Gravitationally Induced Topological Transitions

• Gravitationally induced topological phase transitions describe how gravity alters the global topological states by reconfiguring vacuum sectors, black-hole defects, and domain structures.
• The approach employs higher-curvature corrections in theories like Lovelock and Gauss–Bonnet, analyzing bubble nucleation and free energy shifts to reveal critical transition points.
• These transitions influence black-hole thermodynamics and condensed-matter analogues, offering a unified framework for understanding phase transitions across diverse physical systems.

A gravitationally induced topological phase transition refers to the phenomenon by which the presence of gravity—either as a dynamical field in spacetime or as an effective geometric background—changes the global topological classification of physical states in a system. This can manifest as a transition between different vacuum sectors, black-hole phases, or condensed-matter topological phases, typically accompanied by the nucleation of a “bubble,” a domain wall, or a reconfiguration of topological defects. These transitions are particularly significant in gravitational systems with higher-curvature corrections, where the underlying topological structure of spacetime solutions may fundamentally alter the thermodynamic landscape, phase diagram, and observable properties of both gravitational and condensed-matter systems.

1. Theoretical Frameworks: Gravitational Bubble Transitions

The foundational setting for gravitationally induced topological phase transitions in gravity is the Lovelock action with higher-curvature corrections, notably incorporating both the Einstein–Hilbert and Gauss–Bonnet terms. The action in $d$ spacetime dimensions is: $$\mathcal{I} = \frac{1}{16\pi G_N} \int_{\mathcal M} \left[ -2\Lambda\,\epsilon_{a_1 \cdots a_d}\, e^{a_1}\wedge\cdots\wedge e^{a_d} + \epsilon_{a_1\cdots a_d}\,R^{a_1a_2}\wedge e^{a_3}\wedge\cdots\wedge e^{a_d} + \lambda\,L^2\,\epsilon_{a_1\cdots a_d}\,R^{a_1a_2}\wedge R^{a_3a_4}\wedge e^{a_5}\wedge\cdots\wedge e^{a_d} \right] +\mathcal I_\partial$$ where $\Lambda$ is the cosmological constant, $\lambda$ is the Gauss–Bonnet coupling, and appropriate boundary terms ($\mathcal I_\partial$) ensure a well-posed variational problem.

A new phase transition distinct from the standard Hawking–Page transition arises when the free energy balance between (i) a pure AdS vacuum and (ii) a bubble configuration—in which a spherically symmetric shell nucleates, separating two AdS regions with different effective cosmological constants $\Lambda_\pm$—is reversed above a critical temperature. The inner region contains a black hole, altering the causal and topological structure of spacetime. The matching of the induced metric and generalized junction conditions at the bubble ensure consistency of the global solution (Camanho et al., 2012).

2. Topological Order Parameters and Defect Interpretation

A common theme in gravitationally induced topological phase transitions is the emergence and manipulation of topological invariants or winding numbers characterizing distinct phases. In higher-derivative gravities, black-hole solutions themselves are identified as topological defects in thermodynamic or field-theoretic parameter spaces. For instance, using Duan’s $\phi$-mapping topological current theory, the configuration space is parametrized by an order-parameter field $\phi^a(x^\mu)$, whose zeros correspond to black-hole solutions and whose winding number assigns a Chern index to each phase.

The exchange or annihilation of these defects at bifurcation points in parameter space, e.g., as temperature or pressure is varied, yields a robust mechanism for first-order phase transitions—mirrored in the Maxwell swallowtail construction and reflected in free-energy nonanalyticities. This interpretation also underpins small–large black-hole transitions in Gauss–Bonnet-AdS backgrounds and their analogies with classic condensed-matter transitions, such as Kosterlitz–Thouless vortex-antivortex dynamics (Fairoos, 2023, Rathi et al., 2024).

3. Critical Temperatures and Nucleation Mechanisms

The canonical gravitationally induced topological phase transition is governed thermodynamically by the comparison of free energies. For the gravitational bubble setup, the free energy of the bubble configuration is

$F_{\rm bubble}(T) = M_+ - T S_{\rm BH}$

where $M_+$ is the ADM mass and $S_{\rm BH}$ is the Bekenstein–Hawking–Wald entropy of the black hole inside the bubble. For empty AdS, $F_{\rm AdS}(T) = 0$ in the regularization scheme. A critical temperature $T_c$ is defined by $F_{\rm bubble}(T_c) = 0$. Above $T_c$, the nucleated bubble has negative free energy and dominates the ensemble. The bubble nucleation rate is controlled by the Euclidean action difference: $$\Gamma \sim \exp(-\Delta \hat{\mathcal{I}}) = \exp(-\beta_+ F_{\rm bubble}(T))$$ At nucleation, the cosmological constant effectively jumps and the topology of the spacetime changes—such as a shift in the winding of the Euclidean time–radial surface around a new black-hole singularity (Camanho et al., 2012).

