Spin and Optical Fingerprints of Pressure-Driven Double Topological Phase Transition in Monoclinic BaIn₂As₂

Abstract

Distinguishing topological from trivial phases is central to condensed matter physics, yet resolving distinct nontrivial regimes that share the same Z ₂ classification remains challenging. Here, using first-principles calculations, we show that monoclinic BaIn₂As₂ undergoes two successive pressure-driven topological phase transitions within the same P2/m structural phase, evolving from a topological insulator at ambient pressure to a trivial insulator at 16 GPa and then to a high-pressure topological semimetal at 26 GPa. The two nontrivial phases share the same Z ₂ = 1 classification but exhibit distinct surface spectra and opposite low-energy optical anisotropy. We establish a two-level identification scheme in which spin Hall conductivity provides a fingerprint for separating nontrivial phases from the trivial phase, with possible inverse spin Hall electrical readout, whereas monoclinic symmetry-enabled in-plane reflectance anisotropy resolves the two nontrivial phases. We further propose a diamond-anvil-cell platform combining yttrium-iron-garnet-based spin pumping with polarization-resolved infrared reflectance.

1. Introduction

Topological insulators are commonly classified by the Z ₂ topological invariant into trivial and nontrivial phases. − Experimentally identifying the band topology requires combining probes with different sensitivities. Angle-resolved photoemission spectroscopy directly resolves surface-state dispersions, − while scanning tunneling microscopy can visualize surface quasiparticle interference and local electronic states; however, both are surface sensitive. Transport measurements probe macroscopic conducting channels but can be affected by bulk carriers, disorder, and the Fermi-level position. Complementary optical approaches have therefore attracted increasing attention, with nonlinear photocurrents providing bulk-sensitive signatures of topological band geometry, high-harmonic spectroscopy tracking strong-field dynamical responses associated with topology, and infrared spectroscopy detecting low-energy optical signatures of band inversion and topological phase transitions. An even subtler challenge arises when a single material hosts multiple inequivalent nontrivial phases, sharing the same Z ₂ classification. Throughout this work, we refer to such inequivalent nontrivial phases as topological branches. Developing experimentally accessible fingerprints that can distinguish these branches is, therefore, of fundamental importance.

To identify such topological branches, one must first realize a material platform in which multiple topological regimes can be continuously accessed and compared. Pressure provides a particularly effective route for this purpose, because it can tune interatomic bonding, crystal-field splitting, and the relative ordering of spin–orbit-coupled bands, thereby driving gap-closing and gap-reopening processes without introducing chemical disorder. − This makes narrow-gap Zintl arsenides such as the AIn₂As₂ (A = Ca, Sr, Ba) family especially attractive, as their pressure-sensitive electronic structures link band topology to orbital character, transport behavior, and thermoelectric properties. , Related In₂As₂-based compounds, such as EuIn₂As₂ and SrIn₂As₂, have already been shown to host axion-related and dual-topological electronic structures at ambient pressure. , Moreover, our previous study systematically investigated pressure-induced topological phase transitions in the AIn₂As₂ family. The same family also displays rich pressure responses beyond topology, as SrIn₂As₂ has been reported to exhibit superconductivity under suitable compression. Among these compounds, BaIn₂As₂ is particularly appealing because it differs structurally from the hexagonal P6₃/mmc phases of CaIn₂As₂ and SrIn₂As₂ and instead crystallizes in the low-symmetry monoclinic P2/m structure. , Our earlier study of pressure-induced structural transitions in the AIn₂As₂ family confirmed that monoclinic BaIn₂As₂ remains mechanically stable over a broad compression range, consistent with experimental observations. Together, its structural stability, pressure-sensitive electronic structure, and reduced monoclinic symmetry make BaIn₂As₂ a promising platform for exploring pressure-driven topological phases, while the symmetry-allowed off-diagonal dielectric response provides tensorial optical observables for resolving distinct topological regimes.

