Interaction-Induced Topological Phases

• Interaction-induced topological phases are many-body states where interactions drive new symmetry-protected orders absent in free-fermion systems.
• Key mechanisms include symmetry fractionalization, renormalization of single-particle parameters, and spontaneous symmetry breaking at band touchings.
• These phases have been realized in varied platforms such as superconducting circuits, Floquet systems, and quantum simulators, offering novel experimental insights.

Interaction-induced topological phases are topological phases or topological phase transitions whose defining structure depends essentially on interactions beyond a free-particle band description. In the literature represented here, the phrase covers several distinct but related phenomena: genuinely new interacting symmetry-protected topological phases with no free-fermion analog, interaction-driven conversion of trivial insulators or critical states into topological states, topology of collective or composite excitations such as magnons and doublons, and finite-temperature or environment-assisted topological transitions that have no counterpart in the corresponding noninteracting limit (Ning et al., 2014, Lapa et al., 2014, Wang et al., 2012, Su et al., 14 Mar 2025, Huang et al., 2024, Pavan et al., 2023).

1. Conceptual scope and recurring mechanisms

The phrase does not denote a single mechanism. One recurring mechanism is interaction-enabled symmetry fractionalization: the many-body Hilbert space supports projective edge representations that cannot occur in a free-fermion band problem with the same microscopic symmetry. A second mechanism is interaction-driven renormalization of single-particle structure, where self-energy effects shift orbital energies, masses, or effective dimerizations until a topological invariant changes. A third is spontaneous symmetry breaking at quadratic or critical band touchings, where repulsion opens a topological mass or splits a quadratic crossing into Dirac cones. A fourth is topological reorganization of composite sectors, as in doublon, magnon, or open-system quasiparticle descriptions (Ning et al., 2014, Wang et al., 2012, Zeng et al., 2018, Zhou et al., 2024, Su et al., 14 Mar 2025, Pavan et al., 2023).

A central distinction in the literature is between phases that are truly absent in the noninteracting classification and phases whose noninteracting analog already exists but whose boundaries are shifted by correlations. In the first category are the interacting Haldane insulator in the spinless three-leg ladder and interaction-enabled crystalline phases based on the $\mathbb Z\to\mathbb Z_8$ reduction of class BDI (Ning et al., 2014, Lapa et al., 2014). In the second category are interaction-driven topological transitions in the Kane-Mele-Hubbard family and in mirror-symmetric crystalline chains, where interactions alter the stability or phase boundary of an already existing topological phase rather than creating a new topological class from scratch (Hung et al., 2013, Bhakuni et al., 2022).

This multiplicity of meanings is not terminological noise but a structural feature of the subject. In some settings interactions enlarge the topological landscape; in others they destabilize it; in still others they generate new finite-temperature, Floquet, or dissipative descendants. A precise discussion therefore depends on whether the relevant object is a many-body SPT classification, a Green’s-function invariant, a composite quasiparticle band, or an edge-state anomaly.

2. Symmetry, projective representations, and interaction-enabled classifications

A canonical one-dimensional example is the spinless fermion ladder with $U(1)\rtimes Z_2^T$ symmetry, where the noninteracting classification is trivial but the interacting classification is not (Ning et al., 2014). In that model the protecting symmetry acts as

$$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$

with

$Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$

The relevant cohomology is

$\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$

so one-dimensional interacting SPT phases are classified by two projective classes even though the free-fermion classification is $\mathbb Z_1$. The nontrivial edge realization satisfies

$$M(g_1)M(g_2)=e^{i\theta(g_1,g_2)}M(g_1g_2),\qquad [M(T)K]^2=-1,$$

and this projective minus sign is the mathematical origin of the interaction-induced Haldane insulator (Ning et al., 2014).

