Detection of a pair density wave state in UTe₂

Abstract

Spin-triplet topological superconductors should exhibit many unprecedented electronic properties, including fractionalized electronic states relevant to quantum information processing. Although UTe₂ may embody such bulk topological superconductivity^{1,2,3,4,5,6,7,8,9,10,11}, its superconductive order parameter Δ(k) remains unknown¹². Many diverse forms for Δ(k) are physically possible¹² in such heavy fermion materials¹³. Moreover, intertwined^14,15 density waves of spin (SDW), charge (CDW) and pair (PDW) may interpose, with the latter exhibiting spatially modulating^14,15 superconductive order parameter Δ(r), electron-pair density^16,17,18,19 and pairing energy gap^{17,20,21,22,23}. Hence, the newly discovered CDW state²⁴ in UTe₂ motivates the prospect that a PDW state may exist in this material^24,25. To search for it, we visualize the pairing energy gap with μeV-scale energy resolution using superconductive scanning tunnelling microscopy (STM) tips^{26,27,28,29,30,31}. We detect three PDWs, each with peak-to-peak gap modulations of around 10 μeV and at incommensurate wavevectors P_i=1,2,3 that are indistinguishable from the wavevectors Q_i=1,2,3 of the prevenient²⁴ CDW. Concurrent visualization of the UTe₂ superconductive PDWs and the non-superconductive CDWs shows that every P_i:Q_i pair exhibits a relative spatial phase δϕ ≈ π. From these observations, and given UTe₂ as a spin-triplet superconductor¹², this PDW state should be a spin-triplet PDW^24,25. Although such states do exist³² in superfluid ³He, for superconductors, they are unprecedented.

Main

Bulk Cooper-pair condensates are definitely topological when their superconductive or superfluid order parameters exhibit odd parity^33,34 Δ(k) = −Δ(−k) with spin-triplet pairing. This situation is epitomized by liquid ³He, the only known bulk topological Cooper-pair condensate^35,36. Although no bulk superconductor exhibits an unambiguously topological Δ(k), attention has recently focused on the compound UTe₂ as a promising candidate^{1,2,3,4,5,6,7,8,9,10,11,12}. This material is superconducting below the critical temperature T_c = 1.65 K. Its extremely high critical magnetic field and the minimal suppression of the Knight shift³ on entering the superconductive state both imply spin-triplet superconductivity^1,2. Temperature⁴, magnetic field^4,5 and angular dependence⁵ of the superconductive quasiparticle thermal conductivity are all indicative of a superconducting energy gap with point nodes^4,5,6. In the superconductive phase, evidence for time-reversal symmetry breaking is provided by polar Kerr rotation measurements⁷ but is absent in muon-spin-rotation studies⁸. Furthermore, the superconductive electronic structure when visualized at opposite mesa edges at the UTe₂ (0–11) surface breaks chiral symmetry⁹. Dynamically, UTe₂ seems to contain both strong ferromagnetic and antiferromagnetic spin fluctuations^10,11 relevant to superconductivity. Together, these results are consistent with a spin-triplet and, thus, odd-parity, nodal, time-reversal symmetry breaking, chiral superconductor¹². Figure 1a shows a schematic of the crystal structure of this material, whereas Fig. 1c is a schematic of the Fermi surface in the (k_x, k_y) plane at k_z = 0 (dashed lines; ref. ³⁷). An exemplary order parameter Δ(k) proposed⁵ for UTe₂ is also shown schematically in Fig. 1c (solid lines), but numerous others have been proposed¹², including that of a PDW state^24,25. In theory, this PDW, if generated by time-reversal and surface-reflection symmetry breaking, is a spin-triplet PDW²⁵. Such a state is unknown for superconductors but occurs in topological superfluid ³He (ref. ³²).

Fig. 1: Momentum-space and real-space characteristics of UTe₂.

a, Schematic crystal-lattice structure of UTe₂ oriented to the primary unit cell vectors a,b and c. The (0–11) cleave plane of UTe₂ is indicated schematically by the grey-shaded plane. b, Schematic of elemental identities and atomic sites and unit cell of the (0–11) termination layer of cleaved UTe₂. c, Schematic Fermi surface in the (k_x, k_y) plane at k_z = 0 for UTe₂ is indicated by dashed curves. A schematic example of one possible superconductive order parameter magnitude is indicated by solid curves representing the magnitude of energy gap Δ(k). Here, for pedagogic purposes only, is presented a chiral, spin-triplet superconductor with energy gap nodes along the a axis or k_x axis. d, Typical topographic image T(r) of UTe₂ (0–11) surface measured with a superconducting tip at T = 280 mK (I_s = 0.5 nA, V_s = 30 mV). Inset, measured high-resolution T(r) at low junction resistance (I_s = 3 nA, V_s = 5 mV), clarifying two types of Te atom. Scale bars, 5 nm (main), 1 nm (inset). e, Measured T(q), the Fourier transform of T(r) in d, with the surface reciprocal-lattice points labelled as dashed orange circles, which are consistent with simulated results (Extended Data Fig. 2).

PDW visualization

In general, a PDW state is a superconductor but with a spatially modulating superconductive order parameter^14,15. Absent flowing currents or magnetic fields, a conventional spin-singlet superconductor has an order parameter

$${\Delta }_{{\rm{S}}}({\bf{r}})={\Delta }_{0}{{\rm{e}}}^{{\rm{i}}{\phi }_{{\rm{S}}}}$$

(1)

for which ϕ_S is the macroscopic quantum phase and Δ₀ the amplitude of the many-body condensate wavefunction. A unidirectional PDW modulates such an order parameter at wavevector P as

$${\Delta }_{{\rm{P}}}\left({\bf{r}}\right)=\Delta \left({\bf{r}}\right){{\rm{e}}}^{{\rm{i}}{\bf{P}}\cdot {\bf{r}}}+{\Delta }^{* }\left({\bf{r}}\right){{\rm{e}}}^{-{\rm{i}}{\bf{P}}\cdot {\bf{r}}}$$

(2)

meaning that the electron-pairing potential varies spatially. By contrast, a unidirectional CDW modulates the charge density at wavevector Q such that

$${\rho }_{{\rm{Q}}}({\bf{r}})=\rho \left({\bf{r}}\right){{\rm{e}}}^{{\rm{i}}{\bf{Q}}\cdot {\bf{r}}}+{\rho }^{* }\left({\bf{r}}\right){{\rm{e}}}^{-{\rm{i}}{\bf{Q}}\cdot {\bf{r}}}$$

(3)

The simplest interactions between these three orders can be analysed using a Ginzburg–Landau–Wilson free-energy density functional

$${\mathscr{F}}=\lambda [\,{\rho }_{{\rm{Q}}}{\Delta }_{{\rm{S}}}^{\ast }{\Delta }_{{\rm{P}}}+{\rm{c}}\,.{\rm{c}}.]$$

(4)

representing the lowest-order coupling between superconductive and density wave states.