In this work, we combine first-principles electronic structure calculations, linear-response spin Hall conductivity (SHC), and tensorial optical response calculations to investigate the pressure-driven topological evolution of monoclinic BaIn₂As₂. We find that hydrostatic pressure induces two successive topological phase transitions, while the P2/m structure remains unchanged. The system evolves from a topological insulator at 0 GPa to a trivial insulator at 16 GPa and then to a high-pressure topological semimetal at 26 GPa. In the 26 GPa phase, a nontrivial direct gap persists at the A-point, while the Fermi level crosses nearby valence bands. This evolution is accompanied by a pronounced reconstruction of the spin-Berry curvature (SBC). Although the two nontrivial phases share the same Z ₂ = 1 classification and the same irreducible representations of the inverted bands, they exhibit distinct surface-state morphologies and opposite low-energy optical polarization preferences, thereby forming two inequivalent topological branches. We further established a two-level identification scheme. The SHC provides a spin-transport fingerprint for separating the topologically nontrivial phases from the trivial phase, while the monoclinic-symmetry-enabled polarized-reflectance anisotropy discriminates between the two nontrivial branches. Finally, we propose a possible diamond anvil cell (DAC) readout geometry combining yttrium iron garnet (YIG)-based spin pumping and mid-infrared polarized reflectance for future high-pressure verification of these fingerprints.

2. Results and Discussion

2.1. Two Successive Pressure-Induced Topological Phase Transitions in BaIn₂As₂

BaIn₂As₂ belongs to the In₂As₂-based Zintl family and has been experimentally confirmed to crystallize in the monoclinic P2/m structure (see Figure S1 and Supplementary Note 1 for details). This structural information provides the starting point for examining its pressure-driven topological evolution within a fixed low-symmetry lattice. Figure summarizes the pressure-driven topological evolution of monoclinic BaIn₂As₂. At low pressure, the system is a topological insulator with inverted bands near the A-point. With increasing pressure, the inverted band ordering is suppressed, and the band gap widens, driving the system into a trivial insulating phase. Upon further compression, the gap narrows again and a second band inversion occurs, leading to a high-pressure topologically nontrivial semimetallic phase, similar to those observed in GaGeTe and tunable half-Heusler compounds. , All three regimes retain the same P2/m space group, indicating that the transitions are isostructural and originate from pressure-driven band structure evolution rather than crystallographic phase transitions. The lower panels of Figure show the accompanying evolution of the SBC. In the two topologically nontrivial regimes, the SBC develops a pronounced sign-alternating texture near the A point, consistent with band inversion, whereas in the trivial phase, it becomes weak and lacks such sign reversal.

1.

Schematic of the two successive pressure-driven topological phase transitions and the associated spin Berry curvature reconstruction. The upper panel shows the evolution of band inversion, surface states, and Fermi-level position across the topological insulator (TI), trivial insulator (NI), and high-pressure topological semimetal phases. The lower panel shows spin Berry curvature textures, highlighting a pronounced sign reversal in the two topological phases, in contrast to the trivial phase where this feature is strongly suppressed.

To clarify the microscopic origin of these two topological phase transitions, we examine the orbital-projected band structures along the E–A–Y path at three representative pressures, as shown in Figure a–c. In the normal band ordering near the A point, the valence and conduction bands are mainly contributed by In/As p and s orbitals, respectively. At 0 GPa, spin–orbit coupling (SOC) induces an s-p orbital-character inversion near the A point, giving rise to a topological insulator (TI) phase, as confirmed by the Wilson-loop evolution shown in Figure S2. Around the A point, the inverted valence and conduction bands transform as the A ₅ + A ₆ and A ₃ + A ₄ irreducible representations, respectively. The character table of the little group at the A point in the P2/m structure is shown in Figure d, and the corresponding Seitz symbols, coordinate transformations, and matrix representations are listed in Figure e. At 16 GPa, compression restores the normal p-s band ordering near A, converting the system into a normal insulator (NI). Upon further compression to 26 GPa, the gap decreases again and a second s-p orbital inversion occurs near the A point, giving rise to another topologically nontrivial phase with semimetallic character, denoted as a topological semimetal (TS). In this high-pressure phase, the direct gap associated with the inverted bands remains nontrivial near the A point, while the Fermi level intersects nearby valence bands. Notably, the inverted band-edge states near A at 26 GPa have the same irreducible representations as those at 0 GPa, and the two nontrivial phases share the same Z ₂ = 1 classification, as confirmed by the Wilson-loop evolution in Figure S2. This indicates that the Z ₂ classification alone cannot distinguish these two nontrivial phases, motivating the search for additional spectroscopic fingerprints.