Interaction-enabled crystalline topology sharpens the same idea. In class BDI, the free one-dimensional classification is $\nu\in\mathbb Z$, but interactions reduce it to $\mathbb Z_8$ (Lapa et al., 2014). Imposing inversion maps $\nu\mapsto-\nu$. In the free theory this forces $U(1)\rtimes Z_2^T$0, but after reduction to $U(1)\rtimes Z_2^T$1 the nontrivial solution $U(1)\rtimes Z_2^T$2 becomes allowed. The same logic yields a two-dimensional $U(1)\rtimes Z_2^T$3-symmetric weak topological superconductor with $U(1)\rtimes Z_2^T$4, together with anomalous gapless edges and crystalline defects binding tetrads of Majorana zero modes (Lapa et al., 2014). These phases are interpreted as topological charge-$U(1)\rtimes Z_2^T$5 superconductors whose topology is enabled by quartic Fidkowski-Kitaev interactions rather than by quadratic Bogoliubov-de Gennes band theory.

A concrete platform for this symmetry class is a superconducting topological-insulator surface tuned to $U(1)\rtimes Z_2^T$6, where a time-reversal-like symmetry $U(1)\rtimes Z_2^T$7 with $U(1)\rtimes Z_2^T$8 emerges (Chiu et al., 2015). Vortices and antivortices bind Majorana zero modes of opposite $U(1)\rtimes Z_2^T$9-parity, identified with $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$0 and $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$1 Majoranas. At neutrality, same-type bilinears are forbidden while quartic interactions remain allowed, making the system naturally interaction-dominated. In the proposed realization of the Lapa-Teo-Hughes phase, open chains exhibit a fourfold ground-state degeneracy in the topological regime, while the noninteracting limit is topologically trivial (Chiu et al., 2015).

3. Diagnostics and theoretical formalisms

The earliest diagnostics of interaction-induced SPT order in the examples surveyed here are intrinsically many-body. In the spinless three-leg ladder, the Haldane insulator is identified by a gapped bulk, exponentially decaying local correlations, and a characteristic real-space entanglement spectrum degeneracy: twofold in the infinite system and fourfold in the finite open chains studied numerically, because the two weakly coupled edges become entangled (Ning et al., 2014). In that framework the entanglement spectrum is not an auxiliary observable but the direct signature that the Schmidt states on a cut transform projectively under the protecting symmetry.

For interacting band-topological transitions, the dominant formalism is Green’s-function topology. In the interacting BHZ model, the zero-frequency topological Hamiltonian

$$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$2

is combined with DMFT and a pole-expansion auxiliary-Hamiltonian construction to compute the interacting $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$3 invariant (Wang et al., 2012). For $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$4 and $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$5, the interacting $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$6 index changes from $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$7 to $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$8 at $$U_\theta c_\alpha U_\theta^{-1}=c_\alpha e^{i\theta},\qquad Tc_\alpha T^{-1}=-c_\alpha,$$9, converting a trivial band insulator into a correlated topological insulator. The same work emphasizes that the static Hartree shift captures the onset of band inversion, but the full dynamical self-energy is essential deep in the correlated regime, where the system remains topological even when a naive static effective-mass argument would suggest otherwise (Wang et al., 2012).

A more recent extension replaces the Hermitian topological Hamiltonian by an energy-dependent non-Hermitian quasiparticle Hamiltonian

$Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$0

whose complex eigenvalues encode renormalized dispersion and damping (Krishtopenko et al., 3 Mar 2025). In the disordered BHZ example, the interacting $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$1 invariant is controlled by the sign change

$Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$2

and the methodology shows explicitly why imaginary parts of the self-energy can matter in more general non-Hermitian topological problems (Krishtopenko et al., 3 Mar 2025). This formalism treats disorder, phonons, and genuine many-body interactions on the same Green’s-function footing whenever they can be represented through a self-energy.

The same Green’s-function perspective also underlies numerically exact QMC studies of interaction-modified phase boundaries. In the generalized Kane-Mele-Hubbard models, the $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$3 invariant and spin Chern number are computed directly from the zero-frequency Green’s function, demonstrating that interactions stabilize the QSH phase in $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$4-preserving models and destabilize it when the one-body terms break $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$5 symmetry (Hung et al., 2013). This is an important caution: interaction-driven topology does not always mean a new phase; it may instead mean a non-perturbative shift of an existing topological transition.

