Interacting SPT Matter and Physical Diagnostics

An interacting symmetry-protected topological phase is short-range entangled in the bulk but cannot be deformed to a symmetric product state without closing the gap or breaking the protecting symmetry. Because quasiparticle bands need not exist, the diagnosis must use many-body response, symmetry defects, entanglement, or a boundary anomaly. Intrinsic topological order is excluded: it has deconfined anyons or topology-dependent ground-state sectors.

Required background. Time-reversal band topology supplies the free reference point; invertible responses supplies the short-range-entangled response viewpoint.

Helpful background. Anomaly inflow supplies the obstruction to a standalone symmetric boundary.

Many-body definition and diagnostics

On a closed spatial manifold, a bosonic SPT has a unique gapped ground state in the thermodynamic limit and no nontrivial bulk superselection sectors. Fermionic invertible phases require the spin structure and fermion parity to be included. Two phases are equivalent if connected by a symmetry-preserving gapped path after adding symmetric product degrees of freedom.

Useful diagnostics answer different questions:

• Quantized response: couple the conserved or background symmetry to a probe field and extract a quantized coefficient, including its allowed local counterterms.
• Symmetry defects: insert fluxes, twists, dislocations, or domain walls and measure their fractional charge or projective symmetry action.
• Many-body Berry phase: vary twisted boundary conditions while keeping a unique many-body gap; polarization and pumps are examples.
• Entanglement: test the projective action on Schmidt states or a robust degeneracy pattern, with finite-size and cut dependence controlled.
• Boundary anomaly: show that a local boundary cannot be simultaneously symmetric, trivially gapped, and nondegenerate.

No single diagnostic is universal. A response may vanish for a nontrivial SPT, and a boundary can be gapped by symmetry breaking or, in higher dimensions, by intrinsic topological order.

A one-dimensional example

The spin-1 Haldane phase with $SO(3)$ symmetry has a unique gapped bulk on a ring but spin-$1/2$ edge degrees of freedom on an open chain. Each edge transforms projectively even though the combined finite chain transforms linearly. In matrix-product language, the symmetry acts on the virtual bond by matrices $V_g$ satisfying

$$V_gV_h=\omega(g,h)V_{gh},$$

and the cohomology class of the phase factor $\omega$ cannot change along a symmetric gapped path Pollmann et al. 2012. Coupling the two ends at finite length can split their degeneracy exponentially; this does not remove the bulk phase.

Interactions can also reduce free classifications. Eight copies of the one-dimensional BDI Majorana chain can be symmetrically gapped by quartic interactions, producing the reduction $\mathbb Z\to\mathbb Z_8$ Fidkowski and Kitaev 2011. Thus a band winding number is not automatically an interacting invariant.

Evidence boundaries

Before naming an interacting SPT, specify the symmetry group including antiunitary and spatial actions, the dimension, whether fermions are present, the permitted product states, and the many-body gap. Demonstrate that the diagnostic is quantized or symmetry protected and rule out spontaneous symmetry breaking and intrinsic order. A single-particle topological Hamiltonian constructed from a Green function can be useful only when its zeros, singularities, and interaction assumptions are controlled; it is not a general many-body classification.

Exercise

Why can two spin-$1/2$ edge states on a finite Haldane chain form a nondegenerate singlet without contradicting SPT protection?

Solution

The two projective edge representations combine to a linear representation and may couple through the finite bulk. Their splitting is exponentially small in chain length. Protection says that one isolated edge cannot be realized as a unique, symmetric, trivially gapped zero-dimensional system while the bulk remains unchanged; it does not require exact finite-size degeneracy of both edges together.

References

• Lukasz Fidkowski and Alexei Kitaev, “Topological Phases of Fermions in One Dimension,” Physical Review B 83 (2011) 075103, doi:10.1103/PhysRevB.83.075103.
• Frank Pollmann, Erez Berg, Ari M. Turner, and Masaki Oshikawa, “Symmetry Protection of Topological Phases in One-Dimensional Quantum Spin Systems,” Physical Review B 85 (2012) 075125, doi:10.1103/PhysRevB.85.075125.