Higgs-Leggett mechanism for the elusive \({\phi }_{0}/3={hc}/6e\) oscillation in Little-Parks setup of Kagome superconductor CsV₃Sb₅

Quantization of magnetic flux \({\phi }_{0}={hc}/2e\) is the hallmark of superconductivity (SC) as a global quantum state of electron pair condensation^{1,2,3,4,5,6,7}. As shown first by Little and Parks⁸, in a ring-shaped superconductor the SC transition temperature and resistance between two electrodes oscillate as functions of applied magnetic flux through the ring with period \({\phi }_{0}\). Very recently, in the new vanadium-based Kagome superconductors CsV₃Sb₅^{9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24} a Little-Parks experiment demonstrated clear resistance oscillations of period \({\phi }_{0}/3={hc}/6e\), hinting existence of a possible six electrons (\(6{{{{{\rm{e}}}}}}\)) SC state at intermediate temperatures slightly above the bulk transition point for vanishing zero-field resistance²⁵. This observation raises several questions which may be intimately related with the fundamental understanding on SC²⁶: Does the period \({\phi }_{0}/3\) of magneto resistance imply a condensate of six electrons (\(6{{{{{\rm{e}}}}}}\)) different from the \(2{{{{{\rm{e}}}}}}\) Cooper pairs? How does the Kagome structure play the role in this phenomenon, and is it possible that Cooper pairs associated with the three important reciprocal lattice vectors of the Kagome structure interweave with each other in a unique way to give birth to the never-before-seen period \({\phi }_{0}/3\)? After all, can and in what sense this phase be considered a SC state, and how does the ordinary \(2{{{{{\rm{e}}}}}}\) SC state re-assert itself at lower temperatures where zero-field resistance drops to zero?

Here, we propose a theory based on Ginzburg-Landau (GL) free-energy functional with three SC order parameters for Cooper pairs associated with the three reciprocal lattice vectors connecting M points of the hexagonal Brillouin zone of the material. Based on analysis of the free-energy functional involving the gauge field on a ring geometry mimicking the Little-Parks setup, we find that, in a series of intermediate states, phase of one SC order parameter winds 2π more or less than the other two ones around the ring, which yields local free-energy minima at integer multiples of \({\phi }_{0}/3\) starting from zero applied magnetic flux, responsible for the magneto resistance oscillations with period of \({\phi }_{0}/3\) observed in the recent Little-Parks experiment for CsV₃Sb₅²⁵. As clarified in our present work, the degeneracy associated with a \({{{{{{\rm{Z}}}}}}}_{2}\) chirality defined by phases of the three SC order parameters^{27,28,29,30,31} is crucial for this phenomenon. At intermediate temperatures slightly below the mean-field transition temperature, domain walls (DWs) of \({{{{{{\rm{Z}}}}}}}_{2}\) chirality domains are meta-stabilized by a Higgs-Leggett mode^27,32 where phases and amplitudes of three SC order parameters intervene in a unique way. Expelling these DWs from the system at low temperatures brings the system to its ground state, where the phases of three SC order parameters are locked to each other and behave in the same way as a single SC order parameter, yielding free-energy minima only at integer multiples of \({\phi }_{0}\) and conventional Little-Parks signals with the well-known Higgs mode^33,34 as the responsible mechanism. Therefore, our present theoretical analysis unveils that the period \({\phi }_{0}/3\) of resistance oscillation in the Little-Parks experiment performed for CsV₃Sb₅²⁵ results from the interplay of three \(2{{{{{\rm{e}}}}}}\) SC order parameters hosted by the Kagome structure, rather than a \(6{{{{{\rm{e}}}}}}\) condensate. It is also revealed that half magnetic flux quantization \({\phi }_{0}/2\) is possible which depends on the detailed Josephson-type coupling. The present theory is expected to provide a reasonable starting point for upcoming exploration on the rich physics hosted by the Kagome vanadium-based superconductors. Besides the conceptual advance in unveiling the Higgs-Leggett collective mode of superconductivity, our work also exhibits possible new technological capability of superconductivity. As built-in features rising from the Kagome symmetry, both \({{{{{{\rm{Z}}}}}}}_{2}\) chirality induced double degeneracy and multiple degeneracy associated with kaleidoscopic phase windings in the three Cooper pairs can be exploited for building next-generation SQUID-type qubits superior to the contemporary ones, where top-down nano fabrications are required to produce degenerate bit states.

