Interactive Poster: Superconducting Peierls Instability

01

THE ORGANIZING IDEA

The Peierls analogy

A one-dimensional metal can lower its energy by distorting its lattice at the wavevector connecting its Fermi points.[1] Here the same idea appears at a superconducting edge: Andreev bound states select an edge pairing fluctuation, which condenses into a pair-density wave.

Peierls Instability

Uniform lattice

1. Electronic band

2. Kohn anomaly

Peierls mechanism 1D Fermi points

A 1D metal has Fermi points at \(\pm k_F\). The separation \(2k_F\) is the wavevector of the instability.

Phase transition The lattice dimerizes and becomes insulating

After the Peierls transition: the lattice dimerizes with period \(\pi/k_F\) and a gap opens in the electronic band at \(\pm k_F\), indicating an insulating state

Dimerized lattice

3. Gapped band

Superconducting Peierls Instability

Uniform pair density at edge

1. Andreev bound state

2. Superconducting Kohn anomaly

superconducting analogue ABS zero energy crossings

The edge mode has \(E_{\rm ABS}(\pm k_c)=0\). These \(\pm k_c\) points are the analogues of the \(\pm k_F\) Fermi points.

phase transition The edge PDW forms and gaps out the ABS

The superconducting order parameters modulate along the edge with period \(\pi/k_c\) and a gap opens in the ABS spectrum at \(\pm k_c\)

Pair density wave at edge

3. Gapped ABS

1D band \(\leftrightarrow\) edge ABS \(2k_F\) phonon \(\leftrightarrow\) \(Q_{\rm ABS}=2k_c\) pairing mode lattice distortion \(\leftrightarrow\) edge PDW

Fermi-point analogy

Top: a 1D metal has low-energy states at \(\pm k_F\). Bottom: the superconducting edge has ABS crossings at \(\pm k_c\), so the analogous pairing wavevector is \(Q=2k_c\).

Soft-mode precursor

Top: the Peierls metal softens a phonon at \(2k_F\). Bottom: the uniform superconducting edge softens a pairing fluctuation at \(2k_c\).

Gap opening

Top: the lattice dimerization gaps the metal at \(\pm k_F\). Bottom: the edge PDW gaps the ABS near \(\pm k_c\).

02

A MICROSCOPIC REALIZATION

The 2D extended Hubbard model

We test this idea in a square-lattice extended Hubbard model. In the bulk, onsite and nearest-neighbor attraction produce a mixed \(s+d+ip\) superconducting state.[2][3][4] To focus on the boundary instability, we use an infinite strip: open across \(x\), periodic along \(y\), and cleanly separated into two long edges.

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} \\ &+\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 \(\Delta_{p_y}\).

For an \(x\)-normal edge we use the \(s+d+ip_x\) orientation. This choice fixes the parent edge whose ABS spectrum is analyzed next.

LATTICE GEOMETRY

An infinite strip isolates the edges

Open boundaries across \(x\) expose the edge Andreev states, while periodicity along \(y\) keeps momentum \(k_y\) well defined. This lets us identify the ABS crossings first, then test whether the same edge develops a finite-\(q_y\) pairing instability.

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

Drag to rotate.

UNIFORM STRIP

Uniform order parameters

These are the uniform, \(y\)-invariant order parameters on the strip, computed self-consistently using the Bogoliubov-de Gennes (BdG) method. The next section uses this same uniform BdG state to examine the edge quasiparticle energies.

05

THE BROKEN-SYMMETRY STATE

An edge-bound pair-density wave

The self-consistent BdG solution shows what the instability becomes: the superconducting order parameters develop amplitude modulations localized to the edges. The modulation wavevector is the same one selected by the ABS spectrum and the softened edge mode.

THE PAIR-DENSITY WAVE

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.

The singlet components and \(\Delta_{p_x}\) are approximately \(\cos(2k_c y)\)-like near the edge, while the induced \(\Delta_{p_y}\) component is shifted toward a \(\sin(2k_c y)\)-like profile.

PDW WAVELENGTH

Fourier Transform of the PDW

The edge Fourier spectrum has its dominant finite-\(q_y\) weight at \(q_y=\pm2k_c\), the wavevector connecting the two zero-energy ABS crossings of the uniform strip.

RESULT Translation symmetry is spontaneously broken along the edge. All five order parameters share the same finite wavevector, \(Q_{\rm PDW}=2k_c\), matching the ABS crossings and the soft edge pairing mode. The relative \(\cos\)- and \(\sin\)-like profiles point to the edge symmetry rules behind the mixed PDW.

03

THE EDGE SPECTRUM

ABS crossings and gap opening

The uniform strip has edge-localized Andreev bound states crossing zero energy at \(k_y=\pm k_c\). These crossings act like the boundary version of Fermi points: they identify the wavevector \(Q=2k_c\) that the edge can use to open a low-energy gap.

