
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38513104
202313488
10.1073/pnas.2313488121
research-articleResearch ArticlephysPhysics426
Physical Sciences
Physics
Surface spectroscopy and surface–bulk hybridization of Weyl semimetals
Zhang Xiao-Xiao xxzhang@hust.edu.cn
a b 1 https://orcid.org/0000-0003-3142-9232

Nagaosa Naoto nagaosa@riken.jp
b 1
aWuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
bRIKEN Center for Emergent Matter Science (CEMS), Saitama 351-0198, Japan
1To whom correspondence may be addressed. Email: xxzhang@hust.edu.cn or nagaosa@riken.jp.
Contributed by Naoto Nagaosa; received August 5, 2023; accepted January 30, 2024; reviewed by Peter Armitage and Xi Dai

21 3 2024
26 3 2024
21 9 2024
121 13 e231348812105 8 2023
30 1 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

Weyl semimetal is an epitome of frontier topological quantum materials. Its surface property accessible to photoemission measurements is important and contains rich information about the system. Lacking analysis of the inherent three-dimensional surface–bulk hybrid nature, investigations are often limited to verifying the existence of surface states and miss the hybridization aspect of general bulk-boundary correspondence. Based on an exactly solvable formalism consistently treating the surface and bulk states, we reveal the missing mathematical structure of the surface–bulk hybrid and the hidden information about the topological surface state; singular spectroscopic responses of the surface–bulk merging are highlighted. This underscores the importance of appreciating the intricate surface–bulk hybridization in quantum materials, paving the way for exploring such unique features.

Weyl semimetal showing open-arc surface states is a prominent example of topological quantum matter in three dimensions. With the bulk-boundary correspondence present, nontrivial surface–bulk hybridization is inevitable but less understood. Spectroscopies have been often limited to verifying the existence of surface Fermi arcs, whereas its spectral shape related to the hybridization profile in energy–momentum space is not well studied. We present an exactly solvable formalism at the surface for a wide range of prototypical Weyl semimetals. The resonant surface state and the bulk influence coexist as a surface–bulk hybrid and are treated in a unified manner. Directly accessible to angle-resolved photoemission spectroscopy, we analytically reveal universal information about the system obtained from the spectroscopy of resonant topological states. We systematically find inhomogeneous and anisotropic singular responses around the surface–bulk merging borderline crossing Weyl points, highlighting its critical role in the Weyl topology. The response in scanning tunneling spectroscopy is also discussed. The results will provide much-needed insight into the surface–bulk-coupled physical properties and guide in-depth spectroscopic investigation of the nontrivial hybrid in many topological semimetal materials.

Weyl semimetal
Fermi arc
ARPES
surface state
STS
MEXT | JST | Core Research for Evolutional Science and Technology (CREST) 501100003382 JPMJCR1874 Naoto Nagaosa
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pmcTopological quantum materials, characteristically accompanied by nontrivial topological states at the boundary, are an important frontier of condensed matter physics. Intriguing physical properties are often closely related to the principle of bulk-boundary correspondence and the boundary states. An epitome in three dimensions (3D) is the Weyl semimetal (WSM), which is significantly beyond an extension of the two-dimensional (2D) Dirac physics(1–13). It shows Weyl points (WPs) with linear band crossings as momentum-space Dirac monopoles and the associated chiral anomaly effects (14–21). At its boundary, unusual open-arc surface states appear in the projected surface Brillouin zone (BZ) (12, 13, 22–25). However, such Fermi arc surface states cannot be separated from the bulk at will, e.g., in transport measurements, because they interplay to form an organic whole and hence contribute together. For instance, unique quantum oscillations and quantum Hall effects originate from the Weyl orbit combining arc and bulk states (26–32). An important question arises: What are the mathematical structure and physical effects of the inherent 3D surface–bulk hybrid in WSMs?

How the boundary and bulk are nontrivially connected and hybridized is a less clear but indispensable aspect of the general principle of bulk-boundary correspondence. WSMs provide an ideal platform for such investigation because their 2D surfaces are open to many experimental probes. However, spectroscopy has been often limited to merely verifying the existence of surface states and overlooked the bulk-boundary hybrid and its nontrivial energy–momentum profile. Such information actually lies in the neighborhood of the surface and hides rich information about the whole system, crucial for studying further optical responses, transport, and magnetic properties (25, 33–36). It should be naturally detectable by contemporary spectroscopies, e.g., the angle-resolved photoemission spectroscopy (ARPES) that resolves the important momentum dependence and the scanning tunneling spectroscopy (STS) that encodes the density of states in the tunneling I-V characteristics (37–40). Photoemission is presumably more tunable between surface or bulk sensitive via the penetration depth of variable photon energy, which thus can accommodate sensitively the effect of surface–bulk hybrid (41, 42). Further equipped with spin resolution, these techniques can extract information inherent to the spin channel and magnetism as demonstrated experimentally (37, 38, 43–46).

To this end, we present a general formalism for a wide range of typical WSMs (in terms of symmetry, number of WPs, and type of Weyl fermion) with an exact solution near the surface. The consistently unified surface state and bulk influence substantially broaden the physical reach. Targeting surface ARPES measurements, we find that the surface resonance signal contains much more information about the whole system than usually deemed, hidden but comprehensively encoded in its shape and intensity profile. A by-product is finding unequal numbers of Fermi arcs and WP pairs, contrary to the common belief. Revealing the remarkable structure of surface–bulk hybrid, highly anisotropic singular responses near the loop-like surface–bulk merging borderline in the energy–momentum space are found accompanied by anomalous inverse square root scaling behavior. They provide a concrete prediction and picture of the borderline crucially mediating the surface–bulk hybridization, which serves as a critical transition boundary between surface and bulk effects, together with its own special edge at WPs. Due to the spin-polarized surface state, spin-resolved STS measurement can also achieve a separation of the bulk and surface tunneling contributions.

