Broadened quantum critical ground state in a disordered superconducting thin film

Abstract

A superconductor-insulator transition (SIT) in two dimensions is a prototypical quantum phase transition (QPT) with a clear quantum critical point (QCP) at zero temperature (T = 0). The SIT is induced by a field B and observed in disordered thin films. In some of weakly disordered or crystalline thin films, however, an anomalous metallic (AM) ground state emerges over a wide B range between the superconducting and insulating phases. It remains a fundamental open question how the QPT picture of the SIT is modified when the AM state appears. Here we present measurements of the Nernst effect N, which has great sensitivity to the fluctuations of the superconducting order parameter. From a thorough contour map of N in the B-T plane, we found a thermal-to-quantum crossover line of the superconducting fluctuations, a so-called ghost-temperature line associated with the QPT, as well as a ghost-field line associated with a thermal transition. The QCP is identified as a T = 0 intercept of the ghost-temperature line inside the AM state, which verifies that the AM state is a broadened critical state of the SIT.

Introduction

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) electrons^1,2,3. 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 systems^4,5,6. 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 films^8,9,10,11, crystalline 2D superconductors^12,13,14, and Josephson junction arrays (JJAs)^15,16,17. The origin of the superconductor-metal-insulator transition (SMIT) has been debated^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22} 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 vortices^18,19,20, and recent experiments have detected its convincing evidence^10,15,16. 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 SIT^10,19,20. 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 magnets^23,24. 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 B^1,2,3, 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 T_c0, which leads to an exact determination of T_c0^{26,27,28,29,30,31,32,33}. 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 experimental^{10,24,26,27,28,29,30,31,32,33,34,35,36,37,38} and theoretical works^{39,40,41,42,43,44,45}. 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-)Mo_xGe_1−x with a thickness of 10 nm, which shows the B-induced SMIT^8,9,10, 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 systems^37,46.

Results

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 T_c0( ≡ T_c0,R) and a zero-resistance temperature T_c 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 T_c(B)( ≡ B_c(T)) in a B-T plane (Fig. 2a–c) with solid black circles (see Supplementary Fig. S1). B_c(T) corresponds to a boundary between the vortex-glass phase with zero resistance and the vortex-liquid phase with nonzero resistance. By linearly extrapolating B_c(T) to T = 0, B_c(0) is estimated to be 2.5 T.

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

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 T_cross 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 fit^8,47. e An enlarged view of MR curves at different T. All the MR curves cross at B_MI indicated by a dashed line. N (f) and α_xy( = 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 α_xy with the same colors. A black straight line in (g) shows an example for a linear extrapolation of α_xy to determine B_c2. In the inset of (g), α_xy in the high-B region is enlarged. Vertical dashed lines indicate B_MI. The B dependence of N (h) and α_xy (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.

Contour maps of N (a) and α_xy (b) constructed from Fig. 1f−i. Black circles, crosses, and black triangles denote B_c, T_cross, and B_p determined from R_□, respectively. White triangles and white inverted triangles represent B_N→+0 obtained from N. Red, green, and blue squares show B_c2, B^*, and T^*, respectively, which are determined from α_xy. 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 B_c2(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 B_QCP. Error bars for B^* and T^* represent ranges of B and T in which α_xy(B) at fixed T and α_xy(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 fluctuations^43,44 using the codes provided in ref. ⁴³. Note that the input parameters are only T_c0 = 2.36 K and B_c2(0) = 5.5 T obtained experimentally. The T and B axes are normalized by T_c0 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 α_xy due to strong quantum fluctuations⁴⁴.

Above B_c(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 B_c(0)( ≡ B_SM) 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 k_B the Boltzmann constant. Deviations from the dashed straight lines are marked with open red circles, which correspond to crossover temperatures T_cross 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/k_B = 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 B_c2(0) = 5.5 T (as shown later) is consistent with the collective creep theory^8,47.

Figure 1e shows an enlarged view of the MR curves at different T. The metal-insulator transition field B_MI = 5.56 T is given as a field at which all of the MR curves cross and R_□ is independent of T. Above B_MI, 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 insulator^6,10,11, which is expected to cross over to a strongly insulating state with exponential divergence at much lower temperatures^5,48. 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 α_xy converted from the relation N = R_□α_xy (see Methods). For comparison, the MR curve at each T is also shown. The B and T dependences of N and α_xy are similar to those reported in vortex systems^{10,26,29,33,34,35,36,37,38}. With increasing B, N together with R_□ starts to grow above B_c. Then, N decreases after showing a peak, which is attributed to a decrease in α_xy (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-Mo_xGe_1−x film with T_c0 = 2.58 K used in our previous work¹⁰ but smaller than ${N}_{\max }$ = 8.3 μV/K of a multi-layered a-Mo_xGe_1−x film with T_c0 ~ 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 B_N→+0 with open triangles and open inverse triangles in the low and high-B regions, respectively.

