Symmetry-Protected Topological Phases

A symmetry-protected topological phase is a gapped, symmetry-preserving, invertible phase that cannot be connected to a trivial symmetric product phase by a symmetry-preserving gapped path, but becomes trivial when the protecting symmetry is relaxed. Its bulk has no intrinsic topological order: on a closed space it has no topology-dependent ground-state degeneracy or deconfined anyons. Its nontriviality instead appears through a symmetry obstruction, often as a projective edge action, an anomalous boundary theory, a quantized response, or a nontrivial symmetry-defect process.

Every word in the definition carries an assumption:

• gapped refers to a stable many-body gap in the thermodynamic limit;
• symmetry-preserving requires the symmetry group and its action on microscopic degrees of freedom to be specified;
• invertible means there is a phase that cancels it under stacking;
• trivial is understood after allowing product-state ancillas that transform linearly under the symmetry;
• protected means that the obstruction disappears if the relevant symmetry constraint is removed.

Required background. Phases of Matter in Many-Body Quantum Mechanics and Topology in Quantum Matter supply the gapped-deformation framework, while Schmidt Decomposition supplies the boundary-entanglement capability.

Helpful background. Symmetry-Protected Structure Preview and Unitary Symmetries supply the protection and group-action language, while Matrix Product States Preview supplies the one-dimensional virtual-edge picture.

The Symmetry-Restricted Equivalence Relation

Consider local Hamiltonians $H(s)$ on the same spatial dimension, with $s\in[0,1]$. Let $G$ act through operators $U(g)$, which may be unitary or antiunitary. A symmetry-preserving path satisfies

$$U(g)H(s)U(g)^{-1} = H(s)$$

for every $g\in G$ and every $s$. Two symmetric Hamiltonians are in the same $G$-protected phase when, after adding decoupled trivial degrees of freedom if needed, there is such a local path with a nonzero thermodynamic gap,

$$\inf_{s\in[0,1]} \Delta_\infty(s) > 0.$$

The symmetry is part of the equivalence relation. The same endpoints can be distinct in the space of $G$-symmetric Hamiltonians and equivalent in the larger space where $G$ may be broken.

Hamiltonians, ground states, and circuits

A uniformly gapped local path gives quasi-local continuation of the ground state: the generator has rapidly decaying spatial tails. A finite-depth circuit is a useful exact description of fixed-point representatives,

$$U_{\mathrm{FD}}=\prod_{\ell=1}^{D}\prod_xU_{\ell,x},$$

where depth $D$ and gate range remain bounded as the system grows. Generic gapped states have exponentially small correlations at arbitrarily large separations, so an exact bounded-depth circuit from a product state need not reproduce them. The circuit description is used after coarse graining or to a stated local accuracy; it is not an exact identity for every microscopic ground state.

For onsite unitary symmetry, the allowed circuit must be built from local gates that each commute with the symmetry restricted to their support. Merely requiring the complete circuit to commute with the global symmetry is too weak. A circuit can preserve symmetry on a closed chain through cancellations between neighboring gates while creating a nontrivial SPT. The cluster example below demonstrates this explicitly.

Thus the obstruction is to a locally symmetric disentangling process, consistent with the symmetric gapped-path definition. Fermionic circuits must also preserve parity and graded locality. Antiunitary and spatial symmetries require the corresponding adapted equivalence relation; they cannot be treated as arbitrary onsite unitary matrices. Chiral invertible phases are excluded from the symmetry-trivializable class discussed here.

Stable equivalence

Adding a decoupled symmetric product degree of freedom should not create a new phase. If

$$\lvert\phi_{\mathrm{anc}}\rangle = \bigotimes_x\lvert\phi_x\rangle$$

is a symmetry-preserving product state in allowed linear representations, then

$$\lvert\psi\rangle \sim \lvert\psi\rangle\otimes \lvert\phi_{\mathrm{anc}}\rangle.$$

This is stable equivalence. The ancilla state must itself be invariant up to a phase; a Hilbert space carrying a linear representation can still contain symmetry-breaking product states. Stable equivalence prevents the phase label from depending on how many inert local orbitals are included. Blocking sites or adding allowed ancillary representations may be needed to make a symmetric product reference available. It does not permit adding an already nontrivial lower-dimensional SPT at a boundary for free; that operation can change the boundary realization while leaving the bulk class fixed.

