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OPEN

Bose glass and Fermi glass
Korekiyo Takahashi 1,2,6*, Keiji Nakatsugawa 2,5,7, Masahito Sakoda 1,2, Yoshiko Nanao 3,
Hiroyoshi Nobukane 2,4, Hideaki Obuse 1,2,7 & Satoshi Tanda 1,2,7
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 to Bose glass. Here,
we discovered an experimental renormalization group flow from Fermi glass to Bose glass in terms 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 findthat all our experimental data for Fermi glass fall beautifully into the conventional self-consistent β-function. Surprisingly, however, flows perpendicular to the conventional β-function are observed in
the weakly localized regime of both systems, where localization becomes even weaker. Consequently,
we propose a universal transition from Bose glass to Fermi glass with the new two-dimensional criticalsheet resistance close to R = h/e2.
The electric conductance in the quantum localized regime (a regime where the electrical resistance increases as
the temperature decreases) of two-dimensional (2D) disordered systems has been discussed in terms of Fermi­glass1–7, i.e., Mott ­localization8,9 for strongly correlated systems and Anderson ­localization10–15 for non-interacting­systems16, and Bose g­ lass17,18. 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 m­ odel8,21,22.
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 r­ esistance8,24 h/4e2 and is different from thequantum 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 interms of β-function analysis. The critical sheet resistance, which is the boundary between Bose glass and Fermi
glass, is indicated to be around h/e2 as shown in Fig. 1.
To discuss the universality of this flow, we analyzed manifestly different systems, namely a Nd-based twodimensional 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 the superconducting 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 theevaporation 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 self-consistent β-function. Surprisingly, however, flows perpendicular to the conventional β-function were observed in the weakly
localized regimes of both systems, where localization becomes even weaker. We show that the perpendicular flow
implies the existence of Bose glass by using an experimental β-function and the temperature-derivative equationof the β-function ( β ′-function). Consequently, we can propose that there is a universal 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

1

Department of Applied Physics, Hokkaido University, Sapporo 060‑8628, Japan. 2Center of Education andResearch for Topological Science and Technology, Hokkaido University, Sapporo 060‑8628, Japan. 3School ofPhysics and Astronomy, University of St Andrews, Fife KY16 9SS, Scotland. 4Department of Physics, Hokkaido
University, Sapporo 060‑0810, Japan. 5Research Center for Materials Nanoarchitectonics, National Institute forMaterial Science, Tsukuba 305‑0044, Japan. 6Nomura Research Institute, Ltd., Tokyo 100‑0004, Japan. 7Theseauthors contributed equally: Keiji Nakatsugawa, Hideaki Obuse and Satoshi Tanda. *email: korere@gmail.com
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Figure 1.  Schematic phase diagram for 2D disordered superconducting materials. The analysis in thisairticle used the β-function to show the behavior of the change from Bose glass to Fermi glass before reachingsuperconductivity (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 (red dot). The bi-directional black dotted arrow represents
Paalanen et al.’s ­experiments18.

Figure 2.  The temperature dependence of the electric resistivity (a) Nd2 CuO4−x Fx26, (b) ­Nd2–xCexCuO427, (c)­ d2–xCexPdO428 and (d) ultrathin Pb fi
N­ lms29 (thickness changed from 13.8Å to 42.8Å). Superconductivity–
insulator transition is observed in (a) ­Nd2CuO4–xFx (the inset shows the superconducting state), N­ d2–xCexCuO4and (d) ultrathin Pb films (the inset shows the superconducting state), while in (c) ­Nd2–xCexPdO4 only theinsulating state appears. Sample E of ­Nd2CuO4–xFx, Sample I of N
­ d2CuO4–xFx and Sample F of ultrathin Pb
films will be discussed in detail later. Sample E of ­Nd2CuO4–xFx is shown to be Bose glass and Sample I of
­Nd2CuO4–xFx is shown to be Fermi glass. Sample F of ultrathin Pb films shows a change from Fermi glass toBose glass as the temperature becomes lower.
a β ′-function different from that of the Fermi glass phase, with R between h/e2 and h/4e2, this phenomenon
is considered universal.

