Generalized Strange Correlators

• Generalized strange correlators are overlap-based correlation functions that reinterpret bulk quantum data as effective boundary theories in topological systems.
• They extend traditional strange correlators by incorporating tensor network methods, higher-point functions, and replica-space constructions to diagnose both SPT and intrinsic topological orders.
• Their diagnostic power relies on precise operator choices and overlap formulations, enabling clear differentiation between trivial and nontrivial phases in varied quantum models.

Generalized strange correlators are a family of overlap-based or nonstandard correlation functions that extend the canonical strange correlator

$$C(r,r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle \Omega|\Psi\rangle},$$

originally introduced for short-range entangled and symmetry-protected topological (SPT) states, to broader settings including intrinsic topological order, subsystem symmetries, non-invertible symmetries, mixed states, replica-space observables, and dynamical four-point functions. Across these settings, the common theme is that bulk data are reorganized into an effective boundary, interface, or transfer-matrix problem, so that long-range, quasi-long-range, oscillatory, or anomalous behavior of the generalized correlator diagnoses nontrivial topology, anomalous symmetry action, or unconventional dynamics (You et al., 2013, Bal et al., 2018, Tsuji et al., 2018, Zhang et al., 2022, Sala et al., 11 Jun 2025).

1. Foundational definition and boundary interpretation

The canonical strange correlator is defined between a target ground state $|\Psi\rangle$ and a trivial, symmetry-preserving reference state $|\Omega\rangle$ by

$$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$

Its original interpretation is as a correlation function at the temporal boundary between two states in an imaginary-time path integral. Under a Wick-rotation picture, that temporal boundary is reinterpreted as a spatial interface, so the strange correlator probes the boundary theory separating a trivial phase from a nontrivial one (You et al., 2013, Wierschem et al., 2014).

For $1d$ and $2d$ nontrivial short-range entangled states, the strange correlator either saturates to a constant or decays as a power law, even though both $|\Psi\rangle$ and $|\Omega\rangle$ are individually short-range correlated. In trivial phases, the corresponding interface is gapped, and the strange correlator is expected to decay exponentially. This boundary-based logic underlies its use as a bulk diagnostic of SPT order (You et al., 2013).

The construction is symmetry-sensitive. The reference state must share the protecting symmetries of the target state; otherwise the interface can be trivially gapped and the diagnostic can fail. In projective quantum Monte Carlo formulations, the same correlator can be written as

$$C(r-r')=\frac{\langle\Omega|\phi(r)\phi(r')({\cal H}-{\cal C})^{2m}|\psi\rangle}{\langle\Omega|({\cal H}-{\cal C})^{2m}|\psi\rangle},$$

which makes explicit that strange correlators are evaluated at the ends of the operator string rather than in its middle (Wierschem et al., 2014).

The simplest physical reading is therefore not “an unusual bulk correlator,” but an interface correlator encoded in bulk wave functions. This suggests why the notion generalizes naturally whenever a problem admits a meaningful overlap between a nontrivial state and a simple reference, or an equivalent transfer-matrix reinterpretation.

2. Tensor-network and topological-state-sum generalizations

A major generalization replaces SPT wave functions by string-net or related tensor-network states. For a Levin–Wen string-net ground state $|\Psi_{\mathrm{SN}}\rangle$ and a product state $|\Psi\rangle$0, the overlap

$|\Psi\rangle$1

is itself a tensor-network partition function. In this form, the strange correlator is no longer merely diagnostic: it constructs a $|\Psi\rangle$2 classical partition function whose transfer matrix inherits the matrix product operator (MPO) symmetries of the underlying fusion category. Because the resulting transfer matrix commutes with the full MPO algebra, the strange-correlator partition function is constrained to be either critical or symmetry-broken (Bal et al., 2018).

This topological version makes the relation to conformal field theory explicit. The MPO algebra implements lattice versions of topological conformal defects, while Ocneanu tube-algebra idempotents project onto topological sectors corresponding to distinct anyons in the Drinfeld center. Inserting those idempotents into the strange-correlator network yields different conformal sectors, twisted partition functions, and conformal boundary conditions (Bal et al., 2018).

Concrete realizations include the Fibonacci string-net, where the strange correlator with a $|\Psi\rangle$3-projected product state gives the critical hard-hexagon model, and the Ising string-net, where a suitable strange correlator yields the self-dual critical Ising model. In these cases, lattice spectra extracted from transfer matrices reproduce conformal towers, conformal spins, and defect sectors of the corresponding minimal conformal field theories (Bal et al., 2018).

A further extension uses strange correlators to engineer Kramers–Wannier defects in generalized Ising models built from CSS chain complexes. There the overlap between a topologically ordered cluster-state wave function and a product state reproduces a generalized Ising partition function, while “stitched” strange correlators between different topological regions produce duality defects localized on interfaces. In the $|\Psi\rangle$4 Ising case, this realizes the non-invertible fusion rule

$|\Psi\rangle$5

and analogous constructions extend to $|\Psi\rangle$6 anisotropic plaquette Ising models and $|\Psi\rangle$7 Ising gauge/Ising interfaces (Mana et al., 17 Nov 2025).