Results

Model and Effective Surface Green’s Function.

We consider a two-band Hamiltonian that breaks both the time-reversal and inversion symmetries[1] H(k)=[Dx(1−coskx)+Dz(1−coskz)]σ0−t∑i=x,y,z(coski−cosk¯i)σ1+(tsinky−t2coskx)σ2+t1sinkxσ3.

With appropriate parameter ranges, it gives a minimal pair of WPs k¯±=(0,sin−1t2t,±kw) aligned along the kz-direction as shown in Fig. 1A and at the energy ε0=Dz(1−coskw). Finite Dx,Dz break the particle–hole symmetry and add extra spin-independent dispersion in the xz-plane surface of our main interest; especially Dz accounts for the realistic Fermi arcs with finite curvature, which is common in real materials and crucial, e.g., to many phenomena involving Weyl orbits. We show how this Hamiltonian is formulated and related to more general cases in SI Appendix, section I. Also discussed later, most other representative types of WSMs can be treated.

Fig. 1. Topological surface state on the top surface of a WSM. (A) Schematic of the main model system. Orange coordinates and shapes additionally indicate the momentum space. One WP pair along the kz-axis is projected onto the 2D top surface Brillouin zone (BZ), where a solid Fermi arc state appears. (B1) The orange momentum constraint region of the surface state in the surface BZ. WP charges ± are noted. (B2) Surface state dispersion ε(k) restricted in the constraint region. Projected WP pair is indicated by black dots. Green and red lines are momentum cuts of the surface state at fixed kz or kx, respectively corresponding to Fig. 2 (A and B). Black lines are isoenergy lines of the surface state, i.e., Fermi arcs, corresponding to arc state signals at various Fermi energies in Fig. 3 from low to high energies. Horizontal and vertical dotted lines in (B1) correspond to the selected paths in Fig. 5. Full parameters are given in Fig. 2. Panels (C1 and C2) present a different noncentrosymmetric model with two pairs of WPs, which bears a multiply connected surface state constraint region. Within a certain range of EF, e.g., at the WP energy, there appear two Fermi arcs for one WP pair due to the inner hole structure.

The surface effective Green’s function can be analytically obtained by considering a sample semi-infinite in the −y^-direction. The Hamiltonian of the top surface, if decoupled from the bulk, is given by (summation over α=0,⋯,3 is henceforth understood) h=dα(k)σα, where d0=Dx(1−coskx)+Dz(1−coskz),d1=−t(∑i=x,zcoski−∑i=x,y,zcosk¯i),d2=−t2coskx,d3=t1sinkx. We use k to denote kx,kz-dependence in the surface BZ for simplicity, which is our main focus. The whole system is then cast in a recursive form of h coupled to neighboring layers. With the shorthand notation d¯0=z−d0 for generic complex frequency z and d12=d12+d22, we obtain two exact solutions labeled by r=±1 for the surface Green’s function g(z,k) (Materials and Methods)[2] g=b0σ0+b·σ,b0−b3=−(d¯0−d3)(K+2t2)2d122t2,b1,2=−d1,2(K+2t2)2d122t2,b0+b3=−K2(d¯0−d3)t2,

where[3] K(r)=B+rC

depends on the choice of r with[4] C=B2+4t2F

and B=d2−d¯02−t2,F=d32−d¯02. Momentum and energy dependence is henceforth suppressed for brevity. This solution is found by effectively integrating out the bulk degrees of freedom, and the surface effective Green’s function g will encode the full information of the original WSM system. A related approach via integrating the bulk has been applied to mesoscopic transport calculation and quantum oscillation under magnetic field (28, 47).

Spectral Function.

The information most relevant to spectroscopy can be extracted from the spectral function of Green’s function g(z,k) in Eq. 2. It is given by[5] A(ω,k)=i(gr−gr†)=Aασα,

where the retarded Green’s function gr=g(z→ω+i0+), leading to the charge (α=0) and three spin (α=1,2,3) channels[6] Aα(ω,k)=−2Itr[grσα]=−4limη→0+Ibα(z→ω+iη).

In real systems, η may signify the phenomenological strength of possible relaxation mechanisms, e.g., disorder, phonon scattering, electron correlation, etc. ARPES without spin resolution measures A0 only while spin-resolved ARPES can extract each of Aα with spin-polarized photoelectron detectors. To understand its physical meaning, it is crucial to know how r=±1 in Eq. 3 should be chosen, which turns out to be a nontrivial issue because C of Eq. 4 is not necessarily always positive. It is, however, remarkable to note that the above definition of spectral function holds in general regardless of the sign of C under the square root because Green’s functions can generally be complex.

We plot the above spectral function in Fig. 2. Exemplified in Fig. 2A1 and B1, both the topological surface state and the influence from the bulk states are unified in g and hence in A; the surface state eventually ends and merges into the bulk at a certain borderline in the energy–momentum space. To facilitate and guide the discussion, we point out three crucial aspects of the key quantity C:

The sign of C determines in the energy–momentum space the boundary between bulk contribution (C<0) and the region admissible to surface state (C>0);

The borderline of surface state entails C=0, where the surface state merges into the bulk states;

The choice of r=±1 has to be determined physically according to the positivity of the charge channel spectral function A0, because it directly corresponds to the density of states.