State the symmetry group, not only its name

“Time reversal” or “spin rotation” is not enough information. One must state relations such as

$$\mathcal T^2 = \begin{cases} +I, & \text{integer spin},\\ (-1)^F, & \text{spinful fermions}, \end{cases}$$

and whether charge conservation, translation, reflection, or other symmetries are also required. A localized spin-$1/2$ has $\mathcal T^2=-I$ even in a spin model; “bosonic” does not universally imply $\mathcal T^2=+I$. Projective representations per unit cell can impose separate constraints on the existence of a symmetric gapped phase. The physical fermionic symmetry group $G_f$ contains the central subgroup generated by fermion parity,

$$\mathbb Z_2^F \subset G_f.$$

Quotienting by fermion parity gives a bosonic symmetry group $G_b=G_f/\mathbb Z_2^F$, but distinct extensions of $G_b$ by $\mathbb Z_2^F$ can lead to different fermionic classifications.

Short-Range Entanglement, Invertibility, and Stacking

Different kinds of gapped phases can look similar if one checks only the bulk spectrum:

Phase	Defining distinction	Typical boundary behavior
Symmetry breaking	Several symmetry-related bulk states	Order-parameter textures
SPT	Trivializable after relaxing symmetry	Symmetry obstruction
Intrinsic topological order	Nonlocal sectors and excitations	May support anyons
Symmetry-independent invertible	Nontrivial without optional symmetry	For example, chiral transport

An SPT is not diagnosed merely by the absence of a local order parameter. A featureless paramagnet is also symmetric and gapped; it is the identity SPT. Conversely, a symmetry-enriched topological phase contains intrinsic topological order and is not an SPT even though symmetry acts nontrivially on its anyons.

Stacking defines a composition law

Place two decoupled phases side by side:

$$H_{A\oplus B} = H_A\otimes I_B + I_A\otimes H_B.$$

After allowing symmetric interactions between the layers, the resulting phase is called their stack,

$$[A]+[B] = [A\oplus B].$$

The trivial product phase is the identity $0$. An invertible phase $A$ has an inverse $\bar A$ satisfying

$$[A]+[\bar A] = 0.$$

Under standard assumptions, invertible $G$-protected phases form an Abelian group. The group may be $\mathbb Z$, $\mathbb Z_n$, or a product of such factors. A label is meaningful only with its dimension, statistics, symmetry action, and interaction assumptions.

Forgetting symmetry

There is a conceptual map

$$\mathcal F: \mathrm{Inv}_G^d \longrightarrow \mathrm{Inv}^d$$

that forgets the $G$ action. In a precise modern convention, purely $G$-protected phases lie in the kernel:

$$\mathrm{SPT}_G^d \subseteq \ker\mathcal F.$$

This separates an SPT from a phase that remains nontrivial even after $G$ is removed. The distinction matters for the bosonic $E_8$ state and other chiral invertible phases. In fermionic systems, fermion parity is structural and is not physically discarded; “forgetting symmetry” means forgetting optional symmetry beyond $\mathbb Z_2^F$.

Boundary Obstruction and Anomaly

A nontrivial SPT bulk can be completely featureless on a closed manifold. For onsite protecting symmetries, its sharpest universal signature often appears at an interface with the trivial phase. The boundary cannot generally realize, by itself in the same dimension and with the same microscopic symmetry action, a unique gapped symmetric short-range-entangled state. Spatial symmetries need a separate boundary qualification because an individual surface or end need not preserve them.

Depending on dimension and symmetry, a nontrivial boundary may instead be:

1. gapless while preserving the symmetry;
2. gapped by spontaneously breaking the protecting symmetry;
3. gapped, symmetric, and intrinsically topologically ordered;
4. absorbed by another boundary sector carrying the inverse anomaly.

The third option is available only when the boundary dimension can support intrinsic topological order. A zero-dimensional end cannot hide its anomaly in anyons; it typically carries a protected multiplet or projective symmetry action.

Anomaly inflow

Couple the symmetry to a nondynamical background field $A$. The bulk effective action may change under a boundary-supported gauge transformation:

$$\delta_\lambda S_{\mathrm{bulk}}[A] = \int_{\partial M} \mathcal I_\lambda[A].$$

The boundary partition function transforms oppositely,

$$Z_{\partial}[A^\lambda] = Z_{\partial}[A]\, \exp\!\left( -i\int_{\partial M}\mathcal I_\lambda[A] \right).$$

Only the combined bulk–boundary system is invariant. This is anomaly inflow. Here “anomaly” does not mean that the microscopic symmetry is inconsistent. It means that the boundary symmetry action cannot be regularized as an ordinary standalone local theory in the boundary dimension while retaining all assumptions.

