Bose-glass-Fermi-glass transition
Korekiyo Takahashi1,2,7 , Keiji Nakatsugawa2,6 , Masahito Sakoda1,2 , Yoshiko
Nanao3 , Michio Naito4 , Hiroyoshi Nobukane5,2 , Hideaki Obuse1,2 , Satoshi Tanda1,2
1

Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan
2

Center of Education and Research for Topological Science and Technology,
Hokkaido University, Sapporo, 060-8628, Japan3

School of Physics and Astronomy,

University of St. Andrews, Fife, KY16 9SS, Scotland
4

Department of Applied Physics, Tokyo University ofAgriculture and Technology, Tokyo, 184-8588, Japan
5

Department of Physics, Hokkaido University, Sapporo, 060-0810, Japan
6

International Center for Materials Nanoarchitectonics,

National Institute for Material Science, Tsukuba, 305-0044 Japan
7

Nomura Research Institute, Ltd., Tokyo, 100-0004, Japan(Dated: April 21,2022)

0Abstract
It is known that two-dimensional superconducting materials undergo a quantum phase transition
from a localized state to superconductivity. When the disordered samples are cooled, bosons(Cooper pairs) are generated from Fermi glass and reach superconductivity through Bose glass.
However, there has been no universal expression representing the transition from Fermi glass toBose glass.
Here, we discovered an experimental renormalization group ﬂow from Fermi glass to Bose glass interms of simple β-function analysis. To discuss the universality of this ﬂow, we analyzed manifestly
diﬀerent systems, namely a Nd-based two-dimensional layered perovskite and an ultrathin Pb ﬁlm.
We ﬁnd that all our experimental data for Fermi glass fall beautifully into the conventional selfconsistent β-function. Surprisingly, however, ﬂows vertical to the conventional β-function are
observed in the weakly localized regime of both systems, where localization becomes even weaker.
We show that the vertical ﬂow is evidence for the existence of Bose glass. Consequently, we conﬁrm
a universal quantum transition from Bose glass to Fermi glass with the new two-dimensional criticalsheet resistance close to R□ = h/e2 .

1

MAIN

The electric conductance in the quantum localized regime (a regime where the electrical resistance increases as the temperature decreases) of two-dimensional (2D) disorderedsystems has been discussed in terms of Fermi glass1–7 , i.e., Mott localization8,9 for strongly
correlated systems and Anderson localization10–15 for non-interacting systems16 , and Bose
glass17,18 . The Bose-glass phase is an insulator phase with properties similar to those of
Fermi glass, and can be described as the phase where the 2D bosons are localized as a result of the quenched 2D disorder. Anderson localization has been studied using β-function
analysis10,19 . Mott localization has been studied via Mott variable range hopping conduction(VRH)20 and via Fisher scaling for the boson Hubbard model8,21,22 . Recently, Kapitulnik
et al.23 have shown that there is an anomalous metal state that overturns the conventional
wisdom in the regime below the superconducting critical sheet resistance8,24 h/4e2 and is different from the quantum localized regime. However, a boson-fermion mixture or a transition
from Fermi glass to Bose glass has not been considered in the quantum localized regime.
In this Article, we discovered an experimental renormalization group ﬂow from Fermi
glass to Bose glass in terms of β-function analysis. The critical sheet resistance, which isthe boundary between Bose glass and Fermi glass, is indicated to be around h/e2 as shownin Figure 1. To discuss the universality of this ﬂow, we analyzed manifestly diﬀerent systems, namely a Nd-based two-dimensional layered perovskite and an ultrathin Pb ﬁlm. The
layered perovskite structure of Nd2 CuO4 and Nd2 PdO4 is called a T ′ -structure25 , and it has
an ideal 2D electron conductance because the conducting layer is composed of square planar
units. FIG. 2 shows the temperature dependence of the electric resistance for Nd2 CuO4-x Fx
single crystals26 , Nd2-x Cex CuO4 thin ﬁlms27 , and Nd2-x Cex PdO4 thin ﬁlms28 . Doping causes
a Nd2 CuO4 system to undergo a quantum phase transition from the localized state to thesuperconducting state, i.e. superconductivity - insulating (S-I) transition. Superconductivity is not observed in a Nd2 PdO4 system regardless of the doping amount28 . On the other
hand, the ultrathin Pb ﬁlm29 is grown sequentially in situ by the evaporation of Pb. The
ﬁlm changes to a superconductor from an insulator as the ﬁlm thickness increases (FIG. 2).
We ﬁnd that all experimental data in Fermi glass fall beautifully into the conventional selfconsistent β-function. Surprisingly, however, ﬂows vertical to the conventional β-function
were observed in the weakly localized regimes of both systems, where localization becomes2