Fig. 2. Spin-resolved ARPES signals of the surface of a WSM in (A) the (ω,kx)-plane at kz=π/5 and (B) the (ω,kz)-plane at kx=π/10. Spectral function A0,1,2,3 successively in the charge and three spin channels for the WSM with parameters kw=π/2,t=t1=1,t2=0.4,Dx=0.2,Dz=0.4,η=0.02. A deep blue surface state (resonance intensity clipped to enhance the overall visibility) is seen only and identically in the A0,3 channels, signifying the charge–spin locking between them. Such a surface state in panels (A1 and A4) and (B1 and B4) respectively corresponds to the green and red lines in Fig. 1(B2). Exemplified in the A0 channel, it tangentially merges into the projected bulk states bounded by the dotted lines; for clarity, the full surface state dispersion beyond the momentum constraint region is indicated by dashed lines.

Henceforth, when the sign of a certain quantity is concerned, complex frequency z substituted by real ω is understood.

Surface Resonance Spectroscopy.

We first find the states on-shell at the surface, i.e., the topological surface state. As long as C>0 for some energy–momentum region, the analytic continuation in Eq. 6 matters only in the denominator and gives (Materials and Methods)[7] A1,2=0,A0=A3=−πKt2δ(ω−d0−d3).

When K is finite, this readily implies the surface state dispersion relation[8] εk=d0+d3=∑i=x,zDi(1−coski)+t1sinkx,

which mainly accounts for the circulating chiral states on the top surface, and is modified by the extra dispersions in the 2D surface BZ. As aforementioned and shown in Fig. 1B2, finite Dz bends the Fermi arc toward the kx-direction to make it curved; otherwise, the arc becomes a straight line. This is also consistent with earlier low-energy approximate solutions for simpler models (48–50). Furthermore, the fact that only A0,A3 are finite and identical implies a locking between the charge and spin-Sz channels; we also find the surface state entirely polarized in spin-↑. Such a locking and spinful feature is directly visible in Fig. 2, where the surface band appears identically in the charge and Sz channel.

The resonance at the surface state (henceforth denoted by index “ss”) band Eq. 8 immediately leads to the relation Fss=0, Css=Bss2 and Kss=Bss+r|Bss| with[9] Bss=d122−t2,

where Css≡C|ω=εk and similarly for others. Here, Css≥0 generally satisfies the assumption of Eq. 7 self-consistently except for the special case Css=0. Thus, Css=0 defines a momentum-space borderline of the surface state, i.e.,[10] Bss(k)=0,

which forms a loop that touches the two WPs projected in the 2D BZ, as shown in Fig. 1B1. To have a nonvanishing spectral weight of the surface state in Eq. 7, Kss needs to be finite and negative such that A0>0 as aforementioned. Inside the borderline, i.e., within the constraint momentum-space region[11] Bss(k)<0,

the r=−1 solution is chosen and one finds[12] A0,3ss(ω,k)=2πR(k)δ(ω−εk),

where R(k)=−Bss/t2=(1−d122/t2). Hence, Eq. 11 gives the physically allowed region in the 2D momentum space, within which the surface band Eq. 8 is defined. Outside this region, one has Bss>0, and the only nonvanishing choice r=1 leads to unphysical A0<0, we thus still have r=−1 and hence[13] A0,3ss=0,

which is physically expected beyond the surface state constraint and outside the bulk continuum.

It is intriguing to note that the momentum dependence of the resonance in Eq. 12 can be measured from the intensity R(k) in both charge and spin channels of ARPES, which encodes the information of the surface state constraint function Bss or the Hamiltonian components d1,2. Such information is complementary to the surface band dispersion Eq. 8, which involves d0,3 only and is measured through the surface band shape in ARPES as determined by the δ-function in Eq. 12. Therefore, from the resonance intensity R(k) and the band shape εk, most information of the bulk WSM system can be extracted by merely inspecting surface resonance Eq. 12, realizing an intriguing holography-like situation. For instance, as shown in Fig. 3, since ARPES measures photoemission intensity of occupied bands, at different chemical potentials tunable via gate voltage control, the Fermi arc in the 2D BZ traverses across the constraint region Eq. 11 and serves as a tomography of the surface resonance of the WSM. Here, we only exemplify the behavior of one particular surface. In experiments, different crystalline surfaces can be similarly measured, which will extract more complete information about the constraint and the Hamiltonian, including the anisotropy of the system. In other WSM systems where the surface carries more complex spin structures, in-plane A1,2 can also supply further information.

Fig. 3. ARPES signal log-scaled in the charge channel A0 at several energy cuts showing the Fermi arc surface state (deep or dark blue) merging into the bulk. Panels (A–E), energies from low to high, correspond to the Fermi arcs as black lines in Fig. 1(B2); panel (C) is the case when EF is pinned to the WP. Dashed lines indicate the boundary of the surface state constraint region, corresponding to Fig. 1(B1). The momentum-dependent intensity profile and shape of the arc resonance encode complementary and key information about the WSM system. Parameters same as Fig. 2.