α_xy 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 s_ϕ 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 B_c2 in the vortex liquid phase, s_ϕ decreases linearly as s_ϕ ∝ B_c2 − B and vanishes at B_c2. Therefore, one can determine B_c2 and distinguish ${\alpha }_{xy}^{\phi }$ in B ≤ B_c2 from the other contributions in B > B_c2 by linearly extrapolating α_xy to zero as indicated by a solid straight line in Fig. 1g^10,35. 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 quasiparticles^{10,27,28,29,38}. Thus, α_xy in B > B_c2 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 k_B in many superconductors^33,50,51. 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 s_ϕ represents the vortex transport entropy per film thickness (= 10 nm), ${s}_{\phi }^{{{{{{{{\rm{sheet}}}}}}}}}$ is calculated to be 0.14 k_B supposing a unit layer thickness of ~ 3 Å for a-Mo_xGe_1−x. This value is still close to k_B.

We plot B_c2(T) thus obtained with solid red squares in the B-T plane (Fig. 2a–c). Near T_c0,R, B_c2(T) decreases almost linearly as expected in the Ginzburg-Landau (GL) theory and seems to connect to T_c0,R. With decreasing T, B_c2(T) shows a downward deviation from the linear GL line and saturates to B_c2(T → 0) = 5.5 T. The value of B_c2(0) = 5.5 T coincides with B_MI = 5.56 T within errors, indicating that the insulating phase in this sample corresponds to a Fermi insulator without vortices^4,5. B_c2(T) thus obtained is well reproduced by the Werthamer-Helfand-Hohenberg (WHH) theory⁵² as shown with a dashed line, where the theoretical value of B_c2(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 B_c2(T) in the GL regime to T = 0 (Fig. 2c).

Figure 1h, i show the B dependence of N and α_xy, respectively, at different T above T_c0,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 T_c0,R (Fig. 1f, g). As T increases above T_c0,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 T_c0,R^{26,27,28,29,30,31,32}.

According to the theory based on the Gaussian fluctuations^{40,41,42,43,44}, ${\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}$ above T_c0,R in B ≪ B^* follows the expression ${\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}=({k}_{{{{{{{{\rm{B}}}}}}}}}{e}^{2}/6\pi {\hslash }^{2}){\xi }_{{{{{{{{\rm{GL}}}}}}}}}^{2}B$, where ℏ 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 ξ_GL(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 T_c0 defined from the divergence of ξ_GL(T). By plotting the T dependence of α_xy/B∣_B→0 obtained above T_c0,R, we can determine T_c0 = 2.36 K (see Supplementary Section III-1), which coincides with T_c0,R = 2.36 K. With increasing B at a given T, the correlation length of the amplitude fluctuations is reduced from constant ξ_GL(T) to a cyclotron radius (magnetic length) ${l}_{B}(B)=\sqrt{\hslash /2eB}$ when l_B(B) ≤ ξ_GL(T)^{26,28,29,30,31,32}. Consequently, α_xy exhibits a peak around $B={\phi }_{0}/2\pi {\xi }_{{{{{{{{\rm{GL}}}}}}}}}^{2}(\equiv {B}^{*})$, which is called a ghost (critical) field^{26,28,29,30,31,32,44,53}. This means that a temperature at which B^*(T) → 0 with the divergence of ξ_GL(T) also corresponds to T_c0. To determine B^*(T), we fit α_xy(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 T_c0. The existence of B^*(T) in the normal state is also confirmed by scaling analysis for α_xy/B (see Supplementary Section III-2).

Figure 2a, b show the contour maps of N(T, B) and α_xy(T, B), respectively. Below the B_c2(T) line in the thermal vortex-liquid phase, N and α_xy 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 B_c2(T) line, the contribution of the amplitude fluctuations seen from N and α_xy 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 α_xy(T) is plotted for different B from 4.8 T to 10 T. In the inset of Fig. 3, α_xy(T) above B_c2(0) is enlarged and shown. α_xy(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 B_c2(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 predicted^41,42,43,54. 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 fluctuations^36,39.

