More on the spectral localizer

 At the end of last year Cerjan and Loring put on arXiv a very nice paper studying topological invariants for characterizing metallic systems, based on the properties of a spectral localizer (my summary here). This work was published in Physical Review B a few months ago and was shortly followed by an experimental demonstration of the concept using acoustics in an arXiv preprint, "Revealing Topology in Metals using Experimental Protocols Inspired by K-Theory."

The authors consider an acoustic analogue of the Su-Schrieffer-Heeger model, in which dimerized couplings between nominally-identical resonators create a lattice with a topological band gap. To make the system gapless (metallic), the lattice is coupled to a second gapless array with uniform couplings, forming a two-leg ladder. Because the combined system is gapless, regular topological invariants (based on the existence of an energy gap) cannot be used to predict the existence of robust interface states.

By regularizing the position operator appearing in the spectral localizer (converting the unbounded $X-x_0 I$ to $\tanh ( [X-x_0 I]/\alpha )$, the localizer can be re-interpreted as the Hamiltonian of a related system formed by a suitable modulation of the individual resonators' frequencies; positions $x_0$ where the gap of the spectral localizer vanishes host robust localized modes. The gap closing points are then detected experimentally by measuring the local density of states of the related system described by the localizer Hamiltonian.

One limitation of this approach is that it is based on measuring the properties of a different (but related) system; disorder in the implementation or limitations to the classes of Hamiltonians that can be realized in a particular platform may mean one is unable to faithfully reproduce the localizer of the system of interest. It is therefore interesting to consider how the spectral localizer may be directly measured without construct a separate system, enabling in-situ determination of the topological properties of metallic systems.

One way this might be possible is using nonlinear optical processes such as parametric amplification. To see this, consider the example of one- or two-dimensional lattices with $\textbf{x} = (x,y)$ and the localizer operator

$$ L_{\kappa}(x,y,\omega) = \kappa [(X - x I) \sigma_x + (Y - y I) \sigma_y] + (H - \omega I) \sigma_z, $$

where $\sigma_{x,y,z}$ are Pauli operators, $H$ is the lattice Hamiltonian of interest, and $\kappa$ is a scaling parameter ensuring that the spatial $(x,y)$ and energy $\omega$ scales are comparable. Written in this form, the spectral localizer $L$ resembles a Hamiltonian describing parametric amplification in an inhomogeneously-pumped waveguide lattice [see e.g. Phys. Rev. A 99, 063834 (2019), Nat. Commun. 7, 10779 (2016), or Phys. Rev. X 6, 041026 (2016)].

Specifically, assume the lattice is subject to a two-photon driving term with frequency $\omega_p$ and a site-dependent amplitude described by a diagonal matrix $G$. The two photon drive generates signal and idler photons whose field operators at the $n$th lattice site are $\hat{a}_n$ and $\hat{b}_n$, respectively. Conservation of energy requires their frequencies to satisfy $\omega_a + \omega_b = \omega_p$. For near-degenerate signal and idler photons $\omega_a \approx \omega_b \approx \omega_p/2 \equiv \omega$. The full Hamiltonian describing the parametric amplification is, in a frame rotating at frequency $\omega$,
$$\hat{\mathcal{H}} = \boldsymbol{\hat{a}^{\dagger}} ( H-\omega I ) \boldsymbol{\hat{a}} +  \boldsymbol{\hat{b}^{\dagger}} ( H-\omega I ) \boldsymbol{\hat{b}} + \frac{i}{2} \left(\boldsymbol{\hat{a}^{\dagger}} G \boldsymbol{\hat{b}^{\dagger}} -  \boldsymbol{\hat{a}} G^{\dagger} \boldsymbol{\hat{b}} \right),$$
where $\boldsymbol{\hat{a}} = (\hat{a}_1,...,\hat{a}_N)$ and $\boldsymbol{\hat{b}} = (\hat{b}_1,...,\hat{b}_N)$. The Heisenberg equations of motion $i \partial_t \hat{O} = -[\mathcal{\hat{H}},\hat{O}]$ for the creation and annihilation operators reduce to
$$ i \partial_t \left(\begin{array}{c} \boldsymbol{\hat{a}} \\ \boldsymbol{\hat{b}^{\dagger}} \end{array} \right) = \sigma_z \left( \begin{array}{cc} H - \omega I & iG \\ -iG^{\dagger} & H^*-\omega I \end{array} \right) \left(\begin{array}{c} \boldsymbol{\hat{a}} \\ \boldsymbol{\hat{b}^{\dagger}} \end{array} \right).$$
Assuming a pump profile $G = \gamma [(X - xI) - i (Y - yI)]$, a time reversal-symmetric lattice $H=H^*$, and introducing the field vector $\hat{\psi} = (\hat{a}_1,...\hat{a}_N,\hat{b}_1^{\dagger},...,\hat{b}_N^{\dagger})$, the Heisenberg equations of motion reduce to
$$ i \partial_t \hat{\psi} = L_{i\gamma}(x,y,\omega) \hat{\psi}, $$
corresponding to a spectral localizer with a purely imaginary spatial scale parameter $i\gamma$. The spectrum $\lambda$ of $L_{i\gamma}$ yields the down-converted photon frequencies $\omega_a = \omega + \mathrm{Re}[\lambda]$, $\omega_b = \omega - \mathrm{Re}[\lambda]$, and their gain $\mathrm{Im}[\lambda]$. Thus, properties of the Clifford pseudospectrum may be probed by parametrically driving the lattice of interest, at least for the case of time reversal-symmetric Hamiltonians $H$.

One important difference, however, is that the scale parameter $\kappa$ is purely imaginary under parametric driving. Hence, the spectrum of $L$ comprises purely real eigenvalues (driving weak compared to the detuning) and complex eigenvalues (strong driving). Space and energy are thus treated on an unequal footing, in contrast to the standard Hermitian localizer.

Nevertheless, we can still infer some of the properties of $H$ from the non-Hermitian localizer. Assuming the parametric driving is weak compared to the level spacing of $H$ (which is a reasonable assumption, given that nonlinear optical effects are typically weak), only mode pairs $|\phi_j>$, $|\phi_k>$ of $H$ with resonant energies $\omega_j - \omega = \omega - \omega_k = \nu$ will experience strong amplification. Expanding $L$ in the eigenbasis of $H$ and for compactness defining $Z = X - i Y$ and $z = x-iy$,
$$ L_{i\gamma}(z,\omega) = \left(\begin{array}{cc} \nu & i\gamma (Z_{kj} - z \delta_{kj}) \\ i\gamma (Z_{jk}^* - z^* \delta_{kj}) & \nu \end{array}\right), $$
where $Z_{kj} = <\phi_k| Z | \phi_j>$ and $\delta_{kj}$ is the Kronecker delta function, which appears because different eigenstates of $H$ are orthogonal.

In the case of degenerate down-conversion ($j=k$) the eigenvalues of $L$ reduce to $\lambda = \pm i \gamma |z - <\phi_j|Z|\phi_j>|$. Thus, the gain vanishes when the pump is centred on the resonant mode's centre of mass; degenerate down-conversion will be suppressed, allowing one to infer the mode's frequency and position, and thereby reconstruct the localizer.

More generally, for non-degenerate down-conversion with $j \neq k$, $\lambda = \nu \pm i \gamma <\phi_k|Z|\phi_j>$ and one will detect signal and idler photons with energies $\omega \pm \nu$. The gain is determined by the cross-coupling between the two modes. Recalling $x \sim \partial_k$, this might be sensitive to the Berry connection and/or curvature of near-degenerate modes of $H$, or whatever the real space analogue of these quantities is.

Note that the localizer is most useful for finite disordered systems (since under periodic boundary conditions one can define invariants more easily in terms of the Bloch Hamiltonian). This means it's safe to assume the spectrum of $H$ is non-degenerate.

I'm not sure how well this approach will generalize to other classes of lattices, such as Chern insulators in which time-reversal symmetry must be broken. A similar derivation will hold for $\chi^{(3)}$ nonlinear processes such as spontaneous four wave mixing and third harmonic generation [see e.g. Opt. Exp. 20, 27441 (2012), Nature Photon. 15, 542 (2021), and Phys. Rev. Lett. 123, 103901 (2019)]. Another important problem is to generalize this approach from the simple tight binding approximation to continuous waveguides and photonic crystals, where spatial and spectral correlations of the pump beam affect the efficiency of the down-conversion process.