2.

Orbital-projected band structures and irreducible representations. Orbital-projected band structures of In/As s (red) and p (blue) orbitals near the A point for the (a) 0, (b) 16, and (c) 26 GPa systems. Panel (d) shows the character table of the little group at the A point in the P2/m structure. Panel (e) lists the symmetry operations with their Seitz notation, coordinate transformations, and matrix representations.

The reinversion is accompanied by an anisotropic structural response under compression. The nonuniform bond-length and bond-angle changes, together with charge redistribution around the Ba3/Ba4 and As1–As4 environments, modify the In/As s-p hybridization that controls the band edges; the detailed structural and orbital analysis is provided in Supplementary Note 1.

Symmetry-indicator analysis further confirms the nontrivial–trivial–reentrant–nontrivial evolution from 0 to 16 to 26 GPa. The two nontrivial phases share the same nontrivial NLC classification (1,0,1,1), showing that the symmetry indicators alone do not distinguish the two topological branches (Supplementary Note 1).

A complementary spin-resolved Wilson-loop calculation on the k _z = 0 plane gives C _s = 0 at 0 and 16 GPa and C _s = −2 at 26 GPa. This plane-resolved descriptor provides an additional distinction between the two Z ₂-equivalent topological branches (Figure S5 and Supplementary Note 1).

We further performed HSE06+SOC calculations. The HSE06+SOC results indicate nontrivial topological phases at −4 and 29 GPa, whereas the 16 GPa state remains trivial. Although the pressure points at which the inverted band ordering occurs differ from the PBE+SOC results, which show inversions at 0 and 26 GPa, the conclusion of two successive pressure-driven topological transitions remains valid (Figure S17 and Supplementary Note 6).

Although indistinguishable by the Z ₂ classification alone, the two nontrivial phases display clear differences in their surface electronic structures. Figure shows the surface spectral functions on the (101) and (111) terminations. Both the 0 and 26 GPa phases exhibit surface states connecting the projected valence and conduction bands near the surface projection of the A point (see Figure S3), whereas the trivial 16 GPa phase does not. The key distinction between the two nontrivial phases lies in the energy evolution and connectivity of the near-Fermi surface-state constant-energy contours. On the (101) surface, the 0 GPa phase exhibits a relatively simple surface spectrum. The constant-energy cuts at different energies show a single line-like surface state along the Γ̅–Y̅ direction, whose connectivity remains unchanged (see Figure a). By contrast, at 26 GPa, the surface-state branches split in momentum space and become entangled with nearby projected valence-band-edge states. As the energy of the constant-energy cut increases, the surface states evolve from line-like features along the Y̅–L̅ direction to Fermi arcs and then to closed Fermi pockets (see Figure c). A similar but more pronounced reconstruction appears on the (111) surface. At 0 GPa, the surface spectrum exhibits a linear Dirac-cone-like surface state. The constant-energy contours evolve from low-energy Fermi pockets into line-like Fermi arcs along the M̅–K̅ direction (see Figure d). In contrast, the 26 GPa phase shows curved nonlinear surface states. As the energy increases from low to high values, the constant-energy Fermi contours evolve from simple Fermi pockets into multichannel, petal-like pockets composed of multiple components (see Figure f). These changes in the number, shape, and connectivity of the near-Fermi pockets indicate a Lifshitz-type reconstruction of the surface-state constant-energy contours, marked by contour splitting, reconnection, and the formation of multicomponent pockets. Since the P2/m crystal structure is retained throughout the pressure range, this behavior should be understood as an electronic reconstruction of the topological surface spectrum rather than a crystallographic phase transition. Such pressure-driven reorganization of the surface states provides a microscopic basis for the distinct spin and optical responses discussed below.