In addition, with such a direct characterization of the surface state, we are able to point out yet another important but often overlooked phenomenon: The numbers of Fermi arcs and WP pairs are not necessarily identical. The common belief of a one-to-one correspondence between them assumes that an energy plane cuts a surface state continuum and gives an arc. In contrast, the constraint region Eq. 11 may not be simply connected. As shown in Fig. 1C1 and C2 for a minimal noncentrosymmetric model with two WP pairs (SI Appendix, section II), it can generally be multiply connected with inner holes, then the isoenergy curve is disconnected once it passes a hole, i.e., a forbidden region, even if the Fermi energy EF is at WPs. This leads to more Fermi arcs for one WP pair, depending on EF and also the hole position. In experiments, discretion is necessary when identifying or counting WPs from arc spectroscopy; instead, knowledge of the constraint region would be helpful. A hole, i.e., an inner boundary of the constraint region, is not fundamentally different from the usual outer boundary where arcs end; both are merging borderlines from surface to bulk and our later discussion applies equally well. For instance, an inner hole borderline can also pass WPs as the outer one does. The above phenomenon of excess Fermi arcs is very general and can appear in many other WSMs. In fact, based on the present formalism applied to WSMs with multiple WP pairs, we can accurately analyze more generic but unconventional properties of the projected surface connection pattern (SI Appendix, section IV). Strongly affected by the shape of constraint region and surface state energy surface and especially tuned by EF, the naive association of a Fermi arc with a WP pair is often not the case.

Signatures from Bulk Continuum.

Since Green’s function Eq. 2 has the bulk beneath the top surface integrated out, it should have the bulk state contribution included as well, albeit in an off-resonance manner. As aforementioned, away from the energy–momentum space region that admits possible surface state, we do not necessarily have C≥0. Rewriting Eq. 4 as C=(d¯02−E+2)(d¯02−E−2) with Es=d32+(d12+st)2, we require C<0 and generally find two disconnected regions in the energy–momentum space for s=±1[14] E−s2<(ω−d0)2<Es2.

They exactly correspond to the projection of bulk states onto the surface, which manifests as conduction and valence bands. Within such continuum regions, K of Eq. 3 acquires a finite imaginary part even before the analytic continuation and dominates the spectral function Eq. 5; such a contribution is thus off-resonance, i.e., finite and not peaked as δ-functions as the surface state. Therefore, Es should signify the band edges of the bulk states measured from d0. We exemplify such band edges in Fig. 2A1 and B1. In fact, we note from Hamiltonian Eq. 1 that the bulk eigenenergy (measured from d0) squared reads[15] E2(k)=d122+2|t|d12sin(ky−ϕ)+t2+d32,

where ϕ is the polar angle of the coordinate (td2,td1). Its two extrema are exactly the foregoing E±2. When such band edges accidentally touch each other, the effective Green’s function Eq. 2 will become singular (SI Appendix, section V).

It is noteworthy that the choice r=±1 of solution branches in Eq. 2 becomes in general more complicated outside the surface resonance, including the bulk continuum and the merging borderline. However, one can reach an exceptionally simple criterion at any k and ω, not limited to surface or bulk contribution (Materials and Methods): Find the choice of r that maximizes A0(r). This is how all the spectral functions are evaluated throughout this study.

In Fig. 2, we indeed observe the projection of bulk conduction and valence bands in the surface BZ and they are connected through the surface band in between. In Fig. 3A–C, the Fermi arc similarly connects two bulk Weyl cones separated along kz direction; the bulk state is itself connected at energy higher than the WP in Fig. 3 D and E, which is due to the distortion from d0 and consistent with Fig. 2B1. Also, as aforementioned, the spectral intensity of the projected bulk bands remains independent from η in Eq. 6, in contrast to the surface resonance. From Eq. 2 and Eq. 6, we know that A1,2 and A0−A3 are finite only when C<0 for the bulk continuum region, so they all vanish for the surface state resonance as discussed previously. An immediate implication is that the in-plane spin spectral weight A1,2 only appears in the bulk continuum and respectively follows the profile of d1,2, e.g., both d2 and A2 changes sign at kx=±π/2, as we observe in Fig. 2. d3=t1sinkx seemingly suggests A3(ω,kx)=−A3(ω,−kx), which does not accurately hold in Fig. 2A4 although the trend remains in gross. This is because such bulk symmetry, under surface projection, will be distorted, as exemplified by the chiral surface state obviously asymmetric for kx. In fact, if the opposite surface was included, the overall symmetry could be restored for the whole sample.

Singular Behavior along Merging Borderline.

We inspect how the surface state merges into the bulk and the behavior around the merging points, i.e., the C=0 borderline, which is a twofold restriction in the energy–momentum space as shown in Fig. 1 B and C: the loop-like border of the surface state constraint Eq. 10 in the surface BZ and the energy pinned to the corresponding surface band. Since Css=Bss2>0 except for the borderline, the entire surface band Eq. 8 at any k in the whole surface BZ (dashed line in Fig. 2A1 and B1), not limited to the constraint region Eq. 11, is separated from the bulk states (C<0). Hence, they will tangentially touch each other at the borderline. The real surface state restricted by the constraint thus merges into the bulk continuum at the borderline, as shown in Figs. 2 and 3. Although the surface resonance is outside the bulk continuum, its residual influence, according to Eqs. 2 and 6, enters A0 and A3 of the bulk continuum through the energy denominator (index “bs” for bulk states)[16] (A0+A3)bs=2IKt2(ω−εk).

This means stronger spectral weight in A0 and A3 if close to the full surface state dispersion εk of Eq. 8 regardless of the constraint, as seen in Fig. 2A1, A4, B1, and B4. This is physically understandable as the resonant surface spectral weight will not suddenly disappear but smear into the bulk continuum.

