# Protected superconductivity at the boundaries of charge-density-wave domains

###### Abstract

Solid He may acquire superfluid characteristics due to the frustration of the solid phase at grain boundaries. Here, we show that an analogous effect occurs in systems with competition among charge-density-waves (CDWs) and superconductivity in the presence of disorder, as cuprate or dichalcogenide superconductors. The CDWs breaks apart in domains with topologically protected filamentary superconductivity (FSC) at the interfaces. Transport experiments carried out in underdoped cuprates with the magnetic field acting as a control parameter are shown to be in excellent agreement with the theoretical expectation. At high temperature and low fields we find a transition from CDWs to fluctuating superconductivity, weakly affected by disorder, while at high field and low temperature the protected filamentary superconducting phase appears in close analogy with “glassy” supersolid phenomena in He.

^{†}

^{†}preprint: 0

Electrons in the presence of attractive interactions crossover smoothly from the Bardeen-Cooper-Schrieffer (BCS) limit to the Bose condensation (BC) limit as the strength of the interaction is increased Nozieres1985 (). However, as electrons approach the limit of composite bosons, the tendency to localize in real space also increases. Thus, in analogy with He, a real-space ordered state competes with a momentum-space condensed state. Since the entropy of these states is equally small, phase stability is insensitive to temperature, resulting in a phase boundary nearly parallel to the axis and perpendicular to any blue non-thermal control parameter axis (pressure, strain, magnetic field, doping, etc.).

The scenario changes dramatically in the presence of real-space disorder. It has been known for some time that a polycrystal of He atoms develops superfluidity at the interface and acquires supersolid characteristics (i.e., superfluid-like changes of the moment of inertia coexisting with real-space order). See Ref. Balibar2008 () for a review. It is natural to expect that the analogous phenomenon should occur for real-space fermion pairs Attanasi2008 (). In this work, we consider a simple phenomenological model which allows to study the effect of quenched disorder near a transition from a real-space ordered state of fermion pairs to a superconducting (SC) state. We show that disorder induces filamentary superconductivity (FSC) in the spatially ordered charge-density-wave (CDW) state analogous to the supersolid behavior in He. A finite temperature phase diagram is derived. This theoretical scenario is explored experimentally by transport experiments in LaSrCuO (LSCO) using magnetic field as a tuning parameter for dopings close to the insulator-SC transition. We show that the experimentally derived transition lines are in agreement with the theoretical expectations: At moderate temperatures there is a magnetic field driven transition between the superconductor and the CDW phase, as seen with other probes Gerber2015 (). Lowering the temperature in the CDW phase, a very fragile SC phase appears due to the coherent phase-locking of SC filaments at the interfaces of CDW domains analogous to the supersolid effects seen in He.

## Results

### Theory of disorder-induced filamentary superconductivity on charge-density-waves

We consider an electronic system with an attractive interaction that favors real-space formation of fermion pairs. At low temperatures these pairs can either condense in a SC state or form a CDW. An instructive example to study this competition (or eventual intertwining Fradkin2015 ()) is the negative- Hubbard model in a bipartite lattice. At half-filling the CDW and superconductivity become degenerate. The model supports two variants of the CDW (labeled A and B) differing on which of the two sublattices hosts more charge than the other.

At strong coupling, the negative- Hubbard model can be mapped Micnas1990 () onto a Heisenberg model with pseudospins describing charge degrees of freedom analogous to Anderson’s pseudospins Anderson1958 (), but in real space instead of momentum space. Assuming an ordered state at , the projection of the staggered magnetisation on the xy-plane encodes the superfluid order parameter, while its component encodes the CDW order parameter. A positive (negative) component describes the A-CDW (B-CDW) (Fig. 1). In the Hubbard model, the order parameter has SO(3) symmetry, which reflects perfect degeneracy between CDW and superconductivity. This degeneracy is not generic and gets broken when other interactions beyond the canonical Hubbard model are considered. For example, a nearest-neighbor repulsion (attraction) favors the CDW (SC) and drives the effective model to have Ising (xy) symmetry Micnas1990 ().

It is natural to assume that in a region of parameter space in which charge order and superconductivity are seen to coexist, this near degeneracy is reestablished and a model with near SO(3) symmetry generically describes a situation in which the energy for real-space (i.e. CDW) or momentum-space (i.e. SC) condensation of paired fermions is similar. A related lattice model has been considered by Liu and Fisher to study possible supersolid phases in He Liu1972 () close to the boundary between the crystalline phase and the superfluid phase underling the close analogy with our problem.

