Behavior of the antiferromagnetic phase transition near the fermion condensation quantum phase transition in YbRh2Si2

Abstract

Low-temperature specific-heat measurements on YbRh2Si2 at the second order antiferromagnetic (AF) phase transition reveal a sharp peak at T_N=72 mK. The corresponding critical exponent alpha turns out to be alpha=0.38, which differs significantly from that obtained within the framework of the fluctuation theory of second order phase transitions based on the scale invariance, where alpha=0.1. We show that under the application of magnetic field the curve of the second order AF phase transitions passes into a curve of the first order ones at the tricritical point leading to a violation of the critical universality of the fluctuation theory. This change of the phase transition is generated by the fermion condensation quantum phase transition. Near the tricritical point the Landau theory of second order phase transitions is applicable and gives alpha=1/2. We demonstrate that this value of alpha is in good agreement with the specific-heat measurements. Comment: 7 pages, 6 figures. to be published in Phys. Letters A

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arXiv:0911.3325v2 [cond-mat.str-el] 5 Dec 2009
Behavior of the antiferromagnetic phase transition near the fermion condensation
quantum phase transition in YbRh2Si2
V.R. Shaginyan,1,2, ∗M.Ya. Amusia,2and K.G. Popov3
1Petersburg Nuclear Physics Institute, RAS, Gatchina, 188300, Russia
2Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel
3Komi Science Center, Ural Division, RAS, Syktyvkar, 167982, Russia
Low-temperature specific-heat measurements on YbRh2Si2 at the second order antiferromagnetic
(AF) phase transition reveal a sharp peak at TN = 72 mK. The corresponding critical exponent
α turns out to be α = 0.38, which differs significantly from that obtained within the framework
of the fluctuation theory of second order phase transitions based on the scale invariance, where
α ≃ 0.1. We show that under the application of magnetic field the curve of the second order AF
phase transitions passes into a curve of the first order ones at the tricritical point leading to a
violation of the critical universality of the fluctuation theory. This change of the phase transition
is generated by the fermion condensation quantum phase transition. Near the tricritical point the
Landau theory of second order phase transitions is applicable and gives α ≃ 1/2. We demonstrate
that this value of α is in good agreement with the specific-heat measurements.
PACS numbers: 71.27.+a, 64.60.-i, 64.60.Kw
Key Words: Quantum phase transitions; Heavy fermions; Tricritical points; Entropy
I. INTRODUCTION
Fundamental understanding of the low-temperature
physical properties of such strongly correlated Fermi sys-
tems as heavy fermion (HF) metals in the vicinity of
a quantum phase transition persists as one of the most
challenging objectives of condensed-matter physics. List
of these extraordinary properties are markedly large. Re-
cent exciting measurements on YbRh2Si2at the second
order antiferromagnetic (AF) phase transition extended
the list and revealed a sharp peak in low-temperature
specific heat, which is characterized by the critical ex-
ponent α = 0.38 and therefore differs drastically from
those of the conventional fluctuation theory of second
order phase transitions [1], where α ≃ 0.1 [2]. The ob-
tained large value of α casts doubts on the applicability
of the conventional theory and sends a real challenge for
theories describing the second order phase transitions in
HF metals [1], igniting strong theoretical effort to explain
the violation of the critical universality in terms of the
tricritical point [3, 4, 5, 6].
The striking feature of the fermion condensation quan-
tum phase transition (FCQPT) is that it has profound in-
fluence on thermodynamically driven second order phase
transitions provided that these take place in the non-
Fermi liquid (NFL) region formed by FCQPT [7, 8]. As
a result, the curve of any second order phase transition
passes into a curve of the first order one at the tricritical
point leading to a violation of the critical universality of
the fluctuation theory. For example, the second order
superconducting phase transition in CeCoIn5changes to
the first one in the NFL region [9]. As we shall see, it is
this feature that gives the key to resolve the challenge.
∗Electronic address: vrshag@thd.pnpi.spb.ru
It is a common wisdom that low-temperature and
quantum fluctuations at quantum phase transitions form
the specific heat, magnetization, magnetoresistance etc.,
which are drastically different from that of conventional
metals [10, 11, 12, 13, 14]. Usual arguments that quasi-
particles in strongly correlated Fermi liquids ”get heavy
and die” at a quantum critical point commonly employ
the well-known formula basing on assumptions that the
z-factor (the quasiparticle weight in the single-particle
state) vanishes at the points of second-order phase transi-
tions [15]. However, it has been shown that this scenario
is problematic [16]. On the other hand, facts collected on
HF metals demonstrate that the effective mass strongly
depends on temperature T, doping (or the number den-
sity) x, applied magnetic fields B etc, while the effec-
tive mass M∗itself can reach very high values or even
diverge, see e.g. [12, 13]. Such a behavior is so unusual
that the traditional Landau quasiparticles paradigm does
not apply to it.The paradigm says that elementary
excitations determine the physics at low temperatures.
These behave as Fermi quasiparticles and have a cer-
tain effective mass M∗which is independent of T, x, and
B and is a parameter of the theory [17]. A concept of
FCQPT preserving quasiparticles and intimately related
to the unlimited growth of M∗had been developed in
Refs. [7, 18, 19]. In contrast to the Landau paradigm
based on the assumption that M∗is a constant, the FC-
QPT approach supports an extended paradigm, the main
point of which is that the well-defined quasiparticles de-
termine the thermodynamic and transport properties of
strongly correlated Fermi systems, M∗becomes a func-
tion of T, x, B, while the dependence of the effective
mass on T, x, B gives rise to the non-Fermi liquid be-
havior [8, 20, 21, 22, 23]. Studies show that the extended
paradigm is capable to deliver an adequate theoretical
explanation of the NFL behavior in different HF metals
and HF systems [8, 9, 20, 22, 23, 24, 25].