These tensor-network constructions establish a broad topological meaning of generalized strange correlators: overlaps between nontrivial fixed-point states and simple reference states can generate full classical partition functions, conformal spectra, and defect networks, not just binary phase diagnostics.

3. Dynamical, higher-point, and replica-space strange correlators

A distinct line of generalization replaces equal-time two-point overlaps by higher-point or nonstandard time-ordered correlators. In the Hubbard-model study of out-of-time-ordered correlators (OTOCs), the central object is a bipartite thermal average

$|\Psi\rangle$8

and the corresponding imaginary-time four-point function

$|\Psi\rangle$9

with an unusual $|\Omega\rangle$0 periodicity. Its spectral representation admits analytic continuation to a retarded OTOC, and the Keldysh component obeys an out-of-time-order fluctuation–dissipation theorem

$|\Omega\rangle$1

In that work, these imaginary-time four-point objects are explicitly interpreted as prototypical generalized strange correlators because they combine higher-point structure, bipartite thermal averaging, and nonstandard time ordering (Tsuji et al., 2018).

Replica constructions provide another extension. For a reduced density matrix $|\Omega\rangle$2 of an area-law SPT state, replicating $|\Omega\rangle$3 reconstructs a fixed-point $|\Omega\rangle$4-dimensional SPT wave function, and the bulk strange correlator of that reconstructed state is exactly equal to a twisted Rényi-$|\Omega\rangle$5 correlator of the $|\Omega\rangle$6-dimensional mixed state: $|\Omega\rangle$7 This identifies replica-direction long-range or quasi-long-range order in $|\Omega\rangle$8 with the strange-correlator behavior of the higher-dimensional SPT. In explicit examples, $|\Omega\rangle$9 Haldane-like phases yield long-range order in $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$0, while $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$1 chiral and helical SPTs produce power laws such as $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$2 and $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$3 (Sala et al., 11 Jun 2025).

An even broader analogue appears in the ASEP/DSSYK duality, where moments of the double-scaled SYK transfer matrix are written as overlaps between the ASEP stationary matrix-product state and a product state,

$$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$4

This overlap is argued to be an analogue of the strange correlator in the Levin–Wen/Turaev–Viro correspondence, now transplanted into a $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$5 stochastic and quantum-chaotic setting (Okuyama, 17 Jun 2026).

Taken together, these developments show that generalized strange correlators need not be restricted to equal-time overlaps. They also encompass four-point OTOC-like quantities, replica-space observables, and overlap formulae for transfer-matrix moments whenever the same bulk/interface reinterpretation survives.

4. Symmetry classes and diagnostic applications

The original SPT use of strange correlators remains central. In spin-1 Heisenberg antiferromagnets with single-ion anisotropy, the strange correlator distinguishes the Haldane phase from the trivial large-$$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$6 phase: it saturates to a nonzero constant in the Haldane chain, decays exponentially in the large-$$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$7 chain, and decays algebraically near the Gaussian critical point. The same diagnostic shows that the two-leg spin-1 ladder is topologically trivial, whereas the three-leg ladder is nontrivial, consistent with the even/odd-leg distinction (Wierschem et al., 2014).

Subsystem symmetry-protected topological phases require operator choices adapted to anisotropic subsystem symmetries. In the $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$8 cluster model with line-like subsystem symmetries, strange correlators built from a dimer operator $$C(r-r')=\frac{\langle \Omega|\phi(r)\phi(r')|\Psi\rangle}{\langle\Omega|\Psi\rangle}.$$9 and a plaquette operator $1d$0 detect the transition between the SSPT phase and the trivial paramagnet. The dimer strange correlator is intrinsically anisotropic,

$1d$1

at the solvable point, while the corresponding strange order parameters $1d$2 and $1d$3 are finite in the SSPT phase and vanish in the trivial phase (Zhou et al., 2022).

For $1d$4 fermionic SPTs, fixed-point wave functions show that no single operator family suffices. Bosonic strange correlators diagnose the bosonic cohomology data $1d$5, while fermionic strange correlators built from bond-fermion annihilation operators diagnose the fermion-decoration data $1d$6. The transfer-matrix formulation proves that a combination of bosonic order-parameter strange correlators and fermionic strange correlators fully diagnoses the classification of fixed-point $1d$7 fermionic SPT states (Niu et al., 2023).