Boundary spectra are not individually universal

Bulk topology constrains the boundary anomaly, not every microscopic boundary energy. Symmetric boundary perturbations can move, broaden, or reconstruct states. Additional trivial boundary bands can appear. Opposite anomalous sectors can cancel. A physical edge can also break the protecting symmetry even when the bulk Hamiltonian does not.

Thus the robust claim is not

$$E_{\mathrm{edge}}=0 \quad \text{for every termination}.$$

The robust claim is that no allowed boundary completion removes the total anomaly without one of the permitted alternatives above. Edge and Surface States explains how termination and spectroscopy affect the observed spectrum.

Defects and interfaces

An interface between phases $A$ and $B$ carries the same obstruction as the boundary of

$$[A]-[B].$$

If $[A]=[B]$, a symmetric trivially gapped interface is generally allowed. If not, the difference phase must be accounted for by interface degrees of freedom. Symmetry twists, fluxes, dislocations, and domain walls can likewise expose lower-dimensional SPT data. This dimensional reduction viewpoint is central to decorated-domain-wall constructions and crystalline SPT phases.

One-Dimensional SPT Phases

One dimension provides the cleanest interacting classification. Assume a local bosonic chain with:

• a unique gapped ground state on a periodic chain;
• unbroken onsite unitary symmetry $G$;
• finite correlation length;
• stable equivalence under linear-representation ancillas.

An injective matrix-product state gives a controlled representative or approximation of such a phase; it need not represent every microscopic ground state exactly. If $u_g$ is the onsite physical representation, symmetry acts on an MPS tensor as

$$\sum_j (u_g)_{ij}A^j = e^{i\theta_g} V_g^{-1}A^iV_g.$$

The physical action is linear, but the virtual matrices need only be projective:

$$V_gV_h = \omega(g,h)V_{gh}, \qquad \omega(g,h)\in U(1).$$

Associativity produces a cocycle

Associating $V_gV_hV_k$ in two ways gives

$$\omega(g,h)\omega(gh,k) = \omega(h,k)\omega(g,hk).$$

This is the 2-cocycle condition. Rephasing the virtual action,

$$V_g \longmapsto \beta(g)V_g,$$

changes the factor set to

$$\omega'(g,h) = \frac{\beta(g)\beta(h)} {\beta(gh)} \omega(g,h).$$

The two factor sets describe equivalent projective representations. Their equivalence classes form

$$[\omega] \in H^2(G,U(1)).$$

For the assumptions above, $H^2(G,U(1))$ classifies the one-dimensional bosonic SPT factor (Chen, Gu and Wen; Schuch, Pérez-García and Cirac). Extra one-dimensional representations can encode charge per unit cell or boundary conventions, and translation or symmetry breaking adds further data. Antiunitary and reflection symmetries require twisted versions of the cocycle equations.

Why the two ends carry inverse classes

For a long open interval, the low-energy symmetry action factorizes approximately:

$$U(g) \simeq U_L(g)U_R(g).$$

Each factor can be projective,

$$U_L(g)U_L(h) = \omega_L(g,h)U_L(gh),$$

while the full microscopic action remains linear. Therefore the right edge carries the inverse class,

$$[\omega_R] = -[\omega_L].$$

The pair can combine to a linear representation. Coupling the two ends through the bulk produces a splitting exponentially small in length,

$$\delta E \sim e^{-L/\xi},$$

but a local symmetric perturbation near only one end cannot select a unique state from an irreducible projective multiplet. Attaching an external projective degree of freedom to that end can screen it; the boundary Hilbert space is therefore part of any edge claim.

Stacking multiplies cocycles

For two chains with virtual actions $V_g^{(1)}$ and $V_g^{(2)}$,

$$V_g^{(12)} = V_g^{(1)} \otimes V_g^{(2)}.$$

Their factor sets multiply:

$$\omega_{12}(g,h) = \omega_1(g,h)\omega_2(g,h).$$

This realizes cohomology addition physically. If a class has order $n$, then $n$ copies can be symmetrically disentangled after suitable inter-copy couplings and stable ancillas are allowed.

