Abstract
Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale-invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average C2zT, m, and C4zT and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups \(Pm,P{2}^{{\prime} },P{4}^{{\prime} }\), and \(P{6}^{{\prime} }\) with and without spin-orbit coupling, most of which can support CMP.
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Introduction
The interplay between topology and the Anderson (de)localization has provided an understanding of the quantum Hall transition1,2 and the classification (not including crystalline symmetries)3,4,5 of topological insulators (TIs)6,7,8,9. A remarkable result of this interplay is the delocalization in TIs protected by local symmetries3,5: In the presence of disorder that does not induce a bulk phase transition, the TI surface states10,11,12,13 are guaranteed to be delocalized; In the bulk, a disorder-induced transition between trivial and topological insulators must go through a divergent localization length14,15. Similar behavior occurs also in topological phases that require crystalline symmetry to be preserved, despite the breaking of that symmetry by disorders, as long as the symmetry is preserved on average. Examples include weak topological insulators14,16,17,18,19,20,21,22 and some topological crystalline insulators (TCIs)23,24,25,26,27,28,29,30,31,32,33,34,35 (including higher-order states with hinge modes36,37,38,39,40,41,42). For instance, inversion-symmetry-protected axion insulators26,43,44,45 have hinge modes46, and their phase transitions to trivial insulators must experience a delocalized diffusive metal phase if the disorder respects an average inversion symmetry47,48. Here, as defined in refs. 14,49, average symmetry is the symmetry of an ensemble comprising different disorder realizations on a local Hamiltonian. The average symmetry operation transforms an individual system into another with the same realization probability. Also, we require each system in the ensemble to be self-average. Even though not mathematically proven, most TCIs with protected boundary states are believed to be stable against disorders respecting the crystalline symmetries on average. This can be understood intuitively: Suppose the disorder potential slowly varies in real space. Then, during the transition, the disordered system can be divided into topological and trivial regions. Boundary states between the two types of regions must exist as promised by stable topology, giving rise to the delocalized phase transition.
In this work, we find that such delocalization behavior also generalizes to some topologically trivial states. There are two types of non-atomic states that are not conventional TIs or TCIs—the fragile topological insulators50,51,52,53 and the obstructed atomic insulators (OAIs)30,54,55,56. The former has a Wannier obstruction that can be removed by adding trivial bands and was recently found to hold delocalized critical states57. However, the latter is completely trivial and can be wannierized to molecular orbitals with charge centers away from the atoms. Given OAIs’ localized nature, it would be surprising if they can have delocalized states in the presence of disorder. In this work, we demonstrate that such delocalization does exist and is actually a common feature in transitions between magnetic OAIs. Conventional scenarios for delocalization theories, such as weak anti-localization (requiring time-reversal symmetry) and topological phase transition (requiring stable boundary states) cannot explain the underlying mechanism behind the delocalized states that we find.
Results
A C 2z T-symmetric quantum network model
We first investigate several OAIs protected by the C2zT symmetry without time-reversal symmetry (TRS)58,59,60,61. Later, we will generalize the discussion to other magnetic point groups in subsection “OAI transitions in generic magnetic point groups”. The OAIs belong to the Altland-Zirnbauer symmetry class A62, where all states except the quantum Hall transition point were expected to be localized63. These OAIs are characterized by the \({{\mathbb{Z}}}_{2}\) Real Space Invariant (RSI) δw, which is protected by C2zT64,65 and takes the value δw = 1 if the associated C2zT center w is occupied by an odd number of Wannier functions and zero otherwise. When the δw’s of a system have the same value at all the C2zT centers, they are equivalent to the second Stiefel-Whitney class w260,66.
Two features of a w2 = 1 insulator are worth emphasizing here: First, it can be regarded as a C2zT-protected fragile insulator plus two additional trivializing bands. And second, to tune a w2 = 1 insulator into a w2 = 0 insulator, one has to first close the gap by creating pairs of Dirac points, which are locally stabilized by the C2zT symmetry, then braid60,67,68 these Dirac points with other Dirac points inside the valence bands, and only then reopen a gap by annihilating the Dirac points (see subsection “Regularizing the network model to lattice models” below). As a consequence, when the transition is driven by the variation of one parameter, there is a gapless transition region rather than a transition point. Using a finite-size scaling procedure, we find that this gapless region becomes a critical metal phase (CMP) when disorder is added, provided that the disorder respects C2zT on average. Electronic states in CMP are delocalized and contribute to a scale-invariant conductance in the thermodynamic limit. We also find CMPs with other average symmetries and RSIs, suggesting CMP is a common feature of magnetic OAI transitions.
CMP has been numerically observed in systems with random fluxes69,70,71 or random spin-orbit coupling combined with a magnetic field72. Inspired by these works, we first argue the existence of C2zT-stabilized CMP through a semi-classical percolation theory. We then relate the percolation theory to a quantum network model73 and further map it to lattice models for the C2zT-protected OAIs.