There are two elementary possibilities: (1) if Δ_S(r) and Δ_P(r) are the predominant orders, they generate charge modulations of forms ${\rho }_{{\rm{P}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\Delta }_{{\rm{P}}}+{\Delta }_{-{\rm{P}}}^{* }{\Delta }_{{\rm{S}}}$ and ${\rho }_{2{\rm{P}}}\left({\bf{r}}\right)\propto {{\Delta }_{-{\rm{P}}}^{* }\Delta }_{{\rm{P}}}$, that is, two induced CDWs controlled by the wavevector of the PDW; (2) if Δ_S(r) and ρ_Q(r) are predominant orders, they generate modulations ${\Delta }_{{\rm{Q}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\rho }_{{\rm{Q}}}$, that is, a PDW induced at the wavevector of the CDW. In either case, the PDW state described by equation (2) subsists.

To explore UTe₂ for such physics, it is first necessary to simultaneously visualize any coexisting CDW and PDW states. Recent experimental advances have demonstrated two techniques for visualizing a PDW state. In the first^16,17,18,19, the condensed electron-pair density at location r, n(r), can be visualized by measuring the tip-sample Josephson critical-current squared ${I}_{{\rm{J}}}^{2}\left({\bf{r}}\right)$, from which

$$n({\bf{r}})\propto {I}_{{\rm{J}}}^{2}\left({\bf{r}}\right){R}_{{\rm{N}}}^{2}({\bf{r}})$$

(5)

in which R_N(r) is the normal-state junction resistance. In the second PDW visualization technique^{17,20,21,22,23}, the magnitude of the energy gap in the sample, Δ(r), is defined as half the energy separation between the two superconductive coherence peaks in the density of electronic states N(E). These occur in tunnelling conductance at signed energies Δ₊(r) and Δ₋(r) such that

$$\Delta ({\bf{r}})\equiv [{\Delta }_{+}({\bf{r}})-{\Delta }_{-}({\bf{r}})]/2$$

(6)

This can be visualized using either normal-insulator-superconductor (NIS) tunnelling^20,21,22 or superconductor-insulator-superconductor (SIS) tunnelling from a superconductive STM tip^17,23,29 whose superconductive gap energy, Δ_tip, is known a priori.

CDW visualization in normal-state UTe₂

UTe₂ crystals typically cleave to show the (0–11) surface^9,24, a schematic view of which (Fig. 1b) identifies the key atomic periodicities by vectors a* and b*. At temperature T = 4.2 K, this surface is visualized using STM and a typical topographic image T(r) is shown in Fig. 1d, whereas Fig. 1e shows its power spectral density Fourier transform T(q), with the surface reciprocal-lattice points identified by dashed orange circles. Pioneering STM studies of UTe₂ by Aishwarya et al.²⁴ have recently discovered a CDW state by visualizing the electronic density of states g(r, E) of such surfaces. As well as the standard maxima at the surface reciprocal-lattice points in g(q, E), the Fourier transform of g(r, E), Aishwarya et al. detected three new maxima with incommensurate wavevectors Q_1,2,3, signifying the existence of a CDW state occurring at temperatures up to at least T = 10 K. To emulate this, we measure g(r, V) for −25 mV < V < 25 mV at T = 4.2 K using a non-superconducting tip on the equivalent cleave surface to ref. ²⁴. Figure 2a shows a typical topographic image T(r) of the (0–11) surface measured at 4.2 K. The Fourier transform T(q) features the surface reciprocal-lattice points labelled by dashed orange circles in Fig. 2a, inset. The simultaneous image g(r, 10 mV) in Fig. 2b exhibits the typical modulations in g(r, V) and its Fourier transform g(q, V) in Fig. 2c shows the three CDW peaks²⁴ at Q_1,2,3 labelled by dashed blue circles. Inverse Fourier filtration of these three maxima only shows the incommensurate CDW state of UTe₂. Overall, this state consists predominantly of incommensurate charge-density modulations at three (0–11) in-plane wavevectors Q_1,2,3 that occur at temperatures up to at least 10 K (ref. ²⁴) and with a characteristic energy scale up to at least ±25 meV (ref. ²⁴; Methods and Extended Data Fig. 1).

Fig. 2: Visualizing the normal-state CDW of UTe₂.

a, Typical topographic image T(r) of (0–11) surface measured at 4.2 K with a non-superconducting STM tip (I_s = 1 nA, V_s = −30 mV). Inset, measured T(q), the Fourier transform of the topographic image obtained simultaneously as b. Reciprocal-lattice points labelled in dashed orange circles. Scale bar, 4 nm. b, Differential conductance image g(r, 10 mV) measured at 4.2 K. Scale bar, 4 nm. c, Fourier transform g(q, 10 mV) from g(r, 10 mV) in b. Three incommensurate CDW peaks at Q_1,2,3 labelled by dashed blue circles. d, Measured density-of-states modulations g_Q(r, 10 mV) only at the wavevectors Q_1,2,3. This is a highly typical image of the incommensurate CDW state of UTe₂ (Methods and Extended Data Fig. 1). Te₁ atomic locations of UTe₂ (0–11) surface shown as overlay. The filter size of the inverse Fourier transform is 14 Å. Scale bar, 2 nm.

Normal-tip PDW detection at NIS gap edge

Motivated by the discovery that this CDW exhibits an unusual dependence on magnetic field and by the consequent hypothesis that a PDW may exist in this material^24,25, we next consider direct PDW detection in UTe₂ by visualizing spatial modulations in its energy gap^{17,18,20,21,22,23}. The typical tunnelling conductance signature of the UTe₂ superconducting energy gap is exemplified in Fig. 3a, showing a density-of-states spectrum $N(E={\rm{e}}{\rm{V}})\propto {{\rm{d}}I/{\rm{d}}V|}_{{\rm{N}}{\rm{I}}{\rm{S}}}(V)$ measured using a non-superconducting tip at T = 280 mK and junction resistance of R ≈ 5 MΩ. Under these circumstances, researchers find only a small drop in the tunnelling conductance at energies $| E| \le | {\Delta }_{{{\rm{UTe}}}_{2}}|$ (ref. ⁹) and concomitantly weak energy maxima in N(E) at the energy-gap edges $E\approx \pm {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}$ (Fig. 3a, inset). Hence, it is challenging to accurately determine the precise value of the energy gap ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}$ (Methods and Extended Data Fig. 3). Nevertheless, we fit a second-order polynomial to the two energy maxima in measured N(E, r) surrounding $E\approx \pm {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}$, evaluate the images Δ_±(r) of these energies and then derive a gap map for UTe₂ as ${\Delta }_{{{\rm{U}}{\rm{T}}{\rm{e}}}_{2}}({\bf{r}})\equiv [{\Delta }_{+}({\bf{r}})-{\Delta }_{-}({\bf{r}})]/2$. Its Fourier transform ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{q}}\right)$ presented in Methods and Extended Data Fig. 3 shows three incommensurate energy-gap modulations occurring at wavevectors P_i=1,2,3, consistent with the wavevectors of the CDW modulations discovered in ref. ²⁴. Although this evidence of three PDW states in UTe₂ is encouraging, its weak signal-to-noise ratio owing to the shallowness of coherence peaks implies that conventional ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{NIS}}}$ spectra are inadequate for precision application of equation (6) in this material.