The Haldane Chain as a Prototype

The spin-1 antiferromagnetic chain is the canonical interacting bosonic example. At the Affleck–Kennedy–Lieb–Tasaki point, each spin 1 is represented as two virtual spin-$1/2$ objects projected onto the symmetric onsite subspace. Neighboring virtual spins form singlets. On a periodic chain every virtual spin is paired; on an open chain one spin-$1/2$ remains at each end.

The bulk onsite states transform linearly under $SO(3)$, because integer spin is an ordinary representation of $SO(3)$. A boundary spin-$1/2$ is projective under $SO(3)$. For $\pi$ rotations about orthogonal axes, one may choose

$$V_x = i\sigma_x, \qquad V_z = i\sigma_z,$$

so that

$$V_xV_z = -V_zV_x.$$

The corresponding physical $SO(3)$ rotations commute, but their edge representatives commute only up to a phase. This is the nontrivial element of

$$H^2(SO(3),U(1)) \cong \mathbb Z_2.$$

Two copies have two spin-$1/2$ degrees of freedom at each end. They can combine into an integer-spin singlet, consistent with the $\mathbb Z_2$ stacking law.

Which symmetry protects it?

The odd-integer-spin Haldane phase can be protected by any one of several symmetry choices, including:

• the dihedral group generated by $\pi$ spin rotations;
• time reversal;
• bond-centered inversion;
• full $SO(3)$ spin rotation.

These are different classification problems. Breaking one symmetry does not trivialize the phase if another protecting symmetry is still enforced. Conversely, saying “the Haldane phase is protected” without naming the retained group is incomplete.

Bond-centered inversion exchanges the two ends. It can protect the bulk or entanglement distinction without protecting an independent degeneracy at each physical end: inversion-related boundary fields can remove end degeneracies while preserving the spatial symmetry. Onsite spin rotations or time reversal constrain an individual end differently (Pollmann and colleagues).

String order and edge spin are useful diagnostics, but neither alone is the definition. String order can depend on the chosen symmetry and operator. Edge degeneracy can be screened by added boundary degrees of freedom. The invariant information is the bulk symmetric phase class, accessible through the projective action on Schmidt states or an equivalent many-body invariant. Entanglement Spectrum explains how to extract and interpret those Schmidt multiplets.

A cluster chain with explicit edge operators

Consider an even number $N\geq4$ of qubits with Pauli operators $X_j,Z_j$, grouped into two-site unit cells. Define

$$H_C=-J\sum_{j=2}^{N-1}K_j,\qquad J>0,$$ $$K_j=Z_{j-1}X_jZ_{j+1}.$$

The commuting, independent stabilizers $K_j$ have eigenvalues $\pm1$. The open-chain ground space has dimension $2^N/2^{N-2}=4$, separated by a gap $2J$. Its $\mathbb Z_2\times\mathbb Z_2$ symmetry is generated by

$$G_e=\prod_{j\ \mathrm{even}}X_j,\qquad G_o=\prod_{j\ \mathrm{odd}}X_j.$$

After grouping one even and one odd site per cell, this is an ordinary onsite linear action. Operators acting within the ground space are

$$\begin{aligned} Z_L&=Z_1,&X_L&=X_1Z_2,\\ Z_R&=Z_N,&X_R&=Z_{N-1}X_N. \end{aligned}$$

Multiplying the even and odd stabilizers and setting $K_j=+1$ gives

$$G_e=Z_LX_R,\qquad G_o=X_LZ_R$$

within that space. The left factors anticommute, as do the right factors. The full generators commute because the two minus signs cancel. A local left-end perturbation that commutes with both $X_L$ and $Z_L$ is proportional to the identity on that edge qubit. This is a concrete projective-edge obstruction, not just a count of four states.

The eight-site open chain has six independent stabilizers and two edge qubits. Each global symmetry factors into one Pauli operator at each end. At either end the two factors anticommute, while the full physical symmetry is linear. The fourfold degeneracy is exact at this fixed point; generic symmetric local perturbations can couple the ends only through effects that vanish with their separation while the bulk remains gapped.

On a periodic chain, applying $U_{CZ}=\prod_j CZ_{j,j+1}$ to $|+\rangle^{\otimes N}$ creates the unique cluster ground state, where $CZ=\operatorname{diag}(1,1,1,-1)$. Since $U_{CZ}X_jU_{CZ}^\dagger=Z_{j-1}X_jZ_{j+1}$, the extra $Z$ factors cancel in each global generator around the ring. Thus $U_{CZ}$ commutes with $G_e$ and $G_o$ as a whole. Yet a single $CZ$ gate does not commute with the even- or odd-site $X$ on its support. This globally symmetric circuit is not an allowed locally symmetric disentangler.

