A quantum phase transition (QPT) between competing ground states at zero temperature (T = 0) is a central paradigm seen in a variety of systems such as magnetic materials, cold atom gases, and two dimensional (2D) electrons1,2,3. A prototypical example is a field(B)-induced superconductor-insulator transition (SIT) observed in disordered 2D superconductors, which shows a clear quantum critical point (QCP) of (2 + 1)D quantum systems4,5,6. Since the localization theory prohibits a metallic ground state in 2D systems7, quantum fluctuations drive a direct transition between condensation and localization of electrons, leading to metallic dissipation with a saturated resistance only at the QCP.

However, an anomalous metallic (AM) ground state emerges unexpectedly over a wide B range in weakly disordered amorphous films8,9,10,11, crystalline 2D superconductors12,13,14, and Josephson junction arrays (JJAs)15,16,17. The origin of the superconductor-metal-insulator transition (SMIT) has been debated8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 mainly from the following two viewpoints. One is a mechanism of the metallic dissipation at T = 0. Some of theories predicted that it is attributed to quantum creep of vortices18,19,20, and recent experiments have detected its convincing evidence10,15,16. The more fundamental viewpoint is how the QPT picture is modified in the SMIT. A plausible explanation is that the AM state originates from broadening of the SIT10,19,20. An existence of such a broadened critical ground state, a so-called quantum critical phase, has attracted much attention in heavy fermion compounds and organic magnets23,24. To verify this scenario, the QCP must be found inside the broadened critical state. However, clear experimental evidence is still lacking because the critical scaling analysis for resistance data that is frequently used to determine the QCP of the SIT fails for the SMIT9. Furthermore, there are other possibilities including a two-step transition with two QCPs25, and a single QCP at the metal-insulator transition point17. Therefore, it is indispensable to detect the exact location of the QCP using techniques applicable to 2D films beyond the resistance measurements.

A direct way to detect the QCP is to reveal the quantum critical fluctuations of an order parameter around the QCP. A correlation length of the quantum fluctuations diverges at the QCP as a function of a non-thermal parameter such as B1,2,3, just like a correlation length of the thermal fluctuations diverges at a thermal transition temperature as a function of T. Two types of fluctuations exist in the superconducting order parameter \(\Psi=| \Psi | \exp (i\theta )\). The fluctuations of the amplitude Ψ arise from short-lived Cooper pairs that persist deep inside the normal state, and the fluctuations of the phase θ originate from mobile vortices in a vortex-liquid state. In previous reports for 3D superconductors, the thermal critical behavior of the amplitude fluctuations was detected clearly through the Nernst effect above a mean-field transition temperature Tc0, which leads to an exact determination of Tc026,27,28,29,30,31,32,33. These studies utilized a high sensitivity of the Nernst effect to the fluctuations. The Nernst signals generated by the superconducting fluctuations are much larger than the signals from normal electrons as reported in many experimental10,24,26,27,28,29,30,31,32,33,34,35,36,37,38 and theoretical works39,40,41,42,43,44,45. This is in contrast to the resistivity and magnetic susceptibility, where a weak contribution of the fluctuations is buried in the strong intensity from normal electron scattering and the Pauli susceptibility, respectively.

In this work, we have applied the measurements of the Nernst effect to a 2D film of amorphous (a-)MoxGe1−x with a thickness of 10 nm, which shows the B-induced SMIT8,9,10, down to 0.1 K in the quantum regime. By revealing an evolution of the quantum fluctuations across the SMIT, we have found that the QCP is located inside the AM state. The result verifies the view that the AM state is caused by a broadening of the SIT. This work also shows that the Nernst effect is a very useful method to detect a QCP in superconducting systems as reported for the SIT38 as well as in strongly correlated systems37,46.


