The Baxter–Fendley Zn\mathbb{Z}_n free-parafermion chain is exactly solvable for arbitrary inhomogeneous couplings, and the 2024 analysis of its spatially random ("dirty") version showed that its thermodynamic singularities — the ground-state energy and boundary magnetization — are, independent of nn, identical to those of the random transverse-field Ising chain and are governed by infinite-randomness critical points. Correlation functions, however, are out of reach: even for the homogeneous chain only a few ground-state expectation values — obtained from the Hellmann–Feynman theorem, which identifies the derivative of the ground-state energy with respect to a parameter with the ground-state expectation value of the differentiated operator — are known, and for the disordered chain no exact method exists for correlation functions or their disorder distributions — the authors state they "do not see a way to attack exactly" this problem. The problem posed here is to compute exactly the correlation functions of local clock operators in the random free-parafermion chain, together with their distributions over the disorder and their critical scaling, exploiting or extending the fact that the strong-disorder RG decimation rule is identical, decimation by decimation, to that of the random transverse-field Ising chain, where the analogous distributions were obtained by Fisher and Young.

Background

Consider a one-dimensional chain of length LL with an nn-dimensional Hilbert space on each site, carrying the clock operators Z=diag(1,ω,…,ωn−1)Z=\mathrm{diag}(1,\omega,\dots,\omega^{n-1}) with ω=e2πi/n\omega=e^{2\pi i/n}, and XX the cyclic shift, so that Zn=Xn=1Z^n=X^n=1 and XZ=ω ZXXZ=\omega\,ZX. Baxter's Hamiltonian H=−∑j=1L−1Jj Zj†Zj+1−∑j=1Lhj XjH=-\sum_{j=1}^{L-1}J_j\,Z_j^{\dagger}Z_{j+1}-\sum_{j=1}^{L}h_j\,X_j is a Zn\mathbb{Z}_n-symmetric generalization of the transverse-field Ising chain (TFIC), to which it reduces at n=2n=2; for n≥3n\ge 3 it is non-Hermitian (its spectrum is complex, though the state of interest has real energy). Baxter showed that this chain is exactly solvable for arbitrary inhomogeneous positive couplings, and Fendley interpreted the solution through "free parafermions": a parafermionic generalization of the Clifford (Jordan–Wigner) algebra in which the many-body energies are sums over independently occupied single-particle levels, E{nk}=−∑kωnkεkE_{\{n_k\}}=-\sum_k\omega^{n_k}\varepsilon_k with each nk∈{0,…,n−1}n_k\in\{0,\dots,n-1\}. The single-particle energies are εk=zk−1/n\varepsilon_k=z_k^{-1/n}, where zkz_k are the positive zeros of the polynomial generated by the recursion Pm(z)=Pm−1(z)−z bmnPm−2(z)P_m(z)=P_{m-1}(z)-z\,b_m^{n}P_{m-2}(z) with b≡(h1,J1,h2,…,hL)b\equiv(h_1,J_1,h_2,\dots,h_L).

Pandey and Cowsik (2024) exploited this inhomogeneous solvability for spatially random ("dirty") couplings. Writing the ground-state energy E0=−∑kεkE_0=-\sum_k\varepsilon_k as a contour integral over the ratio PM/PM−1P_M/P_{M-1}, they showed that the controlling random variable is the stationary law of a product of random 2×22\times 2 matrices — precisely the object studied by McCoy and Wu for the random-bond two-dimensional Ising model. They obtained exact results for the thermodynamic singularities of E0E_0 and for the disorder-averaged boundary magnetization (a boundary field HZL†\mathfrak{H} Z_L^{\dagger} preserves solvability), finding that all singularities are independent of nn and identical to those of the random TFIC: the transition at ln⁡h‾=ln⁡J‾\overline{\ln h}=\overline{\ln J} is an infinite-randomness critical point. (An infinite-randomness fixed point is a fixed point of the strong-disorder renormalization group (SDRG) — the real-space RG that at each step eliminates the site or bond with the largest local energy scale — at which the effective disorder strength diverges and the dynamics is activated, ln⁡τ∼Lψ\ln\tau\sim L^{\psi}, here with tunneling exponent ψ=1/2\psi=1/2.) They further showed that the SDRG decimation rule — replacing a strong bond JJ between weak fields h1,2h_{1,2} by an effective field h1h2/Jh_1h_2/J — is identical, decimation by decimation, to that of the random TFIC for every nn; related superuniversality of infinite-randomness fixed points appears for random quantum Potts and clock models (Senthil and Majumdar 1996) and at two-dimensional topological transitions.

