Новый тип топологических фаз: теоретики предсказали кристаллическую антисимметрию, защищаемую калибровочными полями

Abstract Gauge fields are known to transmute linear representations of symmetries into projective ones, which fundamentally modify the algebraic relations among symmetry operators. In this paper, we show that gauge fields can similarly modify the commutation relations between crystalline symmetry operators and the Hamiltonian, introducing a crystalline antisymmetry that anticommutes with the Hamiltonian. This antisymmetry enforces a phase difference between symmetry-related hoppings, a constraint which can be readily realized in artificial systems. Focusing on reflection antisymmetry (RAS), we employ Clifford algebra extensions to classify topological insulators and superconductors with this additional antisymmetry across all ten Altland–Zirnbauer (AZ) symmetry classes. Furthermore, by identifying the required fundamental building blocks for both spinless and spinful systems, we develop a dimensional-extension approach to systematically construct -dimensional systems featuring an additional RAS from -dimensional ones within the target AZ class. As a concrete demonstration, we explicitly build a two-dimensional class-A system exhibiting an additional RAS by stacking and coupling Su–Schrieffer–Heeger chains. Our work provides a theoretical foundation for crystalline-antisymmetry-protected topological phases and facilitates their experimental realization. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Symmetry-protected topological states have been a central research focus in condensed matter physics for decades [1–7]. Following the discovery of topological insulators (TIs) protected by time-reversal symmetry (TRS) [8–13], the systematic exploration of novel topological states culminated in the tenfold Altland–Zirnbauer (AZ) classification of TIs and topological superconductors (TSCs) across various dimensions, based on the three nonspatial symmetries: time-reversal, particle-hole, and chiral [14–19]. While the AZ classification has successfully guided the discovery of numerous exotic topological materials, it is now well established that crystalline symmetries—ubiquitous in solid-state systems—can further enrich the topological landscape. Topological states protected by these crystalline symmetries are known as topological crystalline insulators [20–28], which host robust, symmetry-protected gapless boundary states arising from their nontrivial bulk topology. Moreover, recent studies have shown that the family of topological states can be further enriched by the introduction of a gauge field, which transmutes conventional linear representations of symmetries into projective ones [29–35]. Consequently, the algebraic relations between symmetry operators are fundamentally modified by this gauge field, thereby altering the symmetry constraints imposed on the Hamiltonian. While engineering gauge fields in conventional solid-state systems remains challenging, artificial systems—such as cold atoms [36–42], photonic crystals and acoustic crystals [43–49]—have been demonstrated as promising platforms, where hopping amplitudes can be designed to be positive or negative for the realization of the required gauge structure. Conventionally, crystalline symmetries constrain a system by requiring the symmetry operator to commute with the Hamiltonian : Given that a gauge field can modify the algebraic relations among symmetry operators, a natural question arises: can it similarly alter the algebraic relation between a symmetry operator and the Hamiltonian? If so, one encounters a symmetry termed crystalline antisymmetry, which is defined by an operator that anticommutes with the Hamiltonian: Notably, this scenario contrasts with previous studies, which typically treat crystalline antisymmetry as a combination of crystalline and chiral symmetries [50]. Although TIs and TSCs protected by additional order-two crystalline antisymmetries have been successfully classified [26], their experimental realization remains challenging. In this work, we address this question affirmatively by investigating the constraints imposed by crystalline antisymmetry, which fundamentally dictates a phase difference between symmetry-related hopping processes. This provides a direct pathway for experimental realization in artificial systems, where the antisymmetry condition of equation (2) can be satisfied by engineering hopping phases. Crucially, not all crystalline symmetries admit such a transmutation. For instance, a threefold rotation cannot be consistently transmuted into an antisymmetry; if the generator were an antisymmetry, then the identity element would have to anticommute with the Hamiltonian, which is impossible. Focusing primarily on reflection antisymmetry (RAS), we employ Clifford algebra extensions [18] to classify systems with an additional RAS across all ten AZ classes, agreeing with known results [26]. Notably, for the five chiral classes, the combination of RAS and chiral symmetry (CS) reduces to a conventional reflection symmetry, bringing our classification into exact agreement with previous classifications for reflection-symmetric systems [51]. More importantly, aiming to realize these topological states with an additional RAS, we develop a systematic framework for model construction. We first identify the