To present a scenario that naturally leads to a CMP, we consider a system that is tessellated by three types of randomly sized and shaped insulating regions, whose Chern numbers are 0, 1 and −1, respectively, as shown in Fig. 1a, b. Physically, the fluctuation of Chern numbers could arise from random fluxes. The area fraction of Chern number C is pC. By definition ∑CpC = 1. Since the operation C2zT reverses the sign of Chern numbers, an average C2zT symmetry means p1 = p−1. Then there are two distinct phases. If \({p}_{0} > \frac{1}{2}\), according to the classical percolation theory74, the C = 0 regions form an extensive cluster while the C = ±1 regions form isolated islands and can be continuously shrunk to zero. Thus, \({p}_{0} > \frac{1}{2}\) should correspond to a localized phase (LP). If \({p}_{0} < \frac{1}{2}\), it is instead the C = 0 regions that can be shrunk to zero, and the system is equivalently tessellated by the C = ±1 regions with the same area fraction \(\frac{1}{2}\). As in the quantum Hall transition73, the chiral edge states between the C = ±1 regions connect to an extensive cluster with a fractal dimension and contribute to a scale-invariant conductance in the thermodynamic limit. Thus, \({p}_{0} < \frac{1}{2}\) should correspond to the CMP.
We can simulate the above percolation problem with a quantum network model on the Manhattan lattice75 (Fig. 1c). The red and blue squares are Chern blocks with C = 1, −1, respectively. The chiral edge modes between them and the trivial (white) regions form horizontal and vertical wires, and at each intersection, an electron can go straight or turn either left or right, depending on the type of intersection. The scattering equation at one intersection reads
where ψ3,4 and ψ1,2 are the outgoing and incoming modes, respectively, and θ is the single parameter that determines the probability amplitudes of going straight (\(\cos \theta\)) and turning left or right (\(-i\sin \theta\)). The model in the clean limit has symmetries of the magnetic space group PC4bm (#100.177 in BNS setting) generated by C2zT, C4z, and mxy symmetries76. The symmetry elements of the generators are shown in Fig. 1c. (Notice that C2zT centers do not coincide with C4z centers, and \({C}_{4z}^{2}\cdot {C}_{2z}T\) is a magnetic translation, i.e., a translation followed by time reversal. See Supplementary Information for more details.) One can see that the C2zT operation interchanges the C = ±1 regions. Note that the Manhattan lattice is not the only way to simulate the percolation problem. A Kagome-like network also works, with a localization behavior that is similar to the Manhattan lattice (See Section II.A of Supplementary Information).
Random sizes of the Chern blocks are simulated by the random propagation phases, or, equivalently, random vector potentials, along the bonds between intersections. When \(\theta=\pm \frac{\pi }{2}\), the chiral modes form local current loops surrounding C = ±1 regions, and the C = 0 regions are effectively connected. When θ → 0, the chiral modes are almost decoupled wires, and C = 0 regions are effectively separated. Thus, \(\theta=\pm \frac{\pi }{2}\) and θ → 0 should correspond to the localized limits (p0 = 1) and the CMP limit (p0 = 0), respectively. According to the percolation argument, there will be a critical value θc below (above) which the system enters the critical (localized) phase.
We use the transfer matrix techniques77 of quasi-1D systems, where longitudinal size M is much larger than transversal size L, to calculate the quasi-1D localization length ρ for finite L’s. More on this is summarized in the Method section, and one can read Section V of Supplementary Information for the full technical details. The normalized quasi-1D localization length Λ = ρ/L is an indicator of the (de)localization: Divergent, finite, and vanishing Λ’s in the limit L → ∞ indicate metallic, critical, and localized states, respectively. As shown in Fig. 1e, Λ decreases with L for \(| \theta | > {\theta }_{c}\approx \frac{\pi }{4}\) and is almost independent of L for ∣θ∣ < θc. Hence, \(| \theta | \in \left({\theta }_{c},\pi /2\right]\) and ∣θ∣ ∈ (0, θc) correspond to LP and CMP, respectively, which confirms the percolation argument. Note that error bars in plots of this work represent the standard deviations (SD) of corresponding data points. The conductance is also calculated and shown in Fig. 1f. The β-function \(\beta=d\ln G/d\ln L\) derived from the conductance data vanishes in the thermodynamics limit above some critical conductance Gc = 2 ~ 3e2/h (see subsection “β-function of the CMP” for details). The behavior of β-function further establishes the criticality of CMP and may suggest that the CMP-LP transition is similar to the Berezinskii-Kosterlitz-Thouless transition78,79.
We can define the network model through its Hamiltonian HN rather than through the scattering matrix Eq. (1). The Hamiltonian has only three parameters, velocity v of the chiral modes, the δ-potential λ at each intersection, and the lattice constant 2a.
The subscript d = v, h represents the vertical or horizontal orientations of the wires. l = 0, ± 1 ⋯ distinguishes different parallel wires. ξ is the coordinate inside one wire. The operators ψv,l(ξ) and ψh,l(ξ) act at the real space positions (x, y) = (la, ξ) and (ξ, la), respectively. As established in Section III.B of Supplementary Information, the scattering angle θ is determined by the Hamiltonian parameters as θ = λ/v. The evolution of the band structure as θ changes from the localized limit θ = − π/2 to the other localized limit θ = π/2 is illustrated in Fig. 1g–i. At θ = 0, the network model decouples to vertical and horizontal chiral wires, and the corresponding dispersion becomes quasi-1D (Fig. 1h). The symbols appearing in the figure, Γn, Mn (n = 1 ⋯ 5) and X1, represent irreducible representations (irreps) of PC4bm and are defined in Table 1. It is worth mentioning that because the chiral modes have unbounded energies, this Hamiltonian has an infinite number of bands that are periodic in energy, i.e., En(k) = En+8(k) + 2πv/a. For example, the lowest branch (two connected bands) in Fig. 1g–i is identical to the highest branch. Tracing the evolution (detailed in Supplementary Fig. 18), we find the phase transition process between these two LPs can be depicted by the irrep exchange at the zero energy (dashed lines in Fig. 1g–i)
where a minus (plus) sign means an energy level with the associated irrep crosses the zero energy from below (above) to above (below) during the phase transition. One may notice that the occupied Γ4 state in Fig. 1g also changes into Γ3 in Fig. 1i. However, this change is realized by level exchanges between the lowest branch in the figures and the lower branches beyond the scope of the figures. Since Γ3,4 do not cross the zero energy, they are not counted in the phase transition. We now relate the phase transition (3) to a change in the index δw.