Figure 1a, b show the T-dependent sheet resistance R(T) in different B. R(T) in B = 0 was measured as a function of T, while R(T) in B ≠ 0 was converted from the magnetoresistance (MR) in Fig. 1e and Supplementary Fig. S1. A mean-field transition temperature Tc0( ≡ Tc0,R) and a zero-resistance temperature Tc in B = 0 are 2.36 and 1.40 K, respectively (see Methods). With increasing B, the transition curve shifts to the low-T region. We plot Tc(B)( ≡ Bc(T)) in a B-T plane (Fig. 2a–c) with solid black circles (see Supplementary Fig. S1). Bc(T) corresponds to a boundary between the vortex-glass phase with zero resistance and the vortex-liquid phase with nonzero resistance. By linearly extrapolating Bc(T) to T = 0, Bc(0) is estimated to be 2.5 T.

Fig. 1: Resistance and Nernst effect in a 2D superconductor exhibiting the field-induced SMIT.
figure 1

a The T dependence of R in different B. R in B = 0 was taken as a function of T and R in B ≠ 0 are plotted from MR in (e) and Supplementary Fig. S1. b The high-B region of (a). c An Arrhenius plot of (a). Open red circles denote Tcross at which R deviates from a thermally activated form drawn by dashed lines. d The B dependence of U in the thermally activated form. A black line shows a theoretical fit8,47. e An enlarged view of MR curves at different T. All the MR curves cross at BMI indicated by a dashed line. N (f) and αxy( = N/R) (g) as a function of B at different T (≤2.4 K). The MR curve at each T is also shown, where the lines with different colors correspond to the data of N and αxy with the same colors. A black straight line in (g) shows an example for a linear extrapolation of αxy to determine Bc2. In the inset of (g), αxy in the high-B region is enlarged. Vertical dashed lines indicate BMI. The B dependence of N (h) and αxy (i) at high temperatures (T ≥ 2.2 K). Black curves in (i) are drawn by an ad hoc function to extract B* as a peak field as indicated by colored arrows (see main text).

Fig. 2: Superconducting fluctuations revealed from the Nernst effect.
figure 2

Contour maps of N (a) and αxy (b) constructed from Fig. 1f−i. Black circles, crosses, and black triangles denote Bc, Tcross, and Bp determined from R, respectively. White triangles and white inverted triangles represent BN→+0 obtained from N. Red, green, and blue squares show Bc2, B*, and T*, respectively, which are determined from αxy. A dashed curve is a fit by WHH theory52. c The characteristic temperatures and fields obtained experimentally and plotted in (a, b) are extracted and displayed in the B-T plane. Also, theoretically obtained \({B}_{{{{{{{{\rm{theo}}}}}}}}}^{*}\) and \({T}_{{{{{{{{\rm{theo}}}}}}}}}^{*}\) extracted from (d) are plotted as green and blue lines, respectively. A dotted red line is an extrapolation of the linear GL region of Bc2(T). An orange line is a fit to T* by vertically shrinking the \({T}_{{{{{{{{\rm{theo}}}}}}}}}^{*}\) line (\(\equiv {T}_{{{{{{{{\rm{theo}}}}}}}},{{{{{{{\rm{m}}}}}}}}}^{*}\)). A solid orange circle indicates BQCP. Error bars for B* and T* represent ranges of B and T in which αxy(B) at fixed T and αxy(T) at fixed B exceed 95 % of their peak amplitudes, respectively. d A theoretical result of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T,B)(\equiv {\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}},{{{{{{{\rm{theo}}}}}}}}}(T,B))\) calculated for a 2D superconductor based on the Gaussian fluctuations43,44 using the codes provided in ref. 43. Note that the input parameters are only Tc0 = 2.36 K and Bc2(0) = 5.5 T obtained experimentally. The T and B axes are normalized by Tc0 and \({\widetilde{B}}_{{{{{{{{\rm{c2}}}}}}}}}(0)\), respectively. Green and blue lines represent the theoretically obtained \({B}^{*}(\equiv {B}_{{{{{{{{\rm{theo}}}}}}}}}^{*})\) and \({T}^{*}(\equiv {T}_{{{{{{{{\rm{theo}}}}}}}}}^{*})\) respectively. A blank area near T = 0 indicates negative αxy due to strong quantum fluctuations44.

