--- title: Reentrant Topological Phase Transitions url: https://www.emergentmind.com/topics/reentrant-topological-phase-transitions type: topic --- # Reentrant Topological Phase Transitions Reentrant topological phase transitions are sequences of phase transitions in which a system repeatedly enters and exits distinct topological phases as a function of a control parameter such as disorder strength, chemical potential, quasiperiodic modulation, or interaction strength. These transitions are characterized by the appearance, disappearance, and reappearance of topologically protected boundary modes (such as Majorana zero modes or edge states) and associated changes in bulk topological invariants. Reentrant behavior is particularly prominent in low-dimensional, disorder-prone, or periodically/quasiperiodically modulated systems, and has been realized in fermionic wires, superconducting chains, photonic and cold-atom analogues, and even in black hole thermodynamics in anti-de Sitter spacetimes. ## 1. Theoretical Framework and Model Prototypes Reentrant topological transitions were first rigorously analyzed in models interpolating between prototypical topological classes, such as the Kitaev chain (1D class D topological superconductor with unpaired Majorana end modes) and the bond-alternating Haldane/SSH chain (1D class BDI/Solitary Polyacetylene). The canonical Hamiltonian for spinless fermions on an $N$-site chain with bond alternation and $p$-wave pairing is: \[ H = \sum_{j=1}^{N-1}\left[ -t_j c_j^\dagger c_{j+1} + \Delta c_j c_{j+1} + \text{h.c.} \right],\quad t_j = t\left[1 + (-1)^j \delta\right],\ \delta \in [-1,1] \] The parameter $\Delta$ tunes the $p$-wave pairing, while $\delta$ modulates the bond alternation [1708.00982]. This model exhibits multiple topological phases diagnosed by: - **Majorana number** $\mathcal{M} = \mathrm{sgn}\{\mathrm{Pf}[h(0)]\,\mathrm{Pf}[h(\pi)]\}$. - **Bulk Berry phase** or nonlocal string order parameter as in the Haldane limit. Distinct topological sectors ($\mathcal{M} = \pm 1$) correspond to phases supporting Majorana zero modes or bulk string order, respectively. The phase boundaries are analytically given by $\Delta = \pm\delta$ and $\Delta = \pm 1/\delta$, leading, for fixed $\delta$, to a sequence of $\mathcal{M}=-1 \to +1 \to -1$ as $|\Delta|$ increases—a clear instance of reentrant topological phase transitions. ## 2. Quasiperiodicity, Disorder, and Reentrant Topological Anderson Insulators Quasiperiodic modulation and disorder introduce further complexity, yielding reentrant topological Anderson insulator (TAI) phases and rich phase diagrams not captured by clean models: - In generalized SSH chains with quasi-periodic hopping, \[ H = \sum_{m=1}^{L} v_m (c_{m,B}^\dagger c_{m,A} + \text{h.c.}) + \sum_{m=1}^{L-1} w_m (c_{m+1,A}^\dagger c_{m,B} + \text{h.c.}) \] with $v_m = v + \Delta_m$, $w_m = w + \gamma\Delta_m$, and $\Delta_m = \Delta\cos(2\pi\beta m + \phi)$, the competition between off-diagonal quasiperiodic modulations yields robust reentrant TAI and RTAI phases [2306.06818]. These phases are detected by real-space winding number and the divergence or vanishing of the zero-mode localization length. - In the generalized SSH model with both bounded ($|b|<1$) and unbounded ($|b|\ge1$) quasiperiodic modulation of the intracell hoppings, \[ t_{1,n}^\prime = t_1 + \frac{\lambda \cos(2\pi\alpha n + \theta)}{1 - b\cos(2\pi\alpha n + \theta)} \] the system exhibits two classes of re-entrant phase transitions: traditional TI → trivial → TAI (for bounded), and TAI$_1$ → trivial → TAI$_2$ (for unbounded) as $\lambda$ is increased. All transitions are sharply identified via real-space winding number, Lyapunov exponent, and bulk gap closing [2412.08067]. - In topological ladders and trimer lattices, reentrance manifests as the destruction, reformation, and sometimes reversal of quantized transport properties as disorder or modulation strength is swept [2508.04452, 2601.13760]. ## 3. Majorana Wires, Lattice Modulation, and Reentrant Phases In 1D spin–orbit-coupled wires subject to magnetic fields and proximity-induced superconductivity, introducing periodic or quasiperiodic lattice modulation (via potential or hopping) fragments the spectrum into minibands. Each miniband can host topological superconducting (TS) phases with end Majorana fermions: - The number of topological regions ("windows") as a function of chemical potential or modulation amplitude increases with the fractal miniband structure induced by the modulation [1202.5969, 1308.1167]. - Modulation-induced reentrant transitions are robust to modulation-domain-wall phase jumps and persist under both onsite and hopping modulation. - Majorana zero modes appear and disappear multiple times as $\mu$ is swept, resulting in a sequence of trivial and nontrivial TS phases [1202.5969, 1308.1167]. For multi-channel spinless $p$-wave superconductors, disorder induces an alternation of trivial and nontrivial phases, with the number of alternations equal to the channel number $N$. The critical mean free paths for each transition are $l_n = \xi/(n+1)$, where $\xi$ is the coherence length. The topological invariant $Q$ alternates $(-1)^N$, $(-1)^{N-1}$,... as $l$ crosses these thresholds [1302.2071]. ## 