A simple analytic estimate captures the same edge dispersion:

E_{\rm ABS}(k_y)=\pm[\) \(\tilde{\Delta}_{0}\) \(-\dfrac{\mu}{4t}(\) \(\tilde{\Delta}_{s^\ast}\) \(+\) \(\tilde{\Delta}_{d}\) \()-\) \(\tilde{\Delta}_{d}\) \(\cos k_y]

The tildes denote the values of the order parameters at the edge of the strip.

SUPERCONDUCTING EDGE SPECTRUM

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

\(\Delta\sim{\rm const.}\)

Uniform edge

The uniform edge ABS crosses zero energy at \(\pm k_c\).

\(Q_{\rm PDW}=2k_c\) → edge pairing modulation

Show edge left edge right edge

\(\Delta(y)\sim\cos(2k_c y)\)

Modulated edge

The PDW folds the edge spectrum and opens a gap at the ABS crossings.

Peierls metal
Fermi points \(\pm k_F\) → Superconducting edge
ABS crossings \(\pm k_c\)

NEXT When the PDW forms at \(Q_{\rm PDW}=2k_c\), it gaps the low-energy edge spectrum near these crossings. But the same wavevector appears even before the PDW forms, as a softening of an edge pairing fluctuation: the superconducting analogue of a Kohn anomaly.

04

THE MICROSCOPIC PRECURSOR

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

Before the PDW fully forms, the uniform edge already knows which wavevector it wants. A small edge Cooper-pair fluctuation softens near \(q_y=2k_c\), the superconducting analogue of the soft phonon in a conventional Peierls instability.

EDGE SUSCEPTIBILITY

Does a pairing fluctuation soften?

We start from the uniform \(s+d+ip_x\) strip and allow small edge-localized fluctuations in all pairing channels. Diagonalizing the edge pairing-fluctuation kernel gives the stiffness cost of each optimized fluctuation.

\mathcal K(q_y)=\mathcal K^{(0)}(q_y)-\Pi(q_y).

The quasiparticle response \(\Pi\) lowers the bare pairing stiffness. This is why the right-hand plot searches for dips in \(\lambda_n(q_y)\).

\mathcal K(q_y)v_n(q_y)=\lambda_n(q_y)v_n(q_y).

Each curve on the right is one eigenvalue \(\lambda_n(q_y)\); a negative branch means the uniform edge is unstable to that fluctuation.

QUASIPARTICLE RESPONSE

\begin{aligned} -\Pi_{IJ}(q_y) &= \frac{1}{2N_y} \sum_{k_y,m,n} M_I^{mn} [M_J^{mn}]^\ast \frac{ f(E_m(k_+))-f(E_n(k_-)) }{ E_m(k_+)-E_n(k_-) } . \end{aligned}

\begin{aligned} -\Pi_{IJ}(q_y) &= \frac{1}{2N_y} \sum_{k_y,m,n} M_I^{mn} [M_J^{mn}]^\ast \\ &\quad\times \frac{ f(E_m(k_+))-f(E_n(k_-)) }{ E_m(k_+)-E_n(k_-) } . \end{aligned}

Here \(k_\pm=k_y\pm q_y/2\), and \(M_I^{mn}\equiv M_I^{mn}(k_y,q_y)\) is the edge pairing-fluctuation vertex: it scatters a BdG state at \(k_-\) into one at \(k_+\). The Fermi-function quotient is the same occupation-over-energy structure as the Peierls susceptibility \(\chi_0(q)\); for counterpropagating ABS crossings it enhances \(\Pi\) near \(q_y=2k_c\).

EDGE PAIRING STIFFNESS

The Peierls branch becomes negative at \(q_y=\pm2k_c\)

The highlighted branch of \(\lambda_n(q_y)\) develops cusp-like minima at the ABS nesting wavevectors and drops below zero, signalling a finite-momentum pairing instability of the translation-invariant edge. The low-\(q_y\) structures have different origins: the gold branch is an edge-projected phase mode, while the negative \(q_y=0\) feature is a uniform orientation instability of the idealized \(s+d+ip_x\) strip toward \(s+d+ip_y\), not the Peierls PDW.

01Use the uniform ABS

The relevant low-energy states cross at \(\pm k_c\).

02Scan finite-\(q_y\) pairing

The edge fluctuation is tested along the boundary.

03Find negative stiffness

A negative eigenvalue means the uniform edge is unstable.

The result: the edge Cooper-pair stiffness has cusp-like minima at \(\pm2k_c\) and becomes negative there.