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

The T dependences of α_xy(T) for different B from 4.8 T to 10 T. In the inset, α_xy(T) above B_c2(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 B_c2(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 B_QCP (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 B_p ~ 6.5−7.0 T ( > B_c2(0)) near T = 0 (T = 0.07−0.14 K)^5,43, as found in our previous report¹⁰. Furthermore, the theories^41,42,43,44 show that ${\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}(T)$ above B_c2(0) at T ≪ T^* 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 ξ_qf(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 α_xy(T) above B_c2(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 α_xy at B_N→+0. For α_xy(T) below B_c2(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 B_c2(0). This indicates that the location of B_QCP determined from the amplitude fluctuations is inside the AM state. Thus, B_MI( ≈ B_c2(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 α_xy/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 fluctuations^42,43,44 using the codes provided in ref. ⁴³. Here, we input only two parameters, T_c0 = 2.36 K and B_c2(0) = 5.5 T obtained experimentally. The T and B axes are normalized by T_c0 and ${\widetilde{B}}_{{{{{{{{\rm{c2}}}}}}}}}(0)$, respectively. A theoretical line of B_c2(T) (a dashed line), which is drawn by the WHH theory⁵², is identical to the fitting line of the experimental data of B_c2(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 B_c2(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, B_QCP = 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 B_QCP is also confirmed from the B dependence of ${\alpha }_{xy}^{{{{{{{{\rm{fl}}}}}}}}}/T{| }_{T\to 0}(\propto {\xi }_{{{{{{{{\rm{qf}}}}}}}}}{(B)}^{2})$ above B_c2(0), which shows a divergent behavior toward B_QCP (see Supplementary Section III-3).

Discussion

Finally, let us discuss the origin of the AM state. Generally, the quantum critical regime is defined by ξ_qf(B) > L_θ(T)^1,2,3, where L_θ(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 L_θ(T) diverges as T → 0. However, if L_θ(T) is saturated to a constant value below a certain temperature T_sat for some reasons⁵⁵, the field range satisfying ξ_qf(B) > L_θ(T_sat) 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 L_θ(T) has been observed in the AM state of a JJA¹⁶. In our experiment, the existence of the QCP is clearly detected as B_QCP inside the AM state. Moreover, the quantum critical behavior^37,46,56 is observed not only at B_QCP 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 B_SM and B_MI. It is likely that the thermal-to-quantum crossover (ghost-temperature) line T^*(B) obtained in this work reflects a boundary where ξ_qf(B) = L_θ(T) above T_sat (see Supplementary Section III-4).

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

a A standard picture of QPTs for the B-induced SIT in the B-T plane^{1 --3}. The critical field B_SI of the SIT corresponds to the QCP at T = 0. b A proposed model of quantum criticality in the SMIT^10,20. The AM state stems from broadening of the SIT, which is verified in this work by finding B_QCP (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 ξ_qf(B) > L_θ(T). Gray and orange arrows indicate the direction of the increase in L_θ(T) and ξ_qf(B), respectively.

A possible origin of the saturation of L_θ(T) in superconductors is a coupling between Cooper pairs and a fermionic dissipative bath, which makes the pure SIT unstable^{9,15,16,17,20,57}. Meanwhile, in magnetic materials, the emergence of the quantum critical phase is attributed to magnetic frustration, leading to a possible quantum spin liquid^23,24. 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 → B_QCP) → 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.

Methods

Samples

An a-Mo_xGe_1−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 B_c2(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 V_x and transverse voltage V_y, respectively. The film thus prepared was protected by depositing a thin SiO layer. T_c0,R = 2.36 K and T_c = 1.40 K are determined from R_□(T_c0,R) = 0.5R_□,n = 0.5R_□(3.0 K) and the resistance drop below the measurement limit, respectively. T_c = 1.40 K is reduced compared with T_c = 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 T_c = 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_□ = (V_x/L)/(I_x/l) was obtained using standard four-terminal dc and low-frequency ac (19 Hz) lock-in methods with a bias current of I_x≥ 30 nA in the ohmic regime.

N was obtained from N ≡ E_y/ ∇ T_x = (V_y/l)/(ΔT/L), where ΔT = T_high − T_low is a temperature difference. V_y 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 V_x through Cu wires. An average sample temperature was defined by T_ave = (T_high + T_low)/2. ΔT was fixed to be 30%, ~20%, and ~ 10 % of T_ave at 0.1 K, 0.2−0.6 K, and 0.8−4.3 K, respectively, to obtain measurable V_y 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 V_y 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 = R_□α_xy due to the particle-hole symmetry^40,59. α_xy 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 effect^21,22. 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 noise^21,22.