Fig. 3: Atomic-resolution imaging of the superconductive energy gap.

a, Typical NIS spectrum ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{NIS}}}$ from normal tip to UTe₂ (0–11) surface (I_s = 1 nA, V_s = −5 mV) at T = 280 mK. The inset focuses on the energy range in which the coherence peaks can be detected by conventional normal-tip tunnelling at $E=\pm {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}$. Visualizing the superconducting energy gap ${\Delta }_{{{\rm{UTe}}}_{2}}\left({\bf{r}}\right)$ from such ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{NIS}}}$ imaging at T = 280 mK shows three sets of energy-gap modulations occurring at PDW wavevectors P_i=1,2,3 (Methods and Extended Data Fig. 3). We find no deterministic influence of the residual density-of-states modulations on these PDW energy-gap modulations (Methods and Extended Data Fig. 9). b, Typical SIS spectrum ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ from superconducting Nb tip to UTe₂ (0–11) surface. The blue arrows indicate the convoluted conductance peak located at $\left|{\Delta }_{{\rm{tip}}}+{\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\right|$ (I_s = 3 nA, V_s = 3 mV). The inset focuses on the energy range in which subgap ${{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}({\bf{r}},V)$ peaks can be detected at energies E = A_±(r). c, Typical SIS tunnelling topograph T(r) measured at T = 280 mK. Scale bar, 2 nm. d, Exemplary normalized ${{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}({\bf{r}},V)$ focused on the energy ranges near E₊ and E₋ along the trajectory indicated as the light blue arrow in c. The modulations of the energies E₊(r) and E₋(r) of maximum conductance are clearly seen. The two sets of spectra are calibrated such that the ${{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}(V)$ peaks are particle-hole symmetric. e, Measured energy E₊(r) at which ${{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}({V}_{+})$ maxima occur in c. The UTe₂ empty-state superconductive energy gap is ${{\Delta }_{+}\left({\bf{r}}\right)=| E}_{+}\left({\bf{r}}\right)| -| {\Delta }_{{\rm{tip}}}|$, in which $| {\Delta }_{{\rm{tip}}}|$ is a constant. Scale bar, 2 nm. f, Measured energy E₋(r) at which ${{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}({V}_{-})$ maxima occur in c. The filled-state energy gap is ${{\Delta }_{-}\left({\bf{r}}\right)=| E}_{-}\left({\bf{r}}\right)| -| {\Delta }_{{\rm{tip}}}|$. Scale bar, 2 nm.

Superconductive-tip PDW detection

We turn to a well-known technique for improving the resolution of energy maxima in g(r, E) measurements. By using SIS tunnelling from a tip exhibiting high sharp conductance peaks, one can profoundly enhance energy resolution for quasiparticles^{26,27,28,29,30,31}. Most recently, this has been demonstrated in electronic fluid flow visualization²⁹ microscopy, with effective energy resolution δE ≈ 10 μeV. The SIS current I from a superconducting tip is given by the convolution

$$I(V)\propto {\int }_{o}^{eV}{N}_{{\rm{t}}{\rm{i}}{\rm{p}}}(E-eV){N}_{{\rm{s}}{\rm{a}}{\rm{m}}{\rm{p}}{\rm{l}}{\rm{e}}}(E){\rm{d}}E$$

(7)

Equation (7) demonstrates that using a superconductive tip with high sharp coherence peaks at E_± = ±Δ_tip in N_tip(E) will, through convolution, strongly enhance the resolution for measuring the energies ±Δ_sample at which energy maxima occur in N_sample(E); it will also shift the energy of these features to E = ±[Δ_sample + Δ_tip]. In Fig. 3b, we show the ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ spectrum of a UTe₂ single crystal using a superconducting Nb tip at T = 280 mK. Because the tunnelling current is given by equation (7), the clear maxima in ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ occur at energies ±(Δ_tip + Δ_sample). With this technique, the energy maxima can be identified with resolution better than δE ≤ 10 μeV when T < 300 mK (ref. ²⁹). Here we use it to improve the signal-to-noise ratio of the UTe₂ superconductive energy-gap modulations that are already detectable by conventional techniques (Methods and Extended Data Fig. 3).

The UTe₂ samples are cooled to T = 280 mK, with T(r, V) of the (0–11) cleave surface as measured by a superconductive Nb tip shown in Fig. 3c. Here we see a powerful enhancement in the amplitude and sharpness of maxima in ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ relative to Fig. 3a. Consequently, to determine the spatial arrangements of the energy of the two maxima E₊(r) and E₋(r) surrounding 1.6 meV exemplified in Fig. 3b, we make two separate g(r, V) maps in the sample bias voltage ranges −1.68 mV < V < −1.48 mV and 1.5 mV < V < 1.7 mV, and in the identical field of view (FOV). The sharp peak of each ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ is fit to a second-order polynomial ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}=a{V}^{2}+bV+c$, achieving typical quality of fit R² = 0.99 ± 0.005. The energy of maximum intensity in E₊(r) or E₋(r) is then identified analytically from the fit parameters (Methods and Extended Data Fig. 4). The fine line across Fig. 3c specifies the trajectory of an exemplary series of ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ spectra, whereas Fig. 3d presents the colour map ${\left.{\rm{d}}I/{\rm{d}}V\right|}_{{\rm{SIS}}}$ spectra for both positive and negative energy coherence peaks along this line. Periodic variations in the energies at which pairs of peaks occur are obvious, directly demonstrating that E₊(r) and E₋(r) are modulating periodically but in energetically opposite directions. Using this g(r, V) measurement and fitting procedure (Methods and Extended Data Fig. 4) yields atomically resolved images of E₊(r) and E₋(r). The magnitude of both positive and negative superconductive energy gaps of UTe₂ is then ${\Delta }_{\pm }\left({\bf{r}}\right)\equiv \left|{E}_{\pm }\left({\bf{r}}\right)\right|-{\rm{| }}{\Delta }_{{\rm{tip}}}|$, in which |Δ_tip| is constant. These two independently measured gap maps Δ₊(r) and Δ₋(r) are spatially registered to each other at every location with 27-pm precision so that the cross-correlation coefficient between them is X ≅ 0.92, meaning that the superconducting energy-gap modulations are entirely particle-hole symmetric (Fig. 3e,f, Methods and Extended Data Fig. 5).