3.

Surface-state evolution and constant-energy contours. Surface states on the (101) and (111) surfaces at three representative pressures. The three columns from left to right correspond to the 0, 16, and 26 GPa systems, respectively. The upper panels (a)-(c) show the results for the (101) surface, while the lower panels (d)-(f) show those for the (111) surface. The insets show constant-energy contours at selected energies.

2.2. Spin Berry Curvature and Spin Hall Response across the Topological Phase Transitions

The pressure-driven evolution of the bulk band topology is also manifested in the momentum-resolved spin Berry curvature (SBC). As shown in the lower panels of Figures and S4, the SBC component ${\mathrm{\Omega }}_{xz}^y(\mathbf{k})$ exhibits a pronounced sign reversal near the A point in the two topologically nontrivial phases at 0 and 26 GPa, where band inversion occurs. By contrast, in the trivial insulating phase at 16 GPa, the SBC retains the same sign in the corresponding momentum region. These features provide further evidence for the two successive pressure-driven topological phase transitions in BaIn₂As₂.

The corresponding intrinsic SHC, obtained by integrating the SBC over the Brillouin zone, is shown in the upper panel of the third column of Figure . Both topologically nontrivial phases exhibit a finite spin Hall plateau, whereas the trivial 16 GPa phase shows a nearly vanishing SHC within the bulk-insulating gap. The SHC thus provides a useful spin-transport fingerprint associated with the distinction between the topologically nontrivial phases and the trivial phase and can be electrically probed through inverse spin Hall effect measurements, see Supplementary Note 2 for details. However, it cannot, by itself, discriminate between the two topological branches because they differ mainly in the magnitude of the SHC while sharing the same Z ₂ classification. Resolving this distinction therefore requires a complementary probe such as the polarized-reflectance anisotropy discussed below.

4.

Proposed spin–optical identification strategy. The left panel shows a diamond-anvil-cell geometry with a YIG spin-source layer and a polarized infrared optical window. The middle panel presents an enlarged view of the proposed readout device and the corresponding physical responses. The right panel shows the first-principles spin Hall conductivity and reflectance anisotropy $A_R^{xz}$ as a two-tier identification route. The spin Hall conductivity provides a first-level fingerprint for distinguishing nontrivial and trivial regimes, while $A_R^{xz}$ discriminates between the two nontrivial topological branches. In the ferromagnetic resonance spin-pumping module, YIG denotes yttrium iron garnet, M the YIG magnetization, H the applied magnetic field, J _s the injected spin current, and J _c the charge current generated by the inverse spin Hall effect (ISHE). ISHE provides the electrical readout by converting the injected spin current into a transverse charge current or voltage. LPRR denotes low-energy polarization-resolved reflectance for comparing the E∥x and E∥z responses. TI, NI, and TS denote topological insulator, normal insulator, and topological semimetal, respectively.

2.3. Tensorial Optical Anisotropy as a Phase Discriminator

According to symmetry analysis, in addition to the three nonzero diagonal dielectric-tensor components, monoclinic BaIn₂As₂ also allows a finite off-diagonal dielectric-tensor component ε _zx (see Supplementary Note 3 for details). Since the dielectric response is governed by interband optical transition matrix elements and is sensitive to the orbital composition and hybridization of low-energy electronic states, this tensorial response provides a natural channel for probing differences between the two topological branches. Generalized ellipsometry offers an established experimental approach for determining the full dielectric tensor of low-symmetry crystals, as demonstrated for monoclinic CdWO₄ and Y₂SiO₅ , and the theoretical framework for monoclinic dielectric tensors has been well developed.

