ISSN 0021-3640, JETP Letters.

Pleiades Publishing, Inc.,

Universal T/B Scaling Behavior of Heavy Fermion Compounds

(Mini-review)

V. R. Shaginyan

+∗1)

, A.Z.Msezane

∗

, J.W.Clark

×◦

, G.S.Japaridze

∗

, Y.S.Leevik

∇

Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, 188300 Gatchina, Russia

∗

Clark Atlanta University, Atlanta, GA 30314, USA

McDonnell Center for the Space Sciences & Department of Physics, Washington University, St. Louis, MO 63130, USA

◦

Centro de Investiga¸c˜aoemMatem´atica e Aplica¸c˜oes, University of Madeira, 9020-105 Funchal, Madeira, Portugal

∇

National Research University Higher School of Economics, 194100 St. Petersburg, Russia

Submitted 5 October 2020

Resubmitted 23 October 2020

Accepted 24 October 2020

In our mini-review, we address manifestations of T/B scaling behavior of heavy-fermion (HF) compounds,

where T and B are respectively temperature and magnetic ﬁeld. Using experimental data and the fermion

condensation theory, we show that this scaling behavior is typical of HF compounds including HF metals,

quasicrystals, and quantum spin liquids. We demonstrate that such scaling behavior holds down to the lowest

temperature and ﬁeld values, so that T/B varies in a wide range, provided the HF compound is located near

the topological fermion condensation quantum phase transition (FCQPT). Due to the topological properties of

FCQPT, the eﬀective mass M

∗

exhibits a universal behavior, and diverges as T goes to zero. Such a behavior

of M

∗

has important technological applications. We also explain how to extract the universal scaling behavior

from experimental data collected on diﬀerent heavy-fermion compounds. As an example, we consider the HF

metal YbCo

, and show that its scaling behavior is violated at low temperatures. Our results obtained

show good agreement with experimental facts.

DOI: 10.1134/S0021364020220026

Introduction. Topological approach is a powerful

method to gain information about a wide class of physi-

cal systems. Knowledge of the topological properties al-

lows us to improve a general knowledge about physical

systems without solving speciﬁc equations, which de-

scribe concrete systems and are often very complicated.

As usually, the microscopic approach to a heavy fermion

(HF) metal (for example, computer simulations) gives

only particular information about speciﬁc solids, but not

about universal features, inherent in the wide class of

HF compounds. HF compounds can be viewed as the

new state of matter, since their behavior near the topo-

logical fermion condensation quantum phase transition

(FCQPT) acquire important similarities, making them

universal. The idea of this phase transition, forming ex-

perimentally discovered ﬂat bands, started long ago,

in 1990 [1–3]. At ﬁrst, this idea seemed to be a curi-

ous mathematical exercise, and now it is proved to be

rapidly expanding ﬁeld with uncountable applications

[1–9].

e-mail: vrshag@thd.pnpi.spb.ru

The scaling behavior of HF compounds is a challeng-

ing problem of condensed matter physics [6, 10–13]. It

is generally assumed that scaling with respect to T/B

(temperature-magnetic ﬁeld ratio) is related to a quan-

tum critical point (QCP) that represents the endpoint

of a phase transition being tuned to T =0by such con-

trol parameters as magnetic ﬁeld, pressure, and compo-

sition of the heavy-fermion compounds. As soon as the

tuned endpoint of the phase transition reaches T =0,

it becomes a quantum phase transition (QPT). At QCP

involved quantum ﬂuctuations like valence, magnetism,

etc. can take place and inﬂuence on the properties of

system in question [11, 12]. Fluctuations can also oc-

cur at second-order phase transitions, but in all cases

the temperature range of these ﬂuctuations is very nar-

row [14]; in contrast, T/B scaling can span a few or-

ders of magnitude in T/B [6–8]. An attendant problem

to be addressed by theory stems from the experimen-

tal ﬁnding that scaling behavior can take place without

both QCP realization and eﬀective mass M

∗

divergence

[12]. The divergence of eﬀective mass M

∗

at T → 0

is of crucial importance for understanding technologi-

JETP Letters 1

2 V.R.Shaginyan, A.Z.Msezane, J.W.Clark, G.S.Japaridze, Y.S.Leevik

cal applications of quantum materials. For example, the

divergence leads to the high heat capacity C of quan-

tum material, while under the application of magnetic

ﬁeld both M

∗

and C diminishes. As a result, one can

exploit this property constructing low temperature re-

frigerators. To solve these problems, one needs to have

a reliable theoretical framework for analysis of experi-

mental facts related to the scaling behavior.