Analysis based on GL free-energy functional

As revealed by STM technique¹¹, the vanadium-based Kagome superconductor CsV₃Sb₅ exhibits well-developed pair density wave (PDW) orders^{35,36,37,38,39}. The Bogoliubov-de Gennes (BdG) analysis taking into account the electronic structure on lattices with the hexagonal symmetry indicates that the PDW at the three momenta \({{{{{{\boldsymbol{Q}}}}}}}_{j}\) (\(j={{{{\mathrm{1,2,3}}}}}\)) connecting the three M points of Brillouin zone are deeply related to the nesting of Fermi surfaces (see Fig. 1a)⁴⁰. The SC gap functions with finite momenta yielding the PDW order can be described by \({\Delta }_{\pm {{{{{{\boldsymbol{Q}}}}}}}_{j}}=\Delta {e}^{i{\theta }_{j}}{e}^{\pm i\left({\phi }_{j}/2\right)}\) with \(\Delta\) real and positive corresponding to s-wave pairing⁴¹. The condensation energy takes maximum at \({\phi }_{j}=\pm \pi /2,\) and \({\theta }_{2}-{\theta }_{1}={\theta }_{3}-{\theta }_{2}=\pm 2\pi /3\) (mod \(\pi\)) characteristic of spontaneous time-reversal symmetry breaking, where energy costs by changes in \({\phi }_{j}\) are much larger than those associated with the phase variables of the three SC order parameters θ_j⁴⁰. Therefore, thermal fluctuations and/or responses to external magnetic field in the three SC order parameters are predominant for the vanadium-based Kagome superconductor CsV₃Sb₅, which yields the following Gibbs GL free-energy functional for low-energy and long-range physics (for details see Supplementary Note I)

where due to the hexagonal lattice symmetry \({a}_{j}=a\), \({b}_{j}=b\), \({m}_{j}=m\) for j = 1, 2, 3, and \({\gamma }_{{jk}}=\gamma < 0\) for \(j,k={{{{\mathrm{1,2,3}}}}}\) and \(j\ne k\); \(b,m > 0\) for thermodynamic stability. The three s-wave SC order parameters⁴¹ correspond to the \(1d\) irreducible representation *\({{{{{\rm{S}}}}}}{{{{{{\rm{M}}}}}}}_{1}\) of the space group \(P6/{mmm}\). Here we begin with the first-order, bilinear Josephson-type couplings for simplicity, where \(\gamma < 0\) corresponds to the spontaneous time-reversal symmetry breaking⁴⁰, whereas the second-order, bi-quadratic Josephson-type couplings naturally arising from PDW orders^{35,36,37,38,40} are considered later. The mean-field SC transition point \({T}_{{{{{{\rm{mf}}}}}}}\) of the system including the inter-component Josephson-type couplings is given by \(a+\gamma =0\), below which the order parameters take finite amplitude \(|{\psi }_{j}|=\sqrt{-(a+\gamma )/b}\) and the two coherence lengths and the penetration depth of magnetic field diverge²⁷.