Approaching the merging borderline from the bulk continuum, both the nominator and denominator reduce to zero in Eq. 16, which becomes apparently undetermined. The similar happens in Eq. 7 or Eq. 12 from the surface resonance side as key quantities C,B,K all approach zero. To reveal the singular behavior thereof for any approaching direction, we consider the expansion around the momentum kbl and energy εkbl on the borderline respectively (denoted by index “bl”), i.e., Δk=k−kbl,Δω=ω−εkbl. One can cast quantities in Eq. 3 up to the relevant orders in Δk and Δω[17] B=Bss+F≈α·Δk+F1C≈4t2F1−c(Δωη)2+Δk·Λ·Δk+Δωηγ·Δk,

where all parameters like α=∂kBss|bl and rank-2 tensor Λ are evaluated along the borderline and given in Materials and Methods. F1=−2d3(kbl)f is the linear order term of F, and with notation Δωη=Δω+iη, we denote[18] f=Δωη−∂kε|kbl·Δk.

While F1 accounts for the leading linear effect in C along almost the entire borderline, it vanishes at WPs as per the general model construction; hence, the rest of the quadratic terms will become important. With Eq. 17, the general spectral function around the entire borderline is[19] A0,3(Δω,Δk)≈IB+rCt2f.

We visualize Δω=0 case for representative borderline points away from and at the WP in Fig. 4, where a type-II WSM case is exemplified in comparison when the velocity tilting is large enough to induce the Lifshitz transition (51) (SI Appendix, section VI). In Fig. 5, we additionally visualize the evolution and transition in the proximity of the merging borderline, in which the anomalous behavior across the merging along momentum axes manifests conspicuously.

Fig. 4. ARPES signal A0(Δω=0,Δk) based on Eq. 19 log-scaled in the k-space vicinity of a borderline point at the origin: (A1 and A2) a generic borderline point with kx=−0.2; (B1 and B2) the WP at kz=kw. Dashed lines indicate the upper boundary of the surface state constraint region as introduced in Fig. 1B1. (A1 and B1) Type-I case with parameters same as Fig. 2 and respectively corresponding to the upper merging point in Fig. 3 (B and C); (A2 and B2) type-II case for the same model with larger Dz=1. (A) Deep blue Fermi arc surface state merges into the bulk continuum but is anomalously weakened albeit still divergent upon merging (Fig. 5A), which is hence less discernible partially due to the log scale. (B) Fermi arc ends at WP where the signal is finite (Fig. 5B); (B2) type-II case shows Fermi pockets connected to WP.

Fig. 5. Evolution and transition of ARPES signal A0(ω) in the vicinity of the merging borderline: (A) a generic borderline point at kx∗ corresponding to the Upper Right merging point in Fig. 2A; (B) the WP at kz∗=kw. Respectively along the dotted horizontal and vertical paths in Fig. 1B1, panel (A) selects successive kx cuts with an incremental step δkx=0.35 and panel (B) selects similarly along kz with δkz=0.1. Vertical solid lines indicate the surface state resonance with the height proportional to the intensity of δ-function; their intensity envelope function determined by R(k) is shown as a gray dashed curve vanishing at the borderline point, which is passed by the red signal. From blue to red and to brown signals: (A) The δ-function peak is weakened as ∝|Δk| and eventually merges with the bulk signal to produce the red anomaly as A0∝|Δω|−1/2 at the borderline, after which only bulk state signal is left; (B) The δ-function peak is weakened while the bulk gap shrinks till the WP, where they touch each other to produce a finite signal indicated by the vertical dashed line, after which the bulk gap opens again but without surface state resonance.

We first approach the borderline along the surface state resonance. Following the earlier discussion of surface resonance spectroscopy, this sets F=0 in the first place in Eq. 17 and leads to Bss≈α·Δk,Css≈Bss2. Now Eq. 19 becomes in the limit η→0+[20] A0,3(Δω,Δk)≈−2πα·Δkt2δ(Δω−∂kε|kbl·Δk).

The surface state constraint Eq. 11 also applies and α defines the outward normal direction at the constraint region boundary, which is generally nonzero. Hence, its resonance intensity scales with the momentum normal to the borderline; α·Δk<0 exactly means that Δk needs to be inward the constraint region to have finite signals.

Eq. 20 misses the continuum side and the generic off-resonance behavior outside the bulk continuum. With deviation from the surface state resonance, the full form Eq. 19 captures the correct behavior as shown in Fig. 4, where especially F1 in C becomes the leading linear contribution away from WPs. If only this leading effect is concerned, the formula reduces to[21] A0,3(Δω,Δk)≈IrCt2f≈2r|t|I−2d3(kbl)f.

This formula determines the local merging configuration, including whether the borderline is approached from the bulk continuum or the surface state region near the conduction or the valence band (Materials and Methods). Besides, along the borderline, it exhibits a singularity scaled as A0,3bl∝|Δω−∂kε|kbl·Δk|−12 with Δω,Δk→0. Without much loss of generality, we will refer to it by |Δω|−12-singularity. Therefore remarkably, the borderline is a set of singular branch points with unconventional inverse square root divergence |Δω|−12, in contrast to the usual δ-function due to a simple pole divergence |Δω|−1. It resolves the foregoing undeterminedness in any direction, where the spectral weight changes between resonant δ-function divergence (surface-state side), |Δω|−12-singularity (borderline), and finite (bulk side). Accordingly in Fig. 5A, the blue, yellow, and green signals approach the merging point (red signal) from the surface side while the associated surface state resonance is weakened and transitions to the anomalously peaked resonance at merging; further into the bulk side, such a |Δω|−12-divergence changes to the smooth purple and brown signals. This elucidates what happens in Fig. 3 A and B and Fig. 4A near the merging. The borderline forms a set of critical transition points of the spectral signal in ARPES, regardless of the approaching direction in energy–momentum space. With finite relaxation strength η, the signal variation could become a smoother cross-over, but the scaling behavior, including that of the linewidth, bears the same transition.