In topological current approaches, the criticality corresponds to the bifurcation and interchange of winding numbers between topological defects in parameter space. The stability and sequence of transitions are determined by the sign flips in heat capacity or the curvature of the free energy $F(r_+,T)$ as a function of horizon radius (Fairoos, 2023).

4. Extensions: Analogue Gravity and Condensed-Matter Systems

Gravitationally induced topological phase transitions are not confined to pure gravitational systems. In Dirac and Weyl semimetal systems, uniaxial strain acts as an emergent gravitational field for quasiparticles, modifying band parameters and effectively generating a spacetime metric. The transition from quantum spin Hall to trivial insulating phases, or the creation of analogue event horizons (with corresponding Hawking temperatures),

$$T_{\rm H} = \frac{\hbar}{2\pi k_B} \left| \frac{d}{dz}[v_z(z) - w(z)] \right|_{z = z_h}$$

are consequences of such artificial gravitational effects. The local mass term $m(\varepsilon)$ changes sign at a critical strain, mapping precisely to a topological phase transition in the bulk or thin film (Guan et al., 2016).

In topological superconducting wires, a real gravitational redshift modulates energy scales $\mu(x)$ and $\Delta(x)$ along the system; when $\mu(x) = 2 t(x)$, a topological phase boundary forms, with a trapped Majorana zero-mode at the emergent domain wall. Gravity thus directly induces the local transition between topological and trivial phases, shifting the spatial location of Majorana modes (Wong et al., 19 Jan 2026).

5. Gravitational Induction of Defects and Criticality in Field Theories

In scalar and XY-type models, a weak classical gravitational field couples minimally to the kinetic energy, leading to a multi-valued correction $\Phi_G(x)$ in the wavefunction—a phase term whose nontrivial holonomy signals the presence of topological singularities (vortices). These vortices, quantized via $\oint \partial_\mu \Phi_G dx^\mu = 2\pi n$, create degeneracies and break global symmetries. The critical temperature for gravitationally driven Kosterlitz–Thouless–type transitions (vortex unbinding) is

$T_c \sim m y s^2 n^2$

(where $y$ is a dimensionless gravitational coupling, $m$ the mass, and $s$ the spin magnitude). Gravitational susceptibility $\chi_g$ is universally positive, indicating enhanced order as the gravitational coupling increases (Papini, 2019).

6. Topological Phase Transitions in Lower-Dimensional Gravity

In Jackiw–Teitelboim (JT) gravity coupled to Yang–Mills fields, small-to-large black-hole transitions persist as topological changes of thermodynamic winding number, robust to all orders in gauge coupling. Here, off-shell free energy landscapes and the computation of thermodynamic winding number (following the Duan–Lien–Ge $\phi$-mapping approach) classify black-hole solutions as defects with winding numbers $w_1=-1$ (small BH, unstable) and $w_2=+1$ (large BH, stable). The transition $\mathcal W: -1 \to +1$ underpins a genuinely topological mechanism, paralleled in the boundary Schwarzian action and dual SYK-like quantum mechanical models (Rathi et al., 2024).

7. Parameter Regimes and Universality

The dimensionless couplings, such as the Gauss–Bonnet parameter $\lambda$ or the effective gravitational coupling $y$, control the existence and nature of topological phase transitions. Physical regimes satisfy $0 < \lambda < 1/4$ (to ensure nondegenerate vacua in bubble transitions), with critical temperatures and other characteristics (e.g., wall position in the Kitaev wire, strain in semimetals) determined accordingly. As couplings vanish (e.g., $\lambda \to 0$), the novel phase structure decouples, recovering standard gravitational or condensed-matter thermodynamics.

The universality of gravitationally induced topological transitions is manifest in the unification of mechanisms—bubble nucleation, defect bifurcation, domain wall formation—across gravitational, field-theoretic, and condensed-matter systems. The conservation and exchange of winding number underpin both gravitational and non-gravitational transitions, linking the physics of higher-curvature gravity, black-hole thermodynamics, and topological quantum matter (Camanho et al., 2012, Fairoos, 2023, Rathi et al., 2024, Guan et al., 2016, Wong et al., 19 Jan 2026, Papini, 2019).