# Quantum superconductor-insulator transition: Implications of BKT-critical behavior

###### Abstract

We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical behavior on the two dimensional (2D) quantum superconductor-insulator (QSI) transition driven by the tuning parameter . Concentrating on the sheet resistance BKT behavior implies: an explicit quantum scaling function for along the superconducting branch ending at the nonuniversal critical value ; a BKT-transition line where is the dynamic and the exponent of the zero temperature correlation length; independent estimates of , and from the dependence of the nonuniversal parameters entering the BKT expression for the sheet resistance. To illustrate the potential and the implications of this scenario we analyze data of Bollinger et al. taken on gate voltage tuned epitaxial films of LaSrCuO that are one unit cell thick. The resulting estimates and point to a 2D-QSI critical point where hyperscaling, the proportionality between and and the correspondence between quantum phase transitions in D and classical ones in (D+z) dimensions are violated and disorder is relevant.

###### pacs:

74.40.Kb, 74.62.-c, 74.78.-w^{†}

^{†}preprint: PREPRINT (July 12, 2019)

## I Introduction

A variety of different materials undergo a quantum superconductor- insulator (QSI) transition in the limit of two dimensions (2D) and zero temperature by variation of a tuning parameter including film thickness, disorder, applied magnetic field, and gate voltage.gold (); markovic (); gant (); gold2 () A widespread observable to study this behavior is the temperature dependence of the sheet resistance taken at various values of the tuning parameter . The curves of ) at different resemble the flow to a nearly temperature independent separatrix between superconducting and insulating phase with sheet resistance . This behavior implies a crossing point of the isotherms at different temperatures at which is a characteristic feature of a quantum phase transition and in the present case of a QSI transition. Traditionally the interpretation of experimental data taken close to the 2D-QSI transition is based on the quantum scaling relation with .sondhi (); fisher1 (); fisher (); kim () is the dynamic, the critical exponent of the zero temperature correlation length and a nonuniversal coefficient of proportionality.gold (); markovic (); gant () Given sheet resistance data fits to this scaling form yield estimates for the critical value of the tuning parameter and the exponent product , properties which are insufficient to distinguish different models, to fix the universality class to which the QSI transition belongs, or to clarify the relevance of disorder .

The nature of the 2D-QSI transition has been intensely debated.gold (); markovic (); gant (); gold2 () The scenarios can be grouped into two classes, fermionic and bosonic. In the fermionic case the reduction of and the magnitude of the order parameter is attributed to a combination of reduced density of states, enhanced Coulomb interaction and depairing due to an increase of the inelastic electron-electron scattering rate.maek (); finkel () The bosonic approach assumes that the fermionic degrees of freedom can be integrated out, the mean square of the order parameter does not vanish at , phase fluctuations dominate and the reduction of is attributable to quantum fluctuations and in disordered systems to randomness in addition.fisher1 (); fisher (); weich () This scenario is closely related to the suppression of ferroelectricity,morf () e.g. in SrTiO.alex ()

Here we adopt the bosonic scenario and concentrate on systems which undergo at finite temperature a normal state to superconductor transition with Berezinskii-Kosterlitz-Thouless (BKT) critical behavior,bere (); thoul () originally derived for the 2D xy-model with a two component order parameter and short range interactions. The occurrence of BKT criticality in 2D superconductors also implies that the mean square of the order parameter does not vanish at and with that there are, in analogy to He, condensed pairs (bosons) below and uncondensed ones above . In He and superconductors the order parameter is a complex scalar corresponding to the components in the xy-model. Supposing that in superconductors the interaction of Cooper pairs is short ranged and their effective charge is sufficiently small the critical properties at finite temperature are then those of the 3D-xy (bulk) and 2D-xy (thin films) models,tsbook (); parks () reminiscent to the lamda transition in bulk Hegasp () and the BKT-transition in thin He films.fino (); cho (); crow (); oda () In this context it is important to recognize that the existence of the BKT-transition (vortex-antivortex dissociation instability) in He films is intimately connected with the fact that the interaction energy between vortex pairs depends logarithmic on the separation between them. As shown by Pearl,pearl () vortex pairs in thin superconducting films (charged superfluid) have a logarithmic interaction energy out to the characteristic length = , beyond which the interaction energy falls off as . Here is the magnetic penetration depth and the film thickness. As increases by approaching the diamagnetism of the superconductor becomes less important and the vortices in a clean and thin superconducting film become progressively like those in He films.min (); halp ()