Around WPs, C is fully quadratic in deviations and Eq. 19 takes the form[22] A0,3(Δω,Δk)≈Iα·Δk+r−c(Δωη)2+Δk·Λ·Δk+Δωηγ·Δkt2f,

where α=∂kBss|wp,c=4t2,γ=8t2∂kd0|wp and Λ=αα+4t2(∂kd3∂kd3−∂kd0∂kd0)wp. This formula fully captures the merging configuration around WPs, especially the distinction between type-I and type-II cases, as shown in Fig. 4B, is directly related to the principal values of the tensor Λ (Materials and Methods). Approaching the WPs, Eq. 22 gives A0,3wp=2/|t| with r=−1 chosen. This singles out the WP as a unique removable singularity of spectral weight as shown in Fig. 4B, which purely depends on the hopping amplitude and does not diverge with vanishing η. In Fig. 5B, we observe that the bulk gap closes smoothly and reopens; more specifically, it shrinks to zero while the intensity of surface state resonance is weakened to zero upon reaching the merging point (WP), where it transitions to the red finite WP signal. The surface states entirely end at WPs and WPs simultaneously belong to the bulk continuum; this physically explains why the WP signal eventually does not diverge as surface states do. It thus is distinct from the foregoing surface-borderline-bulk transition and instead admits either a surface–bulk transition as in Fig. 5B (from δ-function divergence to finite) or a borderline-bulk transition (from |Δω|−12-divergence to finite), depending on whether the WP is approached from the majority of surface states or along the borderline. Note that the intensity of |Δω|−12-singularity decreases with the distance |Δk| from the WP along the borderline.

These discussions reveal the strong inhomogeneity along the borderline in terms of both the near-merging behavior and the singularities themselves, reflecting a fundamental aspect of the topology generated between WPs in a WSM. While the borderline can be viewed as a special edge curve of the surface state where distinct phenomena occur, there also exist WPs as the special points on the borderline. The surface/bulk distinction plays an important role and one can approach the borderline from either side. On each side, it exhibits nontrivial energy-dependence and highly anisotropic angle argΔk-dependence for the entire borderline. Such information provides important profiling of the WSM system, as reflected in Figs. 4 and 5, and can be measured in ARPES directly, e.g., through inspecting the endpoint behavior of Fermi arcs in Fig. 3.

STS Measurement.

STS measurement finds differential conductance from varying the bias voltage of the (spin-polarized) tip. It is able to reconstruct the local density of the electronic states and is directly related to the spectral function Eq. 6 integrated over momentum, i.e., Aα(ω)=SBZ−1∫dkAα(ω,k) with SBZ the area of the surface BZ. An extra merit of STS is that it can probe both occupied and empty states by simply changing the sign of bias voltage. This is, however, achievable in ARPES by directly tuning the chemical potential or with pump–probe ARPES (52). In the presence of impurities, quasiparticle interference patterns of STS related to spatially dependent A(ω,r) can also detect the surface Fermi contour (33, 42, 53–55).

Fig. 6 shows the integrated spectral weight in charge and spin channels. The asymmetry with respect to the WP energy is due to the broken particle–hole symmetry in Eq. 1. By virtue of our analytical solution that physically identifies the meaning of the key quantity C, the surface and bulk contributions can also be clearly separated as Asurface and Abulk. Corroborating with our earlier discussion, only A0surface and A3surface are finite while Abulk comes from all channels. The van Hove singularity in Asurface originates from the lowest energy part of the surface band in Fig. 1B2, which can become relatively flat as the band minimum. This particularly happens when the Weyl vector, the separation vector between WPs, is long such that the constraint region covers the surface band minimum. Therefore, it can be used as a phenomenon for estimating the Weyl vector scale important in experiments. Near the WP energy, bulk spectral weight vanishes in every channel while that from the surface state dominantly contributes in a way that manifests the locking between the charge and spin-Sz channels, which can be used as direct experimental evidence for identifying topological arc states. Besides directly accessing charge and spin channels in either ARPES or STS, one can also measure different surfaces of the sample that may bear arc states or not, from which the unique surface contribution can be clearly identified in comparison and even separated if the anisotropy between those surfaces is not too large.

Fig. 6. Spin-resolved STS signals as a function of frequency. Integrated spectral functions A0,1,2,3 in the charge and spin channels are shown in panel (A). Panels (B and C) respectively separate out the topological surface state contribution and the bulk contribution in (A). The dashed line indicates the WP energy. Parameters same as Fig. 2.

