Luttinger surfaces in topological band structures

Introduction—Bulk-boundary correspondence is a fundamental notion that underpins the standard paradigm of topological band structures. According to this principle, topological materials host robust, gapless surface states that uniquely identify the bulk band topology [1, 2, 3, 4, 5, 6]. In fact, in practice, the surface states invariably serve as the most direct experimental smoking guns of the topological phase. When the bulk undergoes a topological phase transition, the surface also experiences reconstruction, appearance or disappearance of states – in short, a Lifshitz transition [7] – in accordance with the bulk-boundary correspondence. If the topological phase is protected by a symmetry, breaking the symmetry generically destroys the surface states and unlocks a path in parameter space that allows one to adiabatically connect topologically distinct phases. In particular, if the symmetry is broken only on the surface, the bulk topology is well-defined but the surface generically remains gapped along this path. As a result, single-particle physics of the surface is expected to vary smoothly across the bulk phase transition and be qualitatively blind to the bulk topology.

The single-particle Green’s function $G(\omega,\mathbf{k})$ is a fundamental tool in physics, crucial for characterizing the spectral properties of single-particle excitations as well as describing their response to external perturbations and behavior under interactions and disorder [8, 9]. In Fermi liquids, $G(\omega,\mathbf{k})$ has poles that directly correspond to the system’s spectrum, with the poles at $\omega=0$ defining the Fermi surface (FS). In contrast, strongly correlated insulating phases like the Mott insulator have $G(\omega,\mathbf{k})$ that lacks poles but has zeros within the energy gap. The locus of the zeros of $G(\omega=0,\mathbf{k})$ is similarly defined as the Luttinger surface (LS) [10]. Since LSs correspond to the complete absence of quasiparticles, they have traditionally been deemed as peculiar analytic features of $G(\omega,\mathbf{k}\mathbf{)}$ indicative of strong correlations, but with little further fundamental or practical value as they are unobservable and do not contribute to physical properties.

Recent breakthroughs, however, have dispelled this notion. Fabrizio proposed a new class of quasiparticles near LSs in strongly correlated systems that produce similar thermodynamic properties as the quasiparticles in a Fermi liquid, such as the linear-in-$T$ specific heat [11]. This was followed by several seminal results within the realm of interacting topological condensed matter. Wagner et al. showed that Green’s function zeros characterize the topology of Mott insulators analogous to how poles characterize band insulators [12]. Jinchao et al. [13] and Bollmann et al. [14] discovered that the zeros of the single-particle Green’s function modify the topological invariant, but the latter showed that the zeros do not affect the quantized topological response. Blason and Fabrizio unified the role of poles and zeros in the topological characterization of generic interacting insulators [15]. In contrast to these works, one of us showed that the boundary of a generic Weyl semimetal (WSM) – a non-interacting topological phase – hosts Luttinger arcs in addition to the well-known Fermi arcs [16]. Nonetheless, both structures connect surface projections of Weyl nodes of opposite chirality. All these results place FSs and LSs on similar footing, and are in similar spirit as a pioneering result by Volovik that the same winding number stabilizes FSs and LSs [3].

In this work, we begin by proving the existence of surface LSs that respect the same bulk-boundary correspondence as the topological surface states (Fig. 1). As a result, a bulk topological phase would trigger a reconstruction or "Lifshitz transition" of the LSs. We then establish two profound consequences of this behavior. First, we show that surface LSs, remarkably, survive symmetry-breaking perturbations on the surface that destroy the topological surface states. This behavior starkly contradicts the prevailing notion that single-particle boundary properties are oblivious to the bulk topology once the surface states have been gapped out. On the other hand, it is similar in spirit, for instance, to the discontinuity in the surface Hall conductivity of a 3D topological insulator (TI) that is tuned across a bulk topological transition with the surface Dirac node gapped out throughout the process [1, 2]. Second, we show that the LSs reveal a disagreement between two routes for obtaining a low-energy surface theory in topological band structures, leading to unusual thermodynamic properties, such as a negative surface specific heat. We prove all these properties using generic topological arguments and substantiate them numerically using lattice models of 3D time-reversal symmetric $\mathbb{Z}_{2}$ TIs and WSMs perturbed by surface magnetism and surface superconductivity, respectively. We close by concretizing the results in the context of Bi₂Te₃/MnBi₂Te₄ heterostructures.

Robust surface LSs—We first show that topological band structures not only host surface LSs that obey the same bulk-boundary correspondence as the surface states, generalizing the prediction of surface Luttinger arcs in WSMs [16], but that the LSs have a surprising robustness – they survive even when surface perturbations gap out the surface states.