Our calculations show that this tensorial optical response is highly sensitive to pressure-driven electronic reconstruction.

To connect the tensorial optical response to an experimentally accessible observable, we further evaluate the normalized polarized-reflectance anisotropy under normal-incidence polarized light.

$$A_R^{xz}(\omega )=\frac{R_{xx}(\omega )-R_{zz}(\omega )}{R_{xx}(\omega )+R_{zz}(\omega )}$$

Here, R _xx (ω) and R _zz (ω) are the reflectances measured with the incident electric field polarized along the in-plane crystallographic x and z directions, respectively, with light propagating along the surface normal y. Thus, $A_R^{xz}(\omega )$ directly compares the two in-plane polarized-reflectance channels, where $A_R^{xz}>0$ denotes x-dominant reflectance, $A_R^{xz}<0$ denotes z-dominant reflectance, and $A_R^{xz}\approx 0$ indicates weak low-energy anisotropy. As an optical observable derived from the xz dielectric sub-tensor, $A_R^{xz}$ captures the low-symmetry tensorial response encoded in ε _xx , ε _zz , and ε _zx . The detailed definition and derivation of $A_R^{xz}$ are provided in Supplementary Notes 3 and 4. As shown in the lower right panel of Figure , $A_R^{xz}$ is positive at low energies for the 0 GPa phase, negative for the 26 GPa phase, and close to zero for the trivial 16 GPa phase. Accordingly, the low-energy optical response changes from x-dominant at ambient pressure to z-dominant at high pressure. This reversal of optical anisotropy provides a clear optical fingerprint for distinguishing the two topological branches.

2.4. High-Pressure Spin–Optical Detection Strategy

We additionally performed HSE06+SOC optical calculations. The dielectric-tensor and polarized-reflectance results in Figures S18 and S19 are consistent with our initial conclusion that the low- and high-pressure topological phases can be distinguished through their in-plane optical anisotropy; further details are provided in Supplementary Note 6.

Combining the above analyses, the complementary sensitivities of the spin Hall and optical anisotropy responses suggest a possible spin–optical readout strategy for tracking the pressure-driven phase evolution, as shown in Figure . The spin Hall channel can probe the transition between nontrivial and trivial regimes through the presence or strong suppression of the inverse spin Hall voltage, whereas the polarized-reflectance channel can distinguish between the two nontrivial topological branches. One possible implementation is to combine a diamond anvil cell with a YIG spin-source layer for spin pumping, lateral Au electrodes for inverse spin Hall voltage readout, and a diamond optical window for normal-incidence polarized infrared reflectance measurements, leveraging established spin-pumping detection in ferrimagnetic-insulator/topological-material heterostructures − and high-pressure optical spectroscopy in DAC experiments. −

The proposed readout should be viewed as a complementary fingerprinting scheme rather than a replacement for conventional transport. Resistivity, Hall effect, magnetoresistance, and quantum oscillation measurements remain essential for pressure-driven topological materials, but their interpretation can involve concurrent bulk and surface contributions. This issue remains relevant even in prototypical Bi₂Se₃ topological insulators, where intrinsic defect-induced bulk conduction and bulk-dominated Shubnikov–de Haas oscillations can complicate the identification of surface-state transport. , Under DAC conditions, electrical transport also depends on sample geometry, pressure environment, and electrode integration. , The ISHE/LPRR scheme therefore adds two independent response channels. ISHE probes the spin-to-charge conversion associated with the calculated spin Hall response, whereas LPRR compares the low-energy E∥x and E∥z reflectances to distinguish the two nontrivial branches.

In such a platform, the electrical channel would monitor the emergence of an inverse spin Hall voltage as an experimentally accessible signature of a finite spin Hall response associated with the topological regime, while the optical channel would determine the realized topological branch by measuring the sign of $A_R^{xz}$ at specific mid-infrared wavelengths. Further details are provided in Supplementary Note 5. This combined spin–optical approach connects the predicted SHC and optical fingerprints with experimentally measurable observables and offers a practical route for mapping pressure-driven topological phase diagrams.