Regularizing the network model to lattice models
In order to see the band topology, we need to regularize the network model to a lattice model. Our strategy is to use the local current loop states (flat bands in Fig. 1g) in the localized limit \(\theta=-\frac{\pi }{2}\) as a basis set and then truncate the basis according to their energies. The minimal model that can reproduce the phase transition in Eq. (3) is constructed from the upper eight consecutive flat bands in Fig. 1g to obtain the Hamiltonian H8B shown in Fig. 2a. (See Section IV.C of Supplementary Information for more details.) It has eight orbitals that are respectively located at the eight corners of the two squares in one unit cell, which correspond to the two Chern blocks in Fig. 1c. The explicit form of H8B can be expressed as
where p, q are the site indices, 〈⋅〉, 〈〈⋅〉〉, and 〈〈〈⋅〉〉〉 represent the green (nearest neighbor), orange (square edges), and dashed black (square diagonals) bonds in Fig. 2a, respectively. The hopping parameters are given by \(\tilde{t}=(\theta+\frac{\pi }{2})\frac{v}{a},t=\frac{\sqrt{2}\pi v}{4a},{t}^{{\prime} }=-\frac{\pi v}{4a}\). The phase factor ϕpq equals \(\frac{3}{4}\pi\) (\(-\frac{3}{4}\pi\)) if the associated hopping is parallel (anti-parallel) to the orange arrows, which are clockwise and anticlockwise for squares formed by the blue and red sites, respectively. For simplicity, we also denote the complex hopping \(t{e}^{\pm i\frac{3}{4}\pi }\) as t± in the following.
We take the Fermi level to be at EF = 0 and focus on that energy. The Hamiltonian H8B reproduces the flat bands in the localized limit \(\theta=-\frac{\pi }{2}\) when \(\tilde{t}=0\), where blue and red squares are decoupled from each other (Fig. 2c). The four flat valence bands are molecular orbitals at the square centers (C4z centers). Since they do not occupy the C2zT centers (squares corners), all the corresponding RSIs δw = 0 and the Stiefel-Whitney class w2 = 0. As \(\tilde{t}\) increases, H8B closes its gap, and when \(\tilde{t}=\frac{\pi v}{2a}\), it reproduces the quasi-1D bands of the network in the decoupled wire limit θ = 0 except for small deviations (non-linear dispersions) as if the wires are weakly coupled. As a consequence of the C4 symmetry, the Dirac points between the gapless bands occur at the same energy. (See Figs. 1h and 2d) As \(\tilde{t}\) continues to increase, H8B reopens a gap (Fig. 2e), and this gap continues to the limit \(\tilde{t}\to \infty\). By tracing the evolution of the energy levels, one can verify that the irrep exchange at E = 0 is indeed the same as Eq. (3). In the limit \(\tilde{t}\to \infty\), electrons form bonding states at the four C2zT centers (per cell), i.e., green bonds in Fig. 2a. Thus, the final state has δw = 1 at all C2zT centers, i.e., w2 = 1. As detailed in the Method section, the phase transition can be further confirmed through the machinery of Topological Quantum Chemistry30,80.
As shown in Fig. 2d, (tilted) Dirac points between the fourth and fifth bands are created in the phase transition process. The evolution of Dirac points is sketched in Fig. 2b, where the trajectory forms a closed path enclosing the underlying Dirac point at X between the third and fourth bands60,67,68. A more detailed discussion is given in Section IV.D of Supplementary Information.
It is worth mentioning that the vector potential disorder we used in the network model is now mapped to a chemical potential disorder in H8B (plus two times weaker hopping disorders that will be omitted). We leave the mathematical analysis in Section IV.F of Supplementary Information and only present a heuristic argument here. The basis of H8B (Fig. 2a) can be thought as wave-packets of the chiral modes that simultaneously have position centers and momentum centers. The position centers, by construction, are located at the square corners. We denote their 1D momentum centers as qc. Then a vector potential A will shift a momentum center qc to qc + A and result in an energy shift vA. Therefore, the resulting disorder potential in H8B should have a large on-site component. We also ignore the correlations among on-site random potentials for simplicity and efficiency. In numerical calculations, we only use uncorrelated on-site disorder and choose the disorder potential equally distributed in [−W/2, W/2]. Test calculations with full projected disorder potentials and correlations show no qualitative difference (see Supplementary Fig. 30).