From these and equivalent data, the UTe₂ superconducting energy-gap structure ${\Delta }_{{{\rm{UTe}}}_{2}}\left({\bf{r}}\right)=\left({\Delta }_{+}\left({\bf{r}}\right)+{\Delta }_{-}\left({\bf{r}}\right)\right)/2$ can now be examined for its spatial variations δΔ(r) by using

$$\delta \Delta ({\bf{r}})\equiv {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)-\left\langle {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\right\rangle $$

(8)

in which $\left\langle {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)\right\rangle$ is the spatial average over the whole FOV. Figure 4a shows measured δΔ(r) in the same FOV as Fig. 3c. The Fourier transform of δΔ(r), δΔ(q), is presented in Fig. 4b, in which the surface reciprocal-lattice points are identified by dashed orange circles. The three further peaks labelled by dashed red circles represent energy-gap modulations with incommensurate wavevectors at P_1,2,3 of the PDW state in UTe₂. Focusing only on these three wavevectors P_1,2,3, we perform an inverse Fourier transform to show the spatial structure of the UTe₂ PDW state in Fig. 4c (Methods). This state seems to consist predominantly of incommensurate superconductive energy-gap modulations at three (0–11) in-plane wavevectors P_1,2,3 with a characteristic energy scale 10 μeV for peak-to-peak modulations.

Fig. 4: Visualizing the PDW state of UTe₂.

a, Measured variations in energy gap δΔ(r) from Fig. 3c. Scale bar, 2 nm. b, Measured δΔ(q) from a. The surface reciprocal-lattice points are labelled by dashed orange circles and the PDW peaks at P_1,2,3 are labelled by dashed red circles. P_1,2,3 are linked by reciprocal-lattice vectors (Extended Data Fig. 10). δΔ(r) and δΔ(q) exhibit superior signal-to-noise ratio as compared with the normal-tip gap map ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$ (Extended Data Fig. 8). c, Inverse Fourier transform filtered δΔ(q) of panel a at P_1,2,3 shows the first visualization of the PDW (filter size is 11.4 Å). The PDW is repeatable in experimental measurements (Extended Data Fig. 6) and also independently evidenced in Methods and Extended Data Fig. 7. d, Image of g_Q(r, −9 mV) of the CDW, measured at T = 4.2 K in the same FOV as panel c from inverse Fourier transform filtered g(r, −9 mV) at Q_1,2,3 (filter size is 11.4 Å). The precision of registration between the CDW and PDW images is 27 pm (Methods and Extended Data Fig. 5). These coincident CDW and PDW images are measured in the energy ranges 10 meV and 10 μeV, respectively, and appear visually distinct, yet their cross-correlation coefficient of −0.68 shows their anticorrelation. The CDW maxima exist at the PDW minima. e, Statistical relationship between δΔ_P(r) and g_Q(r, −9 mV). The δΔ_P(r) and g_Q(r, −9 mV) are strongly anticorrelated spatially. They are approximately negatives of each other. f, Statistics of the relative spatial phase difference δϕ_i between the CDW phase ${\phi }_{i}^{{\rm{C}}}\left({\bf{r}}\right)$ at Q_i and the PDW phase ${\phi }_{i}^{{\rm{P}}}\left({\bf{r}}\right)$ at P_i in the coterminous images g_Q(r, −9 mV) and δΔ_P(r). The spatial phase difference, defined as $| \delta {{\phi }}_{i}\left({\bf{r}}\right)| \equiv | {{\phi }}_{i}^{{\rm{P}}}\left({\bf{r}}\right)-{{\phi }}_{i}^{{\rm{C}}}\left({\bf{r}}\right)|$, between all three CDWs and PDWs at Q_i:P_i is ${\left|\delta \phi \right|}_{{\rm{RMS}}}=0.96{\rm{\pi }}$. g, Coterminous measurement of CDW g_Q(r, −9 mV) and PDW δΔ_P(r) along a trajectory (arrows in panels c and d).

Energy modulations of Andreev resonances

There is an alternative modality of SIS tunnelling, namely, measuring the effects of Andreev reflections. For two superconductors with very different gap magnitudes, when the sample bias voltage shifts the smaller gap edge (UTe₂ in this case) to the chemical potential of the other superconductor, the Andreev process of electron (hole) transmission and hole (electron) reflection plus electron-pair propagation can produce an energy maximum in dI/dV|_SIS (ref. ³⁸), an effect well attested by experiment³⁹. Here, by imaging the signed energies of A_±(r) of two subgap dI/dV|_SIS maxima detected throughout our studies and identified by the green arrows in Fig. 3b, an Andreev-resonance measure of the UTe₂ energy gap is conjectured as ${\Delta }_{{\rm{A}}}({\bf{r}})\equiv \left[{A}_{+}({\bf{r}})-{A}_{-}({\bf{r}})\right]/2$. These data are presented in Methods and Extended Data Fig. 7 and show a Δ_A(r) modulating with amplitude approximately 10 μeV at wavevectors P_1,2 state, further evidencing the UTe₂ PDW state.

Visualizing the interplay of PDW and CDW

Finally, one may consider the two cases of intertwining outlined earlier: (1) Δ_S(r) and Δ_P(r) are predominant and generate charge modulations ${\rho }_{{\rm{P}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\Delta }_{{\rm{P}}}+{\Delta }_{-{\rm{P}}}^{* }{\Delta }_{{\rm{S}}}$ and ${\rho }_{2{\rm{P}}}\left({\bf{r}}\right)\propto {{\Delta }_{-{\rm{P}}}^{* }\Delta }_{{\rm{P}}}$ or (2) Δ_S(r) and ρ_Q(r) are predominant and generate pair density modulations ${\Delta }_{{\rm{Q}}}\left({\bf{r}}\right)\propto {\Delta }_{{\rm{S}}}^{* }{\rho }_{{\rm{Q}}}$. For case (1) to be correct here, a PDW with magnitude 10 μeV coexisting with a superconductor of gap maximum near 250 μeV must generate a CDW on the energy scale 25 meV and exist up to at least T = 10 K. For case (2) to be valid, a normal-state CDW with eigenstates at energies up to 25 meV coexisting with a superconductor of gap magnitude 250 μeV must generate a PDW at the same wavevector and with amplitude near 10 μeV. Intuitively, the latter case seems the most plausible for UTe₂.

To explore this issue further, we visualize the CDW in the non-superconductive state at T = 4.2 K, then cool to T = 280 mK and visualize the PDW in precisely the same FOV. Figure 4c,d shows the result of such an experiment in the FOV of Fig. 3c. The CDW and PDW images are registered to the underlying lattice and to each other with 27-pm precision. Comparing their coterminous images in Fig. 4c and Fig. 4d shows that the CDW and PDW states of UTe₂ appear spatially distinct. Yet, they are actually registered to each other in space, being approximate negative images of each other (Fig. 4e) and with a measured relative phase for all three P_i:Q_i pairs of $| \delta {\phi }_{i}| \cong \pi$ (Fig. 4f, Methods and Extended Data Fig. 10). A typical example of this effect is shown in a line cut across Fig. 4c,d along the Te chain direction, with the directly measured values shown in Fig. 4g. The direct and comprehensive knowledge of CDW and PDW characteristics and interactions presented in Fig. 4 now motivates search for a Ginzburg–Landau description capable of capturing this complex intertwined phenomenology and that reported in ref. ²⁴.