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In the present short communication, we analyze the
specific-heat measurements on YbRh2Si2in the vicinity
of the second order AF phase transition with TN = 72
mK [1]. The measurements reveal that the correspond-
ing critical exponent α = 0.38 which differs drastically
from that produced by the fluctuation theory of second
order phase transitions, where α ≃ 0.1. We show that un-
der the application of magnetic field B the curve TN(B)
of the second order AF phase transitions in YbRh2Si2
passes into a curve of the first order ones at the tricriti-
cal point with temperature Tcr= TN(Bcr). This change
is generated by FCQPT. Near the tricritical point the
Landau theory of second order phase transitions is ap-
plicable and gives α ≃ 1/2 [2]. This value of α is in
good agreement with the specific-heat measurements de-
scribing the data in the entire temperature range around
the AF phase transition. As a result, we conclude that
the critical universality of the fluctuation theory is vio-
lated at the line of the AF phase transitions due to the
tricritical point.
II. FERMION CONDENSATION QUANTUM
PHASE TRANSITION
We start with visualizing the main properties of FC-
QPT. To this end, consider the density functional theory
for superconductors (SCDFT) [26]. SCDFT states that
the thermodynamic potential Φ is a universal functional
of the number density n(r) and the anomalous density
(or the order parameter) κ(r,r1) and provides a varia-
tional principle to determine the densities [26]. At the
superconducting transition temperature Tc a supercon-
ducting state undergoes the second order phase transi-
tion. Our goal now is to construct a quantum phase
transition which evolves from the superconducting one.
Let us assume that the coupling constant λ of the BCS-
like pairing interaction vanishes, λ → 0, making vanish
the superconducting gap at any finite temperature. In
that case, Tc → 0 and the superconducting state takes
place at T = 0 while at finite temperatures there is a
normal state. This means that at T = 0 the anomalous
density
κ(r,r1) = ?Ψ ↑ (r)Ψ ↓ (r1)? (1)
is finite, while the superconducting gap
∆(r) = λ
?
κ(r,r1)dr1
(2)
is infinitely small [8, 9]. In Eq. (1), the field opera-
tor Ψσ(r) annihilates an electron of spin σ,σ =↑,↓ at
the position r. For the sake of simplicity, we consider
a homogeneous electron liquid [8].
the thermodynamic potential Φ reduces to the ground
state energy E which turns out to be a functional of
the occupation number n(p) since the order parameter
Then at T = 0,
κ(p) =
ing E with respect to n(p), we obtain [8, 18]
?n(p)(1 − n(p)) [23, 26, 27, 28]. Upon minimiz-
δE
δn(p)= ε(p) = µ,(3)
where µ is the chemical potential. As soon as Eq. (3)
has nontrivial solution n0(p) then instead of the Fermi
step, we have 0 < n0(p) < 1 in certain range of momenta
pi≤ p ≤ pf with κ(p) =
?n0(p)(1 − n0(p)) is finite in
this range, while the single particle spectrum ε(p) is flat.
Thus, the step-like Fermi filling inevitably undergoes re-
structuring and forms the fermion condensate (FC) when
Eq. (3) possesses for the first time the nontrivial solution
at some quantum critical point (QCP) x = xc. Here pF
is the Fermi momentum and x = p3
the range vanishes, pi→ pf→ pF, and the effective mass
M∗diverges at QCP [8, 18, 20, 22]
F/3π2. In that case,
1
M∗(x → xc)=
1
pF
∂ε(p)
∂p
|p→pF;x→xc→ 0. (4)
At any small but finite temperature the anomalous den-
sity κ (or the order parameter) decays and, as we shall
see, this state undergoes the first order phase transition
and converts into a normal state characterized by the
thermodynamic potential Φ0. At T → 0, the entropy
S = −∂Φ0/∂T of the normal state is given by the well-
known relation [17]
S0[n0(p)] = −2
?
[n0(p)ln(n0(p)) + (1 − n0(p))
× ln(1 − n0(p))]
dp
(2π)3,(5)
which follows from combinatorial reasoning. It is seen
from Eq. (5) that the normal state is characterized by the
temperature-independent entropy S0 [8, 27]. Since the
entropy of the superconducting ground state is zero, we
conclude that the entropy is discontinuous at the phase
transition point, with its discontinuity ∆S = S0. Thus,
the system undergoes the first order phase transition.
The heat q of transition from the asymmetrical to the
symmetrical phase is q = TcS0= 0 since Tc= 0. Because
of the stability condition at the point of the first order
phase transition, we have Φ0[n(p)] = Φ[κ(p)]. Obviously
the condition is satisfied since q = 0.
At T = 0, a quantum phase transition is driven by a
nonthermal control parameter, e.g. the number density
x. To clarify the role of x, consider the effective mass
M∗which is related to the bare electron mass M by the
well-known Landau Eq. [17]
1
M∗=
1
M+
?
pFp1
p3
F
F(pF,p1)∂n(p1,T)
∂p1
dp1
(2π)3.(6)
Here we omit the spin indices for simplicity, n(p,T) is
quasiparticle occupation number, and F is the Landau
amplitude. At T = 0, Eq. (6) reads [29, 30]
M∗
M
=
1
1 − N0F1(x)/3. (7)