Vertical flows in β‑function analysis

We investigate our experimental results with β-function ­analysis10,13, which is applicable from 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 thedimensionless conductance [ g = (/e2 )σ in 2D]. 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

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localized. In other words, if we view the form of the β-function as a renormalization group flow, we can 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 the temperature. The value of exponent p depends on the inelastic scattering mechanism. For
Nd-based 2D layered perovskite systems, p = 1.026. For ultrathin Pb films, p = 2.129. So, the β-function is derived
from the experimental data of the 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;
In a weakly localized regime, the celebrated weak localization theory in ­2D13 predicts the logarithmic systemsize dependence of the dimensionless conductance as L,g = g0 − g1 ln(2)lwith a non-universal constant g0 = 1/(kF l), where kF and l are the Fermi wave number and the mean free path,pand a universal coefficient g1 = 1/π 2 . Substituting Eq. (2) into Eq. (1) with the fact that ln L = − 2 ln T , weobtain the universal scaling relation as follows:p

βEXP. (g) = −2 d ln(g0 + g1 2 ln T)
g1
1=− =− 2 .
pd ln Tg
π g(3)Since p is canceled out in the above calculation, 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 thegeneralized activation ­energy30.
On the other hand, in the self-consistent theory of the β-function, Vollhardt and Wölfle gave a single 2D
system conductivity formula for orthogonal classes from weakly to strongly localized r­ egimes31,32.


1
1
(x
+
1)
ln
+
1
exp(−x),gVW (x) =
(4)
2π 2
x2
Here, x = L/ξ ∝ T −p/2 /ξ , where ξ is the localization length. Furthermore, the β-function from Eq. (4) can
be calculated as follows:22d ln gVW (x)
x
�
� .
= −
+
βVW (g) =(5)d ln xx + 1 �x 2 + 1� ln 1 + 1
x2

We analyzed our experimental data by using the above equations: first, we estimated ξ from Eq. (4). Next we
described the β-function obtained from Eq. (5). Finally we overlaid the βEXP. (g) data of Eq. (1) and the βVW (g)data of Eq. (5), 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 shown in Fig. 3, we discover that the data exhibit different perpendicular flows (indicated by the red arrows)
against the usual Anderson localization β-function curve (indicated by the blue arrows). In the weakly localized
regime, these perpendicularflows are present in both Nd-based 2D layered perovskite and ultrathin Pb films.
In some ultrathin Pb film samples, it is clear that the flow changes from the usual Anderson localization flow to
the perpendicularflow at a certain temperature (e.g., Pb-sample-F surrounded by a dotted circle). ­Nd2CuO4–xFx
single crystals and N­ d2–xCexCuO4 thin films have different perpendicularflows, but N­ d2–xCexPdO4 does nothave them. It is considered to be a phenomenon that is a precursor to the transition from weak localization tosuperconductivity. The discovery of the upturn from the βVW (g) to the perpendicularflow suggests that the single
parameter hypothesis is not s­ ucient33–36. However, this β-function analysis method will suggest us to understandthe causes and the conditions of these perpendicularflows.
To understand the mechanism of perpendicular flows, we again analyze the temperature dependence ofresistivity in more significant detail. The functional forms of these perpendicular flows appear in the weakly
localized regime but are not adapted to −1/g . Therefore, we analyzed two samples before and after the perpendicular flow occurred; namely (a) N­ d2CuO4–xFx (NCOF) sample E in which this perpendicular flow appearsand (b) NCOF sample I in which the perpendicular flow does not appear. As a result, we find a difference in thetemperature dependence of conductivity σ and resistivity ρ(= 1/σ ). As shown in Fig. 4, the ln T dependence of
the NCOF-sample-E resistivity is more suitable than that of the conductivity. In contrast, according to the Andeson localization theory, the NCOF-sample-I conductivity has ln T dependence. Although these are in the same
weakly localized regime, the perpendicular flow is clearly characterized by ρ ∼ ln(1/T) rather than by σ ∼ ln T .
Das et al.37 pointed out that the weak localization of bosons occurring on 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 perpendicular flow canbe regarded as boson localization or the Bose glass phase. These experimental results indicate that the Bose glass
regime has been found.

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Figure 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 solid black line shows βVW (g) — Vollhardt and
Wölfle β-function [(eq. (5)]. The data for Nd-based 2D layered perovskite were plotted using resistance values
from 5 to 25 K. The data for ultrathin Pb films were plotted using resistance values from 4 to 10 K. The color
depends on the sample. If we view 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 perpendicularto βVW (g). In the weakly localized regime, theseflows 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 perpendicularflow
at a certain temperature. The yellow star in each graph indicates g = 1/2π and β = −2/π, which is thedimensionless version of the critical sheet fermion resistance R = h/e2. This star is the boundary value atwhich the perpendicularflow occurs, as determined from Fig. 5 and Appendix B. Therefore, it proves that thephenomenon of these perpendicularflows is at its least apparent when R is smaller than h/e2.