FIG. 1: Schematic phase diagram for 2D disordered superconducting materials. The analysis in
this airticle used the β-function to show the behavior of the change from Bose glass to Fermi
glass before reaching superconductivity (bi-directional red arrow). The critical sheet resistance,
which is the boundary between Bose glass and Fermi glass, is indicated to be around h/e2 (reddot). The bi-directional black dotted arrow represents Paalanen et al.’s experiments18 .

even weaker. We show that the vertical ﬂow is evidence for the existence of Bose glass byusing an experimental β-function and the temperature-derivative equation of the β-function(β ′ -function). Consequently, we can conﬁrm that there is a universal quantum transition
from Bose glass to Fermi glass with the new two-dimensional critical sheet resistance close
to R□ = h/e2 . Since the Bose-glass phase in two types of Nd-based two-dimensional layered
perovskites and Pb ultrathin ﬁlms both exhibit a β ′ -function diﬀerent from that of the Fermiglass phase, with R□ between h/e2 and h/4e2 , this phenomenon is considered universal.
3

(a) Nd2 CuO4-x Fx(b) Nd2-x Cex CuO4(c) Nd2-x Cex PdO4(d) Ultrathin Pb ﬁlms

FIG. 2: The temperature dependence of the electric resistivity 2a (a) Nd2 CuO4-x Fx 26 , (b)Nd2-x Cex CuO4 27 , (c) Nd2-x Cex PdO4 28 and (d) ultrathin Pb ﬁlms29 (thickness changed from
13.8Å to 42.8Å) . Superconductivity-insulator transition is observed in (a) Nd2 CuO4-x Fx (theinset shows the superconducting state), Nd2-x Cex CuO4 and (d) ultrathin Pb ﬁlms (the inset
shows the superconducting state), while in (c) Nd2-x Cex PdO4 only the insulating state appears.
Sample E of Nd2 CuO4-x Fx , Sample I of Nd2 CuO4-x Fx and Sample F of ultrathin Pb ﬁlms will be
discussed in detail later. Sample E of Nd2 CuO4-x Fx is shown to be Bose glass and Sample I of
Nd2 CuO4-x Fx is shown to be Fermi glass. Sample F of ultrathin Pb ﬁlms shows a change from
Fermi glass to Bose glass as the temperature become lower.

VERTICAL FLOWS IN β -FUNCTION ANALYSIS

We investigate our experimental results with β-function analysis10,13 , which is applicablefrom weak to strong localization regimes. The β-function is deﬁned as β(g) ≡ d ln g/d ln L,
where L is the sample size and g is the dimensionless conductance [ g = (ℏ/e2 )σ ]. The
β-function shows the ﬂow of electronic states as a function of size L, and can be used
to determine whether it is delocalized or localized, and even weakly or strongly localized.
With orthogonal class in 2D systems, β(g) < 0 means that the system always belongs to
the localized phase due to Anderson localization regardless of the size L, but this is not
the case for systems with a strong spin-orbit interaction, or for the quantum Hall eﬀect. In
such cases, the form of the β-function changes depending on the state of localization. In
other words, if we view the form of the β-function as a renormalization group ﬂow, we can4understand the universality of the electronic state with the scale transformation.
Experimentally, L is taken to be the cutoﬀ length due to inelastic scattering: L2 = DT −p .
D is the diﬀusion constant, and T is temperature. The value of exponent p depends on the
inelastic scattering mechanism. For Nd-based 2D layered perovskite systems, p = 1.026 , forultrathin Pb ﬁlms, p = 2.129 . So, the β-function is derived from the experimental data ofthe temperature dependence of the conductance as follows:
βEXP. (g) = −2 d ln g.
p d ln T(1)