For comparison, the SSH model has a free-particle chiral constraint. That constraint is not automatically an ordinary onsite many-body symmetry. Its band winding, inversion-quantized polarization and boundary energy must be tied to the actual symmetry being retained.

Free-fermion phases and their interacting relatives

Band and BdG invariants classify quadratic Hamiltonians; a many-body phase allows general local interactions along its deformation path. This can identify phases that were distinct in the free classification, and can introduce phases with no free representative.

The terminology needs care. In the strict definition used here, fermion parity is never discarded. A class-D Kitaev chain remains nontrivial with parity alone, so it is a fermionic invertible phase, not a phase protected only by an optional symmetry. A chiral Chern phase similarly remains nontrivial after optional symmetries are removed. Some literature includes these in a broader use of “fermionic SPT”; always check the convention.

By contrast, the quantum spin Hall and strong topological-insulator distinctions require charge conservation and spinful time reversal. Their free-band examples are developed in Topological Insulators, and the parity-preserving BdG examples in Topological Superconductors.

Class BDI and the Z8 interaction reduction

The class-BDI Majorana chain is the standard warning. Quadratic Hamiltonians have an integer index $\nu$ counting protected end Majoranas. Symmetry-preserving quartic interactions can gap eight end Majoranas without leaving degeneracy, so

$$\mathbb Z \longrightarrow \mathbb Z_8$$

in the interacting classification (Fidkowski and Kitaev). Values of $\nu$ that differ by eight are not distinct interacting phases. Forgetting the optional time reversal leaves the class-D parity index $\nu\bmod2$. Thus the full $\mathbb Z_8$ describes BDI invertible phases, while its even classes $0,2,4,6$ form the $\mathbb Z_4$ subgroup that becomes trivial when time reversal is relaxed. This distinction reconciles the reduction with the strict SPT definition above.

Class DIII and the Z16 interaction reduction

Three-dimensional class-DIII topological superconductors provide a higher-dimensional analogue:

$$\mathbb Z \longrightarrow \mathbb Z_{16}$$

under interactions. These reductions do not mean weak interactions immediately destroy every boundary state. They mean that the global phase-equivalence relation contains symmetric interacting paths absent from the quadratic subspace.

Interactions can also create phases

The opposite phenomenon also occurs. Interactions can produce bosonic or fermionic SPT phases with no free-particle representative. A complete interacting classification therefore cannot be obtained by simply taking a periodic table and adding perturbative corrections.

To compare a free result with an interacting phase:

1. identify the microscopic symmetry group, including fermion parity;
2. determine the free classification if a quadratic description is justified;
3. test which labels survive arbitrary symmetry-preserving interactions;
4. include intrinsically interacting phases not reachable from free fermions.

Interacting SPT Phases

An interacting SPT can have no sharp single-particle bands at all. Its universal data may instead appear in many-body response, symmetry defects, entanglement, or anomalous surfaces.

Bosonic integer quantum Hall phase

Two-dimensional bosons with conserved $U(1)$ charge can form an SPT with no anyonic bulk excitations and a unique ground state on the torus. Its Hall conductivity is an even integer in natural bosonic units:

$$\sigma_{xy} = 2n\frac{q^2}{h}, \qquad n\in\mathbb Z.$$

The minimal nontrivial state has counterpropagating edge modes whose net chiral central charge vanishes but whose charge response cannot be symmetrically removed. The evenness follows from bosonic locality and the absence of intrinsic topological order, not from a free-boson band Chern number.

A minimal Abelian description makes these statements tangible (Lu and Vishwanath; Senthil and Levin):

$$K=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \mathbf t=\begin{pmatrix}1\\1\end{pmatrix}.$$

The integer matrix $K$ encodes braiding and edge commutators, and $\mathbf t$ specifies the coupling to the conserved charge. Here $|\det K|=1$, so all integer quasiparticle vectors are equivalent to local particles and the torus has a single ground sector. The two eigenvalues of $K$ have opposite signs, giving net chiral central charge zero. Choose the orientation in which the charge response is $\sigma_{xy}=(q^2/h)\mathbf t^TK^{-1}\mathbf t$; it equals $2q^2/h$. Reversing orientation reverses this sign. Nonzero charge Hall response can therefore coexist with zero net thermal chirality and no fractional bulk quasiparticles.