The transition from Fermi glass to Bose glass

Next, to investigate the change condition from Bose glass to Fermi glass, we plotted the temperature dependence of βEXP. (g) instead of the resistance, as shown in Fig. 5 with a series of experimental data. Interestingly, thetemperature dependence of βEXP. (g) in the weakly localized regime is very similar to the S–I transition graph.
As the localization of each 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 and ultrathin Pb films.
To understand the change in the slope of βEXP. (g) flows in Fig. 5, we use the temperature-derivative equationof the β-function shown below,′(g) = −
βEXP.

d ln |βEXP. (g)|.
d ln T(6)′(g). Substituting
The change in the β-function flow should appear in the positive or negative sign of βEXP.
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 thesigns are different as follows.
′
βEXP.
(g) =

′
βEXP.
(g) = −

σ1> 0,
σ0 + σ1 ln T

ρ1< 0,
ρ0 + ρ1 ln(1/T)(σ = σ0 + σ1 ln T)

(σ =

1
1
=
).
ρ
ρ0 + ρ1 ln(1/T)

(7)
(8)

In the above calculations, we treat σ as g. While the σ has a dimensionality, this does not change the sign of′
βEXP.
(g). These calculation results confirm that the slope of βEXP. (g) reflects the temperature-dependent func′(g) < 0 and foundtional form of the conductivity and resistivity. To check this, we analyzed all the data for βEXP.

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Figure 4.  Graphs comparing the log T dependence of σ (left) and R (right). We analyzed two samples
before and after the perpendicular flow occurred; namely (a) ­Nd2CuO4–xFx (NCOF) sample E in which thisperpendicular flow appears and (b) NCOF sample I in which the perpendicular flow does not appear. (NCOFsample-E is along the red arrow, and NCOF-sample-I is along the blue arrow in Fig. 3). The solid black line
shows the regression line. The left vertical axis of the graph is σ or R. The graph’s horizontal axis is log T ,and the right vertical axis is the value obtained by subtracting σˆ or Rˆ from the experimental value. For
standardization, we divide by the value obtained by subtracting the minimum value from the maximum valueof the experimental data on the vertical axis. Clearly, NCOF-sample-E is more suitable for the log T dependenceof the resistivity than that of the conductivity, while NCOF-sample-I is the opposite. The perpendicular flow inthe 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. Appendix C shows graphs comparing the log T dependence ofσ and R on other data.
that they were more suitable for ρ ∼ ln(1/T) than σ ∼ ln T . However, in the ultrathin Pb film data in Fig. 5b,′(g) changed from positive to negative when the temperature decreased.
there were samples where the sign of βEXP.
′(g) changed around 6 K. Since the superconducting transition temperature Tc
For Pb-sample-F, the sign of βEXP.
of Pb (in the clean system) is around 7.2 K, it is conceivable that bosons are formed. On the other hand, NCOFsample-E did not exhibit any change of sign. This may 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 the superconducting transition temperature Tc of the ­Nd2Pd1–xCuxO4–xFx (see Appendix A).

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Figure 5.  Graphs of the relationship between the value of βEXP. (g) and temperature T with a series ofexperimental data in a weakly localized regime. Interestingly, the temperature dependence of βEXP. (g) is very
similar to the S–I transition graph. As the localization of each sample 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 (a) Nd-based
2D layered perovskite and (b) ultrathin Pb films. The color depends on the sample. The solid black line showsβVW (g)—Vollhardt and Wölfle β-function [Eq. (5)]. We can see that among the weakly localized samples, those′(g) > 0) fit into βVW (g). However, in the (b) graph, as the localization becomeswith strong localization (βEXP.
weaker, the slope of Pb sample F changes from positive (blue dotted line) to negative (red dotted line) when
the temperature falls below around 6 K. Since the superconducting transition temperature Tc of Pb (in a clean
system) is around 7.2 K, it is conceivable that a boson appears. We also investigated the critical value of βEXP. (g),′(g) goes to zero. βC was obtained by using the slope and the intercept
denoted as βC when the slope of βEXP.
of the vertical axis for each sample at a specific low temperature. We obtained βC = −0.6 ± 0.1 in these twodifferent types of samples. This value is almost the same 0.64 (≃ 2/π) as shown by the yellow star, and has avalue of g = 1/2π when converted using Eq. (3). This value is the dimensionless version of the critical sheet
fermion resistance R = h/e2.
′(g) went to zero.
We also investigated the critical value of βEXP. (g), denoted as βC , when the slope of βEXP.
From Fig. 5, 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 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 toβ = −2/π by using Eq. (3). 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 inFig. 3 as a yellow star, we can see that the phenomenon of these perpendicular flows occurs at least when g islarger than 1/2π . This g = 1/2π value is transformed into the dimensionless version of the critical sheet fermion
resistance R = h/e2. 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 to the 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 ina localized regime.