3. Conclusions

In summary, by combining first-principles calculations of band topology, spin Hall conductivity (SHC), and tensorial optical response, we have shown that monoclinic BaIn₂As₂ undergoes two successive pressure-driven topological phase transitions. Under hydrostatic compression, the system evolves from a topological insulator at ambient pressure to a trivial normal insulator at 16 GPa, and subsequently to a high-pressure topological semimetal at 26 GPa, while retaining the same P2/m crystal structure. In the high-pressure phase, the band inversion at the A point preserves a nontrivial direct gap, whereas the Fermi level intersects nearby valence bands, giving rise to a semimetallic state. The two topological phases at 0 and 26 GPa share the same Z ₂ = 1 classification and the same irreducible representations for the inverted band-edge states near the A point. Nevertheless, they exhibit qualitatively different surface-state morphologies and opposite low-energy optical polarization preferences. The ambient-pressure phase displays a relatively simple Dirac-like surface-state dispersion and a single near-Fermi pocket, whereas the high-pressure phase develops split surface branches, a Lifshitz-type reconstruction of the near-Fermi surface-state contours into a multi-pocket manifold, and a z-dominant optical response. These contrasting signatures support a two-tier framework in which the SHC provides a spin-transport fingerprint for separating topologically nontrivial phases from the trivial phase, while polarized-reflectance anisotropy discriminates between the two nontrivial topological branches. A DAC-compatible spin–optical readout scheme, proposed as a possible experimental outlook, links the predicted fingerprints to experimentally measurable quantities. More broadly, our results establish low-symmetry Zintl phases under pressure as a promising platform for exploring distinct topological regimes beyond conventional bulk-invariant classifications, with tensorial optical responses providing a powerful route for resolving distinctions that are not captured by conventional bulk-invariant classifications.

4. Methods

First-principles calculations were performed using the Vienna Ab initio Simulation Package (VASP) , within the projector-augmented-wave method and the Perdew–Burke–Ernzerhof generalized-gradient approximation, including SOC. The convergence criterion for the total energy was set to 10^–8 eV. A plane-wave cutoff energy of 450 eV was adopted. During structural relaxation, the residual force on each atom was reduced to below 0.01 eV/Å. The Brillouin zone was sampled using an 8 × 16 × 6 Γ-centered Monkhorst–Pack k-point mesh. Because the present study focuses on defect-free BaIn₂As₂ under pressure, all our calculations were performed using the crystallographic unit cell rather than defect supercells. Hybrid-functional studies of monoclinic β-Ga₂O₃ have underscored the importance of accurate band-edge and localized-state descriptions for assigning defect charge states and spectroscopic signatures. − Accordingly, additional HSE06+SOC calculations, using 25% short-range Hartree–Fock exchange and a screening parameter of 0.2 Å^–1, were performed to examine the pressure-driven band ordering. , For the optical calculations, the parameters CSHIFT and NEDOS were set to 0.1 and 2000, respectively. The irreducible representations of the Bloch states were obtained using IRVSP, and the symmetry indicators were evaluated using the VASP2TRACE-related workflow developed for topological-materials diagnosis. , To accurately reproduce the first-principles band structure, a tight-binding model based on maximally localized Wannier functions (MLWFs) , was constructed. Using the surface Green’s function method, − the Wannier charge center (WCC), surface-state, and Fermi-surface (FS) calculations for semi-infinite surfaces were performed with the WannierTools package.

The authors declare that all the relevant data are available within the paper and its Supporting Information file or from the corresponding author upon reasonable request.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c07032.

• Supplementary Notes 1–6, Tables S1–S4, and Figures S1–S19, including structural, Wilson-loop, electronic structure, spin Hall, and optical response analyses (PDF)

The authors declare no competing financial interest.