Non-invertible symmetry-protected topological phases require a further upgrade. In $1d$8 NISPTs, strange correlators are defined between two distinct NISPT matrix-product states, and the appropriate local insertions are “strange charged operators” constructed from the interface algebra, or boundary tube algebra, between the two phases. These strange correlators exhibit long-range order when evaluated between two distinct NISPTs and decay exponentially otherwise. The corresponding string order parameters are built from truncated symmetry operators decorated by endpoint charges determined by the NISPT action tensors (Lu et al., 1 May 2025).

These applications show that generalized strange correlators are strongly representation-dependent. The relevant operator content is global-symmetry charged in ordinary SPTs, directionally constrained in SSPTs, fermionic or bosonic according to supercohomology data in FSPTs, and interface-algebra valued in NISPTs.

5. Mixed states, average symmetries, and open-system extensions

Mixed-state generalizations replace wave-function overlaps by fidelity between density matrices. For average symmetry-protected topological phases, the fidelity strange correlator is defined as

$1d$9

where $2d$0. When $2d$1 and $2d$2 are pure, this reduces to the ordinary strange correlator up to absolute values. In $2d$3 and $2d$4 nontrivial ASPT phases, the fidelity strange correlator is long-range or power-law rather than exponential (Zhang et al., 2022).

In the decorated-domain-wall basis, the fidelity strange correlator becomes a weighted average of pure-state strange correlators. For $2d$5 bosonic and fermionic ASPTs, these weights map to $2d$6 loop models with quantum-corrected loop fugacities. Cluster-state decoration yields loop fugacity $2d$7, Kitaev-chain decoration yields $2d$8, and the strange correlators become watermelon correlators with exact exponents. For example, in the dense phase of the $2d$9 loop model, the $|\Psi\rangle$0-charged fidelity strange correlator decays as $|\Psi\rangle$1; in the $|\Psi\rangle$2 average cluster state, the even-site correlator satisfies $|\Psi\rangle$3 for all separations (Zhang et al., 2022).

Replica-space strange correlators also extend to mixed-state SPTs through Choi doubling and “surgery operators.” If $|\Psi\rangle$4 and $|\Psi\rangle$5 denote vertical and transverse surgery operators, then the lower-dimensional mixed state $|\Psi\rangle$6 supports a twisted Rényi-$|\Psi\rangle$7 correlator

$|\Psi\rangle$8

which is identified with the strange correlator of a fixed-point SPT constructed from the mixed state. This yields a mixed-state topological diagnostic in precisely the same replica-direction sense as in the pure-state entanglement-holography construction (Sala et al., 11 Jun 2025).

These developments make clear that generalized strange correlators survive the transition from pure states to noisy, disordered, open, and average-symmetry settings, provided the overlap structure is replaced by a fidelity or replica-space functional that preserves the interface interpretation.

6. Operator choice, spurious signals, and domain of validity

The effectiveness of a strange correlator depends sharply on operator choice. A systematic procedure is to use bulk-boundary correspondence: identify the gapless edge operator of the phase under study, then choose bulk operators whose strange correlator maps onto that edge correlator. In noninteracting topological insulators and superconductors this leads to particle–hole or Bogoliubov operators tied to edge modes, while in interacting or fractional systems it points to vertex operators, quasiparticle fields, or other boundary primaries (Lepori et al., 2022).

For finite-size analysis, integrated strange correlators are especially useful. The sums

$|\Psi\rangle$9

amplify algebraic tails. If $|\Omega\rangle$0, then

$|\Omega\rangle$1

The modulus-integrated form substantially reduces cancellations and finite-size effects, especially in fermionic systems where oscillations can make the unsigned sum misleading (Lepori et al., 2022).

At the same time, generalized strange correlators are not foolproof. Ill-chosen reference states can induce spurious long-range strange correlators even in trivial SPT phases. In matrix-product-state language, the origin is magnitude-degeneracy of the overlap transfer matrix, and three distinct mechanisms are identified: high-dimensional irreducible representations in the entanglement space, phase mismatch in symmetry representations between target and reference states, and long-range order from symmetry breaking (Gao et al., 7 Dec 2025).

There are also substantive limits. In $|\Omega\rangle$2, a nontrivial SPT can have a short-ranged strange correlator if its boundary is topologically ordered rather than gapless, so a short-ranged result does not universally imply triviality (Wierschem et al., 2014). In the toric code, local two-point strange correlators do not provide a useful diagnostic because gauge invariance trivializes the local construction; more general string-operator or MPO-based observables are required (Lepori et al., 2022). In long-range Kitaev chains with $|\Omega\rangle$3, strange-correlator scaling becomes sub-linear and the standard gapless-edge interpretation breaks down (Lepori et al., 2022).

This suggests that “generalized strange correlator” is best understood as a construction principle rather than a single formula. The principle is to convert topological or anomalous bulk information into an overlap, transfer-matrix, replica-space, or nonstandard four-point object whose asymptotics are controlled by the effective boundary or interface theory. Within that principle, the choice of reference state, symmetry implementation, and operator algebra is decisive.