INTERACTIVE RESEARCH POSTER

A Superconducting Peierls Instability

Why does the amplitude of this multicomponent superconductor oscillate along its edges?

SUMMARY

In a conventional Peierls instability, a metal lowers its energy by developing a lattice distortion which gaps its low-energy electronic spectrum. Here we show that an analogous instability can arise at the boundary of a superconductor, where a spontaneously formed edge pair-density wave (PDW) gaps a low-energy Andreev bound state (ABS) band.

P. Senarath Yapa1,*iD P. Senarath Yapa ORCID: 0000-0002-5031-4695 pramodh.sy@gmail.com 1 Department of Physics and Astronomy, Uppsala University, Box 524, 751 20 Uppsala, Sweden , J. Maciejko2,3iD J. Maciejko ORCID: 0000-0002-6946-1492 2 Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2E1 3 Theoretical Physics Institute & Quantum Horizons Alberta, University of Alberta, Edmonton, Alberta T6G 2E1, Canada , F. Marsiglio2,3iD F. Marsiglio ORCID: 0000-0003-0842-8645 2 Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2E1 3 Theoretical Physics Institute & Quantum Horizons Alberta, University of Alberta, Edmonton, Alberta T6G 2E1, Canada , and A. M. Black-Schaffer1iD A. M. Black-Schaffer ORCID: 0000-0002-4726-5247 1 Department of Physics and Astronomy, Uppsala University, Box 524, 751 20 Uppsala, Sweden
1 UPPSALA UNIVERSITY Uppsala University Department of Physics and Astronomy, Box 524, 751 20 Uppsala, Sweden. Quantum Matter Theory group | 2 UNIVERSITY OF ALBERTA University of Alberta Department of Physics, Edmonton, AB, Canada T6G 2E1. Condensed matter and AMO research | 3 TPI & QUANTUM HORIZONS ALBERTA TPI & Quantum Horizons Alberta Theoretical Physics Institute and Quantum Horizons Alberta, University of Alberta, Edmonton, Alberta T6G 2E1, Canada. Theoretical Physics Institute Quantum Horizons Alberta
01

THE ORGANIZING IDEA

The Peierls analogy

A one-dimensional metal is unstable when a collective mode at \(2k_F\) connects the two Fermi points and opens a gap. In the superconducting case, the one-dimensional spectrum is a dispersive edge ABS with zero-energy crossings at \(\pm k_c\). The soft coordinate is an edge pairing fluctuation; when it condenses, it becomes the PDW.

Step 1 Two low-energy points choose the wavevector

In a one-dimensional metal the relevant momenta are the Fermi points, \(\pm k_F\). At the superconducting edge, the analogous low-energy objects are the ABS zero-energy crossings, \(E_{\rm ABS}(\pm k_c)=0\). The instability wavevector is therefore selected by \(2k_F\leftrightarrow 2k_c\).

Step 2 A collective mode softens at the nesting wavevector

The ordinary Peierls route is a Kohn anomaly, schematically \(\omega_{\rm ph}^2(q)=\omega_{{\rm ph},0}^2-g^2\chi_0(q)\), with the minimum at \(q=2k_F\). In the superconducting problem, the softened coordinate is a pairing fluctuation, \(\Omega_{\rm SC}^2(q_y)=\Omega_{{\rm SC},0}^2-\Pi(q_y)\), with minima near \(q_y=\pm 2k_c\).

Step 3 The ordered state opens the low-energy gap

After the Peierls transition, the lattice dimerizes and gaps the electronic band. Here, the edge pairing mode condenses into a PDW with period \(\pi/k_c\), and that PDW gaps the ABS spectrum at the crossings.

Peierls Instability
Uniform lattice

1. Electronic band

2. Kohn anomaly

Dimerized lattice

3. Gapped band

Superconducting Peierls Instability
Uniform pair density at edge

1. Andreev bound state

2. Superconducting Kohn anomaly

Pair density wave at edge

3. Gapped ABS

1D band \(\leftrightarrow\) edge ABS \(2k_F\) phonon \(\leftrightarrow\) \(2k_c\) pairing mode lattice distortion \(\leftrightarrow\) edge PDW
02

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.

