Global Topological Phase Diagram

• Global topological phase diagram is a systematic mapping of distinct quantum phases defined by invariants like Chern numbers and Z₂ indices, highlighting symmetry-protected phenomena.
• The diagram employs microscopic Hamiltonian models to delineate phase boundaries, multicritical points, and transition nature across parameters such as disorder, chemical potential, and spin-orbit coupling.
• It guides experimental validations via ARPES, STM, and transport studies, aiding in the engineered design of exotic topological states and robust edge modes.

A global topological phase diagram characterizes the organization of topological quantum phases—characterized by quantized invariants and robust, symmetry-protected phenomena—across the full range of relevant physical parameters of a given system (such as interaction strength, disorder, chemical potential, symmetry-breaking fields, or coupling to external environments). Such diagrams reveal the connectivity, boundaries, and multicritical structure of distinct topological phases and their transitions, providing a unified overview of emergent quantum matter. The notion applies across dimensions, symmetry classes, disorder regimes, and both non-interacting and strongly correlated systems.

1. Foundational Concepts and Context

Topological phases are often defined by invariants such as Chern numbers, ℤ₂ indices, and quantized response functions, rather than local order parameters. The global topological phase diagram enumerates all such phases, the critical lines and surfaces separating them, and the nature (e.g., continuous, first-order, multicritical) and universality class of their transitions. The diagram’s axes can represent any combination of experimentally or theoretically tunable parameters: coupling constants, chemical potential, magnetic/electric field, spin-orbit strength, disorder amplitude, temperature, or strain.

Research across quantum Hall systems, topological insulators and superconductors, crystalline topological materials, Mott transitions with nontrivial topology, higher-order topological phases, spin liquids, and engineered lattice models (e.g., Levin–Wen string-net, Kitaev–Heisenberg, cluster-XY, toric code) has led to precise constructions and experimental validations of global phase diagrams (Antonenko et al., 31 Dec 2025, Neupane et al., 2014, Goswami et al., 2016, Kurita et al., 2012, Li et al., 2024).

2. Microscopic Models and Control Parameters

A hallmark of global topological phase diagrams is the explicit mapping from microscopic Hamiltonian parameters to topological regimes. For instance, in the unified analysis of a 2DEG in a perpendicular magnetic field, proximitized by a superconducting Abrikosov vortex lattice, Antonenko, Fu, and Glazman define two dimensionless variables:

• $\delta = \Delta/\omega_c$ (pairing amplitude relative to cyclotron frequency)
• $\tilde{\mu} = \mu/\omega_c$ (chemical potential normalized by cyclotron energy)

The corresponding BdG Hamiltonian is

$$H_{\rm BdG} =\int d^2r\,\begin{pmatrix} \hat c_\uparrow^\dagger(\mathbf r)&\hat c_\downarrow(\mathbf r) \end{pmatrix} \begin{pmatrix} \displaystyle\frac{1}{2m}\bigl(-i\hbar\boldsymbol\nabla - e\mathbf A\bigr)^2-\mu &\Delta(\mathbf r)\ \Delta^*(\mathbf r)& -\displaystyle\frac{1}{2m}\bigl(i\hbar\boldsymbol\nabla - e\mathbf A\bigr)^2+\mu \end{pmatrix} \begin{pmatrix}\hat c_\uparrow(\mathbf r)\\hat c_\downarrow^\dagger(\mathbf r)\end{pmatrix}$$

with vortex lattice supercurrent entering via $\Delta(\mathbf r)$ (Antonenko et al., 31 Dec 2025).

Other paradigmatic axes include tunable spin-orbit coupling $\lambda_{\rm SOC}$ (e.g., in Ce$X$ monopnictides (Kuroda et al., 2017)), band inversion mass $M$, electron interactions $V$, Zeeman fields, disorder strengths $W$ or $\Delta$, strain tensor components $\varepsilon_{ij}$, and temperature $T$.

3. Structure and Features of the Phase Diagram

The global phase diagram typically comprises several discrete topological phases, gapless or critical regions, metallic/diffusive regimes (in the presence of strong disorder), and symmetry-broken or coexisting orders.

In proximitized quantum Hall / vortex lattice systems, the $(\delta, \tilde{\mu})$ diagram exhibits:

• “Fan”-like domes of distinct BdG Chern number $\mathcal{C}=0,2,4,6\dots$, each corresponding to a phase with a fixed number of chiral Dirac edge states.
• Hierarchical splitting of Landau-level transition lines into multiple trajectories, each with gap closings at symmetry-protected momenta due to vortex lattice group symmetries.
• Multicritical structures: for example, at weak pairing, transitions proceed through a sequence of subtransitions with Chern number jumps $|\Delta\mathcal{C}| = 2,4,6,8,12$.
• The trivial superconductor regime ($\mathcal{C}=0$), delimited by overlapping vortex cores for $\delta \gtrsim \sqrt{\tilde{\mu}}$ (Antonenko et al., 31 Dec 2025).