Since we are interested in intertwining, and in the finite temperature phase diagram, we will neglect quantum fluctuations. These become important when the temperature is below the characteristic energies of the problem, which, especially in the FSC region, are very low. Thus we will study a semiclassical model of CDW-SC in the spirit of Ref. Liu1972 () for the supersolid problem.

A generic semiclassical model can be justified by a coarse grained process. Assuming that there is at least short range order in the system we can separate it in regions larger than the lattice spacing but smaller than the correlation length and define a coarse grained ordering field which determines the kind of order in region . We neglect the fluctuations in the strength of fermion pairing which is parametrised by the length of the ordering field so we take . Thus we assume that the CDW consists of localized bosons which in the context of insulator-SC transitions is associated with a Mott-Hubbard bosonic insulating phaseCha1991 (). We also neglect all complications due to unconventional symmetry of the order parameter. In addition, we assume that there are only two possible variants of CDW phases as for the Hubbard model so () encodes the A(B)-CDW. In the Discussion section below, we enumerate possible microscopic origins of the different CDW variants in different materials. Our considerations are, however, independent of these microscopic details. A pure SC state is described by the complex ordering field with , while sideways configurations describe the CDW analog of supersolid behavior as in Ref. Liu1972 (). Fig. 1(a) displays the order-parameter space. We define the semiclassical model on a discrete lattice of cells which is convenient for numerical simulations,

(1) |

Here, describes a short-range stiffness which, for simplicity, we choose to be equivalent for SC correlations and CDW correlations. In the absence of disorder (), the balance between the orders is decided by the parameter . In this case describes a uniform CDW while describes a uniform superconductor. One could as well have used an anisotropic Heisenberg model with the same scope, which at the classical level we are considering would only change minor details. is a random variable that takes into account that charged impurities will locally favour the A- or B-CDW, depending on whether the impurities on the cell have more charge near the A or the B sublattice. We will take the to be random variables with a flat probability distribution between and and, since we are interested in layered systems (cuprates, dichalcogenides), we will consider a two-dimensional (2D) system.

For the model falls into the universality class of the random-field Ising model. As such, for any disorder, it breaks apart in domains of the A- and B-CDW variants. This is obvious for large disorder while for small disorder it follows from Binder’s refinement Binder1983 () of Imry’s and Ma’s arguments Imry1975 (). In the latter case, however, domains can be exponentially large [roughly ].

For one can consider a flat interface between an A-CDW and a B-CDW. Since the only way for the spin to reverse is by passing through the equator, the interface is forced to have the ordering field on the plane and is locally SC as shown schematically in Fig. 1(b). By minimizing the energy, one finds that the SC region has width where is a short-range cutoff of the order of the coarse grained lattice spacing corresponding to the correlation length of the short-range SC (i.e., particle-particle) or CDW (i.e., particle-hole) pairs. Thus, although for the superconductor is globally less stable, it gets stabilized locally because of topological constraints. In the interface both CDW are frustrated so the less stable SC phase prevails as in polycrystalline He.

Adding disorder to the model for makes the uniform CDW to break apart in domains as expected from Imry’s, Ma’s and Binder’s arguments Imry1975 (); Binder1983 (). Figure 2 shows configurations obtained by minimizing the functional Eq. (1) at , with different disorder strengths (see Methods). Blue and red corresponds to the A- and B-CDW respectively. For small and small disorder (upper left corner), one has large domains. As expected, FSC nucleates at the interfaces (light green). From left to right, the local CDW stability (controlled by ) increases. FSC regions become narrower as . Scrolling down the figure, disorder increases, producing a decrease in the size of the CDW domains, which favours the formation of a dense FSC network. The SC phase is uniform along the FSC regions so that at , if the interfaces form a percolative path, the system has zero resistivity.