a Schematic setup for the Little-Parks experiment where Z₂ chirality domains defined by phases of three SC order parameters are separated by DW, which induce distortions in the real-space PDW lattice shown schematically by color. b Free energy as a function of applied magnetic flux penetrating through the superconductivity ring. In the ground states, such as \({[000]}_{{{{{{\rm{g}}}}}}}\), \({[111]}_{{{{{{\rm{g}}}}}}}\), \({[222]}_{{{{{{\rm{g}}}}}}}\) and \({[333]}_{{{{{{\rm{g}}}}}}}\), the three SC order parameters wind in the same way around the ring, which yields free energy minima at the integer multiples of \({\phi }_{0}\). In the metastable states one phase of the three SC components winds one more time than the other two in states such as [100], [211], [322] etc., or one less time than the other two in states such as [011], [122], [233] etc., which intervenes each other and yields the free energy minima separated from each other by \({\phi }_{0}/3\). The metastable states are triply degenerate. Magnetic flux and free energy are in units of \({\phi }_{0}={hc}/2e\) and \({G}_{0}\) = \({a}_{0}^{2}/b\), respectively. Dimensionless parameters are taken as \(a=-0.1\) and \(\gamma =-0.24\). The radius of the ring is taken as \(R=8\).

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In order to simulate the Little-Parks experiment we consider a SC ring with magnetic flux penetrating as shown schematically in Fig. 1(a). For clarity we mainly consider the small ring width as compared with the ring radius (for ring with finite width see Supplementary Note II). Performing numerical calculations we look for stable and/or metastable states of the system when the applied magnetic flux is swept below the mean-field transition temperature \({T}_{{{{{{\rm{mf}}}}}}}\) (with \(a+\gamma \, < \,0\)) (for details see Methods and Supplementary Notes I~III). As displayed in Fig. 1(b), there is a series of ground states with their free-energy minima at integer multiples of magnetic flux quantization \({\phi }_{0}\), such as \({[000]}_{{{{{{\rm{g}}}}}}}\), \({[111]}_{{{{{{\rm{g}}}}}}}\), \({[222]}_{{{{{{\rm{g}}}}}}}\) and \({[333]}_{{{{{{\rm{g}}}}}}}\), where the number \(L={{{{\mathrm{0,1,2}}}}}\), …… refers to the winding numbers of the three SC order parameters around the SC ring. Namely the three SC order parameters wind simultaneously in these ground states although the three phases deviate from each other by \(2\pi /3\) due to the repulsive Josephson-type coupling \(\gamma\, < \,0\) in Eq. (1). In these states the system responds to the external magnetic field in the same way as single-component superconductors⁸; especially the magneto resistance oscillates with applied magnetic flux in the period \({\phi }_{0}\) as seen in the recent Little-parks experiment at low temperatures²⁵.

Unique to the present system, we find another series of metastable states at free energy higher than the ground states which exhibit their free-energy minima at integer multiples of \({\phi }_{0}/3\) starting from the zero applied magnetic flux. Distinguished from the ground states, one phase of the three SC order parameters winds \(2\pi\) more than the other two in states [100], [211], [322] etc., or less than the other two in states [011], [122], [233] etc., which intervene each other and yield the free-energy minima separated from each other by \({\phi }_{0}/3\). We believe these metastable states are responsible for the \({\phi }_{0}/3\) magneto resistance oscillations in the recent Little-Parks experiment at temperatures ranging from 2.9 K to 2.4 K²⁵.

The phase winding in these metastable states is much richer as compared with the ground states. In state [100] depicted in Fig. 2a, b, since the SC order parameter \({\psi }_{1}\) acquires \(2\pi\) phase around the ring, its phase changes faster than the other two SC order parameters, which renders inevitably two DWs: one at the north pole of the ring where the phases of \({\psi }_{1}\) and \({\psi }_{2}\) cross each other, the other one at the south pole of the ring where the phases of \({\psi }_{1}\) and \({\psi }_{3}\) cross each other as shown in Fig. 2b. As addressed in previous works^{27,28,29,30,31} on SC states with three order parameters and \(\gamma < 0\) (see Eq. (1)), a \({{{{{{\rm{Z}}}}}}}_{2}\) chirality can be defined based on the relative phase differences when the Josephson-type coupling takes \(\gamma < 0\) in Eq. (1), where the time reversal symmetry is broken spontaneously. It is clear that in state [100] the two halves of the SC ring correspond to two domains with opposite chirality as depicted by the red and blue colors in the circle at the ring center in Fig. 2b. The DWs are against the Josephson couplings, which raises the free energy of state [100] above the ground state. The system compromises the energy cost by tuning amplitudes of SC order parameters according to the total free-energy expression as can be seen in Fig. 2b for state [100]. When the winding numbers increase, the three phases interweave in magnificent patterns forming beautiful phase kaleidoscopes as displayed in Fig. 2b.