A universal T/B scaling behavior is generated by

quasiparticles belonging to ﬂat bands. These ﬂat bands

can be preformed by van Hove singularities and ﬁnally

formed by inter-particle interaction generating topologi-

cal FCQPT [15]. In narrow electronic bands in which the

Coulomb interaction energy becomes comparable to the

bandwidth, interactions drive the topological FCQPT;

as a result, at T =0ﬂat bands are emerged, see e.g. [6–

8, 15]. Such ﬂat bands in twisted graphene have been ex-

perimentally observed, see, e.g. [3]. Thus, we can safely

use the model of homogeneous HF liquid, since we con-

sider a behavior controlled by ﬂat bands and related

to the scaling of quantities such as the eﬀective mass,

heat capacity, magnetization, etc. As a result, the scal-

ing properties are deﬁned by momentum transfers that

are small compared to momenta of the order of the recip-

rocal lattice length. The high momentum contributions

can therefore be ignored by substituting the lattice for

the jelly model; this observation is in a good agreement

with experimental facts collected on many HF metals

[6, 7, 9]. In our case quasiparticles are well deﬁned ex-

citations, see, e.g. [6, 7, 16], and the divergence of the

eﬀective mass M

∗

is not related to Z → 0 (as it is can

be assumed, see, e.g. [17]), where Z is the quasiparti-

cle amplitude. The divergence is deﬁned by both the

emergence of an extended van Hove singularities, pre-

forming ﬂat bands, and the Coulomb interaction, giv-

ing rise to strong correlations; as a result, at T =0the

electronic dispersion becomes ﬂat at the chemical po-

tential μ, topologically transforming the Fermi surface

into Fermi volume, see e.g. [1, 6, 7, 9, 15, 18–20].

In our mini-review, we show that the fermion con-

densation (FC) theory, which entails the topological

FCQPT, provides the appropriate framework for de-

scribing and analyzing the universal scaling behavior of

HF compounds [1, 2, 4–7, 9]. We predict that T/B scal-

ing behavior can be observed in a wide range of T/B

values, provided the given HF compound is located near

a topological FCQPT. Violation of T/B scaling at the

lowest values of T/B is a signal that the given HF com-

pound is situated before the topological FCQPT on the

T −B phase diagram, and hence exhibits Landau Fermi-

Liquid (LFL) behavior at suﬃciently low temperatures.

We consider the HF metal YbCo

,andshowthat

its scaling behavior is violated at low temperatures. The

results of the FC theory are in good agreement with ex-

perimental observations collected on diﬀerent strongly

correlated Fermi systems like HF metals, quasicrystals

and quantum magnets, holding quantum spin liquids

[6,7,9].Asaresult,theFCtheoryisusefultoolwhen

projecting technological applications of quantum mate-

rials representing by HF compounds.

Scaling behavior of the eﬀective mass near the

topological FCQPT. One of the main experimental

manifestations of the topological FCQPT phenomenon

is the scaling behavior of the physical properties of HF

compounds located near such a phase transition. To un-

derstand this scaling behavior on a sound theoretical

basis, we begin with a brief description of the associ-

ated behavior exhibited by the eﬀective mass M

∗

the framework of a homogeneous HF liquid [6]. This

simpliﬁcation avoids the complications associated with

the anisotropy of solids and focuses of both the ther-

modynamic properties and the non-Fermi-liquid (NFL)

behavior by calculating the eﬀective mass M

∗

(T,B) as

a function of temperature T and magnetic ﬁeld B [6–8],

based on the Landau formula for the quasiparticle eﬀec-

tive mass M

∗

(T,H). The only modiﬁcation introduced

is that the eﬀective mass is no longer approximately

constant but now depends on temperature, magnetic

ﬁeld, and other parameters such as pressure, etc. We

note that the FC theory is a good established theory

based on the density functional theory; in that case,

the Landau functional E[n(p)] and the corresponding

Eq. (1) are also derived in the same frameworks, there-

fore, being exact, see e.g. [6, 7, 9, 18, 4]. Here n(p) is

the quasiparticle distribution function.