Above Bc(0), R(T → 0) saturates to nonzero resistance lower than R(3.0 K) ( ≡ R□,n) as clearly seen in an Arrhenius plot (Fig. 1c), indicating that Bc(0)( ≡ BSM) is a boundary field separating the superconducting phase and the AM state. Dashed straight lines in Fig. 1c represent a thermally activated form \({R}_{\square }={R}_{\square }^{{\prime} }\exp (-U/{k}_{{{{{{{{\rm{B}}}}}}}}}T)\), where \({R}_{\square }^{{\prime} }\) is a coefficient, U an activation energy for vortex motion, and kB the Boltzmann constant. Deviations from the dashed straight lines are marked with open red circles, which correspond to crossover temperatures Tcross from the thermal to the quantum creep regime. They are plotted with crosses in the B-T plane in Fig. 2a, b. U extracted from Fig. 1c is plotted as a function of B in Fig. 1d. The data points are well fitted by \(U={U}_{0}\ln ({B}_{0}/B)\)8,47 with U0/kB = 2.9 K and B0 = 5.0 T as indicated by a straight line. The successful fit with the value of B0 close to an upper critical (crossover) field Bc2(0) = 5.5 T (as shown later) is consistent with the collective creep theory8,47.

Figure 1e shows an enlarged view of the MR curves at different T. The metal-insulator transition field BMI = 5.56 T is given as a field at which all of the MR curves cross and R is independent of T. Above BMI, R shows a logarithmic-like increase with a decrease in T as seen in Fig. 1b. As mentioned in our previous report10, we regard the weak localization behavior as a character of an insulator6,10,11, which is expected to cross over to a strongly insulating state with exponential divergence at much lower temperatures5,48. The weakly localized state is also identified as a dirty metal caused by quantum correction5. From these results, the B-induced SMIT is confirmed in this sample.

Figure 1f displays a Nernst signal N measured as a function of B at different T from 0.1 K to 2.4 K. Figure 1g shows the transverse thermoelectric conductivity αxy converted from the relation N = Rαxy (see Methods). For comparison, the MR curve at each T is also shown. The B and T dependences of N and αxy are similar to those reported in vortex systems10,26,29,33,34,35,36,37,38. With increasing B, N together with R starts to grow above Bc. Then, N decreases after showing a peak, which is attributed to a decrease in αxy (Fig. 1g). The maximum amplitudes of the vortex Nernst signal \({N}_{\max }\) in the B-T range studied is 2.7 μV/K at 1.2 K in 2.8 T. This value is comparable to \({N}_{\max }\) = 2.9 μV/K of the 12 nm-thick a-MoxGe1−x film with Tc0 = 2.58 K used in our previous work10 but smaller than \({N}_{\max }\) = 8.3 μV/K of a multi-layered a-MoxGe1−x film with Tc0 ~ 6 K33. The difference of \({N}_{\max }\) may be due to the difference of effective thickness or dimensionality. In Fig. 2a, b, we plot sensitivity limits of the N signals as BN→+0 with open triangles and open inverse triangles in the low and high-B regions, respectively.