This equation is predicted to behave as follows from weak to strong localized regimes;
With a weakly localized regime, namely the 2D Anderson localization theory with conductance σ = σ0 + σ1 ln T , we obtain the following simple relation:
βEXP. (g) = −2 d ln(σ0 + σ1 ln T )
2 σ1
1
=−
= − 2 ∼ −1/g.
pd ln Tpσ
π g(2)

Since σ1 contains p, the result of the calculation is that p cancels out and βEXP. (g) is related
only to g. With a strong localized regime, namely the VRH theory in Mott 2D, g ∝exp(−εVRH /T 1/3 ), so that, βEXP. (g) ∼ ln g. This εVRH is equal to the generalized activationenergy30 .
On the other hand, in the self-consistent theory of the β-function, Vollhardt and Wölﬂegave a single 2D system conductivity formula for orthogonal classes from weakly to strongly
localized regimes31,32 .
1gVW (x) = 2 (x + 1) ln
2π

(

)
1
+ 1 exp(−x),x2(3)

Here, x = L/ξ ∝ T −p/2 /ξ , where ξ is the localization length. Furthermore, the β-function
from eq.(3) can be calculated as follows:d ln gVW (x)βVW (g) ==−d ln x

(

2
x2
(
)
+ 2
x + 1 (x + 1) ln x12 + 1

)
.

(4)

We analyzed our experimental data by using the above equations: ﬁrst we estimated ξfrom eq.(3), next we described the β-function obtained from eq.(4), ﬁnally we overlaid the5(a) Nd-based 2D layered perovskite(b) ultrathin Pb ﬁlms

FIG. 3: Enlarged view of βEXP. (g) - the experimental β-function in a weakly localized regime.
The inset shows βEXP. (g) from a weak to a strong localized regime. The black solid line showsβVW (g) — Vollhardt and Wölﬂe β-function (eq.(4)). The data for Nd-based 2D layered
perovskite were plotted using resistance values from 5K to 25K. The data for ultrathin Pb ﬁlms
were plotted using resistance values from 4K to 10K. The color depends on the sample. If weview the β-function as a renormalization group ﬂow, the Anderson localization β-function ﬂow
should behave like βVW (g) as indicated with the blue arrows. However, as shown by the redarrows, we discover a diﬀerent ﬂow which is vertical to βVW (g). In the weakly localized regime,these ﬂows are present in both systems. In some ultrathin Pb ﬁlm samples (e.g., Pb-sample-F
surrounded by a dotted circle), it is clear that the ﬂow changed from the usual Anderson
localization ﬂow to a vertical ﬂow at a certain temperature. The yellow star in each graphindicates g = 1/2π and β = −2/π, which is the dimensionless version of the critical sheet fermion
resistance R□ = h/e2 . This star is the boundary value at which the vertical ﬂow occurs, as
determined from FIG. 5 and Appendix B. Therefore, it proves that the phenomenon of these
vertical ﬂows is at its least apparent when R□ is smaller than h/e2 .

βEXP. (g) data of eq.(1) and the βVW (g) data of eq.(4), to verify whether or not they matched.
We see that all the data from weakly to strongly localized regimes fall into βVW (g) (inset
of FIG. 3). However, if we take a closer look at the weakly localized regime, as shownin FIG. 3, we discover that the data exhibit diﬀerent vertical ﬂows (indicated by the red
6arrows) against the usual Anderson localization β-function curve (indicated by the blue
arrows). In the weakly localized regime, these vertical ﬂows are present in both Nd-based
2D layered perovskite and ultrathin Pb ﬁlms. In some ultrathin Pb ﬁlm samples, it is clearthat the ﬂow changes from the usual Anderson localization ﬂow to the vertical ﬂow at a
certain temperature (e.g., Pb-sample-F surrounded by a dotted circle). Nd2 CuO4-x Fx single
crystals and Nd2-x Cex CuO4 thin ﬁlms have diﬀerent vertical ﬂows, but Nd2-x Cex PdO4 does
not have them. It is considered to be a phenomenon that is a precursor to the transition
from weak localization to superconductivity.
To understand the mechanism of vertical ﬂows, we analyze the temperature dependence ofresistivity in greater detail again. The functional forms of these vertical ﬂows appear in the
weakly localized regime, but are not adapted to −1/g. Therefore, we analyzed two samples
before and after the vertical ﬂow occurred; namely (a) Nd2 CuO4-x Fx (NCOF) sample E in
which this vertical ﬂow appears and (b) NCOF sample I in which the vertical ﬂow does notappear. As a result, we ﬁnd a diﬀerence in the temperature dependence of conductivity andresistivity. As shown in FIG. 4, the ln T dependence of the NCOF-sample-E resistivity ismore suitable than that of the conductivity, whereas the NCOF-sample-I conductivity has
ln T dependence according to the Andeson localization theory. Although these are in thesame weakly localized regime, the vertical ﬂow is clearly characterized by ρ ∼ ln(1/T ) rather
than by σ ∼ ln T . Das et al.33 pointed out that the weak localization of bosons occurringon the insulator side of the S-I transition in 2D is characterized by ρ ∼ ln(1/T ). Therefore,
the state of the sample with the vertical ﬂow can be regarded as boson localization, or the
Bose-glass phase. These experimental results indicate that the Bose glass regime has beenfound.