a Free energy for the metastable states replotted from Fig. 1b. b Distributions of phases of the three SC order parameters around the SC ring in the metastable states at the applied magnetic flux marked by circles in (a) where, for example, [211] refers to the state with phase winding 2, 1 and 1 in the SC order parameter \({\psi }_{1},\) \({\psi }_{2}\) and \({\psi }_{3}\), respectively. c Same for (b) except for the ground states at applied magnetic flux of integer multiples of \({\phi }_{0}\). d Distributions of the amplitudes of the three SC order parameters around the SC ring in the metastable states at the applied magnetic flux marked by squares in (a), where the winding numbers jump corresponding to penetration of vortices into the SC ring in one or two out of three SC order parameters. Parameters are taken same as Fig. 1.

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Metastable states should also take place at integer multiples of \({\phi }_{0}\) in order to complete the \({\phi }_{0}/3\) series. Comparing metastable states [000], [111], [222] and [333] with same winding numbers in the three SC order parameters in Fig. 2b with their counterparts in the ground state [000]_g, [111]_g, [222]_g and [333]_g in Fig. 2c, it is clear that in the metastable states there are two DWs between same two SC order parameters, such as \({\psi }_{1}\) and \({\psi }_{2}\), whereas in other metastable states shown in Fig. 2 the two DWs involve three SC order parameters and there is no DW in ground states⁴². There are other possible metastable states including four or more DWs, which take higher free energies and are thus expected at higher temperatures. Nevertheless, their free-energy minima always take place at integer multiples of \({\phi }_{0}/3\), same as the metastable states with two DWs displayed in Fig. 1b and Fig. 2 (see Supplementary Notes IV and V).

Starting from one of the metastable states such as state [100] and increasing the applied magnetic flux, the system remains to the same winding configuration up to a magnetic flux \(\phi /{\phi }_{0}\) = 0.885, which is much higher than \(\phi /{\phi }_{0}\) = 2/3 where the free-energy minimum of state [011] locates. As the price, the system acquires a large free energy which is induced by the large suppression in SC order parameters \({\psi }_{2}\) and \({\psi }_{3}\) as depicted in Fig. 2d. Then, upon a tiny increase of magnetic flux to \(\phi /{\phi }_{0}\) = 0.89 state [100] loses its metastability and the system jumps to state [111] with a smaller free energy accompanied by the recovery of the amplitudes of \({\psi }_{2}\) and \({\psi }_{3}\). Obviously, this is nothing but penetration of two vortices into the SC ring, each carried by \({\psi }_{2}\) and \({\psi }_{3}\). Similarly, around \(\phi /{\phi }_{0}\) = 2.215 two vortices penetrate into the SC ring, both carried by \({\psi }_{1}\).

