Quantum phase transitions in highly crystalline two-dimensional superconductors

Superconductor–insulator transition is one of the remarkable phenomena driven by quantum fluctuation in two-dimensional (2D) systems. Such a quantum phase transition (QPT) was investigated predominantly on highly disordered thin films with amorphous or granular structures using scaling law with constant exponents. Here, we provide a totally different view of QPT in highly crystalline 2D superconductors. According to the magneto-transport measurements in 2D superconducting ZrNCl and MoS2, we found that the quantum metallic state commonly observed at low magnetic fields is converted via the quantum Griffiths state to the weakly localized metal at high magnetic fields. The scaling behavior, characterized by the diverging dynamical critical exponent (Griffiths singularity), indicates that the quantum fluctuation manifests itself as superconducting puddles, in marked contrast to the thermal fluctuation. We suggest that an evolution from the quantum metallic to the quantum Griffiths state is generic nature in highly crystalline 2D superconductors with weak pinning potentials.

Supplementary Figure 1 | Close-ups of the sheet resistance as a function of temperature for ion-gated ZrNCl and MoS2. a, b, Sheet resistance as a function of temperature at zero magnetic field for ZrNCl (a) and MoS2 (b). The black dashed line is a linear fitting before the superconducting transition. The arrows show Tonset at zero magnetic field. c, d, Sheet resistance as a function of temperature in out-of-plane magnetic fields for ZrNCl (c) and MoS2 (d). The applied magnetic fields are 0.1, 0.2, 0.3, 0,5 and 0.7 T, and vary in 0.3 T steps from 0.9 T to 1.8 T and in 0.5 T steps from 2 T to 5 T, and 7 and 8.5 T for ion-gated ZrNCl. They are 0.05, 0.1, 0.2 and 0.4 T, and vary in 0.3 T steps from 0.6 T to 3 T, in 0.5 T steps from 3.5 T to 6 T and in 1 T steps from 7 T to 9 T for ion-gated MoS2.

Supplementary Figure 2 | FSS analysis at different temperatures for ion-gated ZrNCl. a-e,
Sheet resistance as a function of magnetic field at different temperature intervals. f-j, Normalized Rsheet as a function of the scaling variable |B-Bc|(T/T0) -1/zv . Here, the Bc determined the crossing points of Rsheet(B) curves, and T0 is the lowest temperature in each set of the R-B curves.

Supplementary Note 2. Definition of onset of superconducting fluctuations
The resistance drop starts to occur at much higher temperature than Tc MF , reflecting significant contribution of 2D superconducting fluctuation. To complete the B-T phase diagram, we determine the onset temperature of superconducting fluctuation Tonset from the R-T curve at zero and finite magnetic fields as following. In zero magnetic field, the onset was defined as the temperature at

Supplementary Note 3. Application of Griffiths theory to superconductors
The diverging behavior of dynamical critical exponent is nowadays discussed within the context of the theoretical model based on the Griffiths singularity, where the rare ordered regions play a significant role for describing the quantum phase transition. The concept of the rare region was first pointed out by Griffiths in showing the nonanalytic behavior of magnetization in randomly diluted Ising ferrimagnets. In such a system, due to the random quenched disorder such as impurity and defects, the rare spatial regions that are locally in the magnetic persists in non-magnetic bulk.
Although there are many theoretical works describing the Griffiths state and Griffiths singularity using lattice (Ising) systems 6-10 , the related experimental works have been limited within 3dferromagnets, Sr1-xCaxRuO3 11 , Ni1-xVx. 12 and 4f electron systems The possibility of the Griffiths singularity in superconducting systems has been discussed in ultrathin disordered nanowires 13 , but there are no example in 2D systems. Very recently, the quantum Griffiths singularity is experimentally observed in two-kinds of 2D superconductors: Ga thin films 14 and LaAlO3/SrTiO3 (110) polarized-interface 15 . In these two systems, multiple critical points were observed as is reported in our study. To analyze the experimental data, they used the finite size scaling analysis for different temperature regimes (which is explained in Supplementary Note 4), and they found the dynamical critical exponent shows a power-law form. This anomalous scaling behavior, as is observed in the present study, is suggested in the quantum random transverse field Ising model 8,9 , which shows the activated scaling with continuously varying dynamical critical exponent z, when approaching the infinite-randomness QCP. Such critical behavior originates from the special condition of disorder in the statistical treatment, where the average strength of disorder (magnitude of inhomogeneity) increases without limit under coarse graining, that is theoretically equivalent to d < 2 (violation of the Harris criterion 7 ) with d the system dimension and  the correlation length exponent. Therefore, the Griffith state with the diverging z should satisfies this condition. Indeed,  is theoretically predicted as 0.5 for the superconductor-metal transition in a clean 2D superconducting system 16 , the Harris criterion is violated with d = 1. In this case, the introduction of finite quenched disorder can lead to the infinite-randomness QCP, observed as the quantum Griffith singularity. Following this scenario, the product of the critical exponents z is expressed by the activated scaling law 13,[17][18][19] : (4) with the constant C and the 2D infinite-randomness critical exponents of  ≈ 1.2 and ≈ 0.5 20,21 .
Here, Bc* is infinite randomness critical point. This formula is in good agreement with the experimental data of Ga, LAO/STO and our systems.

