Spectral decomposition and high-accuracy Greens functions: Overcoming the Nyquist-Shannon limit via complex-time Krylov expansion

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green’s-functions, is a long-standing problem posing enormous challenges to numerical methods. The paper demonstrates how augmenting the time series with complex-time Krylov states can overcome the Nyquist-Shannon limit and significantly improve the frequency resolution of Green’s functions, even for two-dimensional systems

How it works

The method addresses the limitation imposed by Fourier transforming a time series, where the Nyquist-Shannon theorem limits the frequency resolution according to ∆ω = 2π/T. This is overcome by augmenting the time series with complex-time Krylov states. The general frequency-dependent Green’s function GAB(ω) is regularized by introducing a finite broadening η > 0.

The core of the approach involves recasting the time integration in Eq. (2) by exploiting the group-property of Uˆ(t, ω) and splitting the integration domain into smaller intervals of size T. This leads to defining an operator Sˆ(T, ω) = P∞ p=0 h Uˆ(T, ω + iη)ip, which boosts the evolution on the finite interval [0, T) to all intervals [pT,(p + 1)T), resulting in GAB(ω) = Z T 0 dt⟨φAUˆ(t, ω)Sˆ(T, ω)φB⟩.

Decomposition and Correction Term

The operator Sˆ(T, ω) can be decomposed into the zeroth-order part S˜TAB and a correction KTAB = G˜TAB + KTAB. The key observation is that KTAB accounts for the low-frequency part of GAB. Setting Sˆ ≡ 1 introduces a low-pass filter on the time domain, meaning the Green’s function in frequency space is fully determined from equally spaced samples G˜TAB(ωk). Information about the dependency on smaller frequencies ω ∈ [0, ∆ω] must be encoded in KTAB.

Complex-Time Krylov Space Construction

To evaluate KTAB efficiently using Tensor Networks (TNs), the paper introduces a complex-time Krylov space KDτµ generated by successively evolving the initial state φµ⟩ with step size τ along a complex contour of segments τ = (1−i tan(α))δt. The time-evolution operator for each segment is decomposed as Uˆτ = e−Hδt ˆ tan(α)e−iHδt ˆ, and the Krylov space HDτµ is spanned by normalized states that exhibit a large overlap with the low-energy eigenstates of Hˆ.

The action of the boost operator in this subspace is approximated as GAB(ω) ≈ Z T 0 dt⟨φAUˆ(t, ω + iη)Xr ij=1ψ iB⟩⟨ψ iBSˆ(T, ω + iη)ψ jB⟩⟨ψ jBφB>. This is simplified to Eq. (A.12), which expresses the full Green’s function as GAB(ω) = Z T 0 dt⟨φAUˆ(t, ω + iη)φB⟩ + Z T 0 dt YAB(t, ω + iη)XQ e−i(Eeff−E0−ω−iη)T Q†X†M⃗ 0.

Error Analysis and Convergence

The error analysis involves bounding the approximation using Arnoldi iterations. The error bound for the action of the boost operator is given by Eq. (B.22), which contains an explicit dependency on the complex angle as well as the Arnoldi residual. The dominant contribution D> of leakage into the high-energy subspace is exponentially suppressed in dimension r owing to the finite complex angle α.

The final error bound for the augmented Green’s function is given by Eq. (B.28), which shows that δGKAB(ω + iη) ≲ 2δtη squared e r(1+ln(δtλ˜)-tan(α)δtλ˜))√2πr + C>D>. This bound demonstrates that the complex angle increases the low-energy range of exponential convergence and introduces a second, high-energy regime in which the application of the boost operator is again faithfully captured.

Model Applications

The method was tested on two models. For the critical S−1/2 Heisenberg chain, incorporating KTAB(ω) eliminates artificial oscillations and shows that low-frequency behavior is well captured even for large frequencies. In the two-dimensional Su-Schrieffer-Heeger (SSH) model, the corrected Green’s functions exhibit a distinguished peak right at the ground-state energy ω = E2e0, which is completely washed out in uncorrected simulations.

For isolated two-electron bound states (bipolarons) in the Peierls SSH model, incorporating KTAB(ω) reveals a distinct peak structure indicating dressed two-electron states. This allows the deduction that for weak coupling λ = 0.025 there is no bipolaronic ground state, only unbound two-polaron states with w > 2E1e0. In the strong coupling regime λ = 0.2, the absence of a feature at E2e0 for any considered electron displacements indicates that trial states do not contribute significantly to the two-electron ground state.

Discussion and Outlook

The complex-time Krylov space correction allows for arbitrary frequency resolution, overcoming the Nyquist-Shannon sampling limit. The method is expected to be improved by incorporating entanglement growth along the contour and by using direct sums of Krylov subspaces constructed from different complex contours. The convergence analysis suggests that increasing the frequency of complex-time Krylov space steps, rather than reaching large final evolution times T, is beneficial.

The paper concludes that for practical purposes, evaluating the Arnoldi residual should be preferred to decide whether convergence is sufficient. The derived error bounds provide a threshold to discriminate spectral signals from errors such as replica features. The analysis confirms that exponential convergence independent on the complex angle α occurs as long as δtλ <˜ 1/e, which can help analyze Krylov-space based methods for state preparation on quantum computers.

--- Page 1 ---

The gist

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green’s-functions, is a long-standing problem posing enormous challenges to numerical methods.

Decomposition and Correction Term

The core of the approach involves recasting the time integration in Eq. (2) by exploiting the group-property of Uˆ(t, ω) and splitting the integration domain into smaller intervals of size T. This leads to defining an operator Sˆ(T, ω) = P∞ p=0 h Uˆ(T, ω + iη)ip, which boosts the evolution on the finite interval [0, T) to all intervals [pT,(p + 1)T), resulting in GAB(ω) = Z T 0 dt⟨φAUˆ(t, ω)Sˆ(T, ω)φB⟩.

Model Applications

The method was tested on two models.

1. Overcoming Nyquist-Shannon limit for spectral decomposition: The AI system can compute low-energy spectra of strongly correlated quantum many-body systems with significantly improved frequency resolution by augmenting time series with complex-time Krylov states, as shown by the enormous improvements in accuracy demonstrated for the critical S - 1/2 Heisenberg model.

2. Enhanced computation for two dimensional systems: The system can now compute Green’s functions even for two-dimensional models, overcoming limitations where tensor network methods were previously restricted to one-dimensional systems with small quasi-particle velocities.

3. Precise spectral analysis of bipolarons: The AI system can accurately determine the spectral properties of light bipolarons in the two-dimensional Su-Schrieffer-Heeger model by incorporating the correction term KT AB, which reveals the distinct peak structure indicating dressed two-electron states that was previously washed out.

4. Improved convergence for critical systems: By using complex angles alpha > 0, the system can achieve exponential convergence towards exact Green’s functions even for critical systems like the S - 1/2 Heisenberg chain, which are notoriously challenging for standard Krylov expansion schemes.

5. Accurate determination of ground state energies: The system can distinguish between a bipolaronic ground state and unbound two-polaron states by observing that the lowest eigenvalue is already well-converged towards the two-polaron energy consistent with our findings of the absence of a bipolaron signal.