Conclusions

Notwithstanding such theoretical challenges, in this study, we have demonstrated that PDWs occur at three incommensurate wavevectors P_i=1,2,3 on the (0–11) surface of UTe₂ (Fig. 4b,c). These wavevectors are indistinguishable from the wavevectors Q_i=1,2,3 of the prevenient normal-state CDW at the equivalent surface (Figs. 2c and 4d). All three PDWs exhibit peak-to-peak gap energy modulations in the range near 10 μeV (Fig. 4c,g). When the P_i=1,2,3 PDW states are visualized at 280 mK in the identical FOV as the Q_i=1,2,3 CDWs visualized above the superconductive T_c, every Q_i:P_i pair is spatially registered to each other (Fig. 4c,d), but with a relative phase shift of $| \delta {\phi }_{i}\,| \cong \pi$ throughout (Fig. 4f). Given the premise that UTe₂ is a spin-triplet superconductor¹², the PDW phenomenology detected and described herein (Fig. 4) signifies the entrée to spin-triplet PDW physics.

Methods

CDW visualization in non-superconductive UTe₂

Differential conductance imaging of CDW at T = 4.2 K

At T = 4.2 K and using superconducting tips to study the UTe₂ (0–11) surface, we measure differential tunnelling conductance spectra g(r, V) to visualize the CDW in the normal state of UTe₂. Extended Data Fig. 1a–d shows g(r, V) images V = −7 mV, −15 mV, −23 mV and −29 mV with Fourier transform g(q, V) shown as Extended Data Fig. 1e–h. Three CDW peaks at Q_1,2,3 occur in all g(q, V), representing incommensurate charge-density modulations with energy scale up to approximately 30 meV, consistent with ref. ²⁴.

CDW visualization at incommensurate wavevectors Q _1,2,3

To calculate the amplitude ${g}_{{{\bf{Q}}}_{i}}({\bf{r}})$ of the CDW modulation represented by the peaks at Q_i (i = 1, 2, 3), we apply a two-dimensional computational lock-in technique. Here g(r) is multiplied by the term ${{\rm{e}}}^{{\rm{i}}{{\bf{Q}}}_{i}\cdot {\bf{r}}}$ and integrated over a Gaussian filter to obtain the lock-in signal

$${g}_{{{\bf{Q}}}_{i}}\left({\bf{r}}\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{{\bf{r}}}}\int {\rm{d}}{\bf{R}}g\left({\bf{R}}\right){{\rm{e}}}^{{\rm{i}}{{\bf{Q}}}_{i}\cdot {\bf{R}}}{{\rm{e}}}^{\frac{{\left|{\bf{r}}-{\bf{R}}\right|}^{2}}{2{\sigma }_{{\bf{r}}}^{2}}}$$

(9)

in which σ_r is the cutoff length in the real space. In q space, this lock-in signal is

$${g}_{{{\bf{Q}}}_{i}}({\bf{r}})={{\mathscr{F}}}^{-1}{g}_{{{\bf{Q}}}_{i}}({\bf{q}})={{\mathscr{F}}}^{-1}\,\left[{\mathscr{F}}(\,g({\bf{r}}){{\rm{e}}}^{{\rm{i}}{{\bf{Q}}}_{i}\cdot {\bf{r}}})\cdot \frac{1}{\sqrt{2\pi }{\sigma }_{{\bf{Q}}}}{{\rm{e}}}^{-\frac{{q}^{2}}{2{{\sigma }_{{\bf{Q}}}}^{2}}}\right]$$

(10)

in which σ_Q is the cutoff length in q space. Here ${{\sigma }_{{\bf{r}}}=1/\sigma }_{{\bf{Q}}}$. Next, g_Q(r, V), the inverse Fourier transform of the combined Q_i (i = 1, 2, 3) CDWs, is presented in Extended Data Fig. 1i–l.

To specify the effect of filter size on the inverse Fourier transform, we show in Extended Data Fig. 1m–t the real-space density of states g(r, 10 mV), its Fourier transform g(q, 10 mV) and the evolution of inverse Fourier transform images as a function of the real-space cutoff length σ_r. The differential conductance map g_Q(r, 10 mV) is shown at a series of σ_r, including 10 Å, 12 Å, 14 Å, 18 Å, 24 Å and 35 Å. The distributions of the CDW domains in the filtered g_Q(r, 10 mV) images with cutoff lengths of 10 Å, 12 Å, 14 Å, 18 Å and 24 Å are highly similar. The cutoff length used in Fig. 2d is 14 Å, such that the domains of the CDW modulations are resolved and the irrelevant image distortions are excluded. The same filter size of 14 Å is chosen for all three Q_i vectors. Formally, the equivalent inverse Fourier transform analysis is carried out for Fig. 4c,d but with a filter size of 11.4 Å to filter both the CDW and PDW peaks.

Simulated UTe₂ topography

To identify q-space peaks resulting from the (0–11) cleave-plane structure of UTe₂, we simulate the topography of the UTe₂ cleave plane and Fourier transform. Subsequently, we can distinguish clearly the CDW signal from the structural periodicity of the surface. The simulation is calculated on the basis of the ideal lattice constant of the (0–11) plane of the UTe₂, a* = 4.16 Å and inter-Te-chain distance b* = 7.62 Å. Extended Data Fig. 2a is a simulated T(r) image in the FOV of 14.5 nm. The simulated topography T(r) is in good agreement with experimental T(r) images presented throughout. The Fourier transform, T(q), of the simulated T(r) in Extended Data Fig. 2b shows six sharp peaks, confirming that they are the primary peaks resulting from the cleave-plane structure. Most notably, the CDW peaks in Fig. 2c are not seen in the simulation. They are therefore not caused by the surface periodicity but instead originate from the electronic structure, as first demonstrated in ref. ²⁴.

Normal-tip PDW detection at the NIS gap edge of UTe₂

Initial STM searches for a PDW on UTe₂ were carried out using a normal tip at 280 mK. Extended Data Fig. 3a shows a typical line cut of the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ spectrum taken from the FOV in Extended Data Fig. 3b. There is a large residual density of states near the Fermi level. The gap depth H is defined as the difference between the gap bottom in the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ spectrum and the coherence peak height, that is, $H\equiv {\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{| }}}_{{\rm{NIS}}}(V\equiv {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}})-{\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{| }}}_{{\rm{NIS}}}(V\equiv 0)$. Its modulation is extracted from the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ line cut and presented in Extended Data Fig. 3c; it modulates perpendicular to the Te atom chains.

Conventional, NIS tunnelling discloses superconducting energy-gap modulations as shown in Extended Data Fig. 3a. The superconducting energy gap is defined as half of the peak-to-peak distance in the ${\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{| }}}_{{\rm{NIS}}}$ spectrum (Fig. 3a and Extended Data Fig. 3d). Its magnitude $| {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}|$ is found to lie approximately between 250 µeV and 300 µeV. We measure variations in the coherence peak position from the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ spectrum at each location r. The two energy maxima near ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}$ of each ${\rm{d}}I\,/\,{\rm{d}}V\,{{\rm{| }}}_{{\rm{NIS}}}$ spectrum are fitted with a second-order polynomial function (${R}_{{\rm{RMS}}}^{2}=0.87$). The energy gap is defined as the maxima of the fit, Δ₊ for V > 0 and Δ₋ for V < 0. The total gap map ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})\equiv [{\Delta }_{+}({\bf{r}})-{\Delta }_{-}({\bf{r}})]/2$ is derived from Δ₊ and Δ₋ (Extended Data Fig. 3e). The Fourier transform of ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$, ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{q}}\right)$ (Extended Data Fig. 3f), shows three peaks at wavevectors P_i=1,2,3. They are the initial signatures of the energy-gap modulations of the three coexisting PDW states in UTe₂.

