Non-degenerate surface pair density wave in the Kagome superconductor CsV 3Sb5- application to vestigial orders Yue Yu1 2

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Non-degenerate surface pair density wave in the Kagome superconductor CsV3Sb5-

application to vestigial orders

Yue Yu1, 2

1Department of Physics, Stanford University, Stanford, CA 94305

2Department of Physics, University of Wisconsin, Milwaukee, WI 53201

On the Sb-layer of the Kagome superconductor CsV3Sb5, pair density wave states have been ob-

served. When the high-temperature charge orderings are treated as static backgrounds, these PDW

states exhibit the same wavevector in the eﬀective 2D Brillouin zone. Interestingly, these PDW

states break the same symmetry on the surface. Considering the presence of this non-degenerate

PDW, we investigate the implications for the possible existence of a vestigial charge-4e phase with

a non-zero center-of-mass momentum. To distinguish between diﬀerent vestigial phases, we propose

scanning tunneling microscopy experiments. We aim to provide insights into the nature of the vesti-

gial phases and their distinct characteristics in CsV3Sb5. This research sheds light on the interplay

between PDW states, charge orderings, and superconductivity of the Kagome superconductor.

I. INTRODUCTION

The recently discovered Kagome superconductor

CsV3Sb5(CVS)1,2 is a highly intriguing material that

has attracted signiﬁcant experimental and theoretical

interest due to its exotic charge and superconducting

orderings3.

In its high-temperature phase, the CVS crystalizes

in the P6/mmm space group and exhibits a layered

structure composed of V-Sb sheets intercalated by Cs

atoms1. At a temperature of 94K, scanning tunnel-

ing microscopy (STM) experiments have revealed a 2a0

charge density wave (CDW) state with a 2 ×2 superlat-

tice modulation4–7. Notably, the CDW has been found

to possess a three-dimensional character, with reports of

both 2 ×2×2 and 2 ×2×4 modulations6,8–11.

Moreover, diﬀerent intensity distributions of the CDW

peaks with unusual magnetic response have been ob-

served, indicating the presence of a chiral charge

order7,12,13. Investigations are underway on the time-

reversal symmetry breaking in this system14–22. The dif-

ference in CDW intensity also suggests rotational sym-

metry breaking. The temperature dependence of the ne-

matic transition has been recently explored8,23–27.

Around 60K, an unidirectional CDW with a period-

icity of 4a0is observed on the Sb-layer through STM

measurements5,6,13,28. However, this CDW is not ob-

served on the alkali layer or in the bulk8–11. Supercon-

ductivity is observed in CsV3Sb5at a temperature of

2.3K2, and the nature of its order parameter is still un-

der investigation4,5,29–42.

In the context of the superconducting phase, the pres-

ence of pair density wave (PDW) on the Sb-layer has been

reported by STM probes5. Similar to the 4a0CDW, its

existence on the alkali layer and in the bulk remains un-

conﬁrmed, resulting in unknown 3D wave vectors. Con-

sequently, a comprehensive analysis of the bulk supercon-

ducting phase diagram becomes challenging. Conversely,

the STM results oﬀer valuable insights and provide suﬃ-

cient information to construct the superconducting phase

diagram on the surface. Therefore, studying the surface

phase diagram can serve as a stepping stone towards a

better understanding and analysis of the bulk supercon-

ducting phase.

In this study, our focus is directed towards the surface

superconducting transitions. To simplify the analysis,

we consider the charge orderings that emerges at signif-

icantly higher temperatures than the superconductivity

as a static background. This treatment eﬀectively en-

larges the unit cell and results in a folded Brillouin zone.

Within this folded Brillouin zone, it is observed that all

PDWs exhibit the same wavevector. These PDWs, be-

ing physically equivalent, can be eﬀectively described by

a single order parameter denoted as ∆Q.

The presence of the unusual PDW has signiﬁcant im-

plications for the vestigial phases. Traditionally, vesti-

gial phases are constructed using two independent PDW

order parameters43. However, in the case of the non-

degenerate PDW ∆Qand the uniform SC ∆0in CVS,

both order parameters need to be considered to construct

vestigial phases.

Speciﬁcally, if a charge-4e phase exists, it would be

characterized by the composite order parameter ∆4e=

∆0∆Q. The presence of such a phase can be examined

through experimental techniques other than the Little-

Parks oscillation44. In this study, we will utilize STM sig-

natures to diﬀerentiate between diﬀerent vestigial phases

and provide insights into their distinct characteristics.

This paper will primarily investigate the properties of

the non-degenerate PDW state on the low-temperature

Sb surface. A signiﬁcant focus will be placed on present-

ing a symmetry argument that prohibits the existence

of the conventional uniform charge-4e phase. Further-

more, a comparison of various vestigial phases will be

conducted, emphasizing their distinct STM signatures.