BOSE-GLASS-FERMI-GLASS TRANSITION

Next, to investigate the condition of the change from Bose glass to Fermi glass, we plotted
the temperature dependence of βEXP. (g) instead of the resistance as shown in FIG. 5 witha series of experimental data. Interestingly, the temperature dependence of βEXP. (g) in the
weakly localized regime is very similar to the S-I transition graph. As the localization ofeach sample weakens (from the bottom to the top of FIG. 5), the slope of each set of datachanges continuously from positive to negative in both Nd-based 2D layered perovskite and7ultrathin Pb ﬁlms. To understand the change in the slope of βEXP. (g) ﬂows in FIG. 5, we
use the temperature-derivative equation of the β-function shown below,′
βEXP.
(g) = −

d ln |βEXP. (g)|.
d ln T(5)′
The change in the β-function ﬂow should appear in the positive or negative sign of βEXP.
(g).

Substituting the two equations σ = σ0 + σ1 ln T and σ = 1/ρ = 1/(ρ0 + ρ1 ln(1/T ))(σ, σ0 , σ1 , ρ0 , ρ1 > 0) we see that the signs are diﬀerent as follows.
σ1> 0, (σ = σ0 + σ1 ln T )σ0 + σ1 ln T

(6)

ρ1
1
1
< 0, (σ = =).
ρ0 + ρ1 ln(1/T )ρρ0 + ρ1 ln(1/T )

(7)

′
βEXP.
(g) ≃

′
βEXP.
(g) ≃ −

These calculation results conﬁrm that the slope of βEXP. (g) reﬂects the temperaturedependent functional form of the conductivity and resistivity. To check this, we analyzed′
all the data for βEXP.
(g) < 0 and found that they were more suitable for ρ ∼ ln(1/T ) thanσ ∼ ln T . However, in the ultrathin Pb ﬁlm data in FIG. 5 (b), there were samples where′
the sign of βEXP.
(g) changed from positive to negative when the temperature decreased. For′
Pb-sample-F, the sign of βEXP.
(g) changed around 6K. Since the superconducting transition