At ﬁnite temperatures and magnetic ﬁelds, Landau’s

equation takes the form [6, 8, 21, 22]

∗

(T,H)

+ (1)





σ,σ

, p)

∂n

(p,T,H)

∂p

(2π)

in terms of n

(p) and the quasiparticle interaction

,σ

.HereM can be represented by the bare elec-

tron mass or by the enhanced bare like mass at van

Hove singularity. The single-particle spectrum ε(p,T)

is a variational derivative

(p)=

δE[n(p)]

δn

(p)

, (2)

of the system energy E[n

(p)] with respect to the quasi-

particle distribution (or occupation numbers) n

(p for

JETP Letters

Universal T/B Sca ling Behavior of Heavy Fermion Compounds 3

spin σ, which in turn is related to the spectrum ε

(p)

(p,T)=



1+exp



(ε(p,T) − μ

)



−1

. (3)

In our case, the chemical potential μ depends on spin

due to the Zeeman splitting, μ

= μ ± μ

B,whereμ

is the Bohr magneton. The magnetic ﬁeld B appears in

Eq. (3) via the ratio μ

/T =(μ ± μ

B)/T .Wenote

that (1) and (2) are exact equations [7, 18]. In our case,

the Landau interaction F is ﬁxed by the condition that

the system is situated at a FCQPT. Its sole purpose is

to bring the system to the topological FCQPT point,

where M

∗

→∞at T =0and B =0, altering the

topology of the Fermi surface, transforming it to a vol-

ume such that the eﬀective mass acquires temperature

and ﬁeld dependence [6, 21, 23]. Provided the Landau

interaction is an analytic function, at the Fermi surface

the momentum-dependent part of the Landau interac-

tion can be parameterized as a truncated power series

F = aq

+ bq

+ cq

+ ...,whereq = p

− p

,thevari-

ables p

and p

are momenta, and a, b,andc are ﬁtting

parameters deﬁned by the condition that the system is

at a FCQPT point.

Direct inspection of Eq. (1) shows that at T =0and

B =0, the sum of the ﬁrst and second terms on the

right side vanishes, since 1/M

∗

(T → 0) goes to zero

when the system is located at the FCQPT point. Given

a Landau interaction analytic with respect to momenta

variables, at ﬁnite T the right side of Eq. (1) is propor-

tional F



∗

)

,whereF



is the ﬁrst derivative of

F (q) with respect to q at q → 0. Results for the cor-

responding integrals can be found in textbooks – see

especially [24]. At any rate, we have 1/M

∗

∝ (M

∗

)

and arrive at [6, 7]

∗

(T )  a

−2/3

. (4)

At ﬁnite temperatures, application of a magnetic ﬁeld

B  k

T drives the system to the LFL regime with

∗

(B)  a

−2/3

, (5)

where a

and a

are parameters and k

the Boltz-

mann constant. It follows from Eq. (5) that heat capac-

ity C ∝ M

∗

(B)T of quantum materials strongly de-

pends on B, this property can be used when projecting

e.g. low temperatures refrigerators.

If the system is still located before the FCQPT, the

eﬀective mass is ﬁnite M

∗

= M

,andEq.(5)mustbe

adjusted, since at B → 0 the eﬀective mass M

∗

does

not diverge; thus

∗

(B)  a

+ B)

−2/3

. (6)

From Eq. (6) it follows that at B → 0 the eﬀective mass

becomes M

. Therefore, when B  B

, the eﬀective

mass M

∗

depends on the magnetic ﬁeld in accordance

with Eq. (5), since the contribution coming from B de-

ﬁnes the behavior of M

∗

.Asaresult,atB  B

can replace Eq. (5) by the equivalent equation

∗

(B)  M

+ a

−2/3

. (7)

In the case of the HF liquid, these observations allow

for construction of an approximate solution of Eq. (1)

in the form M

∗

= M

∗

(B,T) that satisﬁes both Eqs. (4)

and (5). Introduction of “internal” scales simpliﬁes the

problem under consideration, allowing us to eliminate

the microscopic structure of the HF compounds under

consideration [6, 7]. To establish such “internal” scales,

we observe that near the FCQPT, the eﬀective mass

∗

(B,T) reaches a maximum M

∗

at a certain tem-

perature T

∝ B. (See later comments on Eqs. (3) and

(5) and Fig. 1). To conveniently measure the eﬀective

Fig. 1. (Color online) Scaling behavior of the dimensionless

eﬀective mass M

∗

versus dimensionless variable (T/B)