required fundamental building blocks by translating the constraints imposed by the RAS into realistic minimal physical systems. Based on these building blocks, we propose a dimensional-extension approach to systematically construct -dimensional systems exhibiting an additional RAS from -dimensional ones within the target AZ class. Our approach consists of three steps. First, using the identified building blocks, we double the -dimensional system by inserting a reflection plane parallel to it. Second, we couple the two systems via RAS-allowed hoppings, forming a bilayer system. Finally, we periodically replicate this bilayer system along the direction perpendicular to the reflection plane and introduce RAS-allowed couplings between adjacent bilayers. Note that all RAS-allowed interlayer couplings can be directly derived from the building blocks. To illustrate this approach, we explicitly construct a two-dimensional (D) class-A lattice system with RAS by stacking and coupling one-dimensional (D) Su–Schrieffer–Heeger (SSH) chains. We also analyze the topological properties of this constructed system. 2. Crystalline antisymmetry constraints We begin by investigating the crystalline antisymmetry associated with a general space group , which can be directly applied to RAS, our primary focus in the subsequent sections. An element can be expressed as , where is an element of a -dimensional point group and denotes a fractional translation vector. A localized state is labeled by the Bravais lattice vector , the sublattice index , and the internal degree of freedom (e.g. spin or orbital) on that sublattice. Under the action of , this state transforms as where is the permutation of the sublattices and is the action of on the internal degrees of freedom. Here, the lattice displacement is given by , where is the position of sublattice within the unit cell. Accordingly, equation (3) can be recast in second quantization as where is the unitary representation of , and summation over repeated indices is implied. Consider a generic tight-binding Hamiltonian of the form where , and is the hopping amplitude from to . Imposing the antisymmetry condition yields the following constraint on the hopping amplitudes: with and . Here, is the hopping matrix between the internal degrees of freedom of sublattices and , while represents the action on the internal degrees of freedom of sublattice . This is a direct consequence of equation (6), which dictates that the amplitudes of symmetry-related hopping processes must carry opposite signs. For spinless systems, this constraint reduces to To capture the topological properties, it is advantageous to employ the periodic Bloch basis defined via the transform which disregards the sublattice structure within the unit cell. From equation (4), we have Substituting equation (9) into equation (5) yields the Hamiltonian in momentum space as where Then, applying the action of to the Hamiltonian in equation (11) via equation (10) and invoking the antisymmetry condition in equation (6), we are led to where the matrix elements of are given by . The constraints in equations (7) and (8) enforce a phase difference between symmetry-related hoppings, which can be readily realized by engineering synthetic gauge fields in artificial systems. Acoustic crystals are particularly advantageous for this due to their highly controllable couplings, despite the inherent challenge of simulating spinful systems [52, 53]. Here, lattice sites are simulated by acoustic resonators and hoppings are mediated by waveguides [54–56]. To control the sign of hopping amplitudes, a bipolar mode is used to mimic a -orbital, with positive and negative air pressures representing the polarization. By changing how the waveguides connect these resonators, one can simulate parallel or anti-parallel -bonds, perfectly analogous to electronic systems. Because the coupling sign is geometrically determined by the connection routing, this well-established technique has been widely utilized to construct topological acoustic systems. 3. Reflection antisymmetry The conventional AZ classification is based on three nonspatial symmetries: TRS, particle-hole symmetry (PHS), and their combination, CS. These symmetries, represented by , , and , respectively, constrain the Hamiltonian in momentum space as follows: where we have adopted the convention . In addition to the AZ symmetries, we now incorporate RAS, beginning with its representation. Notably, the RAS we consider here inherits all the conventional properties of reflection symmetry in both real and momentum spaces, with the only exception that it anticommutes with the Hamiltonian. Specifically, the RAS transforms the coordinates of a -dimensional system as which leads to in momentum space. Thus, equation (13) gives rise to the constraint over the Hamiltonian as with the representation of RAS. Here, we adopt the convention given that these symmetries act on distinct degrees of freedom. Note that replacing with would yield anticommutation relations with the antiunitary symmetries; however, both conventions impose identical constraints on the Hamiltonian. Moreover, the RAS satisfies with the particle spin. In particular, for spinful systems, since reflection can be viewed as spatial inversion followed by a -rotation about the axis perpendicular to the reflection plane. Consequently, the RAS defined in equation (17) acts on the spin degrees of freedom as where are the Pauli matrices. We summarize the classification of TIs and TSCs with an additional RAS in table 1, deferring the detailed derivation via Clifford algebra extensions to Table 1. Classification of TIs and TSCs with an additional RAS. 4. Construction of reflection-antisymmetric systems In this section, we first outline the RAS-constrained building blocks, and then demonstrate how to construct higher-dimensional systems featuring an additional RAS from lower-dimensional ones within the same AZ class. 