temperature Tc of Pb (in the clean system) is around 7.2K, it is conceivable that bosons
are formed. On the other hand, NCOF-sample-E did not exhibit any change of sign. Thismay be attributed to the measurement temperature range. Another set of data measured
up to the high temperature regime was analyzed, and the change was conﬁrmed below thesuperconducting transition temperature Tc of the Nd2 Pd1-x Cux O4-y Fy (see Appendix A).
′
We also investigated the critical value of βEXP. (g), denoted as βC , when the slope of βEXP.
(g)went to zero. From FIG. 5, βC was obtained by using the slope and the intercept of thevertical axis for each sample at a speciﬁc low temperature. We obtained βC = −0.6 ± 0.1
with these two diﬀerent types of samples (see Appendix B). This value is almost equal
to −0.64(≃ −2/π) and becomes the value of g = 1/2π, which we can convert to β =−2/π by using eq.(2). The critical g does not indicate R□ = h/4e2 of the superconducting
critical sheet resistance, but is at least close to R□ = h/e2 of the critical sheet fermion
resistance. Mapping these values in FIG. 3 as a yellow star, we can see that the phenomenonof these vertical ﬂows occurs at least when g is larger than 1/2π. This g = 1/2π value istransformed into the dimensionless version of the critical sheet fermion resistance R□ = h/e2 .
8Based on the above results, at least two conditions are required to generate Bose-glass;
′
′
βEXP.
(g) < 0 and R□ < h/e2 . The point at which the sign of βEXP.
(g) changes is related tothe superconducting transition temperature Tc . This experimental analysis regarding the
β-function clearly demonstrated the criteria for the Fermi-glass to Bose-glass transition. In
other words, it proved the existence of boson formation even in a localized regime.
Finally, we suggest that the fermion-only theory may be insuﬃcient to explain the weakly
localized regime just before reaching superconductivity in 2D disordered superconducting
materials. Fisher’s theory is also insuﬃcient because it is based on boson-only or fermiononly in the strongly localized regime represented by VRH. Therefore, the boson-fermion
mixture state and boson-only state in the weakly localized regime are important areas inclarifying the relationship between statistics (fermion, boson, and anyon) and localization.
We discover an experimental renormalization group ﬂow from Fermi glass to Bose glass byemploying β-function analysis. To discuss the universality of Bose-glass ﬂow, we analyzed
distinctively diﬀerent systems, namely a Nd-based two-dimensional layered perovskite andan ultrathin Pb ﬁlm. What these two systems have in common is that they are both 2D
disordered systems, but the diﬀerence is that a Nd-based 2D layered perovskite is disorderedby doping while an ultrathin Pb ﬁlm is structurally disordered. Furthermore, unlike anultrathin Pb ﬁlm, a Nd-based 2D layered perovskite is a strongly correlated electron system9 .
Although these two systems may have diﬀerent superconductivity mechanisms, the ﬂow ofthe experimental β-function exhibits the same behavior.
The analysis results obtained for the two systems in common are shown in TABLE I.
These results suggest the following mechanism of universal quantum transition. With varying disorder in the weakly localized regime, Anderson localization (σ ∼ ln T ) is observed
when R□ is larger than h/e2 . When R□ is smaller than h/e2 , bosons and vortices are generated and the state becomes Bose glass. Furthermore, when R□ is smaller than h/4e2 ,
fermions disappear and superconductivity occurs. It seems reasonable that Bose glass appears between h/e2 and h/4e2 .
Paalanen et.al.18 identiﬁed Bose glass as the regime where the Hall resistance ρxy has azero or a ﬁnite value when the longitudinal resistance ρxx diverges at low temperatures andin a certain magnetic ﬁeld (the bi-directional black dotted arrow in FIG. 1). On the other
hand, we used the experimental β-function to identify the ﬂow vertical to the 2D Anderson
localized ﬂow as Bose glass (the bi-directional red arrow in FIG. 1). This analysis constitutes
9

TABLE I: Summary of the experimental results. In the weakly localized regime where 2D
Anderson localization dominates, both Fermi-glass and Bose-glass exist. The existence of the
Bose-glass regime is clearly shown by the experimental β-function and the sign of the β ′ -function(the temperature derivative of the experimental β-function). We conﬁrm a universal quantum
transition from Bose glass to Fermi glass with the new two-dimensional critical sheet resistanceclose to R□ = h/e2 at β ′ = 0.

a simple method of identifying weak boson/fermion localization. As an application of theseresults, even if superconductivity does not appear in a 2D material, the discovery of Bose
glass ﬂow in its β-function may provide a hint that could allow us to make the material
superconductive by changing the disorder parameters.

ACKNOWLEDGMENTS

We are grateful to K. Yakubo, K. Ichimura and T. Matsuyama for suggesting the topic
dealt with in this Article. We also thank many of those who attended the Forum of Localisation 2020. We have gained useful insights. Finally, we would like to thank S. Ozeki,
K. Kagawa and K. Inagaki who studied together about 30 years ago, and T. Nakayama for
introducing such a wonderful physics ﬁeld.

1

Anderson, P. W. The Fermi Glass: Theory and Experiment, 353–359.

2

Fleishman, L. & Anderson, P. W. Interactions and the anderson transition. Phys. Rev. B 21,
2366–2377 (1980).