The occurrence of a 2D-QSI transition implies a line of BKT transition temperatures ending at the quantum critical point at where . It separates the superconducting from the insulating ground state. The BKT transition is rather special because the correlation length diverges above as and the sheet resistance tends to zero according to .min (); halp (); tsintf () Approaching the 2D-QSI transition quantum phase fluctuations renormalize , , and . Indeed the BKT-transition line approaches the 2D-QSI transition as because the quantum scaling form exhibits at the universal value of the scaling argument a finite temperature singularity.fisher1 (); kim (); tsbook () Noting that the amplitude of the BKT correlation length should match the divergence of the quantum counterpart, the exponents and should emerge from the amplitude in terms of .tsintf () Accordingly, the nonuniversal functions and entering the BKT expression for the sheet resistance exhibit close to the QSI transition quantum critical properties disclosing the quantum critical exponents , and . The exponents and are characteristic properties of the universality class to which the QSI transition belongs. In addition, given their values the relevance of disorder and the equivalence between quantum phase transitions in systems with spatial dimensions and the ones of classical phase transitions in dimensions can be checked.sondhi (); fisher1 (); tsbook () The occurrence of a BKT transition line also implies: a nonuniversal critical sheet resistance because is nonuniversal; an explicit form of the superconductor branch of the quantum scaling function .

Even though BKT critical behavior is not affected by short-range correlated and uncorrelated disorderharris (); chay () the observation of this behavior requires sufficiently homogeneous films and a tuning parameter which does not affect the disorder. Noting that sample inhomogeneity and vortex pinning are relevant in thickness and perpendicular magnetic field tuned transitions, electrostatic tuning of the 2D-QSI transition using the electric field effect appears to be more promising.tsintf (); paren (); bol (); leng (); bert () Indeed electrostatic tuning is not expected to alter physical or chemical disorder, but changes the mobile carrier density.

In Sec. II we sketch the theoretical background. To illustrate the potential and the implications of finite temperature BKT criticality on the 2D-QSI transition we analyze in Sec. II the sheet resistance data of Bollinger et al.bol () taken on gate voltage tuned epitaxial films of LaSrCuO that are one unit cell thick. Commenting on the difficulties in observing the BKT features in the magnetic penetration depth we consider the data of Bert et al.bert () taken on the superconducting LaAlO/SrTiO interface.

## Ii Theoretical background

Continuous quantum-phase transitions (QPT) are transitions at zero temperature in which the ground state of a system is changed by varying a parameter of the Hamiltonian.sondhi (); tsbook (); parks () The quantum superconductor- insulator transitions (QSI) in two-dimensional (2D) systems tuned by disorder, film thickness, magnetic field or with the electrostatic field effect are believed to be such transitions.gold (); markovic (); gant (); tsintf (); tsbook (); bol (); parks () Traditionally the interpretation of experimental data taken close to the 2D-QSI transition is based on the quantum scaling relation,sondhi (); fisher1 (); fisher (); kim ()

(1) |

where is the resistance per square, the limiting ( and ) resistance. denotes the tuning parameter and a nonuniversal coefficient of proportionality. is a universal scaling function of its argument such that . In addition, is the dynamic and the critical exponent of the correlation length, supposed to diverge as