We calculated the normalized localization length Λ as a function of \(\tilde{t}\) with EF = 0 and W = 2v/a (Fig. 2f). The system is localized when \(| \tilde{t}-\pi v/2a| > {\Delta }_{c}\approx 0.6v/a\), corresponding to the two OAI limits, and becomes critical when \(| \tilde{t}-\pi v/2a| < {\Delta }_{c}\). The criticality has been examined for large transversal sizes up to L = 500 unit cells (2000 atoms). We also calculate Λ at other \(\tilde{t}\)’s and W’s. From these data, we can determine a phase diagram shown in Fig. 2g, where the dashed line separates the CMP inside and the LP outside it. (See Supplementary Fig. 8 for details of large-scale examination and phase boundary determination.) Since H8B does not have chiral or particle-hole symmetries, the choice EF = 0 is not special in terms of symmetries. We have confirmed that CMP also exists when EF ≠ 0 as long as the OAI limits are intact (See Supplementary Fig. 9).
For the CMP in Fig. 2g, if we turn off the disorder, the resulting clean system has a finite density of states (DOS) around the zero energy, which may lead to a large localization length that may exceed the numerically accessible transversal size. To rule out possible finite-size effects, we consider a lattice model \({H}_{8B}^{{\prime} }\) that has the same crystalline symmetries and topology as H8B but vanishing DOS at the zero energy. Such a \({H}_{8B}^{{\prime} }\) can be obtained from H8B by (i) removing diagonal hopping \({t}^{{\prime} }\) and (ii) changing the edge hopping to t± = ( ± Ai − 1)t, where A, chosen as 1.2 hereafter, is an extra parameter that controls the range of the critical phase. (See Section IV.E of Supplementary Information for more details.) The hopping \(\tilde{t}\) (green bond in Fig. 2a) at C2zT center remains unchanged. The \(\tilde{t}=0\) and \(\tilde{t}\to \infty\) limits still represent the two OAI limits with charge centers at the C4z and C2zT centers, respectively. Hence, changing \(\tilde{t}=0\) to \(\tilde{t}=\infty\) will change w2 = 0 to w2 = 1 for the lower four bands as it did in H8B. Fig. 3a–c depict the evolution of the fourth and fifth bands, where the gap only closes at the four (untilted) Dirac points, resulting in a zero DOS at the zero energy.
Parallel to the results of H8B shown in Fig. 2f, g, we show Λ with EF = 0 and fixed W and the phase diagram for \({H}_{8B}^{{\prime} }\) in Fig. 3d, e, respectively, where no qualitative difference for the CMP is found. Thus, the potential finite-size effect of H8B due to large DOS is ruled out. Additionally, we did calculations with EF ≠ 0 and obtained similar results as EF = 0 (See Supplementary Fig. 10).
Local Chern markers
In order to verify the percolation argument directly, we refer to a widely used local topological marker called local Chern marker (LCM)81,82,83, which reformulates the Chern number locally in real space without summing over the whole sample. The LCM of one unit cell is
where R is the position of the unit cell, α indicates the orbitals inside one unit cell, Ac is the area of one unit cell, and \(\hat{P}\) is the projection operator of the occupied states.
Inside a macroscopic region where a gap is well preserved, the LCM converges to the quantized Chern number, whereas around gapless regions such as the boundaries, LCM may strongly fluctuate82. In our models, due to the percolation argument, regions with a single Chern number C (= ±1) never become extended because the fraction pC, as required by the C2zT, is always smaller than \(\frac{1}{2}\) for p0 > 0. Therefore, no macroscopic Chern block is expected for a general point in the CMP. Nevertheless, the microscopically inhomogeneous LCM can still reflect the local topological properties and can be applied to, for example, disordered systems near topological phase transitions84. Figure 4a and c show the topography of LCM of \({H}_{8B}^{{\prime} }\) in the CMP and LP with given disorder configurations, respectively. In the CMP, the sample is dominated by randomly distributed positive and negative Chern cells, consistent with the percolation argument that C = ± 1 regions together are extended through the whole system. In the LP, the LCM almost vanishes everywhere. We also calculate the distribution of the second moment of LCM \(\sqrt{\langle {C}^{2}\rangle }\) over 500 disorder configurations in both the CMP and LP (Fig. 4b, d), where 〈⋅〉 means averaging over all the cells. In the CMP (LP), \(\sqrt{\langle {C}^{2}\rangle } > \frac{1}{2}\) (\( < \frac{1}{2}\)) for most configurations, i.e., the regions with non-zero (zero) Chern numbers dominate. These phenomena confirm our semi-classical percolation picture.
OAI transitions in generic magnetic point groups
For a qualitative understanding of the CMP, we note that a local breaking of C2zT symmetry allows for a local gap of the Dirac nodes. This gap makes the region of the Dirac point carry a spread Berry curvature that integrates to ± π, with the sign being determined by the chirality of the Dirac point multiplied by the sign of the gaping mass. In the cases we considered here, all Dirac points occur at the same energy. Denoting the number of Dirac points by 2N, there are 22N assignments of the signs of the gaping mass, out of which (2N)!/(N!)2 lead to a total Chern number of zero. If the signs of the masses are uncorrelated, as would be expected for random disorder, then \({p}_{0}=\frac{(2N)!}{{2}^{2N}{(N!)}^{2}}\le \frac{1}{2}\), with the equality occurring only for N = 1. In our case N = 2 and \({p}_{0}=\frac{3}{8}\). Thus, for weak disorder, we expect a percolating network of edge states. If the masses are positively correlated, p0 could be further suppressed. To be concrete, \({p}_{0} < \frac{1}{2}\) even in the N = 1 model if there is a higher possibility for the two masses to have the same sign. Under strong disorder, all local Chern insulators become trivial, and the system becomes localized.