3

Freedman, R. & Hertz, J. A. Theory of a fermi glass. Phys. Rev. B 15, 2384–2398 (1977).

10

4Abrahams, E., Kravchenko, S. V. & Sarachik, M. P. Metallic behavior and related phenomena
in two dimensions. Rev. Mod. Phys. 73, 251–266 (2001).

5

Spivak, B., Kravchenko, S. V., Kivelson, S. A. & Gao, X. P. A. Colloquium: Transport instrongly correlated two dimensional electron ﬂuids. Rev. Mod. Phys. 82, 1743–1766 (2010).

6

Popović, D., Fowler, A. B. & Washburn, S. Metal-insulator transition in two dimensions: Eﬀects
of disorder and magnetic ﬁeld. Phys. Rev. Lett. 79, 1543–1546 (1997).

7

Bogdanovich, S. c. v. & Popović, D. Onset of glassy dynamics in a two-dimensional electron
system in silicon. Phys. Rev. Lett. 88, 236401 (2002).

8

Fisher, M. P. A., Grinstein, G. & Girvin, S. M. Presence of quantum diﬀusion in two dimensions:
Universal resistance at the superconductor-insulator transition. Phys. Rev. Lett. 64, 587–590(1990).

9

Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–
1263 (1998).

10Abrahams, E., Anderson, P. W., Licciardello, D. C. & Ramakrishnan, T. V. Scaling theory of
localization: Absence of quantum diﬀusion in two dimensions. Phys. Rev. Lett. 42, 673–676(1979).

11

Abrahams, E. 50 Years of Anderson Localization (World Scientiﬁc, 2010).

12

Anderson, P. W. Four last conjectures (2018). arXiv:1804.11186.

13

Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337(1985).

14

Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).

15

Obuse, H., Subramaniam, A. R., Furusaki, A., Gruzberg, I. A. & Ludwig, A. W. W. Conformal
invariance, multifractality, and ﬁnite-size scaling at anderson localization transitions in two
dimensions. Phys. Rev. B 82, 035309 (2010).

16

Gruzberg, I. A., Read, N. & Vishveshwara, S. Localization in disordered superconducting wires
with broken spin-rotation symmetry. Phys. Rev. B 71, 245124 (2005).

17

Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and thesuperﬂuid-insulator transition. Phys. Rev. B 40, 546–570 (1989).

18

Paalanen, M. A., Hebard, A. F. & Ruel, R. R. Low-temperature insulating phases of uniformly
disordered two-dimensional superconductors. Phys. Rev. Lett. 69, 1604–1607 (1992).

19

Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).

11

20

F.Mott, N.

Conduction in non-crystalline materials.

The Philosophical Magazine:

A Journal of Theoretical Experimental and Applied Physics 19, 835–852 (1969).
21

Fisher, M. P. A. Quantum phase transitions in disordered two-dimensional superconductors.
Phys. Rev. Lett. 65, 923–926 (1990).

22

Haldar, P., Laad, M. S. & Hassan, S. R. Quantum critical transport at a continuous metalinsulator transition. Phys. Rev. B 94, 081115 (2016).

23

Kapitulnik, A., Kivelson, S. A. & Spivak, B. Colloquium: Anomalous metals: Failed superconductors. Rev. Mod. Phys. 91, 011002 (2019).

24

Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional
limit. Phys. Rev. Lett. 62, 2180–2183 (1989).

25

Müller-Buschbaum, H. & Wollschläger, M. Über ternäre oxocuprate. vii. zur kristallstruktur
von Nd2 CuO4 . Zeitschrift für anorganische und allgemeine Chemie 414, 76–80 (1975).

26

Tanda, S., Takahashi, K. & Nakayama, T.

Scaling behavior of the conductivity of

Nd2 CuO4−x−δ Fx single crystals: Evidence for orthogonal symmetry. Phys. Rev. B 49, 9260–
9263 (1994).
27

Tanda, S., Honma, M. & Nakayama, T. Critical sheet resistance observed in high-Tc oxidesuperconductor Nd2−x Cex CuO4 thin ﬁlms. Phys. Rev. B 43, 8725–8728 (1991).

28

Nanao, Y. et al. Crystal growth and metal-insulator transition in two-dimensional layered
rare-earth palladates. Phys. Rev. Materials 2, 085003 (2018).