αxy consists of three contributions: \({\alpha }_{xy}^{\phi }\) from the mobile vortices in the vortex-liquid state, \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}\) from the amplitude fluctuations, and \({\alpha }_{xy}^{{{{{{{{\rm{n}}}}}}}}}\) from normal electrons26. \({\alpha }_{xy}^{\phi }\) is given by \({\alpha }_{xy}^{\phi }={s}_{\phi }/{\phi }_{0}\)26, where sϕ is the transport entropy in the vortex core and ϕ0( ≡ h/2e) the flux quantum. According to the theory49, with increasing B near a crossover field Bc2 in the vortex liquid phase, sϕ decreases linearly as sϕBc2 − B and vanishes at Bc2. Therefore, one can determine Bc2 and distinguish \({\alpha }_{xy}^{\phi }\) in BBc2 from the other contributions in B > Bc2 by linearly extrapolating αxy to zero as indicated by a solid straight line in Fig. 1g10,35. As seen in the high-B region (Fig. 1g inset), the contribution of \({\alpha }_{xy}^{{{{{{{{\rm{n}}}}}}}}}\), which increases in proportion to B, is negligible in amorphous samples due to a very short mean free path of quasiparticles10,27,28,29,38. Thus, αxy in B > Bc2 corresponds to \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}\).

Recent studies of the vortex Nernst effect have suggested that a maximum value of the vortex transport entropy per unit layer \({s}_{\phi }^{{{{{{{{\rm{sheet}}}}}}}}}\) is of the order of kB in many superconductors33,50,51. In our experiment, a maximum amplitude of \({\alpha }_{xy}^{\phi }\) = 30 nA/K at 1.6 K in 0.8 T is converted into \({s}_{\phi }={\phi }_{0}{\alpha }_{xy}^{\phi }=4.5{k}_{{{{{{{{\rm{B}}}}}}}}}\). Since sϕ represents the vortex transport entropy per film thickness (= 10 nm), \({s}_{\phi }^{{{{{{{{\rm{sheet}}}}}}}}}\) is calculated to be 0.14 kB supposing a unit layer thickness of ~ 3 Å for a-MoxGe1−x. This value is still close to kB.

We plot Bc2(T) thus obtained with solid red squares in the B-T plane (Fig. 2a–c). Near Tc0,R, Bc2(T) decreases almost linearly as expected in the Ginzburg-Landau (GL) theory and seems to connect to Tc0,R. With decreasing T, Bc2(T) shows a downward deviation from the linear GL line and saturates to Bc2(T → 0) = 5.5 T. The value of Bc2(0) = 5.5 T coincides with BMI = 5.56 T within errors, indicating that the insulating phase in this sample corresponds to a Fermi insulator without vortices4,5. Bc2(T) thus obtained is well reproduced by the Werthamer-Helfand-Hohenberg (WHH) theory52 as shown with a dashed line, where the theoretical value of Bc2(0) is given by \({B}_{{{{{{{{\rm{c2}}}}}}}}}(0)=0.69{\widetilde{B}}_{{{{{{{{\rm{c2}}}}}}}}}(0)\) using \({\widetilde{B}}_{{{{{{{{\rm{c2}}}}}}}}}(0)\) (\(\equiv {T}_{c0}| {{{{{{{\rm{d}}}}}}}}{B}_{c2}/{{{{{{{\rm{d}}}}}}}}T{| }_{T={T}_{c0}}\)) = 8.0 T obtained from a linear extrapolation of Bc2(T) in the GL regime to T = 0 (Fig. 2c).

Figure 1h, i show the B dependence of N and αxy, respectively, at different T above Tc0,R. Both show a nonmonotonic behavior with a peak at a certain field B*(T). The peak heights are more than one order of magnitude smaller than the ones below Tc0,R (Fig. 1f, g). As T increases above Tc0,R, B* shifts to higher B with a broadening of the peak and a decrease in its height. These behaviors have been observed as common features of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}\) above Tc0,R26,27,28,29,30,31,32.