The above argument can be immediately generalized to generic OAI transitions beyond C2zT symmetry. As a direct verification, we break C2zT in \({H}_{8B}^{{\prime} }\) while keeping mxy and C4zT for the percolation mechanism and making two OAI limits at \(a\tilde{t}/v=0\) and ∞ inequivalent. The unit cell of the modified model \({H}_{8B}^{{\prime\prime} }\) is illustrated by Fig. 5a, where the orange arrows indicate hopping \({t}_{+}^{{\prime} }=(1+Ai)\frac{\pi v}{4a}\) and the purple arrows indicate \({t}_{+}^{{\prime\prime} }=q{t}_{+}^{{\prime} }(q\in {\mathbb{C}})\). The magnetic space group of \({H}_{8B}^{{\prime\prime} }\) reduces to \(P{4}^{{\prime} }{m}^{{\prime} }m\) (#99.165 in BNS setting), and the Wannier centers of OAI limits at \(\tilde{t}=0\) and ∞ are now located at the C2z centers \(2{{{{{{{\rm{c}}}}}}}}[2{{{{{{{{\rm{m}}}}}}}}}^{{\prime} }{{{{{{{{\rm{m}}}}}}}}}^{{\prime} }]\) and mirror planes 4d[m], respectively. The band structure also goes through an evolution of four Dirac points when \(\tilde{t}=0\to \infty\). Figure 5b shows the localization length of \({H}_{8B}^{{\prime\prime} }\) with A = 1.2, q = 0.625e−0.15πi. We can see that CMP indeed survives at \({H}_{8B}^{{\prime\prime} }\).
To further show the generality, in Fig. 6, we enumerate all the minimal OAI transitions protected by magnetic point groups \(Pm,P{2}^{{\prime} },P{4}^{{\prime} }\), and \(P{6}^{{\prime} }\). The details of enumeration are summarized in Section II.B of Supplementary Information. Here, for a given symmetry group, the minimal OAI transition is defined as the OAI transition with minimal band deformation and molecular orbital transition. The band deformation of a generic OAI transition can be viewed as a superposition of minimal band deformations that can be band inversions at high symmetry points or gap closures at generic k-points (see the third column of Fig. 6). Also, since OAI can be wannierized to molecular orbitals (each orbital forming a site symmetry irrep at some Wyckoff position), an OAI transition can be characterized by occupation changes of these orbitals. The minimal molecular orbital transition refers to the minimal occupation change that can realize the corresponding band deformation. For example, the third row of block “P4’-NSOC" in Fig. 6 demonstrates a minimal OAI transition that replaces some occupied band forming irrep X2 at X point by a band with X1 and moves an electron from irrep A at the Wyckoff position 1b to irrep A at 1a.
In terms of band deformation, these transitions can be divided into three categories: quadratic touching from a 2-dim irrep, Dirac points braiding, and immediate band gap closure-reopening. In the former two categories, the band structure is gapless for a finite parameter region in contrast to a single point in the last category. For the first category, if we add slow-varying disorders that mainly open a local gap with a nontrivial Chern number (p0 < 1/2), a CMP is highly possible due to the percolation mechanism protected by the average symmetry. For the second category, as discussed above, the possibility of CMP is higher with more Dirac points and stronger positive Dirac mass correlations. For the third category, the transition can go through a critical point at most since the gapless band structure is necessary for delocalization. The number and correlations of Dirac points also influence the possibility of delocalizing the gapless point.
Discussion
For the first time, our work points out that transitions between trivial states (such as OAIs) without TRS can be critical under simple chemical potential disorders and, through a quantitative mapping, reveals that the criticality is due to a tricolor percolation mechanism of C = 0, ±1 regions. Since the chemical potential disorder is realistic and there are many topologically trivial magnetic materials with symmetries that forbid net (anomalous) Hall conductance80,85, our work also has experimental relevance.
We notice that CMPs in 2D class A systems have been observed in previous works57,86,87,88. Ref. 86 did not report a CMP, yet we find its Fig. 2a may suggest a CMP similar to the one found by ref. 87. Ref. 87 reported a CMP in the Kane–Mele model (\({{\mathbb{Z}}}_{2}\) TI) in the presence of a weak Zeeman field and ascribed the criticality to the change of spin Chern number. Ref. 88 reported a CMP in the Bernevig–Hughes–Zhang model (\({{\mathbb{Z}}}_{2}\) TI) in the presence of a random magnetic field and ascribed the criticality to two coupled quantum Hall transitions. Therefore, these CMPs exist in topological phase transitions with additional weak perturbation terms. However, our CMP exists between two topologically trivial OAIs far from any topological state. The only difference between the two OAIs is the center and representation of Wannier states. It is also worth mentioning that, to exclude possible finite-size effects, we have verified the criticality of CMP up to system sizes L = 2000 for the network model and L = 500 for the lattice model (see Section I of Supplementary Information), which are larger than the system sizes L = 24, 32, 128 used in refs. 86,87,88, respectively.