29

Kagawa, K., Inagaki, K. & Tanda, S. Superconductor-insulator transition in ultrathin Pb ﬁlms:
Localization and superconducting coherence. Phys. Rev. B 53, R2979–R2982 (1996).

30

Hill, R. M. On the observation of variable range hopping. physica status solidi (a) 35, K29–K34(1976).

31

Vollhardt, D. & Wölﬂe, P. Scaling equations from a self-consistent theory of anderson localization. Phys. Rev. Lett. 48, 699–702 (1982).

32

Wölﬂe, P. & Vollhardt, D. SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION:
GENERAL FORMALISM AND APPLICATIONS, 43–71 (World Scientiﬁc, 2010).

33

Das, D. & Doniach, S. Weakly localized bosons. Phys. Rev. B 57, 14440–14443 (1998).

12

(a) Nd2 CuO4-x Fx Sample E in the Bose-glass regime(b) Nd2 CuO4-x Fx Sample I in the Fermi-glass regime

FIG. 4: Graphs comparing the log T dependence of σ□ (left) and R□ (right). We analyzed two
samples before and after the vertical ﬂow occurred; namely (a) Nd2 CuO4-x Fx (NCOF) sample E
in which this vertical ﬂow appears and (b) NCOF sample I in which the vertical ﬂow does notappear. (NCOF-sample-E is along the red arrow and NCOF-sample-I is along the blue arrow in
FIG. 3). The black solid line shows the regression line. The left vertical axis of the graph is σ□or R□ . The horizontal axis of the graph is log T , and the right vertical axis is the value obtainedby subtracting σˆ□ or Rˆ□ from the experimental value. For standardization, we divide by thevalue obtained by subtracting the minimum value from the maximum value of the experimental
data on the vertical axis. Clearly, NCOF-sample-E is more suitable for the log T dependence ofthe resistivity than that of the conductivity, while NCOF-sample-I is the opposite. The verticalﬂow in the β-function indicates ρ ∼ ln(1/T ) rather than σ ∼ ln T . As a result, we ﬁnd a
diﬀerence between the log T dependence of the conductivity and resistivity.

13

(a) Nd-based 2D layered perovskite(b) ultrathin Pb ﬁlms

FIG. 5: Graphs of the relationship between the value of βEXP. (g) and temperature T with aseries of experimental data in a weakly localized regime. Interestingly, the temperaturedependence of βEXP. (g) is very similar to the S-I transition graph. As the localization of eachsample weakens (from the bottom to the top of the graph), the slope of each set of data changes
continuously from positive to negative in both Nd-based 2D layered perovskite and ultrathin Pb
ﬁlms. The color depends on the sample. The black solid line shows βVW (g) — Vollhardt andWölﬂe β-function (eq.(4)). We can see that among the weakly localized samples, those with′strong localization (βEXP.
(g) > 0) ﬁt into βVW (g). However, in the (b) graph, as the localization

becomes weaker, the slope of Pb sample F changes from positive (blue dotted line) to negative(red dotted line) when the temperature falls below around 6K. Since the superconducting
transition temperature Tc of Pb (in a clean system) is around 7.2K, it is conceivable that boson
appears. We also investigated the critical value of βEXP. (g), denoted as βC when the slope of′βEXP.
(g) goes to zero. βC was obtained by using the slope and the intercept of the vertical axis

for each sample at a speciﬁc low temperature. We obtained βC = −0.6 ± 0.1 in these two
diﬀerent types of samples. This value is almost the same 0.64(≃ 2/π) as shown by the yellow
star, and has a value of g = 1/2π when converted using eq.(2). This value is the dimensionlessversion of the critical sheet fermion resistance R□ = h/e2 .