(2) |

The critical sheet resistance separating the superconducting and insulating ground states is determined from the isothermal sheet resistance at the crossing point in vs. tuning parameter at . The existence of such a crossing point, remaining temperature independent in the zero temperature limit, is the signature of a QPT. The data for plotted vs. should then collapse onto two branches joining at . The lower branch stems from the superconducting and the upper one from the insulating phase . This scaling form follows by noting that the divergence of the zero temperature correlation length, , is at finite temperature limited by the length .sondhi () Thus is a finite-size scaling function because . Supposing that there is a line of finite temperature phase transitions ending at the quantum critical point the quantum scaling form (1) exhibits at the universal value of the scaling argument a finite temperature singularity.fisher1 (); kim (); tsbook () The phase transition line is then fixed by

(3) |

Otherwise one expects that sufficiently homogeneous 2D superconductors exhibit at the superconductor to normal state transition BKT critical behavior.tsintf (); bere (); thoul () Note that there is the Harris criterion,harris () stating that short-range correlated and uncorrelated disorder is irrelevant at the unperturbed critical point, provided that , where is the dimensionality of the system and the critical exponent of the finite-temperature correlation length. With and , appropriate for the BKT transition, bere (); thoul () this disorder should be irrelevant. Given a BKT superconductor to normal state transition the sheet resistance scales for asmin (); halp (); tsintf ()

(4) |

allowing to probe the characteristic BKT correlation lengthbere (); thoul (); halp (); tsintf ()

(5) |

While , , and depend on the tuning parameter and are subject to quantum fluctuations the characteristic BKT form of the correlation length and with that of the sheet resistance applies for any . Through standard arguments (see, e.g., Ref. hertz ()) quantum mechanics does not modify universal finite temperature properties. is given bythoul (); tsintf ()

(6) |

with . The parameter is expected to remain constant in the low regime.tsintf () It is related to the vortex core energy in terms of steel () and controls below the Nelson-Kosterlitz jump the temperature dependence of the magnetic penetration depth in terms of .nelson () The amplitude of the BKT correlation length is proportional to the vortex core radiusfino () known to increase with reduced .cho (); oda () Indeed, the zero temperature correlation length diverges as (Eq. (2) and combined with (Eq. (3)) we obtain . Noting that approaches from above the scaling relation

(7) |

should apply,tsintf () making the determination of the exponents and possible. Consistency requires that the resulting agrees with the estimate obtained from the critical BKT line . Other implications concern the universality class of the 2D-QSI transition and the relevance of disorder. Given estimates for and the equivalence between quantum phase transitions in clean systems with spatial dimensions and the ones of classical phase transitions in dimensions can be checked. The fate of a clean critical point under the influence of disorder is controlled by the Harris criterion:harris (); chay () If the zero temperature correlation length critical exponent fulfills the inequality the disorder does not affect the critical behavior. If the Harris criterion is violated, , the generic result is a new critical point with conventional power law scaling but new exponents which fulfill the Harris criterion. Another option is that the disorder destroys the QSI transition.

Finally, given a BKT transition line expression (4) the sheet resistance given by Eq. (5) transforms with Eq. (6) and the scaling variable to

(8) |

valid foe . Close to quantum criticality where reduces to

(9) | |||||

These relations are explicit forms of the quantum scaling function applicable to the superconductor branch. They reveal that the critical sheet resistance is the endpoint of a nonuniversal function and accordingly nonuniversal. Another implication concerns the universality class of the 2D-QSI transition. Supposing that the equivalence between quantum phase transitions in clean systems with spatial dimensions and the ones of classical phase transitions in dimensions applies, the 2D-QSI transition at the endpoint of a BKT line should belong to the finite temperature universality class. denotes an order parameter with two components, including the complex scalar, , of a superconductor.sondhi (); tsbook ()

The BKT - theory of thermally-excited vortex-antivortex pairs also predicts a super-to-normal state phase transition marked by the Nelson-Kosterlitz jump, a discontinuous drop in superfluid density fromnelson (),

(10) |

to zero. The numerical relationship applies for in cm and in K. denotes the thickness of the 2D system, the in-plane magnetic penetration depth, and . In addition there is the prediction that , a measure of the phase stiffness, scales near the endpoint of the BKT transition line astsbook (); parks (); fisher1 (); kim ()