Consider non-interacting electrons in a $d$-dimensional system with a $(d-1)$-dimensional boundary. We label an arbitrary set of degrees of freedom near the boundary as the "surface" and refer to the rest as the "bulk." The full Hamiltonian can be decomposed as

where $H_{S}(\mathbf{k})$ and $H_{B}(\mathbf{k})$ are Bloch Hamiltonians for the surface and the bulk, respectively, $h(\mathbf{k}),h^{\dagger}(\mathbf{k})$ describe the coupling between them, and $\mathbf{k}$ is the momentum in the surface Brillouin zone (sBZ). The full and bulk Green’s functions at complex frequency $z$ are $G(z,\mathbf{k})=\left(z-H(\mathbf{k})\right)^{-1}$, $G_{B}(z,\mathbf{k})=\left(z-H_{B}(\mathbf{k})\right)^{-1}$, while the effective surface Green’s function obtained by integrating out the bulk degrees of freedom is (see App. A):

The LS is defined as the locus of zero eigenvalues of $G_{s}(0,\mathbf{k})$. To see it emerge, we first note that $G_{s}^{-1}(z,\mathbf{k})$ obeys the condition:

Clearly, every pole of $G_{B}(z,\mathbf{k})$ causes $\det G_{s}\left(z,\mathbf{k}\right)$ to vanish provided $\det G\left(z,\mathbf{k}\right)$ stays finite, implying a zero eigenvalue of $G_{s}(z,\mathbf{k})$. Physically, this means surface FSs of $H_{B}(\mathbf{k})$ that disappear upon coupling $H_{B}(\mathbf{k})$ and $H_{S}(\mathbf{k})$ produce surface LSs. At $\mathbf{k}$ points where $G\left(z,\mathbf{k}\right)$ and $G_{B}\left(z,\mathbf{k}\right)$ have equal numbers of poles, the situation is subtler. These poles cancel in Eq. 3 and mandate $\det G_{s}(z,\mathbf{k})$ to be finite and non-zero, hence forcing $G_{s}(z,\mathbf{k})$ to have equal numbers of diverging (pole) and vanishing (zero) eigenvalues, but does not guarantee the presence of either. On physical grounds, though, we expect surface poles of $G(z,\mathbf{k})$, i.e., poles whose corresponding eigenstates have finite weight on the surface in the thermodynamic limit, to be inherited by $G_{s}(z,\mathbf{k})$, since they represent physical surface FSs. We prove this explicitly in Appendix B. To ensure a finite $\det G_{s}(z,\mathbf{k})$, $G_{s}(z,\mathbf{k})$ must then also have an equal number of vanishing eigenvalues, thereby defining a LS that coincides with the surface FS. Thus, in all cases, the LS traces the surface FSs of $G_{B}(z,\mathbf{k})$ and therefore obeys the same bulk-boundary correspondence as the latter.

This construction instantly yields our first result: LSs persists even when surface FSs are destroyed by symmetry breaking perturbations on the surface. Such perturbations gap out the boundary states of the full system, represented by poles of $G(0,\mathbf{k})$ whose eigenfunctions are localized at the boundary. However, since the bulk degrees of freedom remains unaffected, the poles of $G_{B}(0,\mathbf{k})$ persist, and so do the zeros of the $G_{s}(0,\mathbf{k})$. We demonstrate this on a lattice model that realizes 3D TIs and WSMs in next section.

Application to TIs and WSMs—Consider an orthorhombic lattice model defined by the 3D Bloch Hamiltonian [17]:

where $m(\mathbf{k}^{3D})=m_{0}-\sum_{i\in x,y,z}\beta_{i}\cos k_{i}$, $d_{i}\left(\mathbf{k}^{3D}\right)=v_{i}\sin\left(k_{i}\right)$ and $\tau_{i}$ ($\sigma_{i}$) are Pauli matrices in orbital (spin) space. The wavevector is defined as $\mathbf{k}^{3D}\equiv\left(k_{x},k_{y},k_{z}\right)$. By tuning $\bm{\beta}$ and $\theta$, one can induce transitions between a trivial insulator, weak and strong TIs, and WSMs.

We first consider a strong TI, where we expect a 2D Dirac cone and a 2D Luttinger cone on the surface while time-reversal symmetry is preserved. If the symmetry is broken on the surface, we expect the Dirac cone to disappear but the Luttinger cone to survive. To uncover LSs, we calculate the surface Green’s function $G_{s}\left(i\eta+E,k_{x},k_{y}\right)$ [18] recursively (see Appendix E) and locate its zeros (poles) for a given $(k_{x},k_{y})$ by scanning for values of $E$ where the minimum (maximum) singular value of $G_{s}\left(i\eta+E,k_{x},k_{y}\right)$ vanishes (diverges) as $\eta\to 0^{+}$.