Methods
Topological quantum chemistry of the C 2z T model
The CMPs in the C2zT models arise between inequivalent OAIs. To depict the OAI transitions in the clean limit, we can use the tool of topological quantum chemistry, i.e., analyzing the transition of occupied magnetic element band representations (MEBRs)80. In our models, all the OAIs can be wannierized to molecular orbitals centering at the C4z and C2zT centers (Wyckoff positions 2b and 4c). These orbitals respect the site symmetries (\(4{m}^{{\prime} }{m}^{{\prime} }\) and \({2}^{{\prime} }{m}^{{\prime} }m\) for 2b and 4c, respectively) and hence can be characterized by the irreps of site symmetries. According to the topological quantum chemistry, these molecular orbitals will induce bands with MEBRs listed in Table 2. One MEBR is the minimal group of bands formed by a type of molecular orbitals, and the left part of an MEBR notation indicates the site symmetry irrep of the orbitals, e.g., Ab↑G indicates the MEBR formed by s orbitals (trivial irrep A) at positions 2b. Therefore, we can deduce the Wannier centers of OAIs from the occupied MEBRs, depicting the OAI transitions by MEBR transitions.
We start with the network model. Although a network model has infinitely many occupied bands, these bands form an infinite direct sum of MEBRs and can be viewed as a special kind of OAI. The band structure comprises disconnected branches, each containing two bands. One branch forms one of the following four MEBRs defined in Table 2: Ab↑G, Bb↑G, 1Eb↑G, 2Eb↑G, result from effective \(s,{d}_{{x}^{2}-{y}^{2}}+i{d}_{xy}\) (or equivalently \({d}_{{x}^{2}-{y}^{2}}-i{d}_{xy}\)), px − ipy, px + ipy orbitals, respectively. All these orbitals center at the Wyckoff position 2b. Since 2b has multiplicity 2 (two C4z centers per unit cell), each MEBR contains two bands. Also, notice that the band structure comprises infinite repeating units. Each unit contains eight bands, and one can generate the whole band structure by energy translations of a unit with step 2πv/a. Hence, we can focus on the upper eight of the ten bands in Fig. 1g–i. When θ < 0, direct gaps separate different branches, and the lower four of the focused eight bands form a direct sum of two MEBRs: 2Eb↑G ⊕ Ab↑G. When θ = 0, the gaps close, and the band structure is indeed that of a ballistic 1D metal with linear dispersion relations. As θ increases across 0, irreps Γ1, M1 (Γ2, M2) go up (down) across the energy level. Similar irrep exchanges also happen above and below the repeating unit, e.g., the lower two of the focused eight bands will exchange irreps with the lower bands that exceed the scope of Fig. 1g–i. After the transition (θ > 0), the gaps reopen, and the MEBRs of the lower four of the focused eight bands change to 1Eb↑G ⊕ Bb↑G. Hence, θ = ± π/2 correspond to two inequivalent trivial phases, although the molecular orbitals of both phases center at the C4zT centers.
We now turn to the band evolution of H8B (\({H}_{8B}^{{\prime} }\) is similar). See Fig. 2c–e, as \(\tilde{t}\) increases from 0 to πv/a, Γ1 and M1 rise across the Fermi level while Γ2 and M2 fall below the Fermi level. We encounter the same irreps exchange to the network model. Despite similarities near the Fermi level, the MEBR transition of H8B is different from the network model. The lower four bands of H8B change from 2Eb↑G ⊕ Ab↑G to \({{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}\uparrow {{{{{{{\rm{G}}}}}}}}}^{{\prime\prime} }\) during the transition. The Wyckoff position of \({{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}\uparrow {{{{{{{\rm{G}}}}}}}}}^{{\prime\prime} }\) is 4c, which is the C2zT center rather than the C4z center (Table 2). This difference in molecular orbital transition is unavoidable since the transition in the network model involves exchanges of representations between different repeating units, while H8B only has one unit. In the network model with a sufficient number of bands, the four bands below the Fermi level not only exchange representations with the bands above them but also with the bands (in another repeating unit) below them. However, H8B has no band below the lower four bands. Nevertheless, since the origin of this difference is well below the Fermi level, it should not affect the low-energy physics. Therefore, we can expect similar low-energy behaviors between H8B and the network model, which our numerical data confirms.
It is worth mentioning that the transition in H8B changes the position of the MEBRs from C4z-centers (2b) to C2zT-centers (4c). No C2zT center is occupied before the transition, and the RSI δw = 0. Given that there are four C2zT centers per cell and four occupied bands, every C2zT center is occupied by one electron after the transition, and the system has RSI δw = 1. Therefore, the second Stiefel-Whitney class w260,66 must change from 0 to 1. As we have discussed in the second paragraph of subsection “A C2zT-symmetric quantum network model”, the transition process must involve braiding of the Dirac points. In addition, although the lower four bands form \({{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}\uparrow {{{{{{{\rm{G}}}}}}}}}^{{\prime\prime} }\) are always connected in our models, in general cases, \({{{{{{{{\rm{A}}}}}}}}}_{{{{{{{{\rm{c}}}}}}}}\uparrow {{{{{{{\rm{G}}}}}}}}}^{{\prime\prime} }\) can be decomposed into two fragile topological bands (Γ5, M5, X1) and two trivial bands (forming MEBR A2↑G with Wyckoff position \(2a\,(\frac{1}{4}\,\frac{1}{4},0),(\frac{3}{4}\,\frac{3}{4},0)\), i.e., the centers of white squares in Fig. 1c), which is expected from w2 = 1.