Scaling of thermodynamic properties is deﬁned by M

∗

;

see Eq. (15). M

∗

is a function of T

∝ (T/B)

∼

∼ (T/B)/(T/B)

, as it follows from Eq. (14). Solid curve

depicts the scaling behavior M

∗

versus normalized tem-

perature T

as a function of magnetic ﬁeld, given by

Eqs. (8) and (12). Clearly, at ﬁnite T

< 1 the normal

Fermi liquid regime is realized. At T

∼ 1 the system

enters a crossover state, and at growing temperatures ex-

hibits NFL behavior. The LFL, crossover, inﬂection point,

and NFL behavior are indicated by the arrows. The LFL

behavior is additionally shown by the hatched area

mass and temperature versus magnetic ﬁeld B,wein-

troduce the scales M

∗

and T

, generating new vari-

ables M

∗

= M

∗

(normalized eﬀective mass) and

= T/T

(normalized temperature). In the vicinity

JETP Letters

4 V.R.Shaginyan, A.Z.Msezane, J.W.Clark, G.S.Japaridze, Y.S.Leevik

of FCQPT, the normalized eﬀective mass M

∗

) is

well approximated by a universal function [6, 7]

∗

) ≈ c

1+c

8/3

. (8)

Here, T

= T/T

∝ T/B (see Fig. 1) and c

=(1+c

)/(1 + c

),withc

and c

free parameters.

We stress that values of M

∗

and T

are deﬁned by

the microscopic structure of the HF compound under

study, while the normalized values M

∗

and T

demon-

strate the universal scaling exhibited by HF compounds

located near the topological FCQPT, since this scaling

is determined by the nature of both the phase transition

and the model of homogeneous HF liquid; we note that

these observations are in good agreement with experi-

mental facts collected on HF compounds, see e.g. [6–9].

From Equations (5) and (8) it follows that

∗

∝ B

−2/3

∝ T

−2/3

; T/B  T/T

. (9)

The Landau interaction F

σ,σ

(q) appearing in Eq. (1)

can produce the characteristic topological form of the

spectrum ε(p) − μ ∝ (p − p

)

(p − p

),with(p

)

and (p

− p

)/p

 1, leading to M

∗

∝ T

−1/2

and

creating a quantum critical point [25]. The same crit-

ical point is generated by the interaction F (q) as rep-

resented by a non-analytic but integrable-over-x func-

tion with q =



+ p

− 2xp

and F (q → 0) →∞

[6,4,8].BothcasesleadtoM

∗

∝ T

−1/2

,andEq.(4)

becomes

∗

(T )  a

−1/2

. (10)

Inthesameway,weobtain

∗

(B)  a

−1/2

, (11)

in terms of parameters a

and a

Taking into account the fact that Eq. (10) leads to a

spiky density of states (DOS), with the spiky character

fading away under increasing temperature as observed

in quasicrystals [26–28], we note that the general form

of ε(p) produces the behavior of M

∗

given by Eqs. (10)

and (11). This is realized in quasicrystals, which can be

viewed as a generalized form of common crystals [26].

We note further that the behavior 1/M

∗

∝ χ

−1

∝ T

1/2

is in good agreement with the behavior χ

−1

∝ T

0.51

observed experimentally in quasicrystals [28, 26]. Our

result 1/M

∗

∝ T

1/2

is consistent with the robustness of

the exponent 0.51 under hydrostatic pressure [28]. This

robustness is guaranteed by the unique singular density

of states associated with the topological FCQPT, which

survives under application of pressure [26–30]. To de-

velop the consequences of the the solution of Eq. (1) at

ﬁnite B and T near the FCQPT, we construct an ap-

proximate solution by interpolating between the LFL

behavior described by Eq. (11) and the NFL behavior

described by Eq. (10), that models the universal scaling

behavior M

∗

∝ T/B) [26]

∗

) ≈ c

1+c

5/2

, (12)

with c

=(1+c

)/(1 + c

) and c

, c

as ﬁtting param-

eters. Taking into account Eqs. (11) and (12), we arrive

∗

∝ B

−1/2

∝ T

−1/2

. (13)

It follows from Eqs. (3), (8), and (12) that

∝ B; T

∝

∼





. (14)

Here a

is a dimensionless factor, μ

is the Bohr mag-

neton, (T/B)

=(T/B)/(T/B)

,where(T/B)

the point at which M

∗

reaches it maximum value

∗

=1, as illustrated in Fig. 1. Expression (14) shows

that Eqs. (8) and (12) determine the eﬀective-mass scal-

ing in terms of T/B as well. We conclude from Eq. (14)

that since T

∝ B, the curves M

∗

(T,B) merge into a

single curve M

∗

= T/B),withT

= T/T

∝ T/B,

demonstrating the widespread scaling in HF metals (for

example, see [6, 7]). Such behavior is depicted in Fig. 1.