Collective mode involving three Cooper pairs

Why can states with winding numbers distinct in the three SC order parameters be metastable? First of all, there is a double degeneracy in the ground state associated with a Z₂ chirality defined by the relative phases of the three SC order parameters in the present system with repulsive Josephson-type coupling between three SC order parameter (\(\gamma\, < \,0\) in Eq. (1)). In addition, it can be shown analytically that in the system given in Eq. (1) there are two diverging coherence lengths, which accommodate two distinct collective modes²⁷ (see Supplementary Note VI). The collective mode associated with the shorter coherence length \({\xi }_{{{{{{\rm{H}}}}}}}=\hslash /\sqrt{-2\left(a+\gamma \right)m}\) involves only variation in the amplitudes whereas the phase differences remain locked, reminiscent of the Higgs mode known for single-component superconductivity as schematically displayed in Fig. 3a^33,34. The longer coherence length \({\xi }_{{{{{{\rm{HL}}}}}}}=\hslash /\sqrt{-\left(a+\gamma \right)m}\) is accompanied by variation in amplitudes and phases of the SC order parameters simultaneously, which constitutes the Higgs-Leggett mode^27,32 displayed schematically in Fig. 3b as a new collective mode in superconductivity. With the Higgs-Leggett mechanism, the system responds to the applied magnetic flux by enhancing amplitude in one SC order parameter and squeezing the phase difference between the two remaining SC order parameters and simultaneously reducing their amplitudes (see Fig. 2d and Supplementary Note VI). When the phase difference shrinks to zero and then changes sign, a DW between Z₂ chirality domains appears. This renders states with winding numbers distinct in the three SC order parameters, and thus local free-energy minima at integer multiples of \({\phi }_{0}/3\), as revealed above (see Fig. 2). It is noticed that, while the Higgs-Leggett mode is crucial for these metastable, intermediate states, the Higgs mode coexists with the Higgs-Leggett mode as can be exposed by comparing the wavefunctions in Fig. 2d with eigenvectors of the two collective modes (see Eqs. (A21) and (A22) in Supplementary Note VI).

a Higgs mode where only variations in amplitudes of the three SC order parameters are involved, which is essentially same for SC state with a single SC order parameter. b Higgs-Leggett mode where variation in amplitude of one SC order parameter is accompanied by variations in phase differences and amplitudes of the two remaining SC order parameters. The three colored arrows indicate the three complex SC order parameters associated with a Z₂ chiral state induced by the repulsive Josephson-type coupling.

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In stark contrast, in the ground states where only the Higgs mode is permitted, all three SC order parameters carry the same winding number and are suppressed simultaneously (see Supplementary Notes III and VI). In order to accommodate more magnetic fluxes in the SC ring, all three SC order parameters have to be suppressed to zero, which costs a large free energy and thus is unfavorable at intermediate temperatures where the Higgs-Leggett mechanism sets in accompanied by domain walls of Z₂ chirality domains.

Second-order Josephson-type coupling

Instead of the first-order, bilinear Josephson-type coupling in the GL free-energy functional (1), one can consider the second-order, bi-quadratic ones such as \(-\left(\eta /2\right){{\psi }_{j}^{* }}^{2}{\psi }_{k}^{2}\) which were discussed in the context of PDW^35,36,37,38, and \(\eta\, < \,0\) for the vanadium-based Kagome superconductor CsV₃Sb₅⁴⁰ where \(b+\eta\, > \,0\) is presumed for thermodynamic stability (see Supplementary Note VII). In addition to the Z₂ chirality, the ground state gains additional degeneracy as shown in Fig. 4a. Nevertheless, as displayed in Fig. 4b, c, we find that free-energy minima at integer multiples of \({\phi }_{0}/3\) appear in the metastable states, where phase configurations with one SC order parameter winding more or less than the other two ones accompanied by DWs separating Z₂ chiral domains are stabilized by the Higgs-Leggett mechanism, similarly to the case of the first-order Josephson-type coupling. Due to the additional degeneracy in the ground state, a new \(\pi\)-phase kink \({\Pi }_{{jk}}\) appears where SC order parameters \({\psi }_{j}\) and \({\psi }_{k}\) become out-of-phase, as seen in Fig. 4c. However, the fractional magnetic flux quantization is not influenced directly since they do not change the Z₂ chirality. For the second-order Josephson coupling, analytic calculations on coherence lengths and collective modes can be carried out in the same way as for the first-order Josephson coupling, and the results remain the same (see Supplementary Note VI), although one needs to treat multiple degenerate states as displayed in Fig. 4a. This is confirmed numerically as can be seen clearly in Fig. 4b, c.