Discussion and Summary

The main WSM model we focused on can be realized in its own right in minimal magnetic Weyl materials and particularly also captures the surface spin polarization effect directly observable in spin-resolved ARPES and STS. Such discussion singles out the most important physical properties in general magnetic or noncentrosymmetric Weyl materials even with multiple pairs of WPs, since the pairwise connection of projected WPs on a surface is a common and distinguishing feature. In fact, the formalism is general enough to account for more interesting cases, e.g., the time-reversal-breaking WSM with more WP pairs and the inversion-breaking WSM with more pairs such as the one shown in Fig. 1C1 and C2 and WPs not at the same energy (SI Appendix, sections II and III). Importantly, the main features discussed remain intact, despite the distinct shape of the surface state. For noncentrosymmetric Weyl materials, the number of WP pairs has been reduced from twelve in TaAs to four (42, 51, 56, 57) and the minimal two (52, 58); increasingly many magnetic Weyl materials also appear in experiments and theoretical predictions, even with two or the minimal one pair of WPs (12, 13, 25, 59, 60).

As we have already exemplified, all the analysis equally applies to type-II WSMs and the formulae and conclusions are fully general (SI Appendix, section VI). For example, an immediate observation is that although the surface band dispersion Eq. 8 will be strongly tilted by d0, the surface state constraint region Eq. 11 remains intact. Last but not least, the Dirac semimetal case shares similar physics as the signal will be effectively contributed by, e.g., one more pair of WPs, which can be readily captured by adding an extra chirality-reversed copy of the model in our discussion and none of the essential conclusions will change.

Based on a formalism of exactly solved WSM models at the surface, we present a unifying treatment of the topological surface state and the bulk continuum and reveal the crucial hybridization aspect of the bulk-boundary correspondence. This strongly broadens the physical access and enables a comprehensive analysis of the spectroscopic effects of the surface inseparable from the bulk. Important but often overlooked information about the bulk Hamiltonian and surface constraint can be revealed by surface spectroscopy. Unconventional transitions at the surface–bulk merging borderline are found via a systematic analysis of the highly anisotropic singular responses in the energy–momentum space. The fully analytical account and the general prediction will deepen our understanding of topological semimetals from their 3D surface–bulk hybrid nature and push forward the frontier of spectroscopy experiments on topological systems.

Materials and Methods

Solution Method of Model Hamiltonian.

Here, we sketch the method of solving the surface Green’s function. For a sample semi-infinite in the −y^-direction, the Hamiltonian Eq. 1 can be given in a recursive form. Contiguous y-layers start from the top surface and serve as the matrix indices 1,⋯,∞[23] H(kx,kz)=hVV†H,

where h(kx,kz), given in the main text, is the Hamiltonian of the top layer decoupled from the bulk. Also, the inter-y-layer coupling takes the form V=(V,0,0,⋯) with V=−tσ− and σ±=12(σ1±iσ2). We focus on the Green’s function g(k,z)=G11 of the top y-surface with generic complex frequency z and G=(z−H)−1 the original Green’s function of H. Making use of the recursive property of Eq. 23, the self-energy of g for the top surface is given by[24] Σ=VGV†=VgV†.

When we have the bare Green’s function of h as g0=(z−h)−1=(d¯0σ0−d·σ)−1, the exact Dyson equation reads[25] g−1=g0−1−Σ,

which is an equation of b0,b in the effective surface Green’s function g=b0σ0+b·σ. Intuitively, the foregoing form of interlayer coupling V given by σ± hides the information of d1,2, which is “squeezed” but encoded in the surface g. Hence, one finds the intriguing structure of Eq. 2, especially the importance of d0,3 that appears in the energy denominator only for b0+b3. In SI Appendix, section I, we discuss in more detail how such a formalism can be applied to more general Hamiltonians.

We in general face the issue of the choice of r=±1 in calculating Eq. 5, which is now extra complicated by the inevitable branch cut of square root in Eq. 3. First, it is a convenient choice to place the branch cut at argC=π; to account properly for the surface state case of C>0, argC=0 should be avoided; otherwise, it would cause an unphysical exchange between the shaded surface state constraint region in Fig. 1B1 and its complement. Second, as aforementioned, r can be fixed by the positive-definiteness of the charge channel spectral weight A0. Since the off-resonance bulk contribution is purely coming from the imaginary part IK, i.e., rC in Eq. 3, only one r makes A0>0 and hence naturally determines the choice. Slightly away from the surface state resonance and when the analytic continuation in Eq. 6 practically uses a finite η, such uniqueness of r for positive A0 may not hold everywhere, e.g., around the borderline of the constraint region where the surface state merges into the bulk states; this, however, can be resolved by noting that the spectral function varies smoothly away from the exact surface state resonance in Eq. 12. With these considerations combined, we reach the simple but general criterion at any k and ω: Find the choice of r that maximizes A0(r).

Surface State Resonance and Behavior near the Borderline.

According to Eq. 2, the effect of surface state resonance can only enter the component below of the spectral function via the substitution z→ω+iη[26] (A0+A3)ss=limη→0+(ω−εk)IKss−ηRKsst2[(ω−εk)2+η2]/2|ω→εk,

where only the second part with RKss matters and gives the Lorentzian structure, because limη→0+IKss=0 outside the bulk continuum. We thus target the charge–spin-locked A0=A3 in the following. In the general Kss(ω+iη) on surface state resonance, we use Eq. 9 and separate out the contribution due to relaxation η[27] Kss(ω+iη)=Bss+κ+rBss2+κ[2(Bss+2t2)+κ],

where κ(ω)=η(η−2id¯0). When Bss≠0, i.e., away from the surface–bulk borderline, we have[28] Kss(ω+iη)≈(Bss+r|Bss|)+κ(1+rBss+2t2|Bss|),

where we retain the leading contribution of η. Inside the surface state constraint region Eq. 11, (A0+A3)ss is contributed only by Bss+r|Bss| because the second part in the second line of Eq. 28 is at least ∼η2 in its real part; this indicates the choice r=−1 aforementioned and we obtain Eq. 12 in the main text, where δ(ω−εk) is from the resonance ∝η−1 suggested in Eq. 26.

