Interacting Fermionic SPT Phases

Updated 17 June 2026

Inherently interacting fermionic SPT phases are quantum states defined by strong interactions and nontrivial braiding invariants that cannot be achieved by free-fermion or bosonic approaches.
These phases are constructed using finite depth local unitary circuits and commuting projector Hamiltonians, revealing explicit topological invariants like three-loop braiding phases.
Their classification employs group supercohomology and gauging techniques, leading to experimentally observable edge and surface states that signal robust topological order.

Inherently interacting fermionic symmetry-protected topological (SPT) phases are distinct quantum phases of matter that require both strong interactions and the presence of fermionic degrees of freedom for their existence. These phases are not adiabatically connected to any free-fermion SPT or to bosonic SPTs formed from pairwise-bound fermions. Their full classification, topological invariants, and physical properties have been established using a blend of group supercohomology, gauging and braiding statistics, and explicit interacting Hamiltonian constructions.

1. Classification Principles and Diagnostic Invariants

SPT phases in fermionic systems equipped with a global symmetry group $G_f$ (with canonical fermion parity $\mathbb{Z}_2^f\subset G_f$ ) are classified by analyzing the structure of short-range-entangled, symmetric, gapped ground states and possible obstructions in constructing them using free fermions or bosonic composites. For finite Abelian $G_f$ , a powerful approach involves:

Gauging the symmetry: Coupling the fermionic system to a dynamical $G_f$ gauge field and analyzing the resulting topological order, particularly the braiding statistics of symmetry fluxes (vortices) and charges.
Topological invariants: The key diagnostic is a set of Abelian invariants:
- $\Theta_\mu$ : quantized topological spin of a vortex with unit flux along direction $\mu$ ;
- $\Theta_{\mu\nu}$ : Berry phase accumulated when braiding a unit $\mu$ -flux around a unit $\nu$ -flux;
- $\Theta_{\mu\nu\lambda}$ : Berry phase for a commutator sequence of braids involving flux types $\mathbb{Z}_2^f\subset G_f$ 0, $\mathbb{Z}_2^f\subset G_f$ 1, and interleaving $\mathbb{Z}_2^f\subset G_f$ 2.

These invariants are subject to stringent consistency constraints from gauge theory (fusion, associativity, vanishing of chiral central charge $\mathbb{Z}_2^f\subset G_f$ 3), yielding a classification group parameterized by cyclic summands tabulated for each symmetry (Wang et al., 2016). Nontrivial elements in this classification not realized by stacking free-fermion SPTs or group-cohomology bosonic SPTs are identified as inherently interacting fermionic SPTs.

2. Structural Features of Inherently Interacting FSPT Phases

The canonical example in two spatial dimensions is furnished by the symmetry group $\mathbb{Z}_2^f\subset G_f$ 4. The Abelian stacking group for this case is

$\mathbb{Z}_2^f\subset G_f$ 5

hosting 4096 distinct phases, of which precisely half (2048) cannot be realized by free-fermion or bosonic-embedding constructions and are thus intrinsically interacting (Wang et al., 2016).

A distinguishing invariant for these third-kind fermionic SPTs is the three-loop braiding phase

$\mathbb{Z}_2^f\subset G_f$ 6

whereas all free-fermion and BSPT-embedded constructions produce only $\mathbb{Z}_2^f\subset G_f$ 7. Physically, $\mathbb{Z}_2^f\subset G_f$ 8 detects a fractional non-Abelian Berry phase in the commutator braiding of three symmetry fluxes (associated to $\mathbb{Z}_2^f\subset G_f$ 9, and the two $G_f$ 0 factors).

The nontrivial value is generated by a nontrivial 3-cocycle

$G_f$ 1

with the associated three-loop braiding phase $G_f$ 2. Such cohomology data do not admit a free-fermion realization due to the constraints imposed by fermion parity and the structure of quadratic Bogoliubov-de Gennes band theory (Wang et al., 2016).

3. Construction Methods and Explicit Hamiltonians

Explicit lattice realizations for group supercohomology-based intrinsically interacting FSPTs can be obtained via finite depth local unitary (FDLU) circuits and commuting projector Hamiltonians. The generic scheme involves:

First constructing a bosonic "shadow" SPT model with an enlarged symmetry.
Fermionization: mapping the bosonic spins and gauge constraints to fermionic degrees of freedom using spin structure-dependent duality (Ellison et al., 2018).
The resulting fixed-point Hamiltonian is realized via a circuit

$G_f$ 3

where all operators are strictly local, symmetric, and define a non-Gaussian, commuting-projector parent Hamiltonian whose ground state is a short-range-entangled, interacting SPT (Ellison et al., 2018).

In one spatial dimension, block constructions and decorated domain-wall techniques demonstrate that certain symmetry groups (e.g., $G_f$ 4) support SPTs that are impossible to realize using any noninteracting or bosonic model, evidenced by inherently many-body projective edge symmetry algebra (Tantivasadakarn et al., 2018).

For 2D and 3D crystalline (space group or point group) SPTs, real-space block state (dimensional reduction) constructions with explicit gapping terms for Majorana chains on lower-dimensional blocks are used to systematically enumerate and realize all inherently interacting fermionic crystalline SPTs (Zhang et al., 2020, Rasmussen et al., 2018).