a Degenerate ground states for the second-order Josephson-type couplings with \({{{{{{\rm{Z}}}}}}}_{2}\) chirality, where the phase difference between one order parameter and the other two is \(\pm 2\pi /3\) or \(\pm \pi /3\). b Free energy as a function of applied magnetic flux through the ring for the ground (metastable) states with minima at integer multiples of \({\phi }_{0}\) (\({\phi }_{0}/3\)). c Distributions of amplitudes and phases of the three SC order parameters around the SC ring in the metastable states at the applied magnetic flux marked by circles in (b). A new \(\pi\)-phase kink \({\Pi }_{{jk}}\) is observed where the SC order parameters \({\psi }_{j}\) and \({\psi }_{k}\) become out-of-phase, which does not change the \({{{{{{\rm{Z}}}}}}}_{2}\) chirality. Domains with opposite \({{{{{{\rm{Z}}}}}}}_{2}\) chirality as depicted by the red and blue colors are separated by two DW denoted by \({D}_{{jk}}\), and the insets show phase configurations in individual domains. Magnetic flux and free energy are in units of \({\phi }_{0}={hc}/2e\) and \({G}_{0}\) = \({a}_{0}^{2}/b\), respectively. Dimensionless parameters are \(a=-0.2\), \(\eta =-0.15\) and \(R=40\).

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Previous proposals on possible \({{{{{\boldsymbol{4}}}}}}{{{{{\bf{e}}}}}}\) and \({{{{{\boldsymbol{6}}}}}}{{{{{\bf{e}}}}}}\) SC states

Exotic \(4{{{{{\rm{e}}}}}}\) and/or \(6{{{{{\rm{e}}}}}}\) SC states have been proposed previously which emerge after partial thermal melting of parent PDW states with tetragonal and/or hexagonal crystal symmetry in bulk systems in absence of external magnetic field^35,36,38,39, where fractional vortex-antivortex pairs and associated dislocations constitute the elementary topological defects. While all these possibilities are extremely exciting, to our best knowledge, there is no experimental observation on fractional vortices and dislocations associated with charge 6e in bulk systems of CsV₃Sb₅¹¹. As a possible reason for this, the energy costs due to sliding and deformation of PDW order with respect to the underlying material electronic structure turn out to be very large in the vanadium-based Kagome superconductor CsV₃Sb₅⁴⁰ as compared with the energy scale associated with variations in the three SC phase variables, which was not considered explicitly in previous works on PDW orders. In addition, the physics in the Little-Parks experiment²⁵ under concern is dominated by an external magnetic field, and geometrically the superconductor ring is narrow in the radial direction and long in the azimuthal direction, where thermal fluctuations manifest in very different ways, resulting in different topological defects and thus different intermediate states as demonstrated in the present work.

Mean-field and genuine SC phase transitions

In the present work, the fractional flux quantization \({\phi }_{0}/3\) has been clarified below the transition point \({T}_{{{{{{\rm{mf}}}}}}}\) determined by the GL free-energy functional. While these states with phase windings distinct in the three SC order parameters accompanied by DWs of opposite Z₂ chirality are metastable in the GL theory at intermediate temperatures (but below \({T}_{{{{{{\rm{mf}}}}}}}\)) the entropy associated with DWs certainly reduces the total free energy of the system, which stabilizes them thermodynamically as compared with states without DW. At an even lower, genuine transition point \({T}_{{{{{{\rm{c}}}}}}}\) these DWs should be expelled completely from the system due to energy cost, where magnetic flux quantization takes place only at integer multiples of \({\phi }_{0}\) as revealed above. As can be seen in Fig. 1c and Fig. 2i for the experiments on the vanadium-based Kagome superconductor CsV₃Sb₅²⁵, the transition point \({T}_{{{{{{\rm{mf}}}}}}}\) is around 4 K where the bulk resistance starts to drop sharply, and the genuine transition point is estimated as \({T}_{{{{{{\rm{c}}}}}}}\approx 1\) K where the bulk resistances drops to zero; clear signals of fractional magnetic flux quantization of \({\phi }_{0}/3\) appear around 3.0–2.4 K (relatively obscure signals associated with \({\phi }_{0}/2\) are detected for 2.4–1.0 K). Therefore, the present theoretical results explain consistently the intermediate SC states characteristic of fractional magnetic flux quantization observed in experiments.