We note that Eqs. (8) and (14) allow us to describe the

behavior of the strongly correlated quantum spin liquid

(SCQSL) existing in diﬀerent frustrated magnets [6, 7].

Another important feature of the FC state is that

apart from the fact that the Landau quasiparticle ef-

fectively acquires strong dependence on external factors

such as the temperature and magnetic ﬁeld, all the fun-

damental relations inherent in the LFL approach remain

formally intact. In particular, the famous LFL relation

[22, 24, 6, 7],

∗

(B,T) ∝ χ(B, T) ∝

C(B, T)

, (15)

still holds. That is, expression (15) is also valid in

the case of HF compounds located near a topological

FCQPT, where the speciﬁc heat C, magnetic suscepti-

bility χ, and eﬀective mass M

∗

depend on T and B.

Based on Eq. (15), we ﬁnd that the normalized values

of C/T and χ are of the form [6, 7]

∗

(B,T)=χ

(B,T)=



C(B, T)



. (16)

Thus, Equation (16) allows us to reveal the universal

scaling behavior of diﬀerent HF compounds located near

JETP Letters

Universal T/B Sca ling Behavior of Heavy Fermion Compounds 5

the topological FCQPT like HF metals, frustrated insu-

lators with SCQSL, quasicrystals, 2D liquids, etc., see,

e.g. [6, 9]. It is also seen from Eq. (16) that the aforemen-

tioned thermodynamic properties have the same scaling

behavior as depicted in Fig. 1. Moreover, we shall see

below that the thermodynamic properties of HF met-

als, SCQSL of frustrated magnets, and the other HF

compounds exhibit the same typical behavior. Based

on Eq. (8) and Fig. 1, we can construct the general

schematic T − B phase diagram of SCQSL, reported

in Fig. 2. We assume here that at T =0and B =0the

Fig. 2. (Color online) Schematic T −B phase diagram of a

HF compound, with magnetic ﬁeld B as control param-

eter. The hatched area corresponds to the crossover do-

main at T

(B), given by Eq. (14). At ﬁxed magnetic ﬁeld

and elevated temperature (vertical arrow) there is a LFL-

NFL crossover. The horizontal arrow indicates a NFL-

LFL transition at ﬁxed temperature and elevated mag-

netic ﬁeld. The topological FCQPT (shown in the panel)

occurs at T =0and B =0,whereM

∗

diverges

system is approximately located at a FCQPT point. At

ﬁxed temperature the system is driven by the magnetic

ﬁeld B along the horizontal arrow (from the NFL to

the LFL parts of the phase diagram). At ﬁxed B and

elevated T the system moves from the LFL to the NFL

regime along the vertical arrow. The hatched area in-

dicating the crossover between LFL and NFL phases

separates the NFL state from the paramagnetic slightly

polarized LFL state. The crossover temperature T

(B)

is given by Eq. (14).

T/B scaling in heavy fermion compounds. The

experimentally based scaling behavior of M

∗

so de-

rived is displayed in Fig. 1. Explanation of this scaling,

∗

) ∝ C(B, T)/T , presents a serious challenge to

theories of the HF compounds. Most of the current theo-

ries analyze only the critical exponents that characterize

∗

) at T

 1 and thus consider only a part of

the problem, missing the LFL and the transition regime

[6, 7, 13]. This scaling behavior of the eﬀective mass

∗

of HF compounds (or strongly correlated Fermi sys-

tems) is described by Eqs. (8) and (12). It follows then

from Eqs. (15) and (16) that their experimentally ob-

served thermodynamic properties express the universal

scaling behavior revealed by our analysis.