Discussion

Finally, we suggest that the fermion-only theory may be insufficient to explain the weakly localized regime justbefore reaching superconductivity in 2D disordered superconducting materials. Fisher’s theory is also insufficient because it is based on boson-only or fermion-only in the strongly localized regime represented by VRH.

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Table 1.  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 propose a universal transition from Bose glass to Fermi glass with the new two-dimensional
critical sheet resistance close to R = h/e2 at β ′ = 0.

Meanwhile, disordered thin-film superconductors in the weakly multifractal regime can exhibit a S–I transition
and allow for a boson-fermion m­ ixture38–44. The experiment using a disordered superconductor TiN revealed theexistence of Cooper pairs above the superconducting transition temperature Tc45. Therefore, the boson–fermion
mixture state and boson-only state in the weakly localized regime are important areas in clarifying the relationship between statistics (fermion, boson, and anyon) and localization. We discover an experimental renormalization group flow from Fermi glass to Bose glass by employing β-function analysis. To discuss the universalityof Bose glass flow, we analyzed distinctively different systems, namely a Nd-based two-dimensional layered
perovskite and an ultrathin Pb film. What these two systems have in common is that they are both 2D disorderedsystems, but the difference is that a Nd-based 2D layered perovskite is disordered by doping while an ultrathin
Pb film is structurally disordered. Furthermore, unlike an ultrathin Pb film, a Nd-based 2D layered perovskite
is a strongly correlated electron s­ ystem9,46. Although these two systems may have different superconductivity
mechanisms, the flow of the experimental β-function exhibits the same behavior. The analysis results obtained
for the two systems in common are shown in Table 1.
These results suggest the following mechanism of the crossover. With varying disorders 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, fermions disappear when R is smaller than h/4e2, 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 a zero or a finite
value when the longitudinal resistance ρxx diverges at low temperatures and in a certain magnetic field (the bidirectional black dotted arrow in Fig. 1). On the other hand, we used the experimental β-function to identify
the flow perpendicular to the 2D Anderson localized flow as Bose glass (the bi-directional red arrow in Fig. 1).
This analysis constitutes a simple method of identifying weak boson/fermion localization. As an application ofthese results, 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 thedisorder parameters.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding
author on reasonable request.
Received: 29 January 2023; Accepted: 22 July 2023

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Acknowledgements

We are grateful to K. Yakubo, K. Ichimura and T. Matsuyama for suggesting the topic dealt with in this Article. We
thank M.Naito for his help in preparing the experimental N­ d2–xCexPdO4 samples. We also thank many of those
who attended the Forum of Localisation 2020. We have gained useful insights. Finally, we would like to thankS. Ozeki, K. Kagawa and K. Inagaki who studied together about 30 years ago, and T. Nakayama for introducing
such a wonderful physics field.

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Author contributions

Conceptualization, S.T. and K.T.; methodology, S.T., K.T. and K.N.; software, K.T., K.N.; validation, K.T., K.N.
and S.T.; formal analysis, K.T., K.N. and S.T.; investigation, K.T. and Y.N; resources, K.T., Y.N. and M.S.; data
curation, K.T.; writing-original draft preparation, K.T.; writing-review and editing, S.T., H.N., K.T., H.O. andK.N.; visualization, K.T. and K.N.; supervision, S.T.; project administration, S.T.; funding acquisition, S.T. All
authors have read and agreed to the published version of the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary Information The online version contains supplementary material available at https://​doi.​org/​
10.​1038/​s41598-​023-​39285-1.
Correspondence and requests for materials should be addressed to K.T.
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