Krylov complexity and fidelity susceptibility in two-band Hamiltonians

Abstract: We investigate Krylov spread complexity for the ground state of two-band Hamiltonians, where the reference state is a generic state on the Bloch sphere. The spread complexity is obtained by using a purely geometric formulation in terms of Bloch sphere data without constructing the circuit Hamiltonian. For generic reference states, the derivative of the spread complexity is logarithmically divergent at the topological phase transition in the Su-Schrieffer-Heeger (SSH) model. We demonstrate that the derivative of the spread complexity is bounded by fidelity susceptibility for general two-band models, indicating the sensitivity of the spread complexity to any gap closing (topological or trivial). This is illustrated in the massive Dirac Hamiltonian with a trivial gap closing. Finally, we introduce a non-unitary duality in the SSH model between the topological and trivial phases, which manifests itself in the spread complexity and fidelity susceptibility.

• The paper introduces a geometric formalism for Krylov spread complexity using Bloch sphere overlaps to signal quantum phase transitions.
• It establishes a quantitative bound by linking the derivative of complexity to fidelity susceptibility, validated on SSH and massive Dirac models.
• The work extends the framework to non-Hermitian systems, uncovering duality and universal behavior across topological and trivial phase changes.

Krylov Complexity and Fidelity Susceptibility in Two-Band Hamiltonians

Overview and Motivation

This work "Krylov complexity and fidelity susceptibility in two-band Hamiltonians" (2605.18594) presents a rigorous geometric treatment of Krylov spread complexity for ground states in two-band quantum systems, linking it quantitatively to fidelity susceptibility and demonstrating its sensitivity to quantum phase transitions, both topological and trivial. The analysis circumvents the ambiguities and technical obstacles in complexity evaluation by exploiting Bloch sphere geometry, thus avoiding explicit circuit Hamiltonian construction. The implications span condensed matter diagnostics, complexity theory in quantum information, and fundamental studies of quantum geometry across criticality.

Geometric Formalism for Krylov Spread Complexity

In two-band models, the Krylov spread complexity quantifies the minimal average "distance" (in terms of basis expansion) between a reference and a target state, typically the ground state. The geometric formalism takes the reference state as a generic point on the Bloch sphere, parametrized by $(\theta, \phi)$, and the target as a product across momenta of ground states with corresponding Bloch vectors determined by the Hamiltonian. The core result is an analytical expression for the complexity per mode:

$$C_k = \frac{1 - \hat{n}_{\text{ref}}(k) \cdot \hat{n}_{\text{target}}(k)}{2}$$

This circumvents the non-uniqueness and construction difficulties of circuit Hamiltonians, enabling tractable evaluation for any two-band system. The reference state dependence is manifest, influencing both the magnitude and qualitative features of the complexity.

SSH Model: Topological Transition and Complexity Behavior

The Su-Schrieffer-Heeger (SSH) model is employed as the archetypal example of topological phase transitions in one dimension, characterized by gap reopening and associated winding number change. The geometric complexity formulation, integrated over the Brillouin zone, reveals that for generic reference states, the derivative of complexity exhibits a logarithmic divergence at the critical point $t_1 = t_2$, while the absolute complexity can exhibit plateaus or constants depending on the reference (momentum-dependent vs momentum-independent) state choice.

Figure 1: Schematic of the SSH model with two fermion species per site, highlighting the intracell (t₁) and intercell (t₂) hopping structure.

Figure 2: Krylov complexity for the SSH ground state under various choices of reference state, showing plateau and divergence features at the topological transition.

This non-analyticity is robust against target state changes; non-ground-state (excited) targets display analogous singular features in Krylov complexity whenever the band assignment aligns with the gap-closing modes.

Figure 3: Spread complexity and its derivative for piecewise eigenstate targets of the SSH model; singular behavior coincides with gap closing.