Quasi-1D localization length and transfer matrix method
A commonly used physical quantity in research of localization is the quasi-1D localization length ρq−1D. It is defined on a 2D/3D sample prepared in a quasi-1D shape, e.g., a long, thin cylinder with Laxial ≫ Lradius. ρq−1D reflects the decaying rate of eigenstates in the quasi-1D direction, e.g., the axial direction of a long thin cylinder. Since any 1D system is localized under nonzero disorder strength, ρq−1D will always be finite except for a perfectly clean sample. Localization of the original 2D/3D system (in a 2D/3D shape) can be derived from a scaling analysis of the dimensionless quasi-1D localization length Λ = ρq−1D/L, where L is the transversal size of the quasi-1D sample. We denote the localization length of a normally shaped (scales of different directions are similar) and sufficiently large sample as ρ. For a metallic system, ρ in a (normally shaped and sufficiently large) sample is much larger than the sample size. Thus, ρq−1D(L) increases faster than L, i.e., Λ(L) → ∞ in the limit L → ∞. For an insulating system, ρ is finite in a (normally shaped and sufficiently large) sample. Hence, ρq−1D will converge to ρ when L ≫ ρ, i.e., Λ(L) → 0 as L → ∞. In practice, we identify the region where Λ(L) monotonically increases as the metallic phase and where Λ(L) monotonically decreases as the localized phase. If a system contains both localized and extended phases in some parameter space, there will be a critical region (usually a boundary with measure zero) in the parameter space where Λ(L) is independent of (sufficiently large) L.
The transfer matrix method77 is a widely used numerical approach in calculating ρq−1D. Although it has different formulae for (generic) network and lattice models, the basic ideas are the same. A quasi-1D sample is divided into layers with normals along the quasi-1D direction. The amplitudes of an energy-eigenstate on different layers are related by the Schrodinger equation. A (2s × 2s shaped) transfer matrix Tn generally transforms the amplitudes on the (l − r1)th ~ (l + r2 − 1)th layers to those on the (l − r1 + 1)th ~ (l + r2)th layers. Here, \(s\in {{\mathbb{N}}}^{+}\) is proportional to the number of degrees of freedom in one layer. And \({r}_{1},{r}_{2}\in {{\mathbb{N}}}^{+}\) represent that only the (l − r1)th ~ (l + r2)th layers can hop/propagate (in one step) to the l th layer. The values of r1 and r2 depend on the hopping of concrete models. Since the transfer matrix of a (general) network model is determined by the transmission matrix, the amplitudes in the (l + 1)th layer depend only on the lth layer. Hence, r1 = r2 = 1 for general network models (not only ours). For general lattice models, r1 and r2 can take arbitrary finite non-negative integer values. Nevertheless, in our models, r1 = r2 = 1.
To extract ρ1D from the transfer matrix, we can consider a consecutive product of transfer matrices \({O}_{M}=\mathop{\prod }\nolimits_{n=1}^{M}{T}_{n}\). Because of the disorder, some elements in Tn are random variables. According to the Oseledec theorem, the limit \(P=\mathop{\lim }\limits_{M\to \infty }{({O}_{M}^{{{{\dagger}}} }{O}_{M})}^{1/2M}\) exists and has eigenvalues \(\{\exp ({\nu }_{1}),\exp (-{\nu }_{1}),...\exp ({\nu }_{s}),\exp (-{\nu }_{s})\}\) where νi ≥ νi+1 ≥ 0, i = 1, 2. . . s. These (positive) exponents are so-called Lyapunov exponents (LEs). The definition of P indicates that an eigenvector ηi of P with eigenvalue \(\exp (-{\nu }_{i})\) satisfies \(\parallel {O}_{M}{{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}{\parallel }^{2}={{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}^{{{{\dagger}}} }({O}_{M}^{{{{\dagger}}} }{O}_{M})\,{{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}={{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}^{{{{\dagger}}} }{[{({O}_{M}^{{{{\dagger}}} }{O}_{M})}^{1/2M}]}^{2M}\,{{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}\approx {{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}^{{{{\dagger}}} }{P}^{2M}\,{{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}=\parallel \exp (-M{\nu }_{i}){{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\boldsymbol{i}}}}}}}}}{\parallel }^{2}\) for sufficiently large M. Therefore, the smallest LE νs determines the decaying rate of energy-eigenstates (along the quasi-1Dc direction) since any energy-eigenstate is a superposition of eigenvectors of P with eigenvalues \(\exp (-{\nu }_{i})\) (the amplitudes cannot grow exponentially hence \(\exp ({\nu }_{i})\) is excluded). Because of this, one can define the quasi-1D localization length as the inverse of the smallest LE: ρq−1D = 1/νs.
Due to the space limitation, there are three main aspects we cannot explain here: calculating conductance from the transfer matrix, the technique for numerical stability of LE, and the concrete formulae of the transfer matrices of our network and lattice models. Readers interested in these details can refer to Section V of Supplementary Information.
β-function of the CMP
The β-function \(\beta=d\ln G/d\ln L\) is an additional quantity to verify the criticality of the observed delocalized phases. Figure 7a shows the β-function of the network model derived from the finite difference method (we have ignored the default conductance unit e2/h). As we can see, all the data points fall into two parts corresponding to the delocalized and localized phases. A critical conductance Gc = 2~3 divides these two phases. In the localized phase below Gc, data points collapse to one curve, demonstrating the Anderson localization. In the delocalized phases above Gc, the data points distribute around zero below G ~ 5. And above G ~ 5, data from different sizes deviate from each other and become significantly positive. These positive data points result from the finite size effect of the ballistic limit at θ = 0. To justify that the delocalized phase with −π/4 ≲ θ < 0 is indeed a CMP, we have to prove that β(G > Gc, L → ∞) → 0+.