The results are illustrated in Fig. 2 (a) for $k_{y}=0$ and agree precisely with this prediction. Clearly, both the poles and zeros form a cone-like dispersion in the sBZ in the absence of the symmetry-breaking surface perturbation. To break the symmetry on the surface, we introduce a surface magnetization $H_{BTR}\equiv m\sigma_{z}$, which changes the surface Green’s function to $\tilde{G}_{s}=\left(G_{s}^{-1}+H_{BTR}\right)^{-1}$. As shown in Fig. 2(b), the Dirac cone, formed by the poles of $G_{s}$, is immediately gapped, but the Luttinger cone, arising from the zeros of $G_{s}$, persists. In Appendix C, we further confirm the persistence of the Luttinger cone by calculating the Volovik winding number that endows FSs and LSs with topological stability [3].

As a second test case, we consider WSMs, distinguished by their gapless bulk with chiral topological charges known as Weyl nodes, and surface Fermi arcs connecting the projections of Weyl nodes of opposite chiralities onto the sBZ [19, 4]. The Fermi arcs can also be understood as collections of Bloch momenta on the sBZ where $G_{s}\left(z=0,k_{x},k_{y}\right)$ has poles. Similarly, the surfaces of WSMs host Luttinger arcs, defined as regions of the sBZ where $G_{s}\left(z=0,k_{x},k_{y}\right)$ has zeros, which cause the non-interacting surface electrons to effectively acquire strong interactions mediated by bulk states [16]. In contrast to the Dirac point on a TI surface, the Fermi arcs cannot be gapped by a simple translationally invariant magnetization perturbation. However, if the WSM is time-reversal symmetric, the Fermi arcs can be gapped by conventional superconductivity, which gives us an opportunity to inspect LSs under another physically relevant perturbation.

Similarly to the procedure for the TI, we recursively calculate surface Green’s function for the lattice model in the WSM phase. Fermi arcs emerge within the sBZ as illustrated in Fig.3(a), while Luttinger arcs defined by Green’s function zeros appear on top of the Fermi arcs, as shown in Fig.3 (b,d) for two values of $k_{y}$. In the presence of s-wave surface superconductivity, either induced by proximity to another superconductor or intrinsically, as examined in recent works [20, 21, 22, 23], we can write the Bogoliubov-de Gennes (BdG) Hamiltonian as:

where the Hamiltonian $H_{\text{W}}(\mathbf{k}^{3D})$ is defined in the same manner as in Eq. (4), with parameters adjusted to represent the model in the WSM phase. Here, $\Delta=\delta s_{y}\sigma_{y}$ with $s_{y}$ acting on Nambu basis represents the superconducting pairing potential, which only has a non-zero value on the surface of the lattice model. As depicted in Fig.3 (c,e), the Fermi arcs near $\left(k_{x},k_{y}\right)=\left(0,0\right)$ are gapped. However, the Luttinger arcs persist as anticipated.

Surface vs low-energy theory—Consider a gapped non-interacting topological phase on a slab. Conventionally, the gapped, plane waves are referred to as bulk states and the gapless, evanescent waves localized near the surface are called surface states. Besides being parametrically separated in energy, the two classes of eigenstates have vanishing overlap in the thermodynamic limit. Thus, naïvely, integrating out either high energy states and low energy states of the far surface, if any, or integrating out bulk degrees of freedom should produce identical descriptions of the near-surface physics in the thermodynamic limit. This logic also applies to gapless systems such as WSMs if one avoids projections of the bulk Weyl nodes in the sBZ.

We now prove this expectation incorrect. In particular, we show that the two approaches yield distinct values for thermodynamic properties, with the mismatch coming from LSs. Thus, the surface and bulk degrees of freedom are intimately coupled even though energy eigenstates traditionally associated with the surface and the bulk are separated in real space and in energy.

To see this mismatch explicitly, consider the single-particle free energy from a generic Green’s function $\mathcal{G}(z,\mathbf{k})$, $\mathcal{F}=T\sum_{i\omega_{n}}\intop_{\mathbf{k}}\ln\det\mathcal{G}(i\omega_{n},\mathbf{k})$. Writing $\det\mathcal{G}(z,\mathbf{k})$ in terms of its poles $P_{n}(\mathbf{k})$ and zeros $Z_{m}(\mathbf{k})$, $\det\mathcal{G}(z,\mathbf{k})=\frac{\prod_{m}[z-Z_{i}(\mathbf{k})]}{\prod_{j}[z-P_{n}(\mathbf{k})]}$, gives the specific heat:

Clearly, only poles and zeros within $\sim T$ contribute appreciably to $\mathcal{C}(T)$. Evaluating $\mathcal{C}(T)$ at low $T$ using the full ($G$), bulk ($G_{B}$) and effective surface ($G_{s}$) Green’s functions unveils the mismatch. In particular, for each $\mathbf{k}$ point where the bulk plane waves and far-surface states are gapped with a gap $\gg T$ and the near-surface is gapless, setting $\mathcal{G}=G$ yields the specific heat from the near-surface states, $\mathcal{C}=C_{\text{pl}}+C_{\text{far}}+C_{\text{near}}\approx C_{\text{near}}$ as $C_{\text{pl}}$ and $C_{\text{far}}$ are negligible. In contrast, choosing $\mathcal{G}=G_{s}$ includes a negative contribution from the zeros or the LSs: $\mathcal{C}=C_{s}=C_{\text{near}}+C_{\text{LS}}$, where $C_{\text{LS}}<0$. In fact, if the near-surface states are gapped out by a symmetry-breaking surface perturbation, then $C_{\text{near}}/T\to 0$ as $T\to 0$ while $C_{s}<0$. Physically, the bizarre negative specific heat represents the reduction in total specific heat when the surface FSs of $H_{B}(\mathbf{k})$ are gapped out upon coupling the new surface layer $H_{S}$ to $H_{B}$ via $h,h^{\dagger}$. Alternately, it can be viewed as the specific heat due to the appearance of the LS. We demonstrate these points numerically using lattice models for a TI and WSM in Fig. 4; for the latter, we separately calculate the specific heat of the gapless bulk states by applying periodic boundary conditions and subtract it from the total to isolate the Fermi arc contributions. As shown in Figs. 4(a,c), $C_{\text{slab}}/T=(C_{\text{pl}}+C_{\text{far}}+C_{\text{near}})/T>0$ for the TI slab and WSM slab without the near-surface perturbation, while $C_{s}/T=0$. After introducing surface perturbation, $C_{\text{slab}}/T\approx 0$ for both the TI and WSM slabs and $C_{s}/T<0$, as shown in Figs. 4(b) and 4(d). In all cases, the difference, $(C_{s}-C_{\text{slab}})/T$, matches the LS contribution. We assume the far surface is always gapped.

Application to $\text{Bi}_{2}\text{Te}_{3}/\text{MnBi}_{2}\text{Te}_{4}$—To effectively "detect" LSs in experiment, we propose examining the specific heat difference between a TI, $\text{Bi}_{2}\text{Te}_{3}$ [24], and the same TI with a thin film of $\text{MnBi}_{2}\text{Te}_{4}$ [25, 26, 27, 28] deposited on top. Both $\text{Bi}_{2}\text{Te}_{3}$ and $\text{MnBi}_{2}\text{Te}_{4}$ are van der Waals coupled layered materials; given the recent advancements in fabricating van der Waals heterostructures, depositing the latter on top of the former is conceivable. The thin film should ideally have an even number of layers; an odd-layer magnetic $\text{MnBi}_{2}\text{Te}_{4}$ is a Chern insulator with edge states that also yield a contribution to the specific heat which must be subtracted off separately. In addition, the bare TI $\text{Bi}_{2}\text{Te}_{3}$ should be relatively thin so that the surface contribution to the specific heat is distinguishable from the bulk contribution – a nontrivial requirement since bare $\text{Bi}_{2}\text{Te}_{3}$ has bulk Fermi pockets. When $\text{MnBi}_{2}\text{Te}_{4}$ is in its magnetized phase, the surface states of $\text{Bi}_{2}\text{Te}_{3}$ are expected to be gapped out, leading to a reduction in the specific heat. This reduction can alternately be interpreted as the negative specific heat due to the LSs that develop in the $\text{MnBi}_{2}\text{Te}_{4}$ thin film upon coupling to the TI. In this setup, the $\text{MnBi}_{2}\text{Te}_{4}$ thin film functions as the surface in the context of this work.

Summary — We demonstrated that in noninteracting topological phases of matter, LSs, formed by vanishing eigenvalues of the surface Green’s function, coexist with topological surface states. For instance, the surface of a TI hosts both a Dirac cone and a Luttinger cone, while a WSM surface accommodates Fermi and Luttinger arcs. Their coexistence is mandated by no-go theorems that the surface Green’s functions must obey but surface Hamiltonians evade in topological phases. Remarkably, surface perturbations that break the symmetries protecting the topological phase are unable to destroy the Luttinger surfaces even if they gap out the surface Fermi surfaces. We then show that surface LSs are absent in the low-energy theory of topological phases derived by integrating out higher-energy bulk states, but occur in the surface theory through the integration of bulk degrees of freedom. The existence of surface LSs is tightly connected to the bulk topology; thus, the absence of LSs in the low-energy theory of topological phases demonstrates that the surface theory is different from the low-energy theory, although they are naively considered equivalent. Our findings shed new light on the role of Green’s function zeros in topological condensed matter systems.