HOW TO READ THE FIGURES

The cusp gives the wavevector; the eigenvector gives the PDW mixture

The eigenvalue plot identifies the instability: a finite-\(q_y\) edge pairing mode softens exactly at the ABS nesting wavevector. The channel-content plot identifies the leading fluctuation: not one isolated order parameter, but a coherent mixed-symmetry edge mode in the symmetry-even sector.

The extra structure near \(q_y=0\) has a different origin: it reflects uniform phase and orientation physics of the constrained strip. The finite-\(q_y\) cusp is the feature that tracks the superconducting Peierls mechanism.

Together: the ABS select \(Q_{\rm PDW}=2k_c\), the susceptibility supplies the soft-mode precursor, and the self-consistent BdG solution supplies the finite-amplitude edge PDW.

AT \(q_y=+2k_c\)

What is the composition of the softened mode?

The leading softened mode is dominated by \(\mathrm{Re}\,\delta\Delta_{s^\ast}\), with substantial \(\mathrm{Re}\,\delta\Delta_d\) and smaller \(\mathrm{Re}\,\delta\Delta_0\) and \(\mathrm{Im}\,\delta\Delta_{p_x}\) components. The \(p_y\) channel is absent from the leading displayed mode; the symmetry reason is the next step.

06

WHY THIS MIXTURE?

Edge symmetry dictates the PDW symmetry

The susceptibility tells us which collective mode softens. The symmetry structure of the \(s+d+ip_x\) parent explains why that mode mainly mixes \(\Delta_0\), \(\Delta_{s^\ast}\), \(\Delta_d\), and \(\Delta_{p_x}\), while \(\Delta_{p_y}\) is not part of the leading soft eigenvector.

LANDAU PICTURE

The mixed state already ties the even channels together

On the square lattice, \(\Delta_0\) and \(\Delta_{s^\ast}\) are both \(A_1\), \(\Delta_d\) is \(B_1\), and (\(\Delta_{p_x}\), \(\Delta_{p_y}\)) is the \(p\)-wave doublet. The symmetry-allowed coupling

f_{\rm mix}\sim\boldsymbol{\Delta}_s^\dagger\) \(\Delta_d^\ast\) \((\) \(\Delta_{p_x}^2\) \(-\) \(\Delta_{p_y}^2\) \()+{\rm c.c.}

means that, once the uniform edge has nonzero \(\bar{\Delta}_{p_x}\), fluctuations in \(\Delta_0\), \(\Delta_{s^\ast}\), \(\Delta_d\), and \(\Delta_{p_x}\) are linearly coupled. The soft mode is therefore a collective mixed-symmetry fluctuation, not four unrelated instabilities occurring at the same \(q_y\).

EDGE SELECTION RULE

What changes at an edge normal to \(x\)?

uniform edge \(\bar{\Delta}_{p_y}\)\(=0\)

The translation-invariant \(s+d+ip_x\) strip has no background \(p_y\) condensate, so the local Landau term has no direct linear handle on \(\delta\Delta_{p_y}\).

remaining mirror \(M_y:y\to -y\)

\(\Delta_0\), \(\Delta_{s^\ast}\), \(\Delta_d\), and \(\Delta_{p_x}\) are even under this mirror. \(\Delta_{p_y}\) is odd.

allowed near edge \(\Delta_\eta^\ast\partial_y\)\(\Delta_{p_y}\)

The derivative \(\partial_y\) is also odd, so \(\partial_y\)\(\Delta_{p_y}\) is even and may couple to the even channels, but only through a finite-\(q_y\) edge-gradient term.

CONSEQUENCE

\(p_y\) is induced after the leading mode is chosen

leading soft mode \(\delta\Delta_\eta\sim\cos(Q_{\rm PDW}y)\)

\(\eta=0\), \(s^\ast\), \(d\), \(p_x\): the symmetry-even channels oscillate approximately in phase.

edge-gradient response \(\delta\Delta_{p_y}\)\(\sim\sin(Q_{\rm PDW}y)\)

The \(p_y\) texture is phase shifted by \(\pi/2\). It appears in the finite-amplitude BdG PDW as an induced edge component, but it is not the driver of the softened eigenmode.

So the logic is: symmetry fixes which channels can mix strongly, while the ABS susceptibility selects the wavevector \(Q_{\rm PDW}=2k_c\).

07

CONCLUSION

A superconducting Peierls instability

This work identifies a new way for a PDW to form: not from an imposed finite-momentum pairing tendency in the bulk, but from a boundary Peierls mechanism in which dispersive ABS crossings select the condensate modulation wavevector.

The edge susceptibility supplies the soft-mode precursor at \(q_y=2k_c\); symmetry explains the mixed-channel content of that mode; and the self-consistent BdG solution supplies the ordered state, an edge-localized PDW that gaps the ABS.

REFERENCES