4. Field-Theoretic and Topological Response Characterization

Bulk theories of inherently interacting FSPTs are captured by generalized multi-component Chern-Simons or spin-cobordism field theories, rather than by Chern numbers or $G_f$ 5-matrix band theory alone. For unitary finite Abelian symmetries, a representative effective action is given by: $G_f$ 6 with $G_f$ 7 a unimodular integer matrix (det $G_f$ 8). Intrinsically interacting FSPT phases correspond to $G_f$ 9-matrices and symmetry implementations for which certain "root" states (see $G_f$ 0 for $G_f$ 1) cannot be reduced to free-fermion band invariants or bosonic SPTs via condensation or stacking (Ning et al., 2019).

Surface and edge theories manifest "ingappability" (protected gapless edge states or symmetry-protected topological order) when symmetry is preserved. For instance, edge Luttinger liquids with anomalous symmetry action, unremovable by local gap-opening terms, signal the presence of intrinsically interacting bulk SPT order (Sullivan et al., 2019).

5. Generalizations and Extensions: Higher Dimensions, Crystalline, and Cobordism Approaches

Inherently interacting FSPT phases extend to point and space group symmetries, with block states decorated by lower-dimensional interacting FSPTs which cannot exist in bosonic or free-fermion sectors (Zhang et al., 2020, Zhang et al., 2019, Cheng et al., 2018). For example, in spinless fermion systems with wallpaper group $G_f$ 2, the only unobstructed 1D block state involves decorating each mirror axis by a double-Majorana FSPT, resulting in crystalline SPT order not realizable by free bands (Zhang et al., 2020).

The classification and explicit construction of all such phases is accomplished using a generalized cohomology-theoretic decoration framework, efficient computational algorithms for obstruction analysis, and condensation/gauging techniques to diagnose nontrivial loop or junction braiding invariants (Ouyang et al., 2020, Cheng et al., 2017).

For higher dimensional ( $G_f$ 3) and interacting SPTs, spin/pin cobordism theory predicts and classifies inherently interacting fermionic SPTs via the computation of bordism groups and torsion invariants (e.g., $G_f$ 4) which possess nontrivial elements invisible to free-fermion K-theory, thus predicting new interaction-only phases in, for example, 6+1D with $G_f$ 5 symmetry (Kapustin et al., 2014).

6. Constraints, No-Go Theorems, and Physical Signatures

The impossibility of realizing inherently interacting FSPTs in free-fermion or bosonic SPT frameworks stems from the incompatibility of braiding statistics, edge symmetry representations, and stacking rules with those of free models. For example:

No BSPT embedding can yield a three-loop invariant $G_f$ 6—the corresponding defects in BSPT-embedded phases always braid trivially (Wang et al., 2016, Ning et al., 2019).
Free-fermion models, constrained by charge conservation and quadratic Hamiltonians, cannot generate the required cocycle invariants or projective representations witnessed in these phases.
Even stacking all known free and BSPT phases cannot access the distinguishing invariants of these third-kind FSPTs due to the stacking law's additive structure.

Experimental observables include protected (often degenerate) zero-energy edge modes that transform projectively under the symmetry, gapless Luttinger liquid edges inaccessible to free-fermion gap openers, and non-Abelian braiding signatures in surface or defect responses (Ellison et al., 2018, Zhang et al., 2020, Zhang et al., 2019).

7. Representative Table: Inherently Interacting 2D Fermionic SPTs

Symmetry Group	Classification group	Intrinsically interacting factors	Distinguishing invariant(s)
$G_f$ 7	$G_f$ 8	Half of group ( $G_f$ 9 roots)	$\Theta_\mu$ 0
$\Theta_\mu$ 1	See (Wang et al., 2016)	Trivial (all realized by stacking)	—
$\Theta_\mu$ 2	$\Theta_\mu$ 3	$\Theta_\mu$ 4 roots	Projective edge symmetry algebra
Crystalline $\Theta_\mu$ 5	$\Theta_\mu$ 6	1D mirror block	Flux-induced Majorana zero modes

All intrinsically interacting classifications are determined using the gauging/braiding statistical approach, group or supercohomology, and explicit real-space or commuting-projector constructions (Wang et al., 2016, Ellison et al., 2018, Zhang et al., 2020, Tantivasadakarn et al., 2018, Ning et al., 2019, Cheng et al., 2017).

A broad class of SPT phases in fermionic systems fundamentally require strong interactions and cannot be understood, classified, or detected using free-fermion band theory or bosonic SPT paradigms. Their existence is established through an overview of algebraic, topological, and constructive techniques, with uniquely interacting roots diagnosed by nontrivial braiding invariants, projective edge or defect symmetries, and obstruction-free stacking classification. The canonical examples—such as those for $\Theta_\mu$ 7 symmetry in 2D—are distinguished by fractionalized Berry phases and edge or surface phenomena, and continue to guide the theoretical and experimental search for new quantum phases in correlated electron systems (Wang et al., 2016, Ellison et al., 2018, Ning et al., 2019, Zhang et al., 2020, Zhang et al., 2019, Cheng et al., 2018, Rasmussen et al., 2018).