Additionally, the role of CDW disorder in stabilizing ves-

tigial superconducting phases will be discussed, particu-

larly in the context of commensurate systems. The ne-

cessity of CDW disorder for the stabilization of these

vestigial phases will be explored and elucidated.

arXiv:2210.00023v3 [cond-mat.supr-con] 4 Jul 2023

II. SYMMETRY ANALYSIS

In this study, our focus is on the low-temperature

PDW state observed on the Sb-cleaved surface of the

Kagome superconductor CVS. On this surface, multiple

CDW states have been reported above the superconduct-

ing phase. The ﬁrst CDW state, with a 2a0periodicity,

is observed at 94K2,8,12,13. Its wavevectors, shown as

blue dots in Fig.1, preserve the 6-fold rotational symme-

try. The second CDW state, with a 4a0periodicity in

the XY-plane, emerges around 60K5,13. Its wavevectors,

shown as red dots, break the 6-fold rotational symmetry.

Both CDW ordering temperatures signiﬁcantly exceed

the superconducting transition temperature Tc≈2.3K.

FIG. 1: The Bragg peaks from the Kagome lattice structure

(black), 2a0-CDW (blue) and 4a0-CDW (red). Inside the su-

perconducting phase, four additional peaks (black circles at

±Q1,2) are observed in STM.

Deep in the SC phase at 300mK, four additional CDW

ρQiwith peaks at ±Q1,2have been observed in STM5,

which are the signature for PDW ∆Qi. Given the ex-

istence of a uniform SC component ∆0, these CDW

wave vectors Qiare the same as the PDW wave vectors

through ρQi= ∆∗

0∆Qi. The same wave vectors are ob-

served in the spatial variation of the SC gap magnitude5.

The exact critical temperature for the low-temperature

CDW is currently unknown. However, experimental evi-

dence suggests that it is below 4.2K5. The peak at ±Q3

is a higher harmonics of the unidirectional CDW, which

exists even above the SC phase5, so it does not break

additional translational symmetry.

When considering the low-temperature phases (T <

4K) in the presence of the 2a0and 4a0CDW, it is appro-

priate to treat them as static backgrounds. This leads to

a larger unit cell, requiring the Brillouin zone (BZ) fold-

ing. In Fig.2, the original Brillouin zone is depicted as

a solid black hexagon. Upon folding the Brillouin zone

according to the 2a0CDW (blue dots), the resulting Bril-

louin zone (indicated by the blue line) remains a hexagon,

but with a halved size. The PDWs (black circles) are now

folded onto the M points of the resulting Brillouin zone.

FIG. 2: The unfolded Brillouin (black line), folded Brillouin

zone due to 2a0-CDW (blue line) and folded Brillouin zone

due to both 2a0and 4a0-CDW (red line). In the last folded

Brillouin zone, the PDW (black circle) is located at the corner.

Consequently, the PDWs ∆Qiand ∆−Qibecome physi-

cally identical, breaking the same symmetry.

We now proceed to fold the BZ according to the uni-

directional 4a0-CDW (red dots). The resulting folded

Brillouin zone is depicted as a red rectangle. Interest-

ingly, all four PDWs (originally ±Q1,2) are now located

at the corners of the rectangle (i.e. the (π, π) point).

This implies that these PDWs are physically equivalent,

exhibiting the same symmetry breaking.

To summarize, the low-temperature phase observed

on the Sb-layer of CVS features a non-degenerate PDW

∆Q=(π,π)and a uniform superconducting order param-

eter ∆0. This diﬀers from the well-known ‘LO’-like

phase45, which exhibits a doubly-degenerate PDW ∆±Q.

Furthermore, the low-temperature phase in CVS is dis-

tinct from the ‘FF’-like phase46 as the wavevector Qpre-

serves time-reversal symmetry.

The non-degenerate PDW ∆Qintroduces an inter-

esting scenario for vestigial phases. Traditionally, ves-

tigial phases require two independent superconducting

order parameters. In the case of the doubly degener-

ate PDW ∆±Q, the vestigial charge-4e phase arises from

the uniform order parameter ∆4e

0= ∆Q∆−Q43. In this

phase, ∆4e

0is ordered while the corresponding CDW

ρ2Q= ∆Q∆∗

−Qis disordered. The phase diagram is de-

picted in the left panel of Fig.3, where the blue line rep-

resents the superconducting transition and the red line

represents the CDW transition. The remaining symme-

tries in each phase are also indicated. The superconduct-

ing transition breaks a U(1) symmetry, while the CDW

transition can either break an additional U(1) symmetry

if the CDW ρ2Qis incommensurate or a Zmsymmetry if

the periodicity of the CDW is m times the lattice spac-

ing. To support the charge-4e phase, it is necessary for

the superconducting transition to occur at a higher criti-

cal temperature than the CDW transition. In the case of

an incommensurate CDW, this requirement can be met

in a clean system as long as the superconducting phase

has a larger stiﬀness43. However, for a commensurate

CDW, discrete symmetry breaking generally occurs at a

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摘要：

Non-degeneratesurfacepairdensitywaveintheKagomesuperconductorCsV3Sb5-applicationtovestigialordersYueYu1,21DepartmentofPhysics,StanfordUniversity,Stanford,CA943052DepartmentofPhysics,UniversityofWisconsin,Milwaukee,WI53201OntheSb-layeroftheKagomesuperconductorCsV3Sb5,pairdensitywavestateshavebeenob-s...

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Non-degenerate surface pair density wave in the Kagome superconductor CsV 3Sb5- application to vestigial orders Yue Yu1 2

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