Superconductive-tip PDW visualization at the SIS gap edge of UTe₂

Tip preparation

Atomic-resolution Nb superconducting tips are prepared by field emission. To determine the tip gap value during our experiments, we measure conductance spectrum on UTe₂ at 1.5 K (T_c = 1.65 K), in which the UTe₂ superconducting gap is closed. The tip gap $| {\Delta }_{{\rm{tip}}}| \cong 1.37\,{\rm{meV}}$ is the energy of the coherence peak (Extended Data Fig. 4a). The measured spectrum is fitted using a Dynes model⁴⁰. The typical ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}$ measured at 280 mK on UTe₂ (Fig. 3b) shows the total gap value $E=| {\Delta }_{{\rm{tip}}}| +| {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}| \approx 1.62\,{\rm{meV}}$. Therefore, we estimate $| {\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}| \approx 0.25\,{\rm{meV}}$, consistent with a previous report⁹ and the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ shown in Fig. 3a and Extended Data Fig. 3.

SIS tunnelling amplification of energy-gap measurements

To determine the energy of E₊(r) and E₋(r) at which the maximum conductance in ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}(V)$ occurs, we fit the peak of the measured ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}(V)$ spectra using a second-order polynomial fit:

$$g(V)=a{V}^{2}+bV+c$$

(11)

This polynomial closely fits the experimental data. Extended Data Fig. 4b,c shows two typical ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}(V)$ spectra measured at +V and −V along the trajectory indicated in Fig. 3c. The evolution of fits g(V) in Extended Data Fig. 4d,e shows a very clear energy-gap modulation.

Shear correction and Lawler–Fujita algorithm

To register several images to precisely the same FOV, the Lawler–Fujita algorithm is applied to the experimental data. Then, to recover the arbitrary hexagon of the Te lattice, shear correction is applied to correct any image distortions caused by long-range scanning drift during days of continuous measurement.

To correct against picometre-scale distortions of the lattice, we apply the Lawler–Fujita algorithm. Let $\widetilde{T}(\widetilde{{\bf{r}}})$ represent a topograph of a perfect UTe₂ lattice without any distortion. Three pairs of Bragg peaks Q₁, Q₂ and Q₃ can be obtained from Fourier transform of $\widetilde{T}(\widetilde{{\bf{r}}})$. Hence $\widetilde{T}(\widetilde{{\bf{r}}})$ is expected to take the form

$$\widetilde{T}(\widetilde{{\bf{r}}})=\mathop{\sum }\limits_{i=1}^{3}{T}_{i}\cos \left({{\bf{Q}}}_{i}\cdot \widetilde{{\bf{r}}}+{\theta }_{i}\right)$$

(12)

The experimentally obtained topography T(r) may suffer from a slowly varying position-dependent spatial phase shift θ_i(r), which can be given by

$$T({\bf{r}})=\mathop{\sum }\limits_{i=1}^{3}{T}_{i}\cos \left({{\bf{Q}}}_{i}\cdot {\bf{r}}+{\theta }_{i}({\bf{r}})\right)$$

(13)

To get θ_i(r), we use a computational two-dimensional lock-in technique to the topography

$${A}_{{\bf{Q}}}\,({\bf{r}})=\int {\rm{d}}{\bf{R}}T({\bf{R}}){{\rm{e}}}^{{\rm{i}}{\bf{Q}}\cdot {\bf{R}}}{{\rm{e}}}^{-\frac{{\left({\bf{r}}-{\bf{R}}\right)}^{2}}{2{\sigma }^{2}}}$$

(14)

$${A}_{{{\bf{Q}}}_{i}}\left(\,{\bf{r}}\right)={{\mathscr{F}}}^{-1}{A}_{{{\bf{Q}}}_{i}}\left(\,{\bf{q}}\right)={{\mathscr{F}}}^{-1}\,\left[{\mathscr{F}}(T\left({\bf{r}}\right){{\rm{e}}}^{{\rm{i}}{{\bf{Q}}}_{i}\cdot {\bf{r}}})\cdot \frac{1}{\sqrt{2\pi }{\sigma }_{{\rm{Q}}}}{{\rm{e}}}^{-\frac{{q}^{2}}{2{{\sigma }_{{\rm{Q}}}}^{2}}}\right]$$

(15)

$$\left|{A}_{{\bf{Q}}}\left(\,{\bf{r}}\right)\right|=\sqrt{{\left({\rm{Re}}{A}_{{\bf{Q}}}\left({\bf{r}}\right)\right)}^{2}+{\left({\rm{Im}}{A}_{{\bf{Q}}}\left({\bf{r}}\right)\right)}^{2}}$$

(16)

$${\theta }_{i}({\bf{r}})={\tan }^{-1}\frac{{\rm{Im}}{A}_{{\bf{Q}}}\left(\,{\bf{r}}\right)}{{\rm{Re}}{A}_{{\bf{Q}}}\left(\,{\bf{r}}\right)}$$

(17)

for which σ is chosen to capture the lattice distortions. In the Lawler–Fujita analysis, we use σ_q = 3.8 nm⁻¹. Mathematically, the relationship between the distorted and the perfect lattice for each Q_i is ${{\bf{Q}}}_{i}\cdot {\bf{r}}{\boldsymbol{+}}{\theta }_{i}({\bf{r}})={{\bf{Q}}}_{i}\cdot \widetilde{{\bf{r}}}+{\theta }_{i}$. We define another global-position-dependent quantity, the displacement field ${\bf{u}}\left({\bf{r}}\right)={\bf{r}}-\widetilde{{\bf{r}}}$, which can be obtained by solving equations

$${\bf{u}}\left({\bf{r}}\right)={\left(\begin{array}{c}\begin{array}{c}{{\bf{Q}}}_{1}\\ {{\bf{Q}}}_{2}\end{array}\\ {{\bf{Q}}}_{3}\end{array}\right)}^{-1}\left(\begin{array}{c}\begin{array}{c}{\theta }_{1}-{\theta }_{1}\left({\bf{r}}\right)\\ {\theta }_{2}-{\theta }_{2}\left({\bf{r}}\right)\end{array}\\ {\theta }_{3}-{\theta }_{3}\left({\bf{r}}\right)\end{array}\right)$$

(18)

Finally, a drift-corrected topography, $\widetilde{T}\left(\widetilde{{\bf{r}}}\right)$ is obtained by

$$\widetilde{T}\left(\widetilde{{\bf{r}}}\right)=T({\bf{r}}-{\bf{u}}\left({\bf{r}}\right))$$

(19)

By applying the same correction of u(r) to the simultaneously taken differential conductance map g(r), we can get

$$\widetilde{g}\left(\widetilde{{\bf{r}}}\right)=g({\bf{r}}-{\bf{u}}({\bf{r}}))$$

(20)

in which $\widetilde{g}\left(\widetilde{{\bf{r}}}\right)$ is the drift-corrected differential conductance map.