Supplementary Note 4. Finite-size scaling analysis around QPT at different temperature regimes
To investigate the multiple-crossing behavior of magnetoresistance shown in Fig. 2 in the main text, and to test the possible application of quantum Griffiths singularity, we performed the finite size scaling (FSS) law of magnetoresistance for QPT expressed as,

Supplementary Note 5. Quantum metallic states and their crossover to thermal vortex creep
In our previous paper 22 , we found that the zero-resistance state in gate-induced 2D superconductivity in ZrNCl is easily destroyed, once the out-of-plane magnetic field is switched on. In the low temperature region, the resistance becomes finite and independent of temperature.
Such a state is regarded as a quantum metallic state 22 Here U(H) is the activation energy and R' is the fitting parameter. This indicates that the vortex motion is governed by the thermally-activated 2D collective creep 23 .
On the other hand, in the low temperature region, the resistance deviates from the activation behavior and tends to saturate, displaying a crossover from thermal to quantum creep regions. The crossover temperature Tcross is defined as the temperature, where the experimental data deviates from the red dashed line (shown by arrows in Supplementary Figure 4).
Based on the transport data including the analysis with the 2D collective creep model 23  Supplementary Note 6. Crossover from quantum metallic state to quantum Griffiths state Figure 4c in the main text is suggesting that the quantum fluctuation governs the phase evolution particularly at low temperature. The crossover from the quantum metallic state to the Griffiths state is rationalized as follows. Although the ground state predicted in disorder-free 2D superconductors is the vortex lattice phase 25 , which is similar to the 3D disorder-free type-II superconductors 26,27 , the long-range order of a vortex lattice easily becomes unstable because of the fluctuations caused by disorder in the absence of the longitudinal elasticity of vortices. This leads to the imperfect vortex lattice state containing many dislocations even in relatively clean system. At low magnetic fields, the plastic motion of dislocations based on the weak pinning and weak elasticity occurs by the thermal 23 and quantum fluctuation in high and sufficiently low temperatures, respectively, as shown in Figure 4c in the main text. In this situation, the sample consists of the superconducting puddles with short-range ordered vortices and the dissipation regime surrounding them at low temperatures. With increasing magnetic field above Bc2 MF , the dissipation region evolves into the normal state, but it is possible that puddle-like superconducting islands remain at very low temperature because of the effect of the quantum fluctuation stabilized by quenched disorder, resulting in the quantum Griffiths state with rare superconducting regions (Fig. 4c). Thus, the quantum creep region (quantum metallic state) is naturally connected to the quantum Griffiths state at low temperature. This situation can be a consequence of strong quantum fluctuation in the 2D superconductor with the very weak but finite pinning effect, leading to a standout effect of randomness.
The difference of quantum Griffiths state and quantum metallic state is considered to originate from two kinds of fluctuation: the amplitude and phase fluctuation, using the analogy to the case of the thermal fluctuations. In other words, the quantum Griffiths state and quantum metallic state correspond to quantum fluctuations of amplitude and phase of order parameter, respectively. In the quantum metallic state, the individual motion of vortices is suppressed by the elasticity of the vortices (in other words, the vortices locally form lattices), and vortex creep occurs at the connecting region of local lattices, where the elastic energy of vortices is small. Therefore, the phase fluctuation locally occurs in the quantum metallic state. With increasing magnetic field, the vortex-lattice mismatch regions, where the quantum creep frequently occurs, widen and eventually become normal states. However, there remain the locally ordered regions due to the pinning by quenched disorder, which form the puddle-like regions with nonzero order parameter.
This situation may evolve into to a quantum Griffiths state with spatial amplitude fluctuation of order parameter.