4.1. Building blocks for spinless systems Figure 1 depicts the three building blocks required to realize the RAS. For spinless systems, the RAS can be represented as where the Pauli matrices act on the layer degrees of freedom related by reflection, and is the intralayer identity operator. For notational simplicity, the explicit tensor product symbol is suppressed throughout this paper. For the building block in figure 1(a), equations (8) and (22) constrain the Hamiltonian to the form where and denote the hopping amplitude and onsite energy, respectively. This form reveals two fundamental constraints imposed by the RAS: reflection-related sites must have opposite onsite energies, and the hopping phases for reflection-invariant bonds must be . Note that the second constraint implies a phase difference between reflection-related hopping processes. Since such hopping terms explicitly break TRS, they are prohibited when TRS is preserved. Second, for the pair of reflection-related hopping processes shown in figure 1(b), the RAS-allowed Hamiltonian reads where the Pauli matrices act on the two-site subspace of each layer. In this setup, the hopping amplitudes along the blue and red arrows in figure 1(b) are , respectively, indicating a phase difference between the two hopping processes. Crucially, the second term in equation (24) breaks TRS and must therefore be excluded for time-reversal-invariant systems. Third, for the building block as shown in figure 1(c), equations (8) and (22) yield the RAS-allowed Hamiltonian in which the first term breaks TRS. From equations (24) and (25), we arrive at the third constraint: RAS enforces a phase difference between reflection-related hopping processes. This follows from the definition of antisymmetry in equation (2) with , which requires the coefficients of the reflection-related terms to have opposite signs. 4.2. Building blocks for spinful systems In this subsection, we consider the RAS-allowed spinful Hamiltonians for the building blocks illustrated in figure 1. Here, the RAS is represented as satisfying . First, the RAS-allowed Hamiltonian for the building block in figure 1(a) consists of both spin-independent and spin-dependent terms. The spin-independent part is identical to equation (23) for the spinless case, which we rewrite as We now turn to the spin-dependent terms for the building block in figure 1(a). The allowed Hamiltonian, not involving hopping, is given by where is the component of the Zeeman field along . Physically, this implies that for sites related by reflection, the component of the Zeeman field perpendicular to the reflection plane must be opposite, whereas the components parallel to the plane must be identical. This constraint follows directly from the representation in equation (26), according to which commutes with but anticommutes with and . When hopping is considered, the spin–orbit coupling (SOC) allowed by the RAS is given by with the corresponding hopping amplitudes. Here, the first two terms are spin-flipping, while the third term is spin-conserving but introduces a phase difference of between the two spin states. When constructing time-reversal-invariant systems, only the first term in equation (29) is allowed, since the others break TRS. Second, consider the two building blocks illustrated in figures 1(b) and (c), where the hopping processes are related by the reflection. The corresponding RAS-allowed Hamiltonians are given by and respectively. To understand the SOC in equations (30) and (31), we examine the role of the RAS. The RAS constraints in these equations imply that the mirrored SOC associated with acquires a phase shift, while the other terms do not. Although artificial systems are typically spinless, we expect that spinful systems with RAS could be realized by using cold atoms. 4.3. Dimensional-extension approach We now propose a systematic method—termed the dimensional-extension construction—which uses the aforementioned building blocks to construct a -dimensional system with an additional RAS from a-dimensional system in the target AZ class. First, we double the -dimensional system with Hamiltonian by reflection, using the building blocks in figures 1(a) and (b). Note that this step does not introduce any coupling between the original and doubled systems. The resulting bilayer lattice is illustrated in the shaded region of figure 2, where the dashed line indicates the reflection plane. Hence, the RAS is represented as Download figure: Standard image High-resolution image satisfying the algebra of equation (19) in the presence of TRS and/or PHS. From equations (22) and (26), we have and for the spinless and spinful cases, respectively. This procedure yields for the Hamiltonian of the mirrored lattice. Consequently, the Hamiltonian for the reflection-doubled system is given