(11) |

provided that is below the upper critical dimension where hyperscaling holds. Since in the -xy-model the validity of Eq. (11) is in restricted to . Relation (11) is the quantum counterpart of the Nelson-Kosterlitz relation (8). is a dimensionless critical amplitude with the lower boundtsbook (); parks ()

(12) |

dictated by the characteristic temperature dependence of below the Nelson-Kosterlitz jump (Eq. (10)).tsbook (); parks () Combining Eqs. (10) and (11) we obtain

(13) |

where is the superfluid density. The superfluid transition temperature as a function of the superfluid density has been measured in He films for transition temperatures ranging from to K by Crowell et al.crow () They studied He films adsorbed in two porous glasses, aerogel and Vycor, using high-precision torsional oscillator and dc calorimetry techniques. The investigation focused on the onset of superfluidity at low temperatures as the He coverage is increased. Their data yields with in moles/m and in K. Combined with the BKT transition line we obtain .

## Iii Comparison With Experiment

To illustrate the potential and the implications of the outlined BKT scenario we analyze next the data of Bollinger et al.bol () taken on epitaxial films of LaSrCuO that are one unit cell thick. Very large electric fields and the associated changes in surface carrier density enabled shifts in the midpoint transition temperature by up to K. Hundreds of resistance versus temperature and carrier density curves were recorded and shown to collapse onto a single function, as the quantum scaling form (Eq. (1)) for a 2D-QSI transition predicts. The observed critical resistance is close to the quantum resistance for pairs, k. Our starting point is the temperature dependence of the sheet resistance taken at various gate voltages where a BKT transition is expected to occur. As an example we depicted in Fig. 1a the data for V. The observation of BKT-behavior requires that the data extend considerably below the mean-field transition temperature . We estimated it with the aid of the Aslamosov-Larkin (AL) expressionaslamo () for the conductivity, , with , where Gaussian fluctuations are taken into account and is the mean-field transition temperature. The resulting temperature dependence is included in Fig. 1a. It clearly reveals that the data extend considerably below K. To establish and characterize BKT behavior below we invoke Eq. (4) and

(14) |

As indicated in Fig. 1b this relation is used to fix and while is estimated by adjusting Eq. (4) with given and to the sheet resistance data. Comparing the data with the respective lines we observe that the BKT regime is attained and that hat the BKT is almost an order of magnitude lower than the mean-field counterpart. This uncovers a BKT transition from uncondensed to condensed Cooper pairs driven by strong phase fluctuations. On the other hand it is important to recognize that agreement with BKT-criticality is established in a temperature window only. Its upper bound reflects the crossover from BKT- to AL-behavior while the lower bound stems from the rounded BKT-transition. Precursors of this phenomenon are clearly visible in Fig. 1 around K. Here the correlation length is prevented to grow beyond a limiting length , i.e. the linear extent of the homogeneous domains. As a result, a finite size effect and with that a rounded transition occurs.tsintf () Because the BKT correlation length does not exhibit the usual power law divergence of the correlation length as is approached, it is particularly susceptible to such finite-size effect.

To explore the quantum critical behavior latent in , and we performed this analysis of the temperature dependence of the sheet resistance for additional gate voltages. To fix the critical gate voltage we used the empirical gate voltage dependence of the number of mobile holes per one formula unit of Bollinger et al.bol () yielding down to V with and V. The results , are shown in Fig. 2a and Fig. 2b while is depicted in Fig. 3. As can be seen in Fig. 2a the BKT transition line differs substantially from the so called superconducting dome behavior observed in bulk cuprate superconductors. It is approximately given by where is the maximum .pres () Close to the QSI transition where even bulk cuprate superconductors become essentially 2Dparks () it reduces to with and suggests . In any case it differs substantially from the BKT line shown in Fig. 2a yielding the estimate