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 \(\Delta_{p_y}\).

03

THE BROKEN-SYMMETRY STATE

An edge-bound pair-density wave

When the \(s+d+ip_x\) phase is put in an open strip, the self-consistent order parameters develop amplitude modulations localized to the two edges. This is an edge-bound pair-density wave: a translation-breaking superconducting texture, not a bulk 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 WAVE

The 5 order parameters

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 parent strip.

RESULT Translation symmetry is spontaneously broken along the edge. All five order parameters share the same finite wavevector, \(Q_{\rm PDW}=2k_c\).
04

THE EDGE SPECTRUM

The PDW gaps the ABS crossings

The translation-invariant parent strip has edge-localized ABS branches crossing \(E=0\) at \(k_y=\pm k_c\). Once the edge PDW forms at \(Q_{\rm PDW}=2k_c\), it scatters quasiparticles between those two boundary Fermi points and opens a gap in the low-energy edge spectrum.

SUPERCONDUCTING EDGE SPECTRUM

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

uniform \(\Delta(y)\)

Parent edge

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

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

Modulated edge

Show edge dots

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 \(\pm k_c\)
05

THE MICROSCOPIC PRECURSOR

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

The self-consistent BdG solution shows the ordered PDW. The susceptibility calculation asks the complementary question: can we see the instablity of the the uniform edge to a symmetry-allowed pairing fluctuation at the ABS nesting wavevector?

EDGE SUSCEPTIBILITY

Does a pairing fluctuation soften?

We start from the translation-invariant \(s+d+ip_x\) strip and allow small edge-localized fluctuations in the pairing channels. Diagonalizing the edge inverse susceptibility gives the squared frequencies of optimized collective pairing modes as a function of \(q_y\).

\[ \Omega_{\rm SC}^2(q_y)v_n(q_y) = \omega_{{\rm SC},n}^2(q_y)v_n(q_y). \]
01Use the parent edge spectrum

The relevant low-energy states are the ABS crossings at \(\pm k_c\).

02Probe finite-\(q_y\) pairing

A fluctuation at \(q_y=2k_c\) connects the two zero-energy ABS states.

03Look for a negative mode

A negative squared frequency means the uniform edge wants to condense that fluctuation.

The result: the soft branch has cusp-like minima at \(\pm2k_c\) and becomes negative there, the superconducting analogue of a Peierls soft phonon.

EDGE FLUCTUATION FREQUENCIES

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

The highlighted branch of \(\Omega_{\rm SC}^2(q_y)\) develops cusp-like minima at the ABS nesting wavevectors and drops below zero, signalling a soft-mode instability of the translation-invariant edge. The composition of this softened mode is shown below. The Goldstone-like branch is the edge-projected phase mode of the parent condensate. The softened branch being negative at \(q_y=0\) reflects an orientational instability of the \(s+d+ip_x\) strip toward the competing \(s+d+ip_y\).

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. The gold Goldstone-like branch is the edge-projected phase mode of the parent condensate, so it vanishes at \(q_y=0\). The softened branch being negative at \(q_y=0\) reflects a separate orientation instability of the constrained \(s+d+ip_x\) strip toward the competing \(s+d+ip_y\) orientation; it is not the finite-momentum PDW 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

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

means that, once the parent 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 \(x\)?

parent 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

  1. S. Nayak and S. Kumar, J. Phys.: Condens. Matter 30, 135601 (2018).
  2. J. Hutchinson and F. Marsiglio, J. Phys.: Condens. Matter 33, 065603 (2021).
  3. P. Senarath Yapa, X. Guo, J. Maciejko and F. Marsiglio, Physica C 633, 1354719 (2025).
  4. P. Senarath Yapa, J. Maciejko, F. Marsiglio, and A. M. Black-Schaffer, manuscript in preparation.