In Fig. 3(a) we show the zero temperature superfluid stiffness as a function of the tuning parameter for different disorder strengths. We define the stiffness from the second derivative of the energy with respect to a twist in the boundary conditions and compute it using linear response theory and performing a configuration average Attanasi2008 (). For zero disorder (light green) the stiffness jumps from the bare value to zero as the systems changes abruptly from the SC state to the CDW at . However, as disorder becomes finite (blue curve) the stiffness develops a “foot” for positive indicating that FSC is induced in the nominally CDW region (see Fig. 2). For large detuning from phase stiffness is suppressed exponentially, roughly as , with a constant depending on the disorder strength (see inset). This indicates that an ever more fragile SC regime sets in as the tendency to CDW is increased and the filaments forming the network become narrower. We anticipate that the decreasing but finite stiffness will produce a characteristic “foot” in the temperature dependent phase diagram, which we take as the fingerprint of FSC.

Moderately stronger disorder makes the superfluid phase more robust because the network of filaments becomes denser. At some point, however, for very strong disorder, charge localization is favored at every site and the system becomes an insulating charge glass. The different regimes can be seen by plotting isolines of the phase stiffness as shown in Fig. 3(b). We see that, with increasing disorder, FSC is a reentrant phase in the CDW region. Interestingly, for a fixed control parameter , there is an optimum value for disorder to induce superconductivity.

Although this phase diagram is for zero temperature, we can derive a finite temperature phase diagram assuming that in the clean limit the system behaves as an anisotropic Heisenberg model Cuccoli2003a () [Fig. 4 (a)] and in the presence of disorder develops a finite temperature SC phase with a proportional to the phase stiffness. Fig. 4(b) shows schematically the modified phase diagram in the presence of small disorder. Here, we have assumed that is proportional to the stiffness of the case in the filamentary region and interpolates smoothly to the of the clean limit for negative tuning parameter. We have chosen the microscopic parameters so that the energy scale represents typical values of underdoped cuprates. In the FSC phase the CDW domains coexists with superconductivity. In this mixed state, like the ones shown in Fig. 2, the CDW transition gets broadened by the effect of disorder so that the sharp Ising-like transition of the clean case becomes a crossover (gray band) with glassy characteristics in the presence of disorder Miao2017 (). In the case of unidirectional CDW a sharp transition may persist in a nematic channel, as discussed in Refs. Nie2014 (); Capati2015 ().

In order to associate the phase diagram to magnetotransport experiments, we will define below characteristic temperatures from resistivity data and assume they can be used as proxies of the different transition or crossover lines. A guide to the various temperature scales introduced in this work can be found in Table 1.

Since magnetic field is known to tip the balance between superconductivity and CDW Gerber2015 () we will
use as the tuning parameter where corresponds to the field of a clean quantum-critical point (CQCP),
i.e., zero tuning parameter (). Thus, in Fig. 3(a) we associate the abscissa axis with the
magnetic field (increasing from left to right). In the figure, the temperature units of the tuning parameter
were approximately derived from equivalent energy units in the microscopic Heisenberg model. Conversion to
magnetic-field units would require a precise mapping of the models which is beyond our scope. Empirically,
we find that, as an order of magnitude, 5 K of the microscopic model corresponds to 10 T of the experiment.

Temp. | Proxy | Meaning |

scale | ||

for bulk superconductivity | ||

for filamentary superconductivity | ||

Temp. for the onset of robust SC correlations | ||

Temp. for the onset of FSC effects | ||

CDW crossover temperature | ||

Fit | Cutoff temp. above which an exponential | |

suppression of paraconductivity occurs | ||

Fit | Characteristic temp. scale for the width of | |

low- suppression of resistance due to FSC | ||

Temp. at which vanishes | ||

Temp. at which a local maximum in occurs | ||

Temp. at which a local minimum in occurs | ||

Temp. at which an inflection point in occurs |

At zero or low field [, red line in Fig. 4 (b)] the metallic phase directly becomes SC so transport experiments are expected to yield a monotonic decreasing function of temperature, as shown with the red curve in panel (c). We will use the inflection point in this curve [indicated by the arrow in panel (c)] as a function of field as a proxy for , the characteristic temperature below which robust in-plane SC correlations appear. Notice that three-dimensional (3D) zero-resistance superconductivity sets in at a lower temperature, . should not be confused with another inflection point appearing around 270 K at this doping and unrelated to superconductivity Ando2004 (); Pelc2017 (). At intermediate fields, , and at high/intermediate temperatures, when disorder is not yet relevant, the system critically fluctuates between the SC and CDW states, giving rise to a flat resistance that would seemingly extrapolate to a zero-temperature CQCP (located nearly zero tuning parameter). Eventually, however, SC prevails at low and the resistance vanishes (blue lines). At higher fields (, orange lines) the metal first enters a region of disordered (polycrystalline) CDW with an insulating behavior, thus the temperature corresponding to the resistivity minimum serves as a proxy of the CDW crossover temperature , as shown in panel (c). Lowering even more the temperature the resistivity shows an inflection point very close to a sharp maximum followed, by a rapid drop when coherence establishes between the SC filaments. We will use the inflection point (near the maximum) to signal the onset of FSC, , as shown with the arrow in the upper left corner of panel (c). For weak disorder, the intermediate region showing clean quantum-critical behavior (i.e., the resistivity plateau) is around the region in parameter space where , and merge.

### Filamentary superconductivity in LSCO as revealed by magneto-transport

We expect the scenario presented in the previous section to be realized in underdoped cuprates, where superconductivity and CDW are known to compete. In particular, when a strong magnetic field Gerber2015 () or strain Kim2018 () are present, a static charge order is well documented in the underdoped region of the phase diagram, below a temperature Kivelson2003 (); Wu2011 (). This charge-ordered phase is believed to be responsible for the Fermi surface reconstruction at doping values below Doiron-Leyraud2007 (). We thus chose to study the resistivity of LSCO thin films with Sr doping slightly above the minimal doping for superconductivity (typically ), in order to be able to drive gradually the system towards the insulating state by increasing the magnetic field. The resistivity of the thin films was measured at different temperatures under pulsed magnetic field as high as 54 T. The Method section reports on the experimental details of the transport measurements.

Our goal here is to establish a link between the theoretical analysis and transport experiments carried out in the presence of intense magnetic fields. A microscopic computation of transport would be a formidable task as it would require to take into account the quantum nature of quasiparticles, their scattering in the CDW regions, their role in mediating phase coherence between the SC filaments. In addition one should consider the crossover between this low-temperature coherent SC state and the more 2D regime at higher temperature in which Cooper pairs fluctuate as in standard 2D metals (giving rise to the observed regime of rather robust Aslamazov-Larkin (AL) paraconductivity fluctuations Aslamazov1968a (); Aslamasov1968 (); LeridonPRB2007 (); Caprara2009 ()) and quantum fluctuations at low temperatures Cha1991 (). Therefore, we avoid in our approach any microscopic attempt to describe the resistivity experiments. Instead, we find that a relatively simple and physically motivated expression fits the resistivity in the whole temperature and magnetic field range as shown in Fig. 5(a) and (b) and Fig. S1 (full lines) and we choose to make use of it to extract relevant characteristic temperatures in a phenomenological and systematic way.

The fits were obtained considering two independent contributions to the conductivity,

(2) |

The second term on the r.h.s. represents the field independent conductivity in the absence of any SC fluctuation and is characterised by a crossover from the linear high-temperature behaviour of resistivity of the metallic state to the logarithmic insulating-like behaviour taking place at low under strong magnetic fields, that we associate to the formation of the polycrystalline CDW state (charge-ordered state). We therefore assume that this is the high-field behaviour of the system and estimate it by a fit to the higher field data. The resulting is shown with a black line in Fig. 5(b). Subtracting this contribution to the total conductivity allows to identify the contribution of superconductivity (static as well as fluctuating) to transport, as represented by the first term in the right hand side of Eq. (2), which also encodes all the significant magnetic field dependence of transport.

The superconductivity-related resistance shows that at high-temperature SC fluctuations disappear rapidly (exponentially with reduced temperature) above a characteristic temperature , as shown in Fig. 5(c). This rapid suppression of fluctuations was previously observed in YBaCuO LeridonPRL2001 (); LeridonPRL2003 () and LSCO thin films LeridonPRB2007 () and associated to the presence of an energy cutoff Vidal2002 (); Mishonov2003 (); Caprara2005 () whose value is found to be doping-dependent Luo2003 (). This observation has been reproduced recently in various cuprates popcevic2018 ().

At low temperatures and low/intermediate fields the superconductivity-related resistance is expected to display a linear behavior in temperature as follows from 2D AL fluctuations at temperatures close to . We adopt a simple phenomenological form which interpolates between the exponential behavior at high temperature and the AL behavior,

(3) |

Indeed this expression behaves as at low temperature where will be termed the “bulk” critical temperature. The parameters , and are taken to be functions of as explained in Methods.

The fit with this expression is shown with dashed lines in Fig. 5(b). It represents very well the experimental data except for the low temperature region (K) at intermediate fields. The root of the problem becomes clear upon inspection of panel (a). Notice that fluctuating superconductivity produce a visible magnetoresistance below the zero field from K all the way down to . Thus, the AL regime is associated with a very broad regime of fluctuations characteristic of a 2D superconductor. In contrast, as it is clear from Fig. 5(b) and Supplementary Information Fig. S1, a much more rapid variation sets in below K where points at intermediate fields are not fitted by the dashed lines [Eq. (3)]. In other words, it is not possible to fit with a single AL form both the broad fluctuating regime starting at and the low- regime at intermediate fields. This calls for a different mechanism setting in at low , which we attribute to FSC short circuiting an otherwise finite low temperature resistivity. In order to describe this effect we simply add an hyperbolic tangent cutoff to the SC component,

(4) |

The theta in Eqs. (3), (4) ensures that zero resistivity occurs at the maximum among the “bulk” and the filamentary critical temperature parameters, and respectively. Using Eq. (4) we obtain the full line fits of Fig. 5, which now work over the whole temperature range.

Once the optimum surface is obtained in the plane (see Methods), it is possible to perform equal-resistivity plots as shown in Fig. 6. Here we also show the maximum between and (red dashed line) which determines the upper boundary of the region (colored blue). As expected [] is dominant at low [high] field with a noticeable kink at the intersection. In order to extract the characteristic crossover temperatures defined in Fig. 4(c) we now determine the temperatures of the resistivity maximum and minimum by solving . Furthermore, we find the temperature of the resistivity inflection point, .

We first discuss the inflection point at low fields (blue dashed line). At high temperature, the resistivity has a positive curvature [Fig. 3(a)] which at small field is compensated by the onset of SC correlations. Therefore the inflection point is taken as the characteristic onset temperature, for 2D-SC correlations (blue dashed line in Fig. 6). Notice that zero resistance shown by the blue region in Fig. 6 occurs at a much lower temperature (red dashed line) corresponding to 3D-phase coherence. This issue is discussed below.

Coming back to , we see that it gets rapidly suppressed as a function of field and points to a CQCP around T for sample 008 (). See Fig. 6 (a) and (b). The existence of this CQCP (and an associated plateau) was already proposed, together with the existence of a two-stage transition in Ref. Leridon2013 (). As also observed in this previous work, the resistivity per square at this plateau corresponds quite closely to the quantum of resistance indicated by an open circle at the origin in Figs. 5(a) and (b). is also indicated with a white dashed line in Fig. 6. Theory predicts that in the case of a perfectly self-dual insulator-SC transition, the critical resistance should be equal to the quantum of resistance Fisher1990a (); Fisher1990b (); Cha1991 (). Real systems may show deviation from perfect duality and a different critical resistance. Interestingly enough, the present way of plotting data reveals that indeed coincides with the separatrix line over a broad temperature and field range. A study in which the carrier density in a single layer of the same material was tuned with an electric field Bollinger2011 () found at the insulator-SC transition, in good agreement with the critical resistance in the present study. It is worth mentioning that in the latter experiment the lowest temperature measured was 4.3 K, so the FSC observed here was not accessible.

At high fields and below K, the polycrystalline CDW phase becomes relevant. The crossover is characterized by a change from metallic behavior at high temperature to semiconducting behavior at low temperature, justifying the choice as the characteristic CDW onset temperature (black dashed line). We see that approximately mirrors the behavior of and drops dramatically with decreasing field in the critical region.

At low temperatures and intermediate fields a maximum of the resistivity appears preceded by a nearby inflection point [see for example the blue curve at intermediate field in Fig. 5(a),(b)] and Fig. S2. We associate this behavior with the onset of FSC transition . Indeed, as function of field shows the expected characteristic “foot” in the phase diagram of Fig. 6 [see Figs. 3(a) and 4(b)]. Another fingerprint of FSC is that both (dashed blue line) and (dashed black line) are very close in this region.

In order to further check the experimental result we have repeated the analysis on a different LSCO thin film with slightly different Sr content but different growing conditions [see Fig. S7 in Supplemental Material and Fig. 6(c)].

Fig. 6(c) shows the phase diagram for sample 009 with and Fig. S7 shows the fit to the transport data. In this case the drop of is more gradual, which could be related to a more gradual decrease of the stiffness when disorder is increased(compare with the light blue curve in Fig. 3). In any case, since samples 008 and 009 are grown under different conditions and on different substrates (see Methods), the fact that the FSC is observed in both cases pleads for the universality of the phase diagram in this region of doping.

## Discussion

We propose that in systems in which attractive interactions drive CDW and superconductivity with similar energies, disorder can induce FSC when the CDW is more stable in the clean limit. Long-range superconductivity eventually takes place when the temperature is low enough to allow the phase locking between the SC interfaces.

We have taken a simplified model based on a CDW with only two possible variants or “colors”, A/B. We expect that increasing the number of CDW colors does not change substantially our theoretical results. Also in our modeling we do not need to specify the microscopic origin of CDW colors. There are presently several possibilities in the case of cuprates which we now discuss: i) Scanning tunneling microscopyMesaros2016 () have shown that underdoped cuprates are characterized by CDW domains with 4-lattice spacing periodicity separated by discommensurations. This naturally produces 4 colors for the CDW for a given orientation of the unidirectional CDW (see Fig. 3F in Ref. Mesaros2016 ()). ii) A related possibility is that stripes are formed at high temperatures but are metallic and half-filled Lorenzana2002a () and develop a secondary CDW Peierls stability along the stripe which, in strong coupling, can be seen as a lattice of Cooper pairs Bosch2001 (). It is natural to describe this state with an effective negative- Hubbard description along chains with an associate quasidegenerate SC state. iii) Yet another possibility is suggested by a microscopic analysis which finds an incommensurate CDW in oxygen with -wave symmetry which can be rotated to -wave superconductivity Sachdev2013 (); Efetov2013 (). Here, an Ising order parameter controls excess charge in oriented O bonds with respect to oriented O bonds, which can be associated to the two possible colors of our description. iv) Alternatively, one can see the incommensurate nature as consequence of the weak coupling analysis and consider a locally commensurate (strong coupling) version of the theory with superconductivity nucleating at the discommensurations, as in i). More experimental and theoretical work is needed to establish which scenario occurs in a particular material.

The disorder-induced coexistence of superconductivity and CDW can be seen as a form of intertwined order in the sense of Ref. Fradkin2015 (). However, these authors treat pair-density-wave order (a self-organized version of the Fulde-Ferell-Larkin-Ovchinikov state Larkin1964 (); Fulde1964 ()) as the primary order and CDW as a parasitic order. In the present scenario, both CDW and bulk SC order are primary, while FSC is parasitic.

We have used transport data to derive a phase diagram assuming the magnetic field as tuning parameter (Fig. 4, right). We expect similar phase diagrams using lattice strain Kim2018 (), field effect Bollinger2011 (), or simply doping as tuning parameters. Indeed, comparing 6(a) and (c), one concludes that doping plays a similar role to that of magnetic field, since the phase diagram appears rigidly shifted. The advantage of the magnetic field is that, being associated with a small energy scale, a high resolution scan of the crossovers is possible.

The main difference between the theoretical and the experimental phase diagram is that superconductivity in the former is replaced by 2D fluctuating superconductivity in the latter. One should take into account that the theory does not include long-range interactions and quantum fluctuations which are expected to suppress the zero resistance state Emery1995 (). Therefore, we associate to the transition temperature of the model without these effects. With this caveat, the two phase diagrams are in excellent agreement, in particular, the experimental phase diagram clearly exhibits the foot-like behavior indicating FSC.

After the theory part of this work was completed and possted in Ref. Attanasi2008 (), Ref. yu2019 () appeared where a very similar phase diagram was derived in a model of superconductivity competing with incommensurate CDW (rather as commensurate as here) in the presence of disorder. The kind of topological defects considered are different - the latter model does not predict FSC. Nevertheless the fact that the essential physical outcomes of the two approaches are similar pleads in favour of a rather generic character of disorder-induced SC inside an otherwise stable CDW phase.