About half flux quantization ϕ ₀/2

The present theory involving three SC order parameters cannot capture reasonably the half flux quantization \({\phi }_{0}/2\). Then, we have performed the similar analysis for a system of two SC order parameters with the first-order Josephson-type coupling by omitting one SC order parameter in Eq. (1) (see Supplementary Note VIII). We can only find states with the same winding number in the two SC order parameters where the free-energy minima appear at the integer multiples of \({\phi }_{0}\), whereas configurations with winding numbers distinct in the two SC order parameters are unstable. Meanwhile it is shown that, while the Higgs mode remains the same and exhibits a diverging coherence length, the Leggett mode is associated with a finite coherence length (see Supplementary Note VI).

It is worth noticing that, with two SC order parameters coupled by the second-order Josephson-type interactions, a Z₂ chirality appears, and we find metastable states with free-energy minima at integer multiples of \({\phi }_{0}/2\) (see Supplementary Note IX). This is considered to correspond to the magneto resistance oscillations with period of half magnetic flux quantization observed in the recent Little-Parks experiment, although the signals are much more obscure as compared with those of \({\phi }_{0}/3\) and \({\phi }_{0}\)²⁵. It can be shown that in this case there are two diverging coherence lengths, and that the corresponding collective modes can be identified as the decoupled Leggett mode and Higgs mode (see Supplementary Note VI). Half flux quantization was discussed before in the context of triplet superconductivity based on the GL theory⁴³.

Another possible scenario for evolution of quantum oscillations

By including the quartic terms \(\frac{1}{2}\beta ({|{\psi }_{1}|}^{2}{|{\psi }_{2}|}^{2}+{|{\psi }_{1}|}^{2}{|{\psi }_{3}|}^{2}+{|{\psi }_{2}|}^{2}{|{\psi }_{3}|}^{2})\) in the GL equation and tuning parameter \(\beta\), we found two critical values \({\beta }_{{{{{{\rm{c}}}}}}1}=2\left(b+\eta \right)\) and \({\beta }_{{{{{{\rm{c}}}}}}2}=2\left(b-\eta \right)\), with \({\beta }_{{{{{{\rm{c}}}}}}1} < {\beta }_{{{{{{\rm{c}}}}}}2}\) since \(\eta < 0\) (see Supplementary Note X). For small value of \(\beta < {\beta }_{{{{{{\rm{c}}}}}}1}\), such as \(\beta =b/2\) as shown in Fig. S4, the system remains unchanged. For \({\beta }_{{{{{{\rm{c}}}}}}1} < \beta < {\beta }_{{{{{{\rm{c}}}}}}2}\), the states with two finite SC order parameters are stabilized, whereas for \(\beta > {\beta }_{{{{{{\rm{c}}}}}}2}\), states with one finite SC order parameter become stable. This gives rise to another possible scenario for variation of the period of resistance oscillation from \({\phi }_{0}/3\) to \({\phi }_{0}/2\), then to \({\phi }_{0}\). While this sequence of phase transitions reflects the rich physics of GL theory and is itself interesting, it does not fit the physics happening in the Kagome superconductors, since, in the whole temperature region where the \({\phi }_{0}/3\) and \({\phi }_{0}\)-period oscillations in magnetoresistance in Little-Parks setup are observed experimentally, SC is always of three components as indicated by the PDW order of three Q vectors. For \(\beta < -(b+\eta )\) the system loses thermodynamical stability where higher-order terms of SC order parameters should be considered.

Other possible related phenomena

Other related SC phenomena are expected in addition to the one discussed in the present work. For a SC ring with sufficient width where supercurrent is suppressed to zero due to Meissner effect at the center part of superconductor, the magnetic flux trapped by the SC ring should be quantized into integer multiples of \({\phi }_{0}/3\) resulting in plateaus in magnetization curve⁴² (see Supplementary Note XI). In a narrow constriction between two bulk CsV₃Sb₅ crystals, the critical Josephson current is suppressed significantly (to zero in theory) when the two bulks take opposite chirality, as compared to the case where a same chirality occupies the two bulks²⁷. An unconventional intermediate SC state characterized by clustering vortices may also be possible⁴⁴.