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Here N0is the density of states of a free electron gas and
F1(x) is the p-wave component of Landau interaction
amplitude F. When at some critical point x = xc, F1(x)
achieves certain threshold value, the denominator in Eq.
(7) tends to zero so that the effective mass diverges at
T = 0 [29, 30]. It follows from Eq. (7) that beyond the
critical quantum point xc, the effective mass becomes
negative. To avoid unstable and physically meaningless
state with a negative effective mass and in accordance
with Eq. (4), the system must undergo a quantum phase
transition at QCP with x = xc, which is QCP of FCQPT
[7, 8, 18, 20].
?
?
?
?
FIG. 1: Schematic phase diagram of the system driven to
the FC state. The number density x is taken as the control
parameter and depicted as x/xc. The quantum critical point,
x/xc = 1, of FCQPT is shown by the arrow. At x/xc < 1 and
sufficiently low temperatures, the system in the Landau Fermi
liquid (LFL) state is shown by the shadow area. At T = 0
and beyond the critical point, x/xc > 1, the system is at the
quantum critical line depicted by the dash line and shown by
the vertical arrow. The critical line is characterized by the
FC state with finite superconducting order parameter κ. At
Tc = 0, κ is destroyed, the system undergoes the first order
phase transition and exhibits the NFL behavior at T > 0.
Schematic phase diagram of the system which is driven
to FC by variation of x is reported in Fig. 1. Upon
approaching the critical density xc the system remains
in the Landau Fermi liquid (LFL) region at sufficiently
low temperatures [8, 20], that is shown by the shadow
area. At QCP xcshown by the arrow in Fig. 1, the sys-
tem demonstrates the NFL behavior down to the lowest
temperatures. Beyond the critical point at finite tem-
peratures the behavior is remaining the NFL one and is
determined by the temperature-independent entropy S0
[8, 27]. In that case at T → 0, the system is approaching
a quantum critical line (shown by the vertical arrow and
the dashed line in Fig. 1) rather than a quantum criti-
cal point. Upon reaching the quantum critical line from
the above at T → 0 the system undergoes the first order
quantum phase transition, which is FCQPT taking place
at Tc= 0.
At T > 0 the NFL state above the critical line, see
Fig. 1, is strongly degenerated, therefore it is captured
by the other states such as superconducting (for exam-
ple, by the superconducting state in CeCoIn5[9, 24, 27])
or by AF state (e.g. AF one in YbRh2Si2[23]) lifting the
degeneracy. The application of magnetic field B > Bc0
restores the LFL behavior, where Bc0is a critical mag-
netic field, such that at B > Bc0 the system is driven
towards its LFL state [8, 22, 24]. In some cases, for ex-
ample in HF metal CeRu2Si2, Bc0 = 0, see e.g. [31],
while in YbRh2Si2, Bc0 ≃ 0.06 T [32]. In our simple
model of homogeneous electron liquid Bc0is taken as a
parameter.
III.T − B PHASE DIAGRAM FOR YbRh2Si2
VERSUS ONE FOR CeCoIn5
In Fig. 2, we present temperature T/TN versus field
B/Bc0 schematic phase diagram for YbRh2Si2. There
TN(B) is the N´ eel temperature as a function of the mag-
netic field B. The solid and dash lines indicate boundary
of the AF phase at B/Bc0≤ 1 [32]. For B/Bc0≥ 1, the
dash-dot line marks the upper limit of the observed LFL
behavior. This dash-dot line separates the NFL state and
the weakly polarized LFL, and is represented by [8]
T∗
TN
= a1
?
B
Bc0
− 1,(8)
where a1 is a parameter. We note that Eq. (8) is in
good agreement with facts [32]. Thus, YbRh2Si2demon-
strates two different LFL states, where the temperature-
dependent electrical resistivity ∆ρ follows the LFL be-
havior ∆ρ ∝ T2, one being weakly AF ordered (B ≤ Bc0
and T < TN(B)) and the other being weakly polarized
(B ≥ Bc0 and T < T∗(B)) [32].
peratures and fixed magnetic field the NFL state oