Interactive Poster: Superconducting Peierls Instability

01

THE HAMILTONIAN

The 2D extended Hubbard Model

The extended Hubbard model on a square-lattice supports both singlet and triplet order parameters. We focus on the parity-mixed, time-reversal-breaking \(s+d+ip\) state.^{[1][1] Nayak and Kumar (2018)S. Nayak and S. Kumar, J. Phys.: Condens. Matter 30, 135601 (2018).Open reference
[2][2] Hutchinson and Marsiglio (2021)J. Hutchinson and F. Marsiglio, J. Phys.: Condens. Matter 33, 065603 (2021).Open reference
[3][3] Senarath Yapa et al. (2025)P. Senarath Yapa, X. Guo, J. Maciejko and F. Marsiglio, Physica C 633, 1354719 (2025).Open reference}

EXTENDED HUBBARD MODEL

\begin{aligned} \mathcal{H}={}& -t\!\sum_{\langle i,j\rangle,\sigma} \left(c_{i\sigma}^{\dagger}c_{j\sigma}+\mathrm{h.c.}\right) -\mu\!\sum_{i,\sigma} n_{i\sigma} \\[2pt] &+\colorbox{#dce9c2}{\(\displaystyle U\)}\!\sum_i n_{i\uparrow}n_{i\downarrow} +\colorbox{#f2d0e1}{\(\displaystyle V\)}\!\sum_{\langle i,j\rangle} n_{i\uparrow} n_{j\downarrow} . \end{aligned}

Applying a mean-field approximation gives the onsite and bond pairing amplitudes:

onsite\(\Delta_0(i)=\colorbox{#dce9c2}{\(\displaystyle U\)}\langle c_{i\downarrow}c_{i\uparrow}\rangle\) bonds\(\Delta(i,j)=\colorbox{#f2d0e1}{\(\displaystyle V\)}\langle c_{j\downarrow}c_{i\uparrow}\rangle\)

These can be symmetrized into five independent order parameters:

\(\Delta_0\) (onsite \(s\)) \(\Delta_{s^*}\) (extended \(s\)) \(\Delta_d\) (\(d_{x^2-y^2}\)) \(\Delta_{p_x}\) (\(p_x\)) \(\Delta_{p_y}\) (\(p_y\))

BULK PHASE DIAGRAM

Density: \(n_e=0.75\)

U/t

V/t

The marker shows the parameters used below: \(\colorbox{#dce9c2}{\(\displaystyle U\)}/t=-3\) and \(\colorbox{#f2d0e1}{\(\displaystyle V\)}/t=-4\).

This point lies in the \(s+d+ip\) phase, with four nonzero components: \(\Delta_0\) \(\Delta_{s^*}\) \(\Delta_d\) and either \(\Delta_{p_x}\) or \(\Delta_{p_y}\).

02

THE NUMERICAL PUZZLE

An edge-bound pair-density wave

When the \(s+d+ip\) phase is put on a lattice with open boundaries, the order parameters develop amplitude modulation localized to the edges. As these modulations correspond to oscillations in the Cooper pair density, we refer to this as a pair-density wave (PDW).

LATTICE GEOMETRY

An infinite strip

We keep open boundaries across \(x\) and periodic boundaries along \(y\). The blue and red points indicate the two edges along which the order parameters are modulated.

open edge at \(x=1\) open edge at \(x=N_x\) \(x\) across the strip \(y\) along the periodic edge

Drag to rotate.

THE PAIR-DENSITY WAVES

The 5 order parameters

Channel View

Drag to rotate. Scroll to zoom. Double-click to reset the view. Use the dropdown menu to select any of the 5 order parameters.

We use the Bogoliubov-de Gennes (BdG) method to self-consistently calculate the order parameters on this lattice.

PDW WAVELENGTH

Fourier Transform of the PDW

If we Fourier transform the order parameters at the edges along \(y\), we see that its largest peaks are at \(q_y = \pm 2k_c\), which we will identify below. The other subdominant peaks are higher harmonics of this wavevector.

RESULT Translation symmetry is spontaneously broken along the edge. All 5 order parameters modulate at \(\pm 2k_c\); what selects \(k_c\)?

03

THE PEIERLS ANALOGY

The edge PDW is a Peierls instability^{[4][4] Peierls (1955)R. Peierls, Quantum Theory of Solids, Oxford: Clarendon Press (1955). ISBN: 9780192670175.Open textbook}

The Peierls instability is a paradigmatic mechanism for a metal-insulator transition in a one-dimensional chain; a lattice distortion with wavevector \(q=2k_F\) couples the two Fermi points and opens a gap in the electronic spectrum. The superconducting case follows the same logic: the infinite strip has Andreev bound-state (ABS) crossings at \(\pm k_c\), and a Cooper pair density modulation with \(q_\star=2k_c\) scatters the ABS and gaps them out.

SUPERCONDUCTING EDGE SPECTRUM

A \(2k_c\) order parameter modulation gaps the edge Andreev bound states

WITHOUT THE PDW

Uniform order parameter BdG spectrum

Uniform Δ_a

right
edge left
edge

Color encodes localization across \(x\). We find dispersive Andreev bound states (ABS): edge states that cross \(E=0\) at \(k_y=\pm k_c\).

\[ E_{\mathrm{ABS}}(k_y)= \pm\left[ \colorbox{#d6e6bf}{\(\displaystyle \Delta_0\)} -\frac{\mu}{4t}\left( \colorbox{#f7d0db}{\(\displaystyle \Delta_{s^\ast}\)} + \colorbox{#f5c99f}{\(\displaystyle \Delta_d\)} \right) - \colorbox{#f5c99f}{\(\displaystyle \Delta_d\)} \cos k_y \right]. \]

\(q_\star=2k_c\) → order parameter modulation

show states left edge + bulk right edge + bulk

WITH THE PDW

Modulated order parameter BdG spectrum

Modulated Δ_a

right
edge left
edge

The PDW scatters the Andreev bound states and gaps the crossings near \(\pm k_c\). The inset resolves the gap near \(+k_c\).

THE SAME NESTING LOGIC AS AN ORDINARY PEIERLS INSTABILITY

ORDINARY PEIERLS INSTABILITY

A \(2k_F\) lattice distortion gaps a one-dimensional metal

BEFORE THE DISTORTION

Atomic lattice

Electronic states cross \(E_F\) at \(k=\pm k_F\).

\(q=2k_F\) → periodic distortion

AFTER THE DISTORTION

Dimerized lattice

The distortion gaps the nested Fermi points.

Normal Peierls instability: \(q=2k_F\) lattice distortion same nesting logic Superconducting Peierls instability: \(q_\star=2k_c\) Cooper pair density modulation

04

THE MICROSCOPIC PRECURSOR

A Cooper pair Kohn anomaly at \(q_\star=2k_c\)

The BdG calculation shows that a PDW forms. To understand why, we return to the translation-invariant edge and ask which infinitesimal pairing fluctuation is most strongly shaped by the nested Andreev bound states.

EDGE-STATE LINEAR RESPONSE

Which pairing fluctuation is selected by the edge states?

We start from the uniform strip and perturb the five complex order parameters by a weak modulation with wavevector \(q\). The ABS susceptibility tells us where the edge quasiparticles produce a sharp finite-\(q\) response. We then use that ABS response to seed a branch of the full edge-localized amplitude-phase kernel, allowing the fluctuation to choose both its order-parameter mixture and its decay profile into the bulk.

THE PROJECTED ABS SUSCEPTIBILITY

\delta F_{\mathrm{ABS}}^{(2)} = \frac{1}{2}\sum_q \boldsymbol{\Phi}^\dagger(q) \Pi^{\mathrm{ABS}}(q) \boldsymbol{\Phi}(q).

\(\Pi^{\mathrm{ABS}}(q)\) measures how the tracked edge ABS contribute to the quadratic response of a pairing fluctuation with wavevector \(q\).

We use its finite-\(q\) eigenvector as a marker for the ABS-driven Kohn anomaly, then follow the corresponding branch of the microscopic edge kernel.

01 Identify the edge states

From the uniform-strip BdG spectrum, follow one edge-localized band \(E_{\mathrm{ABS}}(k)\) and its Nambu spinor \(u_{\mathrm{ABS}}(k)\).

02 Scatter the ABS

An order parameter fluctuation transfers momentum \(q\), connecting an incoming ABS state at \(k_-=k-q/2\) to an outgoing state at \(k_+=k+q/2\).

03 Add the virtual scattering processes

Sum over all allowed ABS-to-ABS scattering events. This isolates the momentum dependence generated specifically by the edge quasiparticles.

WHY A CUSP APPEARS

An easy virtual scattering process gives a large response

The response is enhanced when a fluctuation connects an occupied state to a nearby empty state. The edge ABS cross zero energy at \(\pm k_c\), so the especially efficient process is

\[ -k_c \; \xrightarrow{\;\;q=2k_c\;\;} \; +k_c . \]

On the lattice, the same physical nesting vector is understood modulo \(2\pi\): \(Q_{\mathrm{ABS}}=2k_c \pmod{2\pi}\).