Other notable structures in different models include:

• Trivial/topological/metal phase triangles or domes (e.g., strain-tuned ZrTe$_5$ (Tajkov et al., 2022), three-dimensional dirty topological superconductors (Roy et al., 2016, Goswami et al., 2016)).
• Sequence of parity inversions at time-reversal-invariant momenta controlling $\mathbb{Z}_2$ indices (Kuroda et al., 2017).
• First- and second-order lines, quantum critical lines, multicritical points, and crossover regimes in interacting and frustrated spin or bosonic systems (Wu et al., 2012, Schulz et al., 2015, Li et al., 2024).

4. Role of Symmetry, Disorder, and Interactions

Topological invariants and phase boundaries are determined by the symmetries of the underlying Hamiltonian. Space-group symmetries (magnetic translation, $C_4$/$C_6$ point groups, mirror, inversion) protect band crossings; breaking these splits critical points and reduces Chern jump multiplicity.

Disorder adds further control axes:

• Weak disorder preserves sharp topological transitions and criticality (“superuniversality” in 3D Dirac systems).
• Above a critical disorder strength, direct transitions between two gapped topological phases are avoided: a diffusive metallic regime intrudes, with scaling exponents changing from superuniversal to class-dependent (Goswami et al., 2016).
• Topological signatures such as quantized quadrupole moments can be extremely fragile to disorder, while surface or higher-order states (e.g., hinge modes in higher-order Weyl semimetals) persist up to moderate disorder (Zhang et al., 2021).

Interactions can fundamentally alter criticality, as in topological Mott transitions where nonanalytic free-energy terms and quantum critical lines violate Landau-Ginzburg-Wilson paradigms, leading to unconventional exponents $\beta, \delta$ (Kurita et al., 2012).

5. Quantized Topological Invariants and Physical Consequences

The phase diagram partitions into regions characterized by distinct quantized invariants:

• Chern numbers $\mathcal{C}$ in 2D (quantizing the Hall conductance, edge states)
• Mirror Chern numbers $n_M$ (in crystalline insulators (Neupane et al., 2014))
• Strong/weak $\mathbb{Z}_2$ indices (time-reversal-invariant topological insulators (Kuroda et al., 2017, Tajkov et al., 2022))
• Higher-order invariants (e.g., quantized quadrupole moments for hinge states in HOWSMs (Zhang et al., 2021))
• Winding numbers (in 1D chains and quasi-1D models (Jafari, 2017, Li et al., 2024))

Transitions occur via closure of the bulk gap, and usually (though not always) are accompanied by changes in edge/boundary or higher-order modes. Key consequences include the emergence or annihilation of protected chiral edge channels, Majorana zero modes, or fractionalized excitations.

6. Experimental Realizations and Detection

Theoretical global phase diagrams have been validated by bulk and surface-sensitive experiments such as ARPES (e.g., in Ce$X$ monopnictides (Kuroda et al., 2017), Pb$_{1-x}$Sn$_{x}$Se (Neupane et al., 2014)), STM spectroscopy (ZrTe$_5$ (Tajkov et al., 2022)), transport (quantized edge/thermal conductance), and multiterminal tunneling.

Tables such as the one below summarize typical phase boundary conditions and signatures:

Model/System	Axes / Tuning Parameters	Phases (Topological label)
2DEG + vortex lattice (Antonenko et al., 31 Dec 2025)	$(\delta, \tilde{\mu})$	$\mathcal{C}=0,2,4,6\dots$ BdG domes
CeX monopnictides (Kuroda et al., 2017)	$(\lambda_{\rm SOC}, M_0)$	Trivial, Strong TI ($\nu$), by inversion
ZrTe$_5$ (Tajkov et al., 2022)	$(\varepsilon_{in}, \varepsilon_{vdw})$	Weak/strong TI, Dirac metal ($\mathbb{Z}_2$)
Toric code (Wu et al., 2012)	$(h_x, h_z)$	Topological, vortex-condensed, charge-condensed
Cluster-XY (Li et al., 2024)	$(\lambda_x, \lambda_y$, $\Gamma)$	SPT, FM, AFM, PM, critical, LL-like

7. Multicriticality, Quantum Lines, and Universality

Many phase diagrams feature robust multicritical points, critical lines or surfaces:

• In 3D disordered topological systems, the massless Dirac point at $m=0$, $\Delta_-=0$ is superuniversal, but disorder-driven multicritical points separate domains with diffusive metal (Goswami et al., 2016).
• In interacting zero-gap semiconductors, $T=0$ quantum critical lines (“QCLs”) persist away from the quantum critical point and are associated with “beyond-LGW” universality (Kurita et al., 2012).
• First-order versus continuous transitions are determined by symmetry and order parameter manifold structure (Schulz et al., 2015, Wu et al., 2012).

Critical exponents, scaling relations, and renormalization-group flows are analytically predictable and confirmed numerically or experimentally in numerous classes. In several systems, transitions out of the topological phase (by symmetry breaking or disorder) occur via non-Landau-Ginzburg scenarios and protected lines or cones in parameter space.

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These features highlight the unifying power of the global topological phase diagram: it encodes the interplay of symmetry, topology, and quantum criticality across broad classes of condensed matter systems, revealing pathways for engineering and detecting exotic phases and robust boundary phenomena (Antonenko et al., 31 Dec 2025, Goswami et al., 2016, Kurita et al., 2012, Li et al., 2024, Kuroda et al., 2017, Zhang et al., 2021).