According to the theory based on the Gaussian fluctuations40,41,42,43,44, \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}\) above Tc0,R in BB* follows the expression \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}=({k}_{{{{{{{{\rm{B}}}}}}}}}{e}^{2}/6\pi {\hslash }^{2}){\xi }_{{{{{{{{\rm{GL}}}}}}}}}^{2}B\), where is the reduced Planck constant, \({\xi }_{{{{{{{{\rm{GL}}}}}}}}}(T)={\xi }_{0}/\sqrt{\ln (T/{T}_{{{{{{{{\rm{c0}}}}}}}}})}\) the GL coherence length, and ξ0 the Bardeen-Cooper-Schrieffer (BCS) coherence length. This means that the initial slope of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(B)\), namely \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/B{| }_{B\to 0}\), is a measure of ξGL(T), which is the correlation length of the amplitude fluctuations below B*. Thus, a temperature at which \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/B{| }_{B\to 0}\) diverges corresponds to Tc0 defined from the divergence of ξGL(T). By plotting the T dependence of αxy/BB→0 obtained above Tc0,R, we can determine Tc0 = 2.36 K (see Supplementary Section III-1), which coincides with Tc0,R = 2.36 K. With increasing B at a given T, the correlation length of the amplitude fluctuations is reduced from constant ξGL(T) to a cyclotron radius (magnetic length) \({l}_{B}(B)=\sqrt{\hslash /2eB}\) when lB(B) ≤ ξGL(T)26,28,29,30,31,32. Consequently, αxy exhibits a peak around \(B={\phi }_{0}/2\pi {\xi }_{{{{{{{{\rm{GL}}}}}}}}}^{2}(\equiv {B}^{*})\), which is called a ghost (critical) field26,28,29,30,31,32,44,53. This means that a temperature at which B*(T) → 0 with the divergence of ξGL(T) also corresponds to Tc0. To determine B*(T), we fit αxy(B) in Fig. 1i with an ad hoc fitting function \(pB\exp (-q{B}^{r})\) (solid curves)38, where p, q, and r are positive fitting parameters. B*(T) thus obtained is indicated with colored arrows in Fig. 1i and plotted in Fig. 2a–c with solid green squares. B*(T) approaches zero toward Tc0. The existence of B*(T) in the normal state is also confirmed by scaling analysis for αxy/B (see Supplementary Section III-2).

Figure 2a, b show the contour maps of N(T, B) and αxy(T, B), respectively. Below the Bc2(T) line in the thermal vortex-liquid phase, N and αxy have large magnitudes, which are due to vortex motion. They are observed down to the lowest temperature 0.1 K in the high-B range of the AM state, indicating that the AM state originates from the vortex liquid due to the quantum fluctuations, namely quantum vortex liquid (QVL), as reported in our previous work10. Meanwhile, above the Bc2(T) line, the contribution of the amplitude fluctuations seen from N and αxy is relatively weak but persists in a wide range of the B-T plane with arc-shaped contour lines (light gray lines).

In the main panel of Fig. 3, the T dependence of αxy(T) is plotted for different B from 4.8 T to 10 T. In the inset of Fig. 3, αxy(T) above Bc2(0) is enlarged and shown. αxy(T) shows a peak at a certain temperature T*(B). With an increase in B, the peak becomes broad, its height decreases, and T* increases. These behaviors closely resemble those of theoretically obtained \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T)\) above Bc2(0) (Fig. 2b in ref. 44), where the peak temperature T*(B) called a ghost temperature separates the thermal and quantum regimes of the amplitude fluctuations. A similar thermal-to-quantum crossover temperature has been theoretically predicted41,42,43,54. Although a seemingly similar behavior was observed earlier in the high-B region of cuprates36, its origin was different. It was interpreted as originating from vortices with large quantum fluctuations36,39.

Fig. 3: Estimation of a ghost temperature T*.
figure 3

The T dependences of αxy(T) for different B from 4.8 T to 10 T. In the inset, αxy(T) above Bc2(0) is enlarged. The ghost temperature T*(B) is estimated from a peak temperature as indicated by colored arrows. It is obtained from the fit lines using an ad hoc function and smoothly connected eye guide lines for the data above and below Bc2(0), respectively, both of which are shown with solid colored curves.