(15) |

in agreement with the value derived by Bollinger et al.bol () using the quantum scaling approach. In contrast to this, from the nonuniversal parameters and shown in Fig. 2b we derive with Eq. (7) in addition

(16) |

As these exponents satisfy the inequality we use the correct scaling argument, because where denotes the schemical potential.fisher1 () The agreement between these values confirms the applicability of the scaling relation (7). Similarly, the nonuniversal parameter exhibits according to Fig. 3 the expected dependence (Eq. (6)). Noting that exceeds the upper critical dimension the critical exponent of the zero temperature correlation length should adopt its mean-field value . However the fate of this clean critical point under the influence of disorder is controlled by the Harris criterion.harris (); chay () If the inequality is fulfilled, the disorder does not affect the critical behavior. If the Harris criterion is violated ( ), the generic result is a new critical point with conventional power law scaling but new exponents which fulfill . Since violates this inequality disorder is relevant and drives the system from the mean-field to an other critical point with different critical exponents as our estimate , fulfilling , uncovers. The resulting 2D-QSI transition with and violates then the equivalence between quantum phase transitions in systems with spatial dimensions and the ones of classical phase transitions in dimensions. In addition the proportionality between and (Eq. (11)) valid below the upper critical dimension does no longer hold because is above . In fact magnetic penetration depth measurements taken on underdoped high quality YBaCuO single crystals revealed .liang () Another option is that the disorder destroys the QSI transition. Given the evidence for a rounded BKT transition (see Fig. 1) and the missing data close to the QSI transition (see Fig. 4) further studies are required to elucidate this option.

To complete the analysis of the data of Bollinger et al.bol () we depicted in Fig. 4 the plot vs.corresponding to the quantum scaling function (Eq. (8)) in terms of . Apparently, the data does not fall completely on the BKT curve indicated by the dashed line. It corresponds to the superconductor branch of the quantum scaling function. Instead we observe a flow to and away from the universal characteristics. As decreases for fixed the crossover to AL-behavior sets in, while the rounding of the transition leads with increasing to a flow away from criticality. The important lesson then is that the quality of the data collapse on a single curve heavily depends on the temperature range of the data entering the plot. Another striking feature is the extended scaling regime. Within the BKT scenario it simply follows from the fact that the scaling form (4) applies along the BKT transition line irrespective of the distance from the QSI transition. In the quantum scaling approach this property remains hidden and merely suggests an extended quantum critical regime. The excellent quality of the piecewise data collapse also reveals that the observance of the substantial variation of (see Fig. 2b) is essential, while in the quantum scaling approach it is fixed by the critical sheet resistance.

To demonstrate these features even more compelling we depicted in Fig. 5a vs.. Apparently the data do not fall even piecewise on a single curve. After all this is not surprising because the quantum scaling form (1) holds close to the critical sheet resistance only and varies substantially in the gate voltage regime considered here (see Fig. 2b). As shown in Fig. 5b and Fig. 6 this behavior allows to estimate from vs. by rescaling in terms of with to achieve piecewise a collapse of the data. The resulting is shown in Fig. 6 and agrees well with V derived from the BKT behavior of the sheet resistance (see Fig. 2b).

A BKT line with a QSI transition at its endpoint was also explored rather detailed at the interface between the insulating oxides LaAlO and SrTiO exhibiting a superconducting 2D electron system that can be modulated by a gate voltage.tsintf (); reyren (); cavigli (); bert () BKT behavior and with that a 2D electron system was established as follows: The the current-voltage characteristicsreyren () revealed at the BKT transition temperature the characteristic BKT form with . halp () Consistency with the characteristic temperature dependence of the sheet resistance (Eq. (4)) was established.tsintf (); reyren (); cavigli () It was also shown that the effective thickness of the superconducting 2D system can be extracted from the magnetic field dependence of the conductivity at .tstool () The gate voltage tuned BKT phase transition line, , derived from the temperature dependence of the sheet resistance at various gate voltages uncovered with Eq. (4) consistency with K pointing to quantum critical behavior (Eq. (3)) with and the critical sheet resistance k.tsintf (); cavigli () Furthermore the estimates and , confirming , have been derived from . tsintf () This suggests that the gate voltage tuned QSI transition of the 2D electron system at the LaAlO/ SrTiO interface belongs to the -xy universality class where hyperscaling and with that the proportionality between and (Eq. (11)) applies. On the other hand because disorder is according to Harris theorem relevant as well.harris (); chay ()

