An exactly solvable one-dimensional spin chain that interpolates between the transverse-field Ising model and the cluster Ising model (Yu and Li, Phys. Rev. B 110, 045119 (2024)) established that tuning between topologically distinct gapless states produces a non-conformal Lifshitz multicritical point with dynamical exponent z=2z=2 between Ising universality classes distinguished by a winding number, and that fidelity susceptibility can map this 'transition of phase transitions'. In their Conclusion and Outlook the authors explicitly left open two linked directions: whether such transitions between topologically distinct quantum critical points occur in higher spatial dimensions and for other symmetry groups such as Z3\mathbb{Z}_3 or U(1)U(1); and what the finite-temperature phase diagram of the interpolating family looks like. Their entanglement section (Sec. III.2.3) raises a third linked question — the quantitative anomalous entanglement-entropy scaling and universal finite-size amplitudes at the z=2z=2 Lifshitz multicriticality of this Hamiltonian, described there as a promising avenue for future research — and their introduction poses a fourth: whether fidelity susceptibility can effectively detect the quantum critical and scaling behaviors in gapless-gapless phase transitions between topologically distinct universality classes. The paper answers this last question for its own exactly solvable Z2\mathbb{Z}_2 chain, leaving open whether the diagnostic remains reliable for such transitions beyond that single solved model.

Background

A continuous quantum phase transition (QPT) is usually a transition between a gapped ordered phase and a gapped disordered phase, described near criticality by a conformal field theory (CFT) with universal critical exponents. Yu and Li (Phys. Rev. B 110, 045119 (2024)) studied a different situation: a transition between two gapless states that carry the same local critical exponents but differ in topology. The vehicle is a spin-1/21/2 chain combining the transverse-field Ising (TFI) Hamiltonian with a cluster interaction, H=λHTFI+(1−λ)HCIH=\lambda H_{\mathrm{TFI}}+(1-\lambda)H_{\mathrm{CI}} with HTFI=−∑jσjxσj+1x−h∑jσjzH_{\mathrm{TFI}}=-\sum_j\sigma^x_j\sigma^x_{j+1}-h\sum_j\sigma^z_j and HCI=−∑jσjxσj+1x+h∑j=1N−2σjxσj+1zσj+2xH_{\mathrm{CI}}=-\sum_j\sigma^x_j\sigma^x_{j+1}+h\sum_{j=1}^{N-2}\sigma^x_j\sigma^z_{j+1}\sigma^x_{j+2}, where σjx,y,z\sigma^{x,y,z}_j are Pauli matrices and λ\lambda and hh are tuning parameters. The three-spin cluster term stabilizes a symmetry-protected topological (SPT) phase: a gapped phase that is trivial as an intrinsically correlated state but becomes distinct when a symmetry (here Z2\mathbb{Z}_2 spin flip and time reversal Z2T\mathbb{Z}^T_2) is enforced, and that hosts protected edge modes under open boundaries. The model is solvable by the Jordan-Wigner transformation, which maps spins to free fermions and allows exact computation of the spectrum, correlation functions, and entanglement. At h=1h=1 the critical line consists of Ising critical points (central charge c=1/2c=1/2), but these fall into classes distinguished by a winding number — a topological invariant of the free-fermion (Majorana) Hamiltonian, given by the winding of the vector (yk,zk)(y_k,z_k) of the Bogoliubov-de Gennes form as kk sweeps the Brillouin zone, which takes distinct half-integer values (1/21/2 and 3/23/2) on the two topologically distinct Ising critical lines and integer values (00 and 22) in the gapped trivial-paramagnetic and cluster-SPT phases; critical points with different winding numbers are 'symmetry-enriched' or 'gapless-SPT-type' (gSPT) critical points and cannot be connected without passing a multicritical point. At λ=1/2\lambda=1/2 on this line the two topological classes meet at a Lifshitz multicritical point: a transition whose gap closes quadratically, so the dynamical exponent (relating energy to momentum, E∼kzE\sim k^{z}) is z=2z=2 instead of the Ising value z=1z=1, and which is therefore not conformally invariant. For h>1h>1 the model shows transitions between the cluster SPT phase and a trivial paramagnet described by a c=1c=1 Gaussian (free-boson) CFT, while for h<1h<1 the chain is ferromagnetically ordered for all λ\lambda. The authors mapped this phase diagram using fidelity susceptibility — χF(λ)=lim⁡δλ→02[1−F(λ,λ+δλ)]/(δλ)2\chi_F(\lambda)=\lim_{\delta\lambda\to 0}2[1-F(\lambda,\lambda+\delta\lambda)]/(\delta\lambda)^2, the leading response of the ground-state overlap F(λ,λ+δλ)=∣⟨ψ0(λ)∣ψ0(λ+δλ)⟩∣F(\lambda,\lambda+\delta\lambda)=|\langle\psi_0(\lambda)|\psi_0(\lambda+\delta\lambda)\rangle| to a change of the tuning parameter, which diverges at quantum critical points and obeys the finite-size scaling form N−dχF(λ)=N2/ν−dfχF(N1/ν∣λ−λc∣)N^{-d}\chi_F(\lambda)=N^{2/\nu-d}f_{\chi_F}(N^{1/\nu}|\lambda-\lambda_c|) with system size NN (ν\nu the correlation-length exponent, dd the spatial dimension) — and extracted z=2z=2, ν=1\nu=1 at the Lifshitz point. Entanglement entropy, the von Neumann entropy SAS_A of a subsystem reduced from the pure ground state, scales at conformal criticality as SA∼(c/3)ln⁡NS_A\sim (c/3)\ln N plus a constant; at z=2z=2 Lifshitz criticality this scaling is known to be anomalous with universal finite-size amplitudes in other topological chains (K. Wang and T. Sedrakyan, SciPost Phys. 12, 134 (2022); K. Wang, Phys. Rev. B 108, L081112 (2023)), but was not computed for the present Hamiltonian. The paper's Conclusion and Outlook states the future questions: critical behavior between topologically distinct critical points in higher dimensions and within different symmetry groups (e.g., Z3\mathbb{Z}_3, U(1)U(1), among others), and finite-temperature phase diagrams; the entanglement thread is raised in their entanglement section (Sec. III.2.3), where the anomalous z=2z=2 scaling and universal finite-size amplitudes are called 'a promising avenue for future research'; and the fidelity-susceptibility question is posed in their introduction, where the authors answer it for their exactly solvable model.