According to the theory43, the quantum amplitude fluctuations below T*(B) are described as puddle-like fluctuations with a size of the quantum correlation length \({\xi }_{{{{{{{{\rm{qf}}}}}}}}}(B)={\xi }_{0}/\sqrt{\ln (B/{B}_{{{{{{{{\rm{QCP}}}}}}}}})}\), which diverges at a QCP denoted by BQCP (see Supplementary Section III-3). In our MR data (Fig. 1e), indeed, the existence of the quantum amplitude fluctuations is suggested by a slight hump around Bp ~ 6.5−7.0 T ( > Bc2(0)) near T = 0 (T = 0.07−0.14 K)5,43, as found in our previous report10. Furthermore, the theories41,42,43,44 show that \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T)\) above Bc2(0) at TT* is proportional to \({\xi }_{{{{{{{{\rm{qf}}}}}}}}}^{2}T\) (see Supplementary Section III-3). This means that an initial slope of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T)\), namely \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/T{| }_{T\to 0}\), at given B is a measure of ξqf(B). Therefore, a field at which \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/T{| }_{T\to 0}\) diverges with T*(B) → 0 corresponds to a QCP determined from the amplitude fluctuations. To determine T*, we fit αxy(T) above Bc2(0) with a function \(p(T-{T}_{0})\exp (-q{T}^{r})\) (colored solid lines) similar to the one mentioned above, where T0 corresponds to the sensitivity limit of αxy at BN→+0. For αxy(T) below Bc2(0), we draw eye-guide lines (colored solid lines). T*(B) thus obtained is indicated with colored arrows in the main panel and the inset of Fig. 3 and plotted in Fig. 2a–c with solid blue squares. With decreasing B, T*(B) approaches zero toward a field below Bc2(0). This indicates that the location of BQCP determined from the amplitude fluctuations is inside the AM state. Thus, BMI( ≈ Bc2(0)) is not a QCP but a crossover point, below which the quantum amplitude fluctuations are transformed into the QVL. To our knowledge, this is the first experiment to determine the ghost temperature T*(B) in the B-T plane and the QCP inside the AM state. The existence of T*(B) is also confirmed by scaling analysis for αxy/T (see Supplementary Section III-4).

Figure 2d shows the contour map of the theoretical values of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T,B)(\equiv {\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}},{{{{{{{\rm{theo}}}}}}}}}(T,B))\) calculated for a 2D superconductor based on the Gaussian fluctuations42,43,44 using the codes provided in ref. 43. Here, we input only two parameters, Tc0 = 2.36 K and Bc2(0) = 5.5 T obtained experimentally. The T and B axes are normalized by Tc0 and \({\widetilde{B}}_{{{{{{{{\rm{c2}}}}}}}}}(0)\), respectively. A theoretical line of Bc2(T) (a dashed line), which is drawn by the WHH theory52, is identical to the fitting line of the experimental data of Bc2(T) (a dashed line in Fig. 2a–c). It is seen from the contour maps that the theoretical values of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}},{{{{{{{\rm{theo}}}}}}}}}(T,B)\) in Fig. 2d reproduce almost quantitatively the experimental data of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T,B)\) in Fig. 2b, both of which persist in the wide range of the B-T plane above Bc2(T). The green and blue lines in Fig. 2c, d represent the theoretical lines of \({B}^{*}(T)(\equiv {B}_{{{{{{{{\rm{theo}}}}}}}}}^{*}(T))\) and \({T}^{*}(B)(\equiv {T}_{{{{{{{{\rm{theo}}}}}}}}}^{*}(B))\) obtained from the B and T dependences of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}},{{{{{{{\rm{theo}}}}}}}}}(T,B)\), respectively. It is found that the experimental data of B*(T) (green squares) in Fig. 2c are in good agreement with the theoretical values of \({B}_{{{{{{{{\rm{theo}}}}}}}}}^{*}(T)\) (a green line) within error bars. By contrast, the experimental data of T*(B) (blue squares) in Fig. 2c clearly fall on a line below the theoretical line of \({T}_{{{{{{{{\rm{theo}}}}}}}}}^{*}(B)\) (a blue line). However, when \({T}_{{{{{{{{\rm{theo}}}}}}}}}^{*}(B)\) is vertically compressed by a factor of 4.6/5.5 as shown with an orange line (\(\equiv {T}_{{{{{{{{\rm{theo}}}}}}}},{{{{{{{\rm{m}}}}}}}}}^{*}(B)\)), the experimental data of T*(B) well fall on the modified theoretical line of \({T}_{{{{{{{{\rm{theo}}}}}}}},{{{{{{{\rm{m}}}}}}}}}^{*}(B)\) (see also Supplementary Section II). From the end point of the orange line, \({T}_{{{{{{{{\rm{theo}}}}}}}},{{{{{{{\rm{m}}}}}}}}}^{*}(B\to {B}_{{{{{{{{\rm{QCP}}}}}}}}})\to 0\), marked with a solid orange circle, BQCP = 4.6 T is determined. To our knowledge, this value is first obtained from the present Nernst-effect measurements that probe the amplitude fluctuations, which could not be obtained from conventional measurements, such as resistance. Furthermore, the criticality at BQCP is also confirmed from the B dependence of \({\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/T{| }_{T\to 0}(\propto {\xi }_{{{{{{{{\rm{qf}}}}}}}}}{(B)}^{2})\) above Bc2(0), which shows a divergent behavior toward BQCP (see Supplementary Section III-3).