by which satisfies given that . The commutation relations in equation (19) ensure that belongs to the same AZ class as . Explicitly, when present, the TRS and PHS for the bilayer system are represented as where and represent the corresponding symmetries of the original system before doubling. Second, interlayer couplings are introduced using the building blocks illustrated in figures 1(a) and (c). Specifically, these interlayer couplings are given by equations (23) and (25) for the spinless case, and equations (29) and (31) for the spinful case. Importantly, the introduced couplings must preserve all AZ symmetries of the target class, that is, any term in these equations that breaks these symmetries must be excluded. This procedure leads to the interlayer coupling , and the resulting Hamiltonian for the bilayer system takes the form Furthermore, the interlayer coupling satisfies the relation a property that naturally follows from the building blocks introduced previously. Third, we construct the -dimensional system by periodically replicating the bilayer structure along , as illustrated in figure 2. The coupling between the newly adjacent layers is then introduced in the same manner as , accounting for the additional reflection plane enforced by this periodicity. This ensures consistency between the RAS and the translational symmetry. Thus, analogous to equation (36), we have This completes our method for constructing a -dimensional system with an additional RAS from a -dimensional system in the target AZ class. The resulting Hamiltonian in momentum space is given by with denoting the momentum parallel to the reflection plane. By virtue of equations (36) and (37), it is straightforward to verify that the Hamiltonian in equation (38) satisfies the RAS condition defined in equation (18). We note that the constructed -dimensional system by our dimensional-extension approach belongs to the same AZ class as the original -dimensional system. As discussed before, the nontrivial topology may have two origins: the -dimensional system and the -dimensional subsystem on the reflection-invariant plane. For example, consider a dimensional hierarchy of for . In this scenario, the original topology is destroyed by the model construction. Nevertheless, since the RAS effectively acts as an additional CS on the reflection-invariant plane, this -dimensional subsystem falls into a different class from the original and constructed systems, meaning it can still exhibit a nontrivial topology. 4.4. An application In this section, we apply our previously proposed dimensional-extension method to the SSH model in class A to construct a D model in the same class that exhibits RAS. As illustrated by the blue chain in figure 3(a), the Hamiltonian of the starting SSH model reads Download figure: Standard image High-resolution image where and are the annihilation operators for the two sublattices. Here, and denote the dimerized hopping magnitudes, is the hopping phase, and are the onsite energies for the two sublattices, as shown in figure 3(a). By the Fourier transformation and , equation (39) leads to the Hamiltonian in momentum space, given by where both TRS and CS are explicitly broken by the hopping phase and the mass term. Using the building blocks in figures 1(a) and (b), we construct the mirrored chain marked in red in figure 3(a). The resulting Hamiltonian for the double-chain system is obtained as which clearly satisfies with . To introduce inter-chain coupling, we employ the building block of figure 1(a), yielding hopping amplitudes of between reflection-related sites as illustrated in figure 3(a). In addition, we alternately insert the building block of figure 1(c) into the double chains, which gives rise to next-nearest-neighbor hoppings with amplitudes along the yellow and green arrows, respectively. Next, we periodically replicate the double-chain system along and introduce the building block of figure 1(a) between the newly adjacent chains, as shown in figure 3(b). This results in inter-chain hopping amplitudes of . Finally, after the construction shown in figure 3(b), the Hamiltonian in equation (38) becomes where and . According to table 1, the topological nature of this class is characterized by . Since TRS is broken, the first topological invariant is the Chern number, given by where is the -form Berry curvature associated with the Hamiltonian in equation (42). To be precise, the Berry curvature is defined as , where the Berry connection is given by with the state of the -th valence band [3, 17, 57]. Figure 3(c) shows the phase diagram of the constructed system for , , and , where the red, green, and purple regions correspond to Chern numbers of , , and , respectively. The numerical calculation of in figure 3(c) is performed using the gauge-invariant discretization method [58]. In practice, the Brillouin zone is discretized into a mesh of plaquettes. Summing the accumulated phases around all plaquettes yields the Chern number in equation (43). The phase boundaries, where the bulk gap closes, separate distinct topological domains. Furthermore, the RAS constraint in equation (18) yields which indicates that the RAS acts as a CS for the Hamiltonian along the reflection-invariant line . Consequently, belongs to class AIII in one dimension, which features topology according to the conventional AZ classification. Therefore, the second topological invariant for the gapped Hamiltonian can be defined as which counts the number of chiral edge states crossing at . The mathematical origin of this D winding number, which classifies systems with an effective CS, is detailed in the standard literature of topological classification [3, 17, 57]. Figure 3(d) displays the energy spectrum with the open boundary along . Since this boundary condition preserves RAS, a single chiral edge state crosses at the reflection-invariant point, confirming . Note that this zero-energy crossing is symmetry-protected: because the RAS enforces the mapping , the edge state at the reflection-invariant point is strictly pinned to when . 