What is missing is correlations. In the random TFIC (n=2n=2), Fisher's SDRG and the subsequent analysis of Fisher and Young yielded not only thermodynamics but also the distributions of energy gaps and of end-to-end spin correlations. For the free-parafermion chain nothing comparable exists: even for the homogeneous chain only a few ground-state expectation values are known — single-site and nearest-neighbour values obtained via the Hellmann–Feynman theorem (the exact identity equating the derivative of an eigenvalue with respect to a coupling with the expectation value of the differentiated operator in that eigenstate; Liu, Henry, Batchelor and Zhou 2019) — and general two-point functions of the clean chain are open. Moreover, Fendley's parafermionic raising and lowering operators are highly nonlinear in the Jordan–Wigner-type parafermions, so the determinant/Pfaffian machinery of free-fermion correlations (a Pfaffian, the square root of the determinant of an antisymmetric matrix, is what reduces free-fermion and Ising correlation functions to finite determinants) is unavailable. The authors state explicitly that they "do not see a way to attack exactly" the calculation of distributions of correlation functions in the disordered chain, while suggesting that the TFIC-identical SDRG should be exploited for this purpose.

Problem Statement

Compute, exactly, the correlation functions of local operators in the random-coupling Baxter–Fendley free-parafermion chain and their probability distributions over the disorder, in the thermodynamic limit at and on either side of the infinite-randomness critical point ln⁡h‾=ln⁡J‾\overline{\ln h}=\overline{\ln J}.

Concretely, for H=−∑j=1L−1JjZj†Zj+1−∑j=1LhjXjH=-\sum_{j=1}^{L-1}J_j Z_j^{\dagger}Z_{j+1}-\sum_{j=1}^{L}h_j X_j with independent positive random couplings (open boundary conditions, as required by the exact solution), the target quantities are two-point correlation functions of the natural local clock operators — for example order-parameter correlations of the ZjZ_j variables such as ⟨Zi†Zj⟩\langle Z_i^{\dagger}Z_j\rangle and their boundary analogues such as end-to-end correlations, or correlations of the XjX_j variables — in the low-energy (ground-state) regime: their typical and disorder-averaged scaling forms, the full distributions over the disorder, and the dependence (or independence) of these results on nn. Throughout, expectation values are biorthogonal ground-state expectation values ⟨Φ0∣O∣Ψ0⟩\langle\Phi_0|O|\Psi_0\rangle between the real-energy right ground eigenvector ∣Ψ0⟩|\Psi_0\rangle of HH and its biorthogonal partner ⟨Φ0∣\langle\Phi_0| (normalized so that ⟨Φ0∣Ψ0⟩=1\langle\Phi_0|\Psi_0\rangle=1); at n=2n=2 these coincide with ordinary Hermitian ground-state expectation values.

Any exact route is admissible: (a) strong-disorder RG, exploiting the fact that the decimation rule is identical, decimation by decimation, to that of the random transverse-field Ising chain, so that the Fisher–Young analysis should generalize once the Zn\mathbb{Z}_n operator algebra under decimation is controlled; (b) the free-parafermion operator algebra together with the polynomial recursion for the single-particle energies; or (c) stationary distributions of products of random 2×22\times 2 matrices with operator insertions, extending the technology that yielded the ground-state energy and the boundary magnetization.

The completion boundary is: exact, verifiable results — closed forms, exact integral or functional equations, or controlled exact-asymptotic scaling statements — for at least one physically central class of such correlators, together with their disorder distributions and critical scaling, with the nn-dependence characterized. Exact results for the clean chain alone, without the disorder distributions, would constitute substantial partial progress but would not complete the problem.

References

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