Lattice registration of UTe₂ energy gap ${{\boldsymbol{\Delta }}}_{{{\bf{UTe}}}_{{\bf{2}}}}{\boldsymbol{(}}{\bf{r}}{\boldsymbol{)}}$

We measure two separate ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}({\bf{r}},V)$ maps separated by several days and in two overlapping FOVs, with energy ranges −1.68 meV < E < −1.48 meV and 1.5 meV < E < 1.7 meV. Therefore, we obtain two datasets, T₊(r) with the simultaneous ${{{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}}_{+}({\bf{r}},V)$ at positive bias and T₋(r) with the simultaneous ${{{\rm{d}}I/{\rm{d}}V| }_{{\rm{SIS}}}}_{-}({\bf{r}},V)$ at negative bias.

After the shear and Lawler–Fujita corrections are applied, the lattice in the corrected topographs of T₊(r) and T₋(r) become nearly perfectly periodic. Next, we perform rigid spatial translations to register the two topographs to the exact same FOV with a lateral precision better than 27 pm. Extended Data Fig. 5a,b shows two topographs of registered T₊(r) and T₋(r). Cross-correlation (XCORR) of two images I₁ and I₂, X(r, I₁, I₂) at r is obtained by sliding two images r apart and calculating the convolution,

$$X({\bf{r}},{I}_{1},{I}_{2})=\frac{\int {I}_{1}^{\ast }({{\bf{r}}}_{1}){I}_{2}({\bf{r}}+{{\bf{r}}}_{1}){\rm{d}}{{\bf{r}}}_{1}}{\sqrt{\int {|{I}_{1}({{\bf{r}}}_{1})|}^{2}{\rm{d}}{{\bf{r}}}_{1}\int {|{I}_{2}({{\bf{r}}}_{2})|}^{2}{\rm{d}}{{\bf{r}}}_{2}}}$$

(21)

in which the denominator is a normalization factor such that, when I₁ and I₂ are exactly the same image, we can get X(r = 0, I₁, I₂) = 1 with the maximum centred at (0, 0) cross-correlation vector. Extended Data Fig. 5c shows that the maximum of XCORR between T₊(r) and T₋(r) coincides with the (0, 0) cross-correlation vector. The offset of the two registered images are within one pixel. The multiple-image registration method is better than 0.5 pixels = 27 pm in the whole FOV and the maxima of the cross-correlation coefficient between the topographs is 0.93. All transformation parameters applied to T₊(r) and T₋(r) to yield the corrected topographs are subsequently applied to the corresponding ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}({\bf{r}},V)$ maps obtained at positive and negative voltages.

Particle-hole symmetry of the superconducting energy gap ${{\boldsymbol{\Delta }}}_{{{\bf{UTe}}}_{{\bf{2}}}}{\boldsymbol{(}}{\bf{r}}{\boldsymbol{)}}$

The cross-correlation map in Extended Data Fig. 5f provides a two-dimensional measure of agreement between the positive and negative ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}\left(V\right)$ energy-maxima maps in Extended Data Fig. 5d,e. The inset of Extended Data Fig. 5f shows a line cut along the trajectory indicated in Extended Data Fig. 5f. It shows a maximum of 0.92 and coincides with the (0, 0) cross-correlation vector. Thus, it shows that gap values between positive bias and negative bias are highly correlated.

PDW visualization at incommensurate wavevectors P _1,2,3

The inverse Fourier transform analysis for PDW state in Fig. 4c is implemented using the same technique described here in Methods. The filter size chosen to visualize the PDW is 11.4 Å. The inverse Fourier transform of the CDW in Fig. 4d is calculated using an identical filter size of 11.4 Å.

Independent PDW visualization experiments

To confirm that the PDW discovered is present in several FOVs, we show a typical example of the gap modulation Δ₊(r) from one different FOV in Extended Data Fig. 6. The ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}({\bf{r}},V)$ map is measured in the voltage region surrounding the positive Nb-UTe₂ energy maxima near 1.6 meV. The spectra in this FOV are fitted with a second-order polynomial and shear corrected as described here in Methods. The resulting gap map, δΔ₊(r), is presented in Extended Data Fig. 6b. The Fourier transform of this map, δΔ₊(q), is presented in Extended Data Fig. 6c. δΔ₊(q) features the same PDW wavevectors (P₁, P₂, P₃) reported in the main text.

Energy modulations of subgap Andreev resonances

Surface Andreev bound states must occur in p-wave topological superconductors⁴¹. Moreover, based on the phase-changing quasiparticle reflections at the p-wave surface, finite-energy Andreev resonances should also occur in the junction between a p-wave and an s-wave superconductor⁴² and are observed in UTe₂. Inside the SIS gap, we measure the ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}\left({\bf{r}},V\right)$ map in the energy range from −500 µeV to 500 µeV. The map is measured in the FOV in Extended Data Fig. 7a, the same FOV as in Figs. 3 and 4. Three conductance peaks are resolved at approximately −300 µeV, 0 and 300 µeV, annotated with green arrows in the typical subgap spectrum in Extended Data Fig. 7b. The energy maximum of the positive subgap states between 200 µeV to 440 µeV is assigned as A₊. The energy maximum of the negative subgap states between −440 µeV to −200 µeV is assigned as A₋. The averaged energy of the Andreev subgap states is defined as ${\Delta }_{{\rm{A}}}({\bf{r}})\equiv \left[{A}_{+}({\bf{r}})-{A}_{-}({\bf{r}})\right]/2$, which ranges from 300 µeV to 335 µeV (Extended Data Fig. 7c). Fourier transform of the subgap energies Δ_A(q) exhibit two sharp peaks at the PDW wavevectors P₁ and P₂ (Extended Data Fig. 7d).

In the case of two superconductors with very different gap magnitude, when the sample bias voltage shifts the smaller gap edge to the chemical potential, the Andreev process of electron (hole) transmission and hole (electron) reflectional plus electron-pair propagation may produce an energy maximum in dI/dV|_SIS at the voltage of smaller gap energy. Hence, the observations in Extended Data Fig. 7d may be expected if the UTe₂ superconducting energy gap is modulating at the wavevectors P₁ and P₂. Extended Data Fig. 7e shows that the energy of the Andreev states modulate in space with a peak-to-peak amplitude near 10 µeV (see histogram in Extended Data Fig. 7f).

Enhancement of signal-to-noise ratio using superconductive tips

Superconducting STM tips provide an effective energy resolution beyond the Fermi–Dirac limit. They have therefore been widely used as a method of enhancing the energy resolution of STM spectra^{26,27,28,29,30,31}.