Fidelity Susceptibility as a General Bounds and Criticality Probe

Fidelity susceptibility, a well-established phase transition probe, quantifies the sensitivity of the ground state to infinitesimal changes in tunable parameters. The authors derive a general inequality linking the derivative of Krylov complexity to fidelity susceptibility components:

$$|\partial_\lambda C(\lambda)| \leq 4\pi\sum_{i=1}^3|\mathcal{Q}_i|\sqrt{\chi_F^i(\lambda)}$$

Any divergence in complexity's derivative is thus matched or exceeded by a divergence in fidelity susceptibility. This establishes complexity as a detector of gap closings, not limited to topology-changing transitions.

Massive Dirac Hamiltonian: Trivial Gap Closings

The massive Dirac Hamiltonian, structurally similar to SSH but lacking topological phase transitions (winding number remains zero), demonstrates that complexity's derivative shows logarithmic divergence at gap closings (chemical potential $\mu=0$) without topological change. Fidelity susceptibility diverges even more strongly (power-law), confirming the theoretical bound.

Figure 4: Tight-binding chain representing the massive Dirac Hamiltonian; gap closing is not associated with a topological transition.

Figure 5: Cooper pair box realization of massive Dirac physics, mapping external flux to momentum and voltage to the staggered potential.

Duality in SSH Model and Complexity Mapping

A non-unitary duality is uncovered in SSH Hamiltonian parametrizations, relating the topological and trivial phases by $r \leftrightarrow 1/r$ transformations in coupling ratios. This duality is mirrored in both the complexity and fidelity susceptibility, enabling computation in one phase to yield results in the dual phase by explicit analytic relations.

Figure 6: Dual SSH models under intracell coupling $t$ and intercell couplings $-rt$ and $-t/r$, illustrating duality mapping.

The ratio

$$R(\lambda) = \frac{|\partial_\lambda C^{(i)}(\lambda)|}{4\pi|\mathcal{Q}_i|\sqrt{\chi_F^i(\lambda)}}$$

is shown to saturate to $$C_k = \frac{1 - \hat{n}_{\text{ref}}(k) \cdot \hat{n}_{\text{target}}(k)}{2}$$0 at large $$C_k = \frac{1 - \hat{n}_{\text{ref}}(k) \cdot \hat{n}_{\text{target}}(k)}{2}$$1 for both SSH and massive Dirac systems, indicating structural universality for a broad class of two-band Hamiltonians.

Figure 7: Ratio of nonzero component of derivative of complexity to fidelity susceptibility; minimum at critical point, saturation to $$C_k = \frac{1 - \hat{n}_{\text{ref}}(k) \cdot \hat{n}_{\text{target}}(k)}{2}$$2 at large parameter.

Non-Hermitian Extensions

The developed machinery is extended to the non-Hermitian SSH model under periodic boundary conditions using biorthogonal Krylov constructions. Complexity retains sensitivity to gap closing points, with non-analytic behavior at critical parameter values governed by $$C_k = \frac{1 - \hat{n}_{\text{ref}}(k) \cdot \hat{n}_{\text{target}}(k)}{2}$$3, the non-Hermitian parameter. The authors note that open boundary conditions alter the criticality landscape, requiring more intricate numerics.

Figure 8: Krylov complexity for the non-Hermitian SSH model; non-analytic cusps at gap closing under periodic boundary conditions.

Practical and Theoretical Implications

The geometric formalism for complexity provides a robust, reference-state-dependent diagnostic for gap closings in quantum systems. It unifies complexity and fidelity susceptibility, offering quantitative bounds and universality. The explicit mapping between trivial and topological phases via duality and the extension to non-Hermitian realms opens new probes for quantum criticality. These tools are relevant in quantum simulation, condensed matter diagnostics, and studies of non-equilibrium quantum dynamics. Future directions include generalization to higher dimensions, exploration of lower bounds, and resolving complexity behavior for open-system, non-Hermitian models.

Conclusion

The paper establishes Krylov spread complexity, computed purely from Bloch sphere overlaps, as a sensitive, quantitatively bounded indicator of quantum criticality in two-band Hamiltonians. Its behavior and bounds are governed by fidelity susceptibility, with universality in singularity structure and dual mappings across topological phase transitions. The results apply to both Hermitian and non-Hermitian models, reinforcing the geometric content of complexity and its integration into condensed matter and quantum information frameworks.