14

Appendix A: βEXP. (g) and T with Nd2 Pd1-x Cux O4-y Fy

We analyzed special substances in Nd2 Pd1-x Cux O4-y Fy ﬁlms grown on CaF2 . The Ndbased high-temperature superconductor Nd2 CuO4 develops superconductivity as a resultof appropriate reduction treatment. We investigated the temperature dependence of theresistance of Nd2 Pd1-x Cux O4 in which this Cu was replaced with Pd (FIG. 6, left panel).
If the proportion of Cu is large (x = 0.84), it is expected that bosons will be generated ata certain temperature and the βEXP. (g) slope will change. The analysis result is shown on
the right in FIG. 6. At about 20K (red arrow) the experimental data moved away from′βVW (g) and there was a positive and negative change in βEXP.
(g). This temperature is almost

the same as the critical temperature Tc of the Nd-based high-temperature superconductor.
FIG. 6 conﬁrms the transition from Fermi-glass (yellow dots) to Bose-glass (green dots) on
Nd-based 2D layered perovskite as well as ultrathin Pb ﬁlms.

′
Appendix B: boundary value when βEXP.
(g) = 0 is in the weakly localized regime′
So, if there is a boundary between positive and negative βEXP.
(g) in the weakly localized′(g) = 0, what is the value of βEXP. (g) and the conductivity g?
regime, in short when βEXP.

For each graph in FIG. 5, we regarded βEXP. (g) as a linear function of T in a particulartemperature range such as βEXP. (g) = aT + b (where a is the slope of βEXP. (g) and b is theintercept of βEXP. (g)). FIG. 7 shows a graph in which a and b of each graph in FIG. 5
are mapped, and the βEXP. (g) value at T = 0 is obtained. From the experimental data in
FIG. 7, we obtained βC = −0.6 ± 0.1 in these two diﬀerent types of samples. This value isalmost equal to 0.64(= 2/π) and becomes the value of g = 1/2π to which we can convert
β = −2/π using eq.(2). This g is a dimensionless version of the critical sheet fermion′
resistance R□ = h/e2 . The boundary of βEXP.
(g) = 0 separates the Bose-glass and Fermi-

glass regimes. In other words, this result reveals that the condition for the occurrence of
weakly localized ρ ∼ ln(1/T ) is g > 1/2π. This means that superconductivity cannot occur
above R□ = h/e2 , so the results of these analyses are consistent.

15

FIG. 6: The left panel shows the temperature dependence of the resistance of
Nd2 Pd1-x Cux O4-y Fy ﬁlms grown on CaF2 (x = 0.43 ∼ 1.00). The right panel shows therelationship between βEXP. (g) and temperature T with Nd2 Pd0.16 Cu0.84 O4 . Among the localized
samples, this sample was the closest to the superconducting state. To calculate βEXP. (g) requires
the dependence of the value of exponent p on the mechanism of inelastic scattering, but since it
was not measured, it was calculated as p = 1.5. The black solid line shows βVW (g). It was at
about 20K (red arrow) that the experimental data moved away from βVW (g) and the sign of′(g) changed. This temperature is almost the same as the critical temperature Tc of the
βEXP.
′
Nd-based high-temperature superconductor. No change in βEXP.
(g) was observed in samples

other than x = 0.84. This graph conﬁrms the transition from Fermi-glass (yellow dots) to
Bose-glass (green dots) as the temperature decreases.

16

(a) Nd-based 2D layered perovskite systems(b) ultrathin Pb ﬁlms′
FIG. 7: The above graphs are the analysis to used to ﬁnd a boundary value when βEXP.
(g) = 0in the weakly localized regime. For each graph in FIG. 5, we regarded βEXP. (g) as a linear
function of T in a particular temperature range such as βEXP. (g) = aT + b. The horizontal axis ais the slope of βEXP. (g) and the vertical axis b is the intercept of βEXP. (g). These graphs showed aand b of each graph in FIG. 5, and the βEXP. (g) value at T = 0 was obtained. From theexperimental data obtained for the Nd-based 2D layered perovskite systems, we obtained
βEXP. (g) = −0.54 ± 0.03. And from the experimental data for the ultrathin Pb ﬁlms, we obtained
βEXP. (g) = −0.66 ± 0.01. From the experimental data in FIG. 7, we obtained βC = −0.6 ± 0.1 inthese two diﬀerent types of samples. This value is almost equal to 0.64(≃ 2/π) and becomes avalue of g = 1/2π, to which we can convert β = −2/π using eq.(2). This g is the dimensionless′version of the critical sheet fermion resistance R□ = h/e2 . The boundary of βEXP.
(g) = 0separates the Bose-glass and Fermi-glass regimes.

17