To comment the difficulties in observing the BKT features in the magnetic penetration depth as well as the relation between and , we reproduced in Fig. 7 the data of Bert et al.bert () for a gate voltage tuned superconducting LaAlO/SrTiO interface in terms of vs. . is defined here as the temperature at which the diamagnetic screening drops below the noise level corresponding to a detectable of cm. A glance at Fig. 7 reveals that this is just the regime where the universal quantum behavior applies, indicated by the dashed and solid lines, corresponding to the lower bound (12) and the behavior derived from the He data of Crowell et al.,crow () respectively. Nevertheless the data reveal the flow to the 2D-QSI transition which is attained at much lower ’s. Otherwise the data points resemble the outline of a fly’s wing,parks () remarkably similar to the vs. plots of the bulk superconductors YCu-123, Tl-1212,bern () and Tl-2201nied (), covering nearly the doping regime of the so called superconducting dome extending from the underdoped to the overdoped limit . According to the generic plot vs. , where is the anisotropy and denote the in-plane and c-axis correlation length, these cuprates become nearly 2D in the underdoped limit.parks () In any case, Fig. 7 shows that in this plot the universal QSI behavior is attained at comparatively low values only. Accordingly, in the QSI regime of interest the Nelson-Kosterlitz jump given by Eq. (10) becomes very small and appears to be beyond present experimental resolution.bert () On the contrary in the temperature dependence of the sheet resistance is the BKT critical regime accessible because ,tstool () even though small compared to that in the LaSrCuO films where .

## Iv Summary and discussion

In sum, we sketched and explored the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical behavior on the quantum critical properties of a two dimensional (2D) quantum superconductor-insulator transition (QSI) driven by the tuning parameter . It was shown that the finite temperature BKT scenario, implies in terms of the characteristic temperature dependence of the BKT correlation length an explicit quantum scaling function for the sheet resistance along the superconducting branch ending at the nonuniversal critical value . This scaling form fixes the BKT-transition line and provides estimates for the quantum critical exponent product . In addition, independent estimates of , and follow from the dependence of the nonuniversal parameters and entering the characteristic BKT expression for the sheet resistance . This requires that the BKT critical regime where phase fluctuations dominate is attained and the finite temperature BKT relation for the sheet resistance applies for any . The last condition is satisfied because the BKT expression for the sheet resistance is simply related to the characteristic temperature dependence of the BKT correlation length. Quantum fluctuations enter via the nonuniversal parameters and disclosing close to the respective quantum critical behavior. Even though BKT critical behavior is not affected by short-range correlated and uncorrelated disorderharris (); chay () the observation of this requires sufficiently homogeneous films and a tuning parameter which does not affect the disorder. Noting that in the magnetic field tuned case there is no BKT-line, thickness and gate voltage tuned 2D-QSI transitions appear to be promising candidates. In any case the scenario outlined here requires a line of BKT-transitions with a quantum critical endpoint and sheet resistance data which attain the BKT critical regime.