To better quantify the signal-to-noise ratio improvement of the measured energy-gap modulations, we compare the fitting quality of the superconducting gap maps obtained using a normal tip (Extended Data Fig. 3) and a superconducting tip (Fig. 4). The fitting quality is defined using the coefficient

$${R}^{2}\left({\bf{r}}\right)=1-\frac{{\sum }_{i=1}^{N}{\left[g\left({\bf{r}},{V}_{i}\right)-{\rm{d}}I/{\rm{d}}V\left({\bf{r}},{V}_{i}\right)\right]}^{2}}{{\sum }_{i=1}^{N}{\left[g\left({\bf{r}},{V}_{i}\right)-\bar{g}\left({\bf{r}}\right)\right]}^{2}}$$

(22)

in which ${\rm{d}}I\,/\,{\rm{d}}V\left(V\right)$ is the measured spectrum, g(r, V) is the fitted spectrum and $\bar{g}\left({\bf{r}}\right)$ is the averaged fitted spectrum. Extended Data Fig. 8a shows a typical spectrum measured using a superconductive tip, ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}$ from the FOV in Fig. 3c. Extended Data Fig. 8d is a typical ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ spectrum measured using a normal tip from the FOV in Extended Data Fig. 3. The energy-maximum noise level is decisively lower in ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{SIS}}}$ spectra than in ${\rm{d}}I\,/\,{\rm{d}}V{| }_{{\rm{NIS}}}$ spectra and the fitting quality ${R}_{{\rm{SIS}}}^{2}$ is substantially higher than ${R}_{{\rm{NIS}}}^{2}$.

Extended Data Fig. 8b,c shows maps of the fitting parameter R² calculated from fitting the dI/dV|_SIS energy-maxima map obtained using a superconductive tip, that is, the ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})$ images presented in Fig. 3e,f. Extended Data Fig. 8e,f shows maps of R² calculated from the coherence peak fitting of dI/dV|_NIS obtained using a normal tip, that is, the ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}({\bf{r}})$ images presented in Extended Data Fig. 3e. Comparing these R² quality-of-fit parameter maps, we find that a much larger fraction of normal-tip coherence peak maps have poor correspondence with the fitting procedures used. For superconducting tips, the root-mean-square values of the fitting parameter, ${R}_{{\rm{RMS}}}^{2}$, are 0.98 and 0.99 for the positive and negative coherence peak fitting, respectively. The normal-tip ${R}_{{\rm{RMS}}}^{2}$ values are 0.87 and 0.86 for the positive and negative coherence peak fitting, respectively. The superconducting tip therefore demonstrably achieves a marked signal-to-noise ratio enhancement for evaluation of ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$ images.

As the signal-to-noise ratio is increased in the SIS-convoluted coherence peaks measured using a superconducting tip, it has been possible to resolve the UTe₂ energy-gap modulations of order approximately 10 μV. Fundamentally, the energy resolution is associated with the ability of the superconductive tip to resolve the energy at which the dI/dV|_SIS coherence peak reaches its maximum amplitude. Consequently, we determine our energy resolution to be 10 μV.

Thus, the same superconductor energy-gap modulations in ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$ of UTe₂ can be observed using either a superconducting tip or a normal tip. However, the former substantially increases the SIS conductance at $\left|E\right|={\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}+{\Delta }_{{\rm{tip}}}$ and allows for considerably better imaging of these energy maxima and thus ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$.

Interplay of subgap quasiparticles and PDW

Here we show simultaneous normal-tip-measured modulations of the UTe₂ subgap states and ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$ at T = 280 mK, to study their interplay. Extended Data Fig. 9a shows the integrated differential conductance from −250 μV to 250 μV, ${\sum }_{-250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{r}},E\right)$. Inverse Fourier transform of the three wavevectors Q_1,2,3 from ${\sum }_{-250\,{\rm{\mu }}{\rm{V}}}^{250\,{\rm{\mu }}{\rm{V}}}g\left({\bf{r}},E\right)$ and P_1,2,3 from the simultaneous ${\Delta }_{{{\rm{UTe}}}_{{\rm{2}}}}\left({\bf{r}}\right)$ in Extended Data Fig. 3e are compared in Extended Data Fig. 9c,d. Clearly, from the highly distinct spatial structure of these images, there is no one-to-one correspondence between the subgap density-of-states modulations and the simultaneously measured PDW energy-gap modulations in UTe₂. Overall, there is a very weak anticorrelation, with a cross-correlation value of −0.23 ± 0.05 that is not inconsistent with coincidence. Hence we demonstrate that there is no deterministic influence of the subgap density-of-states modulations on the PDW energy-gap modulations in superconducting UTe₂.

Visualizing the interplay of PDW and CDW in UTe₂

The analysis of phase difference between PDW and CDW at three different wavevectors is shown in Extended Data Fig. 10. The inverse Fourier transforms of each CDW and PDW wavevector demonstrate a clear half-period shift between the two density waves (Extended Data Fig. 10a–f). This shift motivates the statistical analysis of the phase difference. The phase map of ${g}_{{{\rm{Q}}}_{1}}({\bf{r}},-9\,{\rm{mV}})$, ${\phi }_{1}^{{\rm{C}}}\left({\bf{r}}\right)$, and the phase map of ${\Delta }_{{{\rm{P}}}_{1}}({\bf{r}})$, ${\phi }_{1}^{{\rm{P}}}\left({\bf{r}}\right)$, are calculated. The phase difference between two corresponding maps is defined as $| \delta {\phi }_{1}| ={\phi }_{1}^{{\rm{C}}}\left({\bf{r}}\right){\boldsymbol{-}}{\phi }_{1}^{{\rm{P}}}\left({\bf{r}}\right)$ for the P₁:Q₁ wavevectors. Identical procedures are carried out for P₂:Q₂ and P₃:Q₃. The histograms resulting from this procedure show that the statistical distributions of the phase shift $|\delta {\phi }_{i}{\rm{|}}$ are centred around π (Extended Data Fig. 10j–l). Although the distribution varies, this π phase shift reinforces the observation of the spatial anticorrelation between CDW and PDW.

As shown in the inset of Extended Data Fig. 10g, the three PDW wavevectors are related by reciprocal lattice vectors: P₂ = P₁ − G₃ and P₃ = G₁ − P₁. Nevertheless, the three UTe₂ PDWs seem to be independent states when analysed in terms of the spatial modulations of the amplitude of the P_1,2,3 peaks from Fig. 4 using equation (16). The amplitude of P_1,2 has a domain width beyond 10 nm in the real space (Extended Data Fig. 10g,h). The amplitude of P₃ is short-ranged, of which the averaged domain width is approximately 5 nm (Extended Data Fig. 10i). The one-pixel shift of P₃ from the central axis is within the error bar of experimental measurements. The spatial distributions of the three PDWs are negligibly correlated with cross-correlation values of their amplitude of X(P₁, P₂) = −0.3, X(P₁, P₃) = 0.9 and X(P₂, P₃) = 0.28. The weak cross-correlation relationships indicate that the three PDWs are independent orders.