When θ = 0, the network degenerates to decoupled parallel chiral wires in four directions (\(\pm \hat{x},\pm \hat{y}\)). Since there are L channels in one direction, the ballistic conductance \({G}_{\max }(L)=L\). Hence, for a given system size L, the β-function should terminate at the point (G = L, β = 1), which is confirmed numerically (the inset of Fig. 7a). Due to the finite size effects, the ballistic limit will induce a diffusive metal phase with θ → 0 corresponding to the positive region of β-functions for large G. Further, the data suggest a linear hypothesis of β-function for large G, i.e., β(G, L) = G/L when G ≫ Gc. To verify this hypothesis, we can view it as a differential equation and test its solution (as a hypothesis of conductance) on conductance data. To be concrete, β(G, L) = G/L implies an ansatz of conductance G(θ, L)−1 = G(θ, ∞)−1 + L−1 when G ≫ Gc. Here, the fitting parameter G(θ, ∞) is the conductance in the thermodynamic limit for a given θ. As illustrated in Fig. 7b, this ansatz fits our conductance data very well when G > 5. Although we have not understood the mechanism behind the linear β-function, we can conclude from the above results that β(G > Gc, L → ∞) → 0+, proving the criticality of CMP.
Localized OAI transition in class AI
One should not conclude from the above results that CMP exists in any transition between inequivalent 2D OAIs. Due to the localized nature of OAI, it is commonly believed that disorders should localize general transitions between inequivalent 2D OAIs. We can obtain a localized OAI transition by modifying our lattice model H8B. Recall Eq. (4), H8B is described by four real parameters \(t,{t}^{{\prime} },\tilde{t}\), and ϕpq. The only parameter breaking TRS is ∣ϕpq∣ = 3π/4. If we take ∣ϕpq∣ = 0 or π, the system will respect TRS even if on-site disorders are present. Now, the previous C2zT centers are promoted to C2z centers. According to ref. 64, in the presence of TRS, the Wyckoff position with site symmetry 2 has a \({\mathbb{Z}}\)-valued RSI δ = m+ − m− where m+ (m−) is the occupation number of orbital even (odd) under C2z. Hence, when ∣ϕpq∣ = 0 or π, tuning \(\tilde{t}\) from 0 (δ = 0) to + ∞ (δ = − 1) still drives a transition between inequivalent OAIs. On the other hand, due to the weak localization effect in the presence of spinless TRS, the system must be localized regardless of \(t,{t}^{{\prime} }\), and \(\tilde{t}\). In Fig. 8, we take \(\tilde{t}=\frac{\pi v}{2a}\), corresponding to the middle of the CMP in Fig. 2f, and tune ∣ϕpq∣ from 3π/4 to π. We can see a CMP-LP transition in Fig. 8. For all the ∣ϕpq∣ scanned, the band structure in the clean limit is gapless near the Fermi level EF = 0, i.e., the clean system stays at the metallic intermediate state of the OAI transition but will be localized by disorders. Although the LP shrinks with weaker disorder strength W, ∣ϕpq∣ = π is always localized.
Data availability
We provide all the raw data in Supplementary Dataset. Source data are provided with this paper.
Code availability
Codes required for reproducibility are available upon request to the authors.
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Acknowledgements
We are grateful to Roni Ilan, Ryuichi Shindou, Chen Wang, Chui-Zhen Chen, Jian Li, and Anton Akhmerov for helpful discussions. Z.-D.S. and F.-J.W. were supported by National Natural Science Foundation of China (General Program No. 12274005), Innovation Program for Quantum Science and Technology (No. 2021ZD0302403), National Key Research and Development Program of China (No. 2021YFA1401903). B.A.B. was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 101020833), the ONR Grant No. N00014-20-1-2303, the Schmidt Fund for Innovative Research, Simons Investigator Grant No. 404513, the Gordon and Betty Moore Foundation through the EPiQS Initiative, Grant GBMF11070 and Grant No. GBMF8685 towards the Princeton theory program. Further support was provided by the NSF-MRSEC Grant No. DMR-2011750, BSF Israel US foundation Grant No. 2018226, the Princeton Global Network Funds. R.Q. is supported by the Simons Foundation award 990414, the U.S. Department of Energy (DOE) under award DE-SC0019443, and the NSF MRSEC DMR-2011738
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Zhi-Da Song, Ady Stern, and B. Andrei Bernevig are conceived of this work. Zhi-Da Song proposed the network model and the percolation argument. Fa-Jie Wang completed the mapping from the network model to the lattice model and ran the numerical calculations. Zhen-Yu Xiao proposed the calculation on local Chern marker. Raquel Queiroz helped analyze the β-function. Ady Stern proposed the connection between the number of Dirac points and the percolative conductivity. All the authors contributed to the writing.
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Wang, FJ., Xiao, ZY., Queiroz, R. et al. Anderson critical metal phase in trivial states protected by average magnetic crystalline symmetry. Nat Commun 15, 3069 (2024). https://doi.org/10.1038/s41467-024-47467-2
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DOI: https://doi.org/10.1038/s41467-024-47467-2
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