Symmetric gapped surfaces with topological order

A three-dimensional SPT surface need not remain gapless. Strong interactions can generate a symmetric gapped surface with anyons. The symmetry then acts on those anyons in a pattern impossible in a strictly two-dimensional system with the same microscopic assumptions. This is anomalous symmetry enrichment.

Such a surface does not convert the bulk SPT into intrinsic three-dimensional topological order. The bulk remains invertible. The anyons live only in a chosen boundary termination, and a different boundary may be gapless or symmetry breaking.

Gauging as a diagnostic

For a finite onsite symmetry, one can promote symmetry twists to dynamical gauge fluxes. Gauging turns the symmetry-dependent distinction between SPTs into a distinction between topologically ordered phases, detectable through the braiding or statistics of their flux excitations.

Gauging is a diagnostic transformation, not a claim that the original SPT contained deconfined anyons. Before gauging, symmetry defects are extrinsic and require branch cuts or background fields.

Group Cohomology Preview

Group cohomology organizes a large and important family of bosonic SPT phases with onsite symmetry. The cleanest finite-group statement is formulated in spacetime dimension $d+1$ for a system with $d$ spatial dimensions.

Why d plus one appears

An $n$-cochain assigns a $U(1)$ phase to $n$ group arguments. In this multiplicative notation, the coboundary operation satisfies

$$\delta(\delta\beta)=1.$$

An $n$-cocycle obeys $\delta\omega_n=1$, while an $n$-coboundary has the form $\omega_n=\delta\beta_{n-1}$. Write $Z^n$ for the group of degree-$n$ cocycles and $B^n$ for its subgroup of coboundaries. Cohomology is the quotient

$$H^n(G,U(1))=Z^n/B^n.$$

For the bosonic onsite-unitary construction,

$$[\omega_{d+1}] \in H^{d+1}(G,U(1))$$

labels a family of $d$-dimensional SPT fixed points. The shift by one occurs because the cocycle weights a $(d+1)$-dimensional spacetime history, not merely a spatial configuration.

Background-field response

For a finite group, a flat background $G$ gauge field can be represented by a map

$$A: M^{d+1} \longrightarrow BG,$$

where $BG$ is the classifying space. A multiplicative cocycle can be written $\omega_{d+1}=e^{2\pi i\nu_{d+1}}$, with additive $\nu_{d+1}\in\mathbb R/\mathbb Z$. Its topological partition-function factor is

$$Z_{\omega}[M,A] = \exp\!\left[ 2\pi i \int_{M^{d+1}} A^*\nu_{d+1} \right].$$

Changing the cocycle by a coboundary leaves the closed-manifold phase unchanged; the action differs by a boundary term on an open manifold. On a manifold with boundary, that term is precisely where the anomalous boundary transformation appears.

Antiunitary symmetry and twisted coefficients

For antiunitary symmetry, the coefficients are twisted because antiunitary elements complex-conjugate $U(1)$ phases. One writes schematically

$$H^{d+1}(G,U(1)_T).$$

For continuous compact groups, Borel group cohomology or equivalently suitable cohomology of $BG$ supplies the relevant refinement. These statements require care about topology on $G$ and about which background bundles are allowed.

What group cohomology does not classify

The formula $H^{d+1}(G,U(1))$ must not be presented as the complete classification of every SPT. Its original domain is bosonic phases with onsite internal symmetry, and even there it misses known beyond-cohomology phases in some dimensions. It does not by itself include:

• fermionic statistics and spin-structure dependence;
• all antiunitary or orientation-reversing subtleties;
• all crystalline symmetries;
• symmetry-independent chiral invertible phases;
• every bosonic SPT beyond the cohomology construction.

Fermionic supercohomology captures an important subset of fermionic SPTs. Cobordism and generalized-homology approaches provide broader frameworks. In modern language, an interacting invertible phase defines a deformation class of reflection-positive invertible field theories with specified tangential and symmetry structure. The mathematics is powerful, but the physical input remains essential: dimension, statistics, symmetry extension, orientation or spin structure, and locality.

Responses and mathematical descriptions

At scales larger than the correlation length, a gapped bulk has no propagating low-energy excitations, yet its partition function can depend on background symmetry fields and geometry. Group cocycles provide explicit examples. Broader cobordism and invertible-field-theory constructions also incorporate orientation, spin or pin structure and gravitational response (Kapustin; Freed and Hopkins). Specifying those structures is part of the physical question, not a technical detail that can be omitted.