In the present context, the HF compounds are taken

to represent strongly correlated Fermi sytems as realized

in HF metals, high-T

superconductors, quasicrystals,

SCQSL of frustrated magnets and two-dimensional liq-

uids like

He. One can expect that HF compounds with

their extremely diverse composition and microscopic

structure would demonstrate very diﬀerent thermody-

namic, transport, and relaxation properties. To reveal

the universal scaling behavior of HF compounds, irre-

spective of speciﬁc properties of individual compounds,

we have introduced internal scales to measure the corre-

sponding thermodynamic properties, as is done when we

consider the scaling behavior of the eﬀective mass M

∗

This uniform behavior arises from the fact that HF com-

pounds are located near a topological FCQPT, gener-

ating their uniform scaling behavior with respect to the

eﬀective mass M

∗

[6, 31, 9] (see Fig. 1). As an example,

Fig. 3 displays the universal T/B scaling behavior of the

HF metal CeCu

6−x

and SCQSL of the frustrated

insulator herbertsmithite ZnCu

(OH)

[32, 33]. The

existence of such universal behavior, exhibited by vari-

ous and very distinctive strongly correlated Fermi sys-

tems, supports the conclusion that HF compounds rep-

resent a new state of matter [31, 9]. In contrast to the

situation for an ordinary quantum phase transition, this

scaling, induced by the topological FCQPT, occurs up

to high characteristic temperature T

, T<T

∼ 100 K,

since the NFL behavior is deﬁned by quasiparticles

(with M

∗

given by Eqs. (8) and (12)), rather than by

ﬂuctuations, or by Kondo lattice eﬀects [6, 7, 9].

Some remarks are in order here. A strongly corre-

lated Fermi system can be situated after the topolog-

ical FCQPT, i.e., on the ordered side deﬁned by the

quantum critical line (QCL), as shown in the schematic

phase diagram 4. As it is shown in Fig. 4, FCQPT can

be tuned by dimensionless control parameters: normal-

ized pressure P/P

, composition x/x

, magnetic ﬁeld

B/B

. Here we assume that the critical magnetic ﬁeld

> 0.InthecaseofB

=0, see Fig. 2, demonstrating

that at T =0and B =0the system is located at the

topological FCQPT. Note that there can be two critical

magnetic ﬁelds B

and B

,asitisincaseoftheHF

metal Sr

[34, 15].

As it is seen from Fig. 4, at T =0the crossover re-

gion is absent, and the FC state is separated from the

LFL region by the ﬁrst order phase transition [6], for

JETP Letters

6 V.R.Shaginyan, A.Z.Msezane, J.W.Clark, G.S.Japaridze, Y.S.Leevik

Fig. 3. (Color online) Universal B/T scaling of strongly

correlated Fermi systems. Scaling behavior of the

HF metal CeCu

6−x

with x =0.1 is ex-

tracted from data (measured at diﬀerent ﬁeld values

B =0.05, 0.1, 0.3, 0.6, 0.9, 1.05 T) in [32], and that of

ZnCu

(OH)

(measured at diﬀerent ﬁeld values B =

=0.5, 1.0, 3.0, 5.0, 7.0, 10.0, 14.0 T), from data in [33]. At

B/T  1 the systems demonstrate NFL behavior with

χ ∝ M

∗

as given by Eq. (4), i.e., T

2/3

χ ∝ const. At

B/T  1 the systems demonstrate LFL behavior with

χ as given by Eq. (5), a decreasing function of B/T (see

Eq. (12)). The LFL, crossover, inﬂection point, and NFL

behavior are indicated by the arrows. The broken lines in-

dicate the asymptotic dependencies in the limits of small

(B/T)

(the NFL behavior) and large (B/T)

(the LFL

behavior). The theoretical prediction is represented by

solid curve

the FC state is characterized by special quantum topo-

logical number, being a new type of Fermi liquid [2].

At T>0 there is the crossover rather than a phase

transition [6]. One may expect that the T/B scaling is

caused by features not related to the presence of QCP

and the divergence of M

∗

(see e.g. [12]). On the other

hand, if the system in question is located before FC-

QPT, as indicated by the dash-dot arrow in Fig. 4, it

exhibits LFL behavior even in the absence of a mag-

netic ﬁeld B at low T → 0. At elevated magnetic ﬁelds

reaching B  B

, Eqs. (5) and (11) are valid and the

scaling behavior returns to that given by Eqs. (8) and

(12). Thus, to witness the presence of both the scaling

behavior and divergence of the eﬀective mass in mea-

surements on HF compounds, one has to carry out mea-

surements at suﬃciently low temperatures and magnetic

ﬁelds. For instance, the HF metal CeRu

exhibits

NFL behavior at low temperatures (down to 170 mK)