4. One-dimensional correlated topological phases

The spinless three-leg ladder provides one of the cleanest explicit realizations of a fermionic SPT phase induced by interactions (Ning et al., 2014). At filling $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$6, the noninteracting model supports only a metal or a trivial band insulator. Turning on the Hubbard-like term $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$7 and the Heisenberg-like exchange $Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$8 produces three phases: metal, trivial insulator, and Haldane insulator. In the large-$Te^{i\hat N\theta}=e^{-i\hat N\theta}T.$9 regime the three orbital states on each rung become an effective spin-1 degree of freedom, so $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$0 generates the analog of the spin-1 Haldane chain; the numerical result is that the topological phase survives finite charge fluctuations and therefore becomes a genuinely fermionic Haldane insulator (Ning et al., 2014).

A different one-dimensional route is the interaction-induced topological metal of anisotropic quantum wires and coupled helical edges (Kainaris et al., 2016). There the low-energy theory contains a gapless charge mode but a gapped neutral mode, so the state remains metallic while all single-particle excitations are gapped. Two strong-coupling phases appear: a trivial phase and a topological SDW phase. The topological phase supports a boundary zero mode with wavefunction

$\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$1

normalizable only for $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$2, and it is markedly more robust to time-reversal-preserving disorder because the leading single-impurity backscattering is suppressed to second order (Kainaris et al., 2016).

Interactions can also act on topological critical lines rather than on fully gapped states. In the extended SSH chain with next-nearest-neighbor hopping, the noninteracting system contains topologically trivial and topologically nontrivial critical phases distinguished by half-integer winding numbers (Zhou et al., 2024). Any nonzero nearest-neighbor repulsion gaps these critical states, producing either a BI or a TI, but near the multicritical point the BI itself can be driven into a TI. For the representative cut $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$3, the thermodynamic-limit critical interaction for the BI $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$4 TI transition is reported as $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$5, making the model a direct example of an interaction-induced topological insulator emerging from the trivial side of a critical manifold (Zhou et al., 2024).

The one-dimensional setting also supports more unusual extensions. In the SSH-Hubbard chain at finite temperature, there is no conventional thermodynamic phase transition, yet a quantized non-local bulk topological order parameter,

$\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$6

undergoes a finite-temperature transition driven by defects that require both interaction and thermal activation (Huang et al., 2024). The mechanism is the proliferation of fluctuating mass domain walls carrying localized zero modes; it has no analogue in the noninteracting finite-temperature SSH chain. In the extended Hubbard chain with attractive nearest-neighbor interaction and positive pair hopping, interactions generate phases with a doubly degenerate entanglement spectrum and edge-spin signatures, including a partially polarized pseudospin phase in which topological spin-sector features coexist with long-range $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$7-wave superconductivity (Rausch et al., 2020). By contrast, the one-dimensional mirror-symmetric topological crystalline insulator studied in a zigzag ladder illustrates the opposite regime: short-range repulsion leaves the mirror-protected dimer insulator intact, while a sufficiently long-range repulsion drives an interaction-induced transition into a trivial CDW (Bhakuni et al., 2022).

5. Two-dimensional electronic, magnetic, and superconducting realizations

In two dimensions, interactions can convert trivial band insulators into quantum spin Hall states. In the time-reversal-invariant Hofstadter-Hubbard model, the phase sequence at finite spin mixing is

$\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$8

so a local Hubbard interaction can first induce a QSH phase out of a trivial staggered band insulator and then destroy it through magnetic ordering at stronger coupling (Kumar et al., 2016). The zero-frequency self-energy enters through the topological Hamiltonian, but the work emphasizes that in some parameter regimes the dynamical part of the self-energy is essential and a static Hartree interpretation is not sufficient (Kumar et al., 2016).

Quadratic band touchings are especially susceptible to interaction-induced topology. On the checkerboard lattice, weak nearest-neighbor repulsion destabilizes the quadratic band touching toward a spontaneous QAH phase with $\mathcal H^2(U(1)\rtimes Z_2^T,U(1))=\mathbb Z_2,$9, while stronger interaction drives first a bond-ordered critical phase with two Dirac points and then a nematic CDW phase (Zeng et al., 2018). On the extended Lieb lattice, the quadratic band crossing point is marginally unstable against infinitesimal repulsions; the ordered descendants include QAH, QSH, charge nematic, and nematic-spin-nematic phases, depending on which interaction channel dominates (Tsai et al., 2011). These studies established the now-standard theme that a finite density of states at a quadratic touching allows weak repulsion to generate a topological mass rather than merely renormalizing a semimetal.