To study the property in the vicinity of the borderline between the surface state and the bulk, it suffices to focus on A0,3 according to Eq. 2. Importantly, the full effect up to the quadratic order in Δω,Δk is relevant to our discussion. We separate B=Bss+F using Eq. 9. Up to the leading order, we first find Bss≈α·Δk, where α=∂kBss|bl=2∑i=1,2di∂kdi|bl, dependent on d1,2 that determines the constraint region Eq. 11, defines the normal direction at the boundary of the constraint region and is in general finite along the entire borderline. Second, we have F≈−d¯02+d32+∂kF·Δk+12Δk·∂k2F·Δk|bl=F1+F2 with[29] F1=−2d3(kbl)Δωη+β·Δk=−2d3(kbl)fF2=−(Δωη)2+Δk·Γ′·Δk+1cΔωηγ·Δk,

where β=∂kF|bl=2d3∂kε|bl, γ=8t2∂kd0|bl, c=4t2, and the rank-2 Hessian tensor Γ′=12∂k2F|bl=Γ+d3∂k2ε|bl with Γ=∂kd3∂kd3−∂kd0∂kd0|bl. Then, we have[30] B≈α·Δk+F1=α′·Δk−2d3(kbl)Δωη

up to the linear order with α′=α+2d3∂kε|bl, which suffices for our purpose. This leads to[31] C≈cF1−c′(Δωη)2+Δk·Λ′·Δk+Δωηγ′·Δk

with c′=c−4d3(kbl)2, Λ′=cΓ′+α′α′, and γ′=γ−4d3(kbl)α′. In the vicinity of WPs, Eqs. 30 and 31 become[32] Bwp≈α·ΔkCwp≈−c(Δωη)2+Δk·Λ·Δk+Δωηγ·Δk

with Λ=cΓ+αα and all parameters evaluated at the WPs. Note that we use primed notation explicitly for parameters defined for generic borderline points, which is omitted in Eq. 17 for brevity. With these definitions, Eq. 2 gives[33] A0−A3≈I2f(K+2t2)t2(t2+α·Δk)≈I2(fK+2iηt2)t4.

Besides the unimportant η-linear term, its leading order is generally nonlinear in Δω,Δk and higher order than K, which is negligible compared to A0−A3 below. Hence, we have identical A0,A3 within the leading order around the entire borderline and can obtain Eq. 19 and what follows.

Determine the Merging Configuration.

Given the crucial role of C in Eqs. 19, 21, and 22, it is clarifying to understand its physical meaning, which exactly determines the local merging configuration. Along the borderline except WPs, regarding Eq. 21, C≈4t2F1 is transparent per the surface/bulk distinction of C≷0: changing the signs of Δω,Δk, i.e., deviating in opposite directions away from the borderline, switches between surface and bulk contributions. For instance, setting Δk=0, one finds C=−8t2d3(kbl)Δω. Since d3 is the chiral part of the surface band Eq. 8, d3(kbl)>0 means the upper point merging into the bulk conduction band, then Δω≷0 is respectively above (bulk continuum) or below (surface state region) the merging point; d3(kbl)<0 similarly accounts for merging into the valence band. Setting Δω=0, one finds C=8t2d3(kbl)∂kε|bl·Δk. The factor ∂kε|bl fixes the local chirality, i.e., the sign of the slope, of surface dispersion at the merging point: If it is increasing, say, with respect to kx, and enters the conduction band (d3(kbl)>0), its left (Δkx<0) must be in the bulk continuum with C<0 while the right (Δkx>0) is outside. These are all observed in, e.g., Fig. 2A and also SI Appendix, Fig. S10A for the type-II case. Therefore, compared to Δω deviation involving sgn(d3(kbl)), one has an extra sign of ∂kε|bl along some Δk direction; together they fully determine the local merging configuration.

In the very vicinity of the WPs, as shown in Eq. 22, now any Δω causes C<0, reflecting that pure energy deviation from a WP always enters the bulk Weyl cone however tilted, as shown in, e.g., the type-II SI Appendix, Fig. S10B. When Δω=0, Λ determines the sign of C and is positive-definite unless the velocity tilting ∂kd0|wp is large enough. This directly corresponds to the distinction between type-I and type-II WSMs as shown in Fig. 4B, since Λ with a negative eigenvalue means C<0 along a certain range of Δk direction, i.e., such momentum deviation enters immediately the bulk electron/hole pockets once it leaves the WP. For the present model with |t2|<|t|, the two eigenvalues of the principal axes of Λ are λ1=4t2t12,λ2=4t2sin2k¯z(t2−t22−Dz2) respectively along kx,kz. Hence, large Dz and deviation Δkz can enter the bulk continuum while Δkx does not in the spectroscopy.

Supplementary Material

Appendix 01 (PDF)

This work was supported by JSPS KAKENHI (No. 18H03676) and JST CREST (No. JPMJCR1874). X.-X.Z. was partially supported by RIKEN Special Postdoctoral Researcher Program.

Author contributions

X.-X.Z. and N.N. designed research; X.-X.Z. performed research; X.-X.Z. and N.N. analyzed data; and X.-X.Z. and N.N. wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.

Supporting Information

Reviewers: P.A., Johns Hopkins University; and X.D., Hong-Kong University of Science and Technology.
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