and small magnetic ﬁelds (B  0.02 mT) comparable

with the magnetic ﬁeld of the Earth [35]. Measurements

carried out under application of magnetic ﬁelds have

led to the incorrect statement that CeRu

demon-

Fig. 4. (Color online) Schematic diagram of temperature

versus these dimensionless control parameters: normalized

pressure P/P

, composition x/x

, magnetic ﬁeld B/B

We assume that B

> 0,ifB

=0, see the phase di-

agram 2. The solid black line indicates the topological

FCQPT point (orange circle). At T =0and beyond the

quantum critical point (to the left of the orange circle), the

system is on the quantum critical line (QCL) implicating

a ﬂat band, as indicated by the red-dashed arrow. At any

ﬁnite temperature T<T

and at elevated P/P

> 1,

x/x

> 1, B/B

> 1, the system enters the crossover

and, then, the LFL region. The blue dash-dot arrow points

to the system as situated before the topological FCQPT,

where at T → 0 it exhibits LFL behavior with eﬀective

mass M

∗

= M

. At elevated magnetic ﬁelds, the behavior

of the eﬀective mass is given by Eq. (7), and the scaling

behavior is restored at B>B

strates LFL behavior at low temperatures (see [35] and

references therein). We note that if the critical magnetic

ﬁeld B

is ﬁnite, then the scaling behavior occurs versus

T/(B − B

) [31].

Violation of scaling behavior. Now we consider

the statement that the scaling behavior can be ob-

served without the presence of both QCP and diver-

gent eﬀective mass M

∗

[12]. The T/B scaling behaviors

experimentally observed in measurements of the mag-

netization dM/dT on the HF metals YbCo

and

β − YbAlB

[12, 11] are displayed in Figs. 5 and 6.

As follows from Eqs. (12), (13), and (15), the function

1/2

dM/dT can be represented as



∂χ(B

,T)

∂T

∝ M

∗



∗

(y)

ydy, (17)

where y = T/B. Taking into account Eq. (13), we obtain

∗

∝ B

−1/2

and Eq. (17) then reads

1/2

∝



∗

(y)

ydy, (18)

JETP Letters

Universal T/B Sca ling Behavior of Heavy Fermion Compounds 7

Fig. 5. (Color online) YbCo

: Scaling behav-

ior of the dimensionless normalized magnetization

1/2

dM(T,B)/dT )

versus dimensionless (T/B)

measured at diﬀerent ﬁeld values B =0.05, 0.1, 0.2, 0.3,

0.5 T. The data are extracted from measurements; see

Fig. 4 of [12]. The LFL, crossover and NFL behaviors

are indicated by the arrows and hatched areas. The

theory is represented by the solid curve, describing

very well the scaling behavior of (B

1/2

dM(T,B)/dT )

obtained in measurements on β −YbAlB

[11] (see Fig. 6).

The LFL and NFL behaviors of (B

1/2

dM(T,B)/dT )

are represented by the labels B

−3/2

T and T

−3/2

respectively

thus, B

1/2

dM/dT is a function of the only variable T/B,

with M

∗

(y) is given by Eq. (12).

As seen from Figs. 5 and 6, Eq. (18) and the corre-

sponding calculations, represented by the solid curve,

are in good agreement with the data [12, 11, 8]. To

calculate the LFL behavior of dM/dT taking place at

(T/B)

 1, i.e., T  B, we use the well-known rela-

tion dM/dT = dS/dB. Taking into account Eq. (11), we

obtain dS/dB = M

∗

(B)T/dB ∝ B

−3/2

T ,asshownin

Figs. 5 and 6. At (T/B)

 1, i.e., B  T , the system

exhibits NFL behavior. Using Eq. (10), we arrive at



χ(T )

dB ∝



∗

(T )

dB ∝ T

−3/2

B. (19)

This theoretical result is in good agreement with ex-

perimental observations [11, 12]. Accordingly, we con-

clude that the fermion-condensation theory correctly de-

scribes the scaling behavior, showing good agreement

with the data.