Problem Statement

Establish whether the transition between topologically distinct quantum critical points — the gapless-gapless 'transition of phase transitions' demonstrated in the one-dimensional, Z2\mathbb{Z}_2-symmetric, exactly solvable interpolating chain H=λHTFI+(1−λ)HCIH=\lambda H_{\mathrm{TFI}}+(1-\lambda)H_{\mathrm{CI}} of Yu and Li (Phys. Rev. B 110, 045119 (2024)), whose paradigm object is the z=2z=2 Lifshitz multicritical point at (λ,h)=(1/2,1)(\lambda,h)=(1/2,1) between Ising critical classes of different winding number (the winding number being the topological invariant of the free-fermion, Majorana Hamiltonian that distinguishes the critical classes) — generalizes beyond that single solved model, and whether the diagnostics used there remain valid. Concretely, the future directions stated by the source authors are: (i) construct and analyze critical behavior between topologically distinct critical points in higher spatial dimensions and within other symmetry groups (for example Z3\mathbb{Z}_3 or U(1)U(1)), including the classification of the relevant topologically distinct critical points and the nature of possible multicriticalities; (ii) construct the finite-temperature phase diagram of the interpolating family, determining how the zero-temperature topological and multicritical distinctions manifest at nonzero temperature; and (iii) work out quantitatively the entanglement-entropy scaling at the z=2z=2 Lifshitz multicriticality of this Hamiltonian, including any anomalous scaling form and universal finite-size amplitudes. Their introduction poses, and answers only for the exactly solvable Z2\mathbb{Z}_2 chain, the further question (iv) whether fidelity susceptibility (the leading response of the ground-state overlap F(λ,λ+δλ)=∣⟨ψ0(λ)∣ψ0(λ+δλ)⟩∣F(\lambda,\lambda+\delta\lambda)=|\langle\psi_0(\lambda)|\psi_0(\lambda+\delta\lambda)\rangle| to a change of the tuning parameter, χF(λ)=lim⁡δλ→02[1−F(λ,λ+δλ)]/(δλ)2\chi_F(\lambda)=\lim_{\delta\lambda\to 0}2[1-F(\lambda,\lambda+\delta\lambda)]/(\delta\lambda)^2, which diverges at quantum critical points) effectively detects and correctly scales gapless-gapless transitions that change topological invariants beyond that single solved model. Items (i)-(iv) share one scientific objective — the generality and diagnostic accessibility of transitions between topologically distinct gapless states. Each of (i)-(iv) is an independently attackable sub-question, and a rigorous resolution of any single one of them — not all four jointly — is what the task requires for completion; item (i) is resolved by a rigorous construction or analysis in any single one of the settings it names — one higher spatial dimension, or one of the symmetry groups Z3\mathbb{Z}_3 or U(1)U(1) (these settings are illustrative, not an exhaustive set of requirements).

References

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