4. Moiré Modulation, Entanglement, and Universality Extended SSH models with commensurate moiré modulations of intracell hopping realize reentrant topological transitions; analytic renormalization shows that the effective intracell coupling undergoes oscillatory crossings with the fixed intercell hopping, generating repeated $0\leftrightarrow 2$ (and more complex) transitions in the winding number. In systems with next-nearest-neighbor $w$, phase boundaries are computed as zeros of $\det q(k)$ in the reduced moiré Brillouin zone [2601.09997]. These transitions exhibit remarkable consistency in bulk-boundary correspondence: the number of zero-energy edge modes, the entanglement spectrum degeneracies, the central charge extracted from CFT scaling of entanglement entropy (with $c = \Delta W$ matching winding number jumps), and the critical exponents (e.g., $|\Delta_c(L)-\Delta_c(\infty)| \propto L^{-1}$) remain invariant across all reentrant topological transitions, suggesting a universal class [2601.09997]. ## 5. Non-Hermitian Lattices and Reentrant Topology Non-Hermitian generalizations, such as the complexified Aubry–André–Harper model, display reentrant transitions driven by the Hermiticity-breaking parameter $g$: \[ H = \sum_j t (c_j^\dagger c_{j+1} + \text{h.c.}) + V e^{i(2\pi\alpha j + g)} n_j + V e^{-i(2\pi\alpha j - g)} n_j \] The sequence of phases, as $g$ is increased, is: - Extended ($\langle\mathrm{IPR}\rangle \to 0$) $\to$ Mixed (mobility edge, partial winding) $\to$ Localized ($\langle\mathrm{IPR}\rangle > 0$, full winding $W=2$) $\to$ Mixed $\to$ Reentrant Extended ($W=0$) [2306.11084]. The topological content is captured by the spectral winding number $W(\varepsilon_b)$, which counts the encirclements of a base point in the complex energy plane and jumps as the system passes through reentrance. ## 6. Reentrant Transitions in Higher-dimensional and Gravitational Systems In gravitational analogues, specifically in Born–Infeld–massive gravity and dRGT theories in topological Anti-de Sitter black holes, reentrant phase transitions (RPTs) appear in the extended black hole thermodynamics: - Equation of state $P(r_+,T)$ admits up to $N$ critical points for $d \ge 4 + N$, allowing for $N$-fold reentrant RPTs and triple points [1707.04101, 2009.07980]. - As temperature is lowered at fixed pressure, the black hole entropy can reenter previously exited branches, characterized by jumps in the Gibbs free energy and supported by swallowtail diagrams in $G(T)$. In higher dimensions, these RPTs are nontrivially richer than classical Van der Waals transitions and have no direct analogy in condensed-matter models. ## 7. Experimental Realizations and Physical Mechanism Reentrant topological phases have been predicted and are feasible to observe in various platforms: - Cold atom systems with engineered SSH-like couplings, Aubry–André potentials, momentum-lattice engineering, and wavepacket dynamics enable direct measurement of winding numbers and localization-delocalization transitions [2306.06818, 2412.08067, 2508.04452]. - Photonic waveguide arrays, modulated nanowires, and Rydberg-atom arrays with programmable hoppings support creation and readout of reentrant edge states [2412.08067, 2601.13760]. - In driven, active matter systems (topological flocks with nonreciprocal alignment), reentrant band formation is triggered by the emergence of instability windows as a function of noise and dissenter fraction [2412.11871]. The central mechanism of reentrance is the interplay of distinct localization/gapping mechanisms—band folding due to periodicity or disorder, emergent minibands and minigaps, or the oscillatory renormalization of effective Hamiltonian parameters. As control parameters are swept, the system successively fulfills then violates the topological gap-opening conditions, resulting in repeated entrance to and exit from topologically nontrivial states. --- **Selected Models and Their Reentrant Features** | Model Type | Key Feature | Reference | |-----------------------|--------------------------------|--------------------| | Kitaev–Haldane Chain | Bond alternation, U(1) symmetry drives $\mathcal{M}=-1\rightarrow+1\rightarrow-1$ | [1708.00982] | | Quasiperiodic SSH | TAI–trivial–RTAI transitions, real-space winding | [2306.06818], [2412.08067] | | 1D Rashba Wire | Band folding induces repeated TS windows | [1202.5969], [1308.1167] | | Mosaic Trimer Lattice | Disorder-induced reentrant topology | [2601.13760] | | Moiré Extended SSH | Universality and entanglement across reentrances | [2601.09997] | | Non-Hermitian AAH | Spectral winding, deloc–loc–deloc | [2306.11084] | | Multichannel $p$-wave | $N$ reentrant transitions via disorder | [1302.2071] | | AdS Black Holes | RPTs, triple points, $N$-fold transitions | [1707.04101], [2009.07980] | | Topological Flocks | Dual-band reentrant microphases | [2412.11871] | Reentrant topological phase transitions thus represent a robust and universal phenomenon in diverse quantum and classical settings, deeply rooted in the interplay of symmetry, modulation, and disorder, and provide new platforms for engineering and studying topologically protected phenomena. Source: https://www.emergentmind.com/topics/reentrant-topological-phase-transitions