THE PROJECTED ABS SUSCEPTIBILITY

\[ \Pi^{\mathrm{ABS}}_{\alpha\beta}(q) = \frac{1}{N_y}\sum_k \lambda_\alpha(k,q)\lambda_\beta^*(k,q)\, \chi_{\mathrm{ABS}}(k,q), \] \[ \chi_{\mathrm{ABS}}(k,q) = \frac{n_F[E_{\mathrm{ABS}}(k_+)]-n_F[E_{\mathrm{ABS}}(k_-)]} {E_{\mathrm{ABS}}(k_+)-E_{\mathrm{ABS}}(k_-)} . \]

The scalar factor \(\chi_{\mathrm{ABS}}\) becomes large when a fluctuation connects nearby occupied and empty ABS states. The vertex vector \(\boldsymbol{\lambda}\) records which order parameters participate in that scattering process.

What do we calculate? The BdG solution fixes \(E_{\mathrm{ABS}}(k)\), \(u_{\mathrm{ABS}}(k)\), the crossing momentum \(k_c\), and the scattering vertices. Near \(q=\pm2k_c\), the ABS-projected response identifies the Kohn-anomaly-like direction. We then track that same collective branch through the variational edge kernel, rather than reselecting an unrelated eigenvector at every momentum.

ABS-SEEDED EDGE-KERNEL BRANCH

The tracked branch develops cusps at \(q=\pm2k_c\)

The curve follows one continuous eigenbranch of the variational edge kernel, seeded by the ABS-projected response at \(q_\star\simeq\pm2k_c\). The cusps mark the Kohn-anomaly-like softening generated by scattering between the two ABS crossings.

AT \(q_\star\simeq\pm2k_c\)

Which fluctuations occur together?

The bars show the phase-fixed eigenvector of the tracked ABS-seeded branch at \(q_\star\simeq\pm2k_c\), separated into real and imaginary fluctuations of each \(\Delta_a\). The thumbnails underneath show the corresponding edge-localized real-space profiles, including the smaller \(p_y\) component, scaled by the plotted coefficients.

HOW TO READ THE TWO FIGURES

The cusp identifies the mechanism; the eigenvector identifies the soft order parameter mixture

WHAT ARE THE VERTICES?

For each order parameter \(a\) and real or imaginary fluctuation \(\eta=R,I\), the vertex is the matrix element of the corresponding order parameter fluctuation between the two ABS states:

\lambda_{a,\eta}(k,q) = u_{\mathrm{ABS}}^\dagger(k_+)\, \Lambda_{a,\eta}(k,q)\, u_{\mathrm{ABS}}(k_-).

The code obtains \(\Lambda_{a,\eta}\) from the BdG pairing block for \(\Delta_0,\Delta_{s^\ast},\Delta_d,\Delta_{p_x},\Delta_{p_y}\). A large \(\lambda_{a,\eta}\) means that this component efficiently scatters one ABS state into the other.

Because each contribution to \(\Pi^{\mathrm{ABS}}\) is an outer product \(\boldsymbol{\lambda}\boldsymbol{\lambda}^\dagger\), one ABS process can coherently mix several order parameters.

HOW TO READ THE EIGENPROBLEM

\Pi^{\mathrm{ABS}}(q)\,v_n(q) =\pi_n^{\mathrm{ABS}}(q)\,v_n(q).

Projected ABS response: identifies the finite-\(q\) direction that couples efficiently to the nested edge states.

Tracked edge-kernel branch: follows that same collective mode continuously in \(q\), so the left plot does not jump between unrelated eigenvectors.

PRECURSOR VERSUS ORDERED STATE

The Kohn-anomaly-like cusp is the microscopic precursor, not by itself a proof that the translation-symmetry breaking state is the lower energy configuration. It shows that integrating out the ABS strongly softens a finite-\(q\) pairing fluctuation at the level of linear response. The 2D BdG calculation shows the nonlinear result: the edge PDW with the same nesting wavevector is indeed the lower free-energy configuration.

1D METAL\(2k_F\) Peierls response EDGE SC\(2k_c\) pairing response

Together, the plots show: the nested ABS generate cusps at \(\pm2k_c\) on a specific edge-localized pairing branch. The associated eigenvector is a mixed fluctuation dominated by real \(s_0\), real \(s^\ast\), real \(d\), and an imaginary \(p_x\) component, with a smaller real \(p_y\) contribution.

05

CONCLUSION

A superconducting Peierls instability

The edge PDW is an ABS-nesting-driven superconducting Peierls instability, whose microscopic precursor is a pairing-channel Kohn anomaly at \(q_\star=2k_c\) modulo \(2\pi\).

The susceptibility identifies the preferred infinitesimal fluctuation. The BdG calculation shows that the broken-translation-symmetry state is the lower free-energy solution.

REFERENCES