5. Summary In this work, we investigate crystalline antisymmetry, with a primary focus on RAS. We demonstrate that RAS fundamentally enforces a phase difference between symmetry-related hopping processes. This phase constraint can be directly engineered in artificial systems via synthetic gauge fields, offering a highly tunable route for experimental realization. Using Clifford algebra extensions, we obtain the classification of TIs and TSCs with an additional RAS. Moreover, we identify the essential building blocks required to realize RAS in lattice systems and develop a dimensional-extension approach for model construction. This approach provides a systematic procedure for constructing higher-dimensional systems exhibiting an additional RAS from lower-dimensional ones within the target AZ class. To demonstrate the practical utility of our approach, we explicitly construct a D lattice system with RAS in class A by stacking and coupling SSH chains. Unlike boundary states protected by internal AZ symmetries, which remain unaffected, the gapless boundary states protected by the RAS are sensitive to boundary conditions. Specifically, for boundaries that preserve the RAS, the system will host robust boundary modes pinned exactly to at the reflection-invariant momenta of the surface Brillouin zone. Since gauge fields can be readily realized in artificial systems, our work paves the way for the experimental realization of topological states protected by crystalline antisymmetries in artificial systems. Acknowledgments This work is supported by National Science Foundation of China (Grants No. 12275126) and Innovation Program for Quantum Science and Technology (Grants No. 2024ZD0300101). Data availability statement No new data were created or analyzed in this study. Appendix: The derivation of table 1 We now turn to the classification of topological insulators (TIs) and topological superconductors (TSCs) in the presence of an additional reflection antisymmetry (RAS), using the method of Clifford algebra extension developed in [18]. Note that a more systematic K-theory approach to the classification with additional order-two symmetries can be found in [26], which generalizes the Teo–Kane framework [57]. However, such a treatment lies beyond the scope of the present work. For a given Altland–Zirnbauer (AZ) class, the generic Dirac Hamiltonian takes the form where the mass term and the Dirac matrices satisfy the anticommutation relations Consequently, these operators can be used to generate a Clifford algebra. For real classes, the presence of the antiunitary symmetries (time-reversal symmetry (TRS) and/or particle-hole symmetry (PHS)) imposes the constraints and according to equations (14) and (15), respectively. Moreover, to account for the antiunitarity of and , we introduce an operator that serves as the imaginary unit, such that In contrast, for complex classes, the absence of antiunitary symmetries renders the introduction of unnecessary. Specifically, for class AIII, the chiral symmetry (CS) defined in equation (16) leads to implying that can be considered as an additional operator to generate the Clifford algebra. Furthermore, according to equation (18), the RAS imposes the relations where for spinless and spinful systems, respectively. Consequently, for real classes with , it naturally follows that . For classes C and D, however, are both possible. For example, a spinless -wave superconductor belongs to class D, as does a superposition of spinful -wave superconductors without TRS. Note that for complex classes, there is no physical distinction between and due to the gauge freedom and the absence of antiunitary symmetries. Hence, we may take without loss of generality. Here, we briefly outline the method of Clifford algebra extension. The initial Clifford algebra is generated by the Dirac matrices , along with the operator and the relevant AZ symmetry operators. Introducing an additional mass term , satisfying and all aforementioned (anti) commutation relations, yields an extended Clifford algebra. The topological classification is then determined by the connectivity of the corresponding classifying space, which is characterized by its zeroth homotopy group, . For complex classes, the extension corresponds to the classifying space . For real classes, the extensions and correspond to the classifying spaces and , respectively [59]. We summarize the resulting classification of TIs and TSCs for all AZ classes with an additional RAS in table 1. Notably, in the presence of CS, acts as a reflection symmetry. Consequently, for classes AIII, BDI, DIII, CI, and CII, our classification matches that of [51] with an additional reflection symmetry.