Now we turn to the magnetic Gr¨uneisen parameter

mag

(T )=−(dM/dT )/C in the NFL regime, i.e., at

B  T , with results reported in Fig. 7a. It is seen that

mag

(T ) has an inﬂection point at T = T

inf

, signaling

that there is LFL behavior at lower temperatures rather

than a divergence (see also Fig. 1). Thus, we assume that

Fig. 6. (Color online) β − YbAlB

: Scaling behav-

ior of the dimensionless normalized magnetization

1/2

dM(T,B)/dT )

versus the dimensionless normal-

ized (B/T)

at diﬀerent magnetic ﬁelds (0.31 mT ≤ B ≤

≤ 2T) [8, 7]. The data are extracted from measurements

on β−YbAlB

[11]. The LFL behavior, crossover, and NFL

behavior are indicated by the arrows. Additionally, the

LFL behavior and the crossover are shown by the hatched

areas. At (T/B)

 1 the NFL behavior is marked by

the label T

−3/2

B.At(T/B)

 1 the LFL behavior is

marked by the label TB

−3/2

. The theory is represented

by the solid curve. Notation is speciﬁed in the caption of

Fig. 5

the HF metal YbCo

is located before the topologi-

cal FCQPT, as indicated by the dash-dot arrow in Fig. 4.

We note that the inﬂection point takes place at too

low temperatures and magnetic ﬁelds, and it could not

make a visible impact on the scaling behavior reported

in Fig. 5, that is, one needs to carry out measurements

at suﬃciently low T and B to clarify a possible violation

of the scaling behavior. We suggest that YbCo

can

be tuned to FCQPT by the application of pressure or by

doping, as it is done in the case of CeCu

6−x

, while

experimental facts show that B

=0and the application

of magnetic ﬁeld drives YbCo

from its QCP [12].

The LFL behavior of YbCo

at T → 0 is supported

by the measurements of Γ

mag

(T ), which exhibits diver-

gent behavior T

−5/2

at the interval T

≤ T ≤ 0.8 K, as

it is seen from Fig. 7, where T

is the crossover temper-

ature. At T ≤ T

the magnetic Gr¨uneisen parameter

mag

(T ) does not follow the behavior indicated by the

straight line because YbCo

enters the crossover re-

gion (see Figs. 1 and 7). At the interval T

≤ T ≤ 0.8 K

one has Γ

mag

(T )=−(dM/dT )/C ∝ T

−3/2

/T = T

−5/2

since at T ≤ 0.8 KandB =0the heat capacity C

demonstrates LFL behavior, namely C(T ) ∝ T [12]. It is

seen from Fig. 7 that at T  0.15 K, YbCo

exhibits

JETP Letters

8 V.R.Shaginyan, A.Z.Msezane, J.W.Clark, G.S.Japaridze, Y.S.Leevik

Fig. 7. (Color online) YbCo

: Magnetic Gr¨uneisen pa-

rameter Γ

mag

(T )=−(dM/dT )/C versus B for values

shown in the legend. The data are taken from [12]. (a)

mag

(T ) versus a logarithmic temperature scale. The ap-

proximate location of the inﬂection point at temperature

inf

is indicated by the arrow. (b) Γ

mag

(T ) is shown on

a double-logarithmic plot. The solid line displays a T

−5/2

dependence at B =0.05 T. At T = T

YbCo

en-

ters the crossover, see Fig. 2, and the dependence T

−5/2

vanished

LFL behavior induced by the application of magnetic

ﬁeld B =0.1. This behavior qualitatively resembles that

occurring at B =0.05 T.

Thus, we predict that at lower temperatures,

mag

(T ) will also exhibit LFL behavior. The mea-

surements of dM/dT depicted in Fig. 8 support this

conclusion: at T ≥ T

, one ﬁnds dM/dT ∝ T

−3/2

(see

also Figs. 5 and 6), while at T ≤ T

the divergent

behavior disappears. It is seen from Fig. 8 that dM/dT

deviates from a straight line for T ≤ T

,enteringthe

crossover region and ﬁnally exhibiting LFL behavior

(see Figs. 1 and 4). We note that the same behavior is

seen in the frustrated magnet ZnCu

(OH)

,which

hosts a quantum spin liquid and demonstrates LFL

behavior at T<400 mK [36]. The scaling behavior is

Fig. 8. (Color online) The −dM/dT NFL behavior at ﬁxed

ﬁeld B =0.05 T. The solid line indicates T

−3/2

depen-

dence on a double-logarithmic plot. The experimental data

are taken from [12]. The temperature T

at which the sys-

tem enters the crossover region is indicated by the arrow.