It has been proposed that in some underdoped cuprates long-range SC order Himeda2002 (); Li2007 (); Berg2007 () is frustrated by a peculiar symmetry of the SC state. It is not clear at the moment if this effect contributes also to the difference between and in the present samples. One can reverse the argument and argue that FSC is particularly unsuited for 3D phase locking, as the filaments in one plane will in general not coincide with the filaments in the next plane, thus frustrating Josephson coupling.

The deduced phase diagram is not peculiar of the sample analyzed in Fig. 6. Remarkably, an almost identical phase diagram has been derived by completely different techniques in a different material, namely specific heat measurements Kacmarcik2018 () in YBaCuO suggesting that this phase diagram is a quite generic feature of underdoped cuprates.

The model presented here is very general and applies to other systems as well, where the balance between CDW and SC can produce a topologically protected intertwined order. A particularly interesting model system is the Cu-intercalated dichalcogenide 1T-TiSe. In this system scanning tunneling microscopy Yan2017 () shows a commensurate CDW in the undoped system with domain walls appearing upon Cu intercalation. Simultaneously with the latter, superconductivity appears too. The link between CDW discomensurations and superconductivity emerges also from magnetoresistance experiments in gated 2D materials Li2016a (); Li2018 () and from X-ray experiments under pressure Joe2014 (). FSC has also been inferred by phenomenological analyses of transport in LaAlO/SrTiO heterostructuresCaprara2013 (), in ZrNCl and some dichalcogenides (TiSe, MoS)Dezi2018 () This adds to the case of He and suggest that the phenomenon at hand is very general. In solids, remarkably, melting occurs first at the surface Frenken1985 (). Thus, when a polycrystal is driven just below the melting temperature, the less stable liquid phase nucleates at the interface, producing a classical analog of FSC/superfluidity and underlying again the generality of the phenomenon.

## Methods

### Numerical Simulations

In order to obtain the stiffness as a function of disorder and tuning parameter (Fig. 3), the energy of the model was minimized for configurations in which the random fields were chosen with a flat probability distribution between and , using a steepest descent algorithm. Then the stiffness was found using linear response and mapping to a resistor network. Result for each parameter were averaged over 200 different configurations. More detail on the computations can be find in Ref. Attanasi2008 ().

### Transport experiments

The thin film for sample 008 was deposited onto STO substrates at KU Leuven, using dc magnetron sputtering as described in Ref. LeridonPRB2007 (). For this sample, the resistance as function of magnetic field for different temperatures was measured in KU Leuven high pulsed magnetic field facilities, using four probe measurements on an epitaxial film of thickness nm, patterned in strips of 1 mmm. High-field pulses up to 49 T were applied from 1.5 to 300 K perpendicularly to the ab-plane of the c-axis oriented films. We therefore obtained a set of data. The 009 thin film was grown by pulsed laser deposition in IIT Kanpur on LSAO substrate. Similar data as a function of field up to 54 T and temperature from 1.5 to 300 K was obtained at LNCMI Toulouse high field facility.

### Fitting procedure

in Eq. (2) was obtained by fitting the resistivity at the highest field measured (typically T, black data in Fig. 5) and maintained for all the fields measured in that sample. For typically we used a linear term plus a polynomial in up to .

Once was fixed, fits were done for the full set of data in the plane
simultaneously minimizing the total square error. To define the 2D fitting functions, we took the parameters in
the fit namely , ,, to be polynomials in of degree 3,1,3,2 respectively. For the
parameter we used a Lorentzian in centered at . Because scans where done in field
at fixed temperatures, the data used has very high resolution in field (nearly 1300 field values between 0 and 50 T)
and much lower resolution in temperature (16 temperatures with higher resolution at low temperatures).

## Acknowledgments

J.L is very much indebted with Andrea Cavagna for important discussions at the early stages of this work. J.L. and S.C. thank all the colleagues of the ESPCI in Paris for their warm hospitality and for many useful discussion while this work was done. The work at the KU Leuven has been supported by the FWO Programmes and Methusalem Funding by the Flemish Government. Research at IIT Kanpur has been supported by the J.C. Bose National Fellowship (R.C.B.). Part of this work has been founded by EuroMagNET II under the EU contract number 228043. S.C. and M.G. acknowledge financial support of the University of Rome Sapienza, under the Ateneo 2017 (prot. RM11715C642E8370) and Ateneo 2018 (prot. RM11816431DBA5AF projects. Part of the work was supported through the Chaire Joliot at ESPCI Paris.

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