Finally, let us discuss the origin of the AM state. Generally, the quantum critical regime is defined by ξqf(B) > Lθ(T)1,2,3, where Lθ(T) ~ T−1/z is a dephasing length and z is a dynamical critical exponent1. In the standard QPTs including the SIT, with decreasing T, the field range of the quantum critical regime is reduced to zero toward the QCP as illustrated in Fig. 4a because Lθ(T) diverges as T → 0. However, if Lθ(T) is saturated to a constant value below a certain temperature Tsat for some reasons55, the field range satisfying ξqf(B) > Lθ(Tsat) remains nonzero at T = 0 as in Fig. 4b20, resulting in the quantum critical ground state that spans over an extended field range. Indeed, the saturation of Lθ(T) has been observed in the AM state of a JJA16. In our experiment, the existence of the QCP is clearly detected as BQCP inside the AM state. Moreover, the quantum critical behavior37,46,56 is observed not only at BQCP but also in the wide range of B inside the AM state in the present sample (see Supplementary Fig. S4) as well as in our previous report10. From these results, we conclude that the AM state originates from the broadening of the SIT. This is consistent with the theoretical prediction19 that there exists a broadened quantum ground state between BSM and BMI. It is likely that the thermal-to-quantum crossover (ghost-temperature) line T*(B) obtained in this work reflects a boundary where ξqf(B) = Lθ(T) above Tsat (see Supplementary Section III-4).

Fig. 4: Schematic models of quantum criticality in SIT and SMIT.
figure 4

a A standard picture of QPTs for the B-induced SIT in the B-T plane1 --3. The critical field BSI of the SIT corresponds to the QCP at T = 0. b A proposed model of quantum criticality in the SMIT10,20. The AM state stems from broadening of the SIT, which is verified in this work by finding BQCP (solid orange circle) as a field that the experimental T*(B) line approaches as indicated by a solid blue line. In both (a, b), the quantum critical regime is defined by the area where ξqf(B) > Lθ(T). Gray and orange arrows indicate the direction of the increase in Lθ(T) and ξqf(B), respectively.

A possible origin of the saturation of Lθ(T) in superconductors is a coupling between Cooper pairs and a fermionic dissipative bath, which makes the pure SIT unstable9,15,16,17,20,57. Meanwhile, in magnetic materials, the emergence of the quantum critical phase is attributed to magnetic frustration, leading to a possible quantum spin liquid23,24. Indeed, ref. 19 theoretically suggests that the critical AM state has characteristics analogous to the quantum spin liquid. Thus, some frustration effect on the Cooper pairs from a dissipative bath can be an origin of the AM state.