Interaction-driven topology also appears in correlated superconductors with magnetic self-organization. In a low-dimensional orbital-selective Mott insulator motivated by iron-based systems, increasing the Hubbard interaction above a critical value $\mathbb Z_1$0 drives the system from a trivial superconducting state into a topological superconducting phase with spiral spin order, an emergent triplet pairing amplitude, and Majorana edge states (Herbrych et al., 2020). The mechanism is explicitly interaction-generated: the Hubbard term reorganizes the magnetic sector into a chiral spiral, which then converts an $\mathbb Z_1$1-wave pairing field into an effective topological superconducting state.

Bosonic and spin-wave settings extend the concept beyond fermionic electronic phases. In the Kagome ferromagnet with DMI and pseudodipolar interaction, the isotropic Heisenberg model supplies Dirac points and a flat band, but the topology of the magnon bands is generated by anisotropic exchange interactions (Ni et al., 23 Dec 2025). DMI acts primarily as complex hopping, while PDI introduces anomalous pairing terms in the bosonic BdG Hamiltonian, producing qualitatively different phase diagrams and high-Chern-number phases. The resulting thermal Hall and magnon Nernst responses can even reverse sign with temperature in specific topological regions, linking transport anomalies directly to interaction-generated Berry curvature redistribution (Ni et al., 23 Dec 2025).

6. Floquet, open-system, and programmable implementations

Periodic driving enlarges the space of interaction-induced phases by allowing topology to emerge in composite sectors with no static single-particle analogue. In the periodically driven extended Bose-Hubbard model on a square lattice, the single-particle problem is second-order topologically trivial, but strong onsite interaction binds two bosons into doublons whose effective motion realizes a BBH-like lattice (Su et al., 14 Mar 2025). The Floquet operator is

$\mathbb Z_1$2

and the two-boson sector supports both normal Floquet second-order corner states and an anomalous Floquet second-order phase with doublon corner states in the $\mathbb Z_1$3 gap. The topology is therefore interaction-induced twice over: first by doublon formation, and second by Floquet quasienergy structure (Su et al., 14 Mar 2025).

Open-system couplings can also induce topology. In the SSH chain coupled to local harmonic-oscillator baths, integrating out the environment produces retarded effective interactions between hopping events and an effective non-Hermitian subsystem Hamiltonian (Pavan et al., 2023). If the bath couples to the intra-cell bond, topology is weakened; if it couples to the inter-cell bond, a trivial SSH chain can be driven into the topological phase. The transition is identified in QMC by the bimodality of the polarization distribution $\mathbb Z_1$4 under open boundary conditions and in CPT by the interacting Green’s-function winding number

$\mathbb Z_1$5

showing that dissipation, if suitably engineered, can create rather than merely destroy topology (Pavan et al., 2023).

Programmable quantum simulators supply a complementary route. In interacting superconducting quantum circuits, a two-qubit Hamiltonian with XY exchange,

$\mathbb Z_1$6

was used to map the Berry curvature and Chern number of the ground-state manifold directly (Roushan et al., 2014). The interaction-induced effect is a new $\mathbb Z_1$7 phase that appears when the coupling $\mathbb Z_1$8 becomes dominant; the transition is interpreted geometrically as motion of parameter-space monopoles relative to the measurement manifold (Roushan et al., 2014). Although this is topology in Hamiltonian parameter space rather than in a bulk material, it demonstrates experimentally that interactions can generate new quantized topological sectors in a controllable many-body system.

Taken together, these developments show that interaction-induced topological phases are not a narrow correction to band topology but a broad many-body phenomenon. They encompass projective edge symmetries unavailable to free fermions, self-energy-driven changes of Green’s-function invariants, spontaneous topological masses at quadratic band crossings, topological descendants of critical and finite-temperature states, interaction-generated topology of bosonic excitations and bound pairs, and designer phases in driven, dissipative, and programmable quantum platforms.