Interactive Poster: Superconducting Peierls Instability

01

THE HAMILTONIAN

A square-lattice superconductor with five ways to pair

The extended Hubbard model supports onsite, bond-singlet, and bond-triplet order parameters. We focus on the parity-mixed, time-reversal-breaking \(s+d+ip\) state.

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 n_j . \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\)

There are 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 parameters used below lie 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

The edge spontaneously develops a pair-density wave

Rotate the three-dimensional surface, switch pairing channels, and compare the full magnitude with the deviation from its bulk value. The modulation is localized near the open boundaries.

LATTICE GEOMETRY

A strip isolates the two superconducting edges

We keep open boundaries across \(x\) and periodic boundaries along \(y\). The blue and red columns are the two edges whose order parameters may reconstruct.

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

Drag to rotate the representative strip window.

UNRESTRICTED BdG TEXTURE

Interactive order-parameter surface

Channel View

Drag to rotate. Scroll to zoom. Double-click to reset the view.

EDGE FOURIER TRANSFORM

The nonlinear PDW selects \(2k_c\)

The selected \(\Delta_a\) channel follows the surface dropdown. The remaining channels are retained as faint context curves.

NONLINEAR BdG RESULT Translation symmetry breaks along the edge, and all five pairing components acquire modulations.

03

THE PEIERLS ANALOGY

The edge PDW mirrors a Peierls distortion

In a one-dimensional metal, a distortion with wavevector \(q=2k_F\) couples the two Fermi points and opens a gap. The superconducting edge follows the same logic: the uniform strip has Andreev bound-state crossings at \(\pm k_c\), and a pairing-field modulation with \(q_\star=2k_c\) reconstructs and gaps them. Here \(k\) denotes a quasiparticle momentum, while \(q\) denotes a modulation wavevector.

SUPERCONDUCTING EDGE SPECTRUM

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

BEFORE THE PDW

Uniform-strip BdG spectrum

Uniform Δ_a

right
edge left
edge

The edge-localized ABS cross at \(\pm k_c\). Color encodes localization across \(x\).

\(q_\star=2k_c\) → pairing-field modulation

show states left edge + bulk right edge + bulk

AFTER THE PDW

Reconstructed BdG spectrum

Modulated Δ_a

right
edge left
edge

The relaxed PDW reconstructs the edge spectral weight and gaps the crossings near \(\pm k_c\). The inset resolves the gap near \(+k_c\).

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]. \]

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.

ordinary Peierls: \(q=2k_F\) lattice distortion same nesting logic superconducting edge: \(q_\star=2k_c\) pairing-field modulation

04

THE MICROSCOPIC PRECURSOR

A pairing-channel Kohn anomaly appears at \(q_\star=2k_c\)

The unrestricted BdG calculation shows that a PDW forms. To understand why, we return to the translation-invariant edge and ask how it responds to a tiny pairing-field modulation.

INTEGRATING OUT THE EDGE QUASIPARTICLES

Which pairing-field modulation is softened by the ABS?

We start from the uniform strip and ask how its free energy changes under a tiny order-parameter wiggle with wavevector \(q\). Each of the five complex pairing fields has an amplitude-like and a phase-like fluctuation, giving a ten-component vector \(\boldsymbol{\Phi}(q)\).

THE QUADRATIC EDGE KERNEL

\delta F^{(2)} = \frac{1}{2}\sum_q \boldsymbol{\Phi}^\dagger(q) \underbrace{\left[M_0(q)+\Pi(q)\right]}_{\Gamma^{-1}(q)} \boldsymbol{\Phi}(q).

\(M_0(q)\) is the smooth restoring background of the pairing fields.

\(\Pi(q)\) is the fermionic pairing-susceptibility bubble generated by integrating out the BdG quasiparticles.

The plots below isolate \(\Pi^{\mathrm{ABS}}(q)\): the part generated by one tracked edge ABS band.

01 Track the ABS pole

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 an edge state

A pairing-field wiggle transfers momentum \(q\), connecting an incoming ABS state at \(k_-=k-q/2\) to an outgoing state at \(k_+=k+q/2\).

03 Sum the virtual processes

Keep the ABS pole on both legs of the bubble. This exposes 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 modulation 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 modulation connects nearby occupied and empty ABS states. The vertex vector \(\boldsymbol{\lambda}\) records which pairing channels participate in that scattering process.

What is microscopic here? The saved BdG solution fixes \(E_{\mathrm{ABS}}(k)\), \(u_{\mathrm{ABS}}(k)\), the crossing momentum \(k_c\), every vertex \(\lambda_\alpha(k,q)\), and the complete \(10\times10\) matrix \(\Pi^{\mathrm{ABS}}(q)\). We now diagonalize that matrix.

MOMENTUM-DEPENDENT PROJECTED ABS RESPONSE

A sharp pairing-channel cusp at \(q=\pm2k_c\)

Each projected-bubble eigenvalue is shown relative to its own \(q=0\) value: \(\delta\pi_n^{\mathrm{ABS}}(q)=\pi_n^{\mathrm{ABS}}(q)-\pi_n^{\mathrm{ABS}}(0)\). This removes the constant baseline without changing the location of the finite-\(q\) cusps. Only one collective branch develops the pronounced anomaly.

AT \(q_\star=+2k_c\)

Which pairing components wiggle together?

The leading projected-ABS eigenvector is a mixed-channel perturbation. Each \(\Delta_a\) has an amplitude-like and a phase-like coefficient direction, shown by the two colors.

HOW TO READ THE TWO FIGURES

The cusp identifies the mechanism; the eigenvector identifies the soft pairing-field mixture

WHAT ARE THE VERTICES?

For each pairing symmetry \(a\) and amplitude- or phase-like direction \(\eta=R,I\), the vertex is the matrix element of the corresponding pairing-field wiggle 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 pairing channels.

HOW TO READ THE EIGENPROBLEM

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

Eigenvalue \(\pi_n^{\mathrm{ABS}}(q)\): the strength of the \(n\)-th collective response. The left plot shows its change relative to \(q=0\).

Eigenvector \(v_n(q)\): the mixture of pairing components in that response. The right plot shows the leading mode at \(q_\star=2k_c\).

PRECURSOR VERSUS ORDERED STATE

The Kohn-anomaly-like cusp is the microscopic precursor, not by itself a proof of spontaneous order. It shows that integrating out the ABS strongly softens a finite-\(q\) pairing fluctuation. The unrestricted two-dimensional BdG calculation independently supplies the nonlinear result: a lower-free-energy edge PDW forms at the same nesting wavevector.

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

Together, the plots show: the projected ABS susceptibility develops sharp cusps at \(\pm2k_c\), and the associated eigenvector reveals which order-parameter components participate first.

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 unrestricted BdG calculation supplies the nonlinear broken-symmetry state.