To conclude, by constructing the thorough contour map of the Nernst signal in the B-T plane, we reveal an evolution of the superconducting fluctuations in the disordered 2D superconductor exhibiting the SMIT. We found the thermal-to-quantum crossover line (ghost-temperature line) T*(B) associated with the QPT, as well as the ghost-field line B*(T) associated with a thermal transition. The T = 0 QCP determined from T*(B → BQCP) → 0 is located inside the AM state, indicating that the AM state is a broadened critical state of the SIT. Our work also shows the usefulness of the Nernst effect to study QPTs in superconducting systems.



An a-MoxGe1−x film with x = 0.77 and a thickness of 10 nm was prepared by rf-sputtering onto a 0.15 mm thick glass substrate. The film thickness is comparable to ξ0 ~ 8 nm estimated from \({\xi }_{0}^{2}={\phi }_{0}/(2\pi {B}_{{{{{{{{\rm{c}}}}}}}}2}(0))\) with Bc2(0) = 5.5 T. To obtain a homogeneous amorphous film, the substrate was mounted on a water-cooled holder rotating with 240 rpm during sputtering. Ag electrodes were deposited 4.2 mm (= L) and 5.6 mm (= l) apart for longitudinal Vx and transverse voltage Vy, respectively. The film thus prepared was protected by depositing a thin SiO layer. Tc0,R = 2.36 K and Tc = 1.40 K are determined from R(Tc0,R) = 0.5R□,n = 0.5R(3.0 K) and the resistance drop below the measurement limit, respectively. Tc = 1.40 K is reduced compared with Tc = 6.2 K for the 330 nm thick film58. The broadened transition curve indicates 2D nature of superconductivity, similarly to the 12 nm thick film with Tc = 2.13 K used in our previous work10.

Resistivity and Nernst-effect measurements

All measurements were carried out using a sample holder installed in a dilution refrigerator10. The magnetic field B was applied perpendicular to the film plane. R = (Vx/L)/(Ix/l) was obtained using standard four-terminal dc and low-frequency ac (19 Hz) lock-in methods with a bias current of Ix≥ 30 nA in the ohmic regime.

N was obtained from N ≡ Ey/ Tx = (Vy/l)/(ΔT/L), where ΔT = Thigh − Tlow is a temperature difference. Vy was measured with a nanovoltmeter. To set ΔT, a heat current was applied from a heater on one side of the glass substrate to the other side glued on a heat bath. ΔT was measured by two RuO2 thermometers, which were thermally isolated from the bath but thermally coupled to the two Ag electrodes of Vx through Cu wires. An average sample temperature was defined by Tave = (Thigh + Tlow)/2. ΔT was fixed to be 30%, ~20%, and ~ 10 % of Tave at 0.1 K, 0.2−0.6 K, and 0.8−4.3 K, respectively, to obtain measurable Vy keeping ΔT as low as possible. At each measurement step, we switched on and off ΔT to subtract the background thermoelectric voltage observed at ΔT = 0 from Vy at ΔT ≠ 0. To correct misalignment of heat flow, we extracted an antisymmetric contribution of N with respect to magnetic field reversal. N from the superconducting fluctuations is approximated by N = Rαxy due to the particle-hole symmetry40,59. αxy is a thermodynamic quantity proportional to the transport entropy, while R reflects electron scattering and viscosity of vortices.

We checked an effect of external noise using low-pass RC filters with a cutoff frequency of 100 kHz, which is adequate to eliminate the noise effect21,22. Although the filters were inserted between the sample and a nanovoltmeter as well as between the sample and a current bias source at room temperature, the results with and without the filters were almost the same. Note that, in the Nernst-effect measurements, the sample was electrically isolated from the current bias source, which is a main possible origin of external noise21,22.