For electronic topological insulators, the electromagnetic angle $\theta$ is periodic modulo $2\pi$ under the appropriate charge and spacetime assumptions. Time reversal changes its sign, allowing $\theta=0$ or $\pi$. An ordinary gapped time-reversal-breaking surface of the $\theta=\pi$ phase has a half-integer Hall offset; intrinsic surface topological order requires a different description. The derivation and limitations are given in Topological Insulators. This response distinguishes the familiar free representative but does not by itself classify every interacting phase with the same symmetries.

Identifying an SPT in a calculation or experiment

Start by specifying the microscopic symmetry and checking that the bulk remains gapped and does not spontaneously break it. A symmetric finite-size cat state can conceal thermodynamic symmetry breaking. A finite correlation length alone does not prove a spectral gap; use gap scaling or other justified information about the Hamiltonian as well.

For a one-dimensional MPS, converge system size and bond dimension, extract the symmetry action on the Schmidt space, and determine its projective multiplication law. For commuting generators $g,h$, the matrix

$$\mathcal C_{g,h}=V_gV_hV_g^{-1}V_h^{-1}$$

is unchanged by rephasing either generator. In the cluster or Haldane example it is $-I$ at one edge, while a linear representation gives $+I$. Multiplet counting alone is weaker: accidental degeneracies and ordinary linear irreducible representations can also give repeated Schmidt values.

At a physical boundary, identify the degrees of freedom and any attached spins or extra layers. Deliberately breaking a protecting symmetry can test the proposed mechanism, but the disappearance of a peak alone does not establish a bulk class. Approximate symmetry gives only approximate protection; a small symmetry-breaking term can split an edge multiplet already at first order.

A phase label needs its assumptions. Dimension, bosonic or fermionic statistics, the actual symmetry extension, and permitted interactions determine the classification. Ordinary group cohomology and the free tenfold way each answer a narrower question than “all topological phases.”

Exercises

1. Why the cluster edge cannot split locally

For the open cluster chain, use the edge factors of $G_e$ and $G_o$ to constrain a Hermitian left-edge perturbation $h_L=aI+bX_L+cY_L+dZ_L$. Then remove $G_o$ while retaining $G_e$. Can the edge split?

Solution

The left factors are $Z_L$ and $X_L$. Commutation with $Z_L$ forces $b=c=0$; commutation with $X_L$ then forces $d=0$. Thus $h_L=aI$ when both symmetries are enforced. If only $G_e$ remains, $dZ_L$ is allowed and splits the doublet by $2|d|$. The protection concerns the symmetry of the local perturbation, not the accidental zero of a particular boundary energy.

2. Derive the projective cocycle condition

Starting from

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

derive the 2-cocycle condition by associating $V_gV_hV_k$ in two ways. Then show how $\omega$ changes under $V_g\mapsto\beta(g)V_g$.

Solution

The first association gives

$$\begin{aligned} (V_gV_h)V_k &= \omega(g,h)V_{gh}V_k\\ &= \omega(g,h)\omega(gh,k)V_{ghk}. \end{aligned}$$

The second gives

$$\begin{aligned} V_g(V_hV_k) &= \omega(h,k)V_gV_{hk}\\ &= \omega(h,k)\omega(g,hk)V_{ghk}. \end{aligned}$$

Associativity therefore requires

$$\omega(g,h)\omega(gh,k) = \omega(h,k)\omega(g,hk).$$

With $V'_g=\beta(g)V_g$,

$$\begin{aligned} V'_gV'_h &= \beta(g)\beta(h)\omega(g,h)V_{gh}\\ &= \frac{\beta(g)\beta(h)} {\beta(gh)} \omega(g,h)V'_{gh}. \end{aligned}$$

Thus

$$\omega'(g,h) = \frac{\beta(g)\beta(h)} {\beta(gh)} \omega(g,h),$$

which differs by a coboundary and represents the same cohomology class.

3. Why two Haldane chains can cancel

Stack two chains whose left edges each carry spin $1/2$. Construct a local $SO(3)$-invariant coupling that removes the edge degeneracy, and relate the result to the $\mathbb Z_2$ classification.