To illustrate the potential and the implications of this scenario we analyzed data of Bollinger et al.bol () taken on gate voltage tuned epitaxial films of LaSrCuO that are one unit cell thick. Evidence for dominant phase fluctuations and BKT-critical behavior was established in terms of the temperature dependence of the sheet resistance revealing a large critical regime extending substantially above the lowest attained temperature K. From the nonuniversal parameters and disclosing the respective quantum critical properties we derived for the quantum critical exponents the estimates: from , and from , yielding . Thus, in contrast to the standard quantum scaling approach, providing an estimate for only, the BKT scenario uncovered , and from the quantum critical behavior disclosed in and . Additional evidence for was established from the comparison of the scaled data with the explicit scaling BKT scaling form of the superconductor branch. We observed that the scaled data does not fall entirely on the BKT curve. Instead a flow to and away from the universal characteristics occurred. As decreases for fixed a crossover to AL-behavior sets in, while the rounding of the transition leads with increasing to a flow away from criticality. The important lesson then is that the quality of the data collapse on a single curve heavily depends on the temperature range of the data entering the plot. Another striking feature is the extended scaling regime. Within the BKT scenario it follows from the fact that the explicit scaling form of the sheet resistance applies along the entire BKT transition line irrespective of the distance from the QSI transition. In the quantum scaling approach this property remains hidden and merely suggests an extended quantum critical regime. The piecewise excellent quality of the data collapse also reveals that the provision of the substantial variation of is essential, while in the quantum scaling approach it is fixed by the critical sheet resistance. Supposing that the equivalence between quantum phase transitions with spatial dimensions and the ones of classical phase transitions in dimensions applies, the 2D-QSI transition at the endpoint of a BKT line should belong to the finite temperature universality class. Our estimate and with that critical behavior where . However the fate of this clean critical point under the influence of disorder is controlled by the Harris criterion.harris (); chay () If the inequality is fulfilled, the disorder does not affect the critical behavior. If the Harris criterion is violated ( ), the generic result is a new critical point with conventional power law scaling but new exponents which fulfill . Since violates this inequality disorder is relevant and drives the system from the mean-field to an other critical point with different critical exponents as our estimate, , consistent with , uncovers. The resulting 2D-QSI transition with and violates then the equivalence between quantum phase transitions in systems with spatial dimensions and the ones of classical phase transitions in dimensions. In addition the proportionality between and (Eq. (11)), valid below the upper critical dimension , does no longer hold because is above . In fact magnetic penetration depth measurements taken on underdoped high quality YBaCuO single crystals revealed .liang () Another option is that the disorder destroys the QSI transition. Given the evidence for a rounded BKT transition (see Fig. 1) and noting that the analyzed data do not extend very close to the QSI transition (see Fig. 4) further studies are required to elucidate this option. In any case, our estimate , the evidence for BKT behavior at finite temperature and the Harris theorem imply the presence and relevance of disorder at the QSI transition and its irrelevance at finite temperature. Further evidence for the importance of disorder follows from the temperature dependence of the sheet resistance in the insulating phase. Considering of Bollinger et al.bol () we observe consistency with the Mott variable range hopping model in 2D. The conductivity exhibits the characteristic temperature dependence which applies to strongly disordered systems with localized states.mott ()

A previous and analogous analysis of the sheet resistance data of the superconducting LaAlO/ SrTiO interface revealed the critical sheet resistance k and the exponents , and .tsintf () In this case the gate voltage tuned QSI transition of the 2D electron system at the LaAlO/ SrTiO interface has a finite temperature counterpart in dimensions namely the -xy model where hyperscaling and with that the proportionality between and (Eq. (11)) applies. On the other hand implies according to Harris theoremharris (); chay () that disorder is relevant as well.

To comment on the BKT-features in the temperature dependence of the magnetic penetration depth we considered the data of Bert et al.bert () for a gate voltage tuned superconducting LaAlO/SrTiO interface in terms of vs. . The data reveals the flow to the universal relationship (11) but much lower must be attained to reach quantum critical regime. However in this low regime is the Nelson-Kosterlitz jump given by Eq. (10) very small and appears to be beyond present experimental resolution.bert () Otherwise the data points resemble the outline of a fly’s wing,parks () remarkably similar to the vs. plots of the bulk superconductors YCu-123, Tl-1212,bern () and Tl-2201nied (), covering nearly the entire doping regime in the so called superconducting dome extending from the underdoped to the overdoped limit.

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