Numerical simulations

In order to capture the period-\({\phi }_{0}/3\) oscillations of magneto resistance observed in the vanadium-based Kagome superconductor, we consider the GL free energy functional with three SC order parameters \({\psi }_{j}(j=1,2,3)\) associated with the three reciprocal lattice vectors connecting M points of the hexagonal Brillouin zone of the material as given in Eq. (1). Performing the variational analysis with respect to \({\psi }_{j}^{* }\), we obtain the GL equations:

with \(j=1,2,3\). For numerical calculations, we use dimensionless quantities: \(a\) and \(\gamma\) in units of \({a}_{0}\), length in units of \({\xi }_{1}={{\hslash }}/\sqrt{-2m{a}_{0}}\), order parameter \({\psi }_{j}\) in units of \({\psi }_{0}=\sqrt{-{a}_{0}/b}\) , \({{{{{\bf{A}}}}}}\) in units of \(\hslash c/2e{\xi }_{1}\), and free energy in units of \({G}_{0}\) = \({a}_{0}^{2}/b\), where \({a}_{0}\) is a typical energy. In this work, we take \(a=-0.1\) and \(\gamma =-0.24\) referring to a temperature below the transition point \({T}_{{{{{{\rm{mf}}}}}}}\), except otherwise noticed. For case of the second-order Josephson-type coupling, we replace the term \(\gamma {\psi }_{k}\) in Eq. (2) with \(\eta {\psi }_{k}^{2}{\psi }_{j}^{* }\), where \(\eta =-0.15\) is taken in unit of \(b\) for numerical calculations.

The simulation is implemented using finite difference method^45,46 for ring geometry of the sample with polar coordinate. For rings with small width as compared to the penetration depth, the demagnetization effect can be neglected. The gauge \(\nabla \cdot {{{{{\bf{A}}}}}}=0\) is taken with \({{{{{\bf{A}}}}}}={{{{{{\bf{e}}}}}}}_{\varphi }{Hr}/2\) for the uniform magnetic field \(H\). The boundary conditions for \({\psi }_{j}\) correspond to zero total current density normal to the sample surface.

Starting from suitable initial configurations of SC order parameters, we obtain ground state solutions and metastable solutions by the iterative relaxation method. To find the metastable states with two DWs carrying integer multiples of \({\phi }_{0}/3\), we should put initial guesses within the attractive basins of the final solutions. As an example, in order to obtain state [100] which carries magnetic flux of \({\phi }_{0}/3\) we set \({\psi }_{1}\left(\varphi \right)=\sqrt{{n}_{0}}{e}^{i\varphi },{\psi }_{2}\left(\varphi \right)=\sqrt{{n}_{0}}{e}^{i2\pi /3}\) and \({\psi }_{3}\left(\varphi \right)=\sqrt{{n}_{0}}{e}^{i4\pi /3}\) where \({n}_{0}\) is the ground state amplitude in absence of the applied magnetic flux. Two DWs separating two different chiral domains are formed after a few iteration steps from the initial guess during simulations. The algorithm is iterated until changes in SC order parameters between two steps are less than \({10}^{-6}\).

Next, we sweep slightly up/down the applied magnetic flux and recalculate the distribution of amplitudes and phases of SC order parameters with those of the previous solution as the initial guess, and reach the new solution. Repeating this process gives one of the parabolic curves in the diagram of free energy functional. Although they are not the ground states, we find that the solutions with two DWs and winding numbers distinct in the three SC order parameters are stable against small perturbations. As the applied magnetic flux deviates largely from the value where a local minimum of free energy is located, the solution jumps discontinuously to a new one with a different set of winding numbers. This results in the series of metastable states with local free energy minima at integer multiples of \({\phi }_{0}/3\).

Note added in proof: After finishing the present work in May, 2022 (arXiv: 2205.08732), we became aware of the following related works^47,48.