Solution

Let $\mathbf s_1$ and $\mathbf s_2$ be the two edge spins. The local coupling

$$H_{\mathrm{edge}} = J\,\mathbf s_1\cdot\mathbf s_2, \qquad J>0,$$

is $SO(3)$ invariant. Since

$$\frac12\otimes\frac12 = 0\oplus1,$$

the antiferromagnetic coupling selects a unique spin singlet. In projective language the two nontrivial factor sets multiply to a coboundary:

$$[\omega]+[\omega] = 0 \quad \text{in}\quad \mathbb Z_2.$$

The same cancellation occurs at the right edge. With suitable symmetric bulk interchain couplings, the two-copy stack is adiabatically connected to a trivial phase.

4. Eight Majoranas can have a unique symmetric state

Let $\{\gamma_a,\gamma_b\}=2\delta_{ab}$ for $a=1,\ldots,8$, with $\mathcal T\gamma_a\mathcal T^{-1}=\gamma_a$ and $\mathcal Ti\mathcal T^{-1}=-i$. Define

$$\begin{aligned} Q_1&=\gamma_1\gamma_2\gamma_3\gamma_4,\\ Q_2&=\gamma_1\gamma_2\gamma_5\gamma_6,\\ Q_3&=\gamma_1\gamma_3\gamma_5\gamma_7,\\ Q_4&=\gamma_1\gamma_2\gamma_7\gamma_8. \end{aligned}$$

Show that $H=-u\sum_{r=1}^4Q_r$, with $u>0$, has a unique gapped ground state in the 16-dimensional eight-Majorana Hilbert space. Why is the corresponding bilinear mechanism forbidden?

Solution

Each quartic product is Hermitian, squares to $I$, preserves parity and is time-reversal invariant. Every pair of $Q_r$ shares two Majoranas, so the four operators commute. Their four binary support vectors are independent: no nonempty product is proportional to $I$. Therefore the simultaneous $+1$ projector

$$P_+=\frac1{16}\prod_{r=1}^4(I+Q_r)$$

has trace one; only its identity term has nonzero trace. The unique ground state has energy $-4u$ and a definite total parity. Selecting that parity sector preserves its uniqueness; the opposite sector contains excited states. Changing one $Q_r$ eigenvalue costs $2u$ in the full Hilbert space. A Hermitian bilinear is $i\gamma_a\gamma_b$, which changes sign under $\mathcal T$ and is forbidden. This example shows how quartic interactions evade a quadratic obstruction; the full $\mathbb Z_8$ bulk classification additionally requires the interacting-path argument of Fidkowski and Kitaev.

5. A charge-protected edge with no fractional bulk particles

For the bosonic Hall example, verify $|\det K|=1$, $\mathbf t^TK^{-1}\mathbf t=2$ and zero signature. A local edge perturbation $\cos(\boldsymbol\Lambda^TK\boldsymbol\phi)$ has integer $\boldsymbol\Lambda$; a charge-preserving gapping vector must obey

$$\mathbf t^T\boldsymbol\Lambda=0, \qquad \boldsymbol\Lambda^TK\boldsymbol\Lambda=0.$$

Is there a nonzero solution? What changes if charge conservation is relaxed?

Solution

The matrix is its own inverse, has determinant $-1$ and eigenvalues $+1,-1$. It therefore has one torus ground sector, Hall coefficient $2q^2/h$ in the stated orientation and net chiral central charge zero. Charge neutrality requires $\boldsymbol\Lambda=(m,-m)^T$. Its quadratic form is $-2m^2$, which vanishes only for $m=0$. No nonzero charge-preserving null vector gaps these two counterpropagating modes. If charge conservation is removed, the primitive vector $(1,0)^T$ is null and supplies a gapping term. This is how the boundary obstruction depends on the optional $U(1)$ symmetry.

Continue studying

Matrix Product States and Entanglement Spectrum develop the one-dimensional computational tools. Bulk–Boundary Correspondence compares free indices with interacting boundary possibilities. Topological Order explains the contrasting role of intrinsic anyons and nonlocal ground-state sectors.

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Further reading

• C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of Topological Quantum Matter with Symmetries,” Reviews of Modern Physics 88, 035005 (2016), doi:10.1103/RevModPhys.88.035005.
• T. Senthil, “Symmetry-Protected Topological Phases of Quantum Matter,” Annual Review of Condensed Matter Physics 6, 299–324 (2015), doi:10.1146/annurev-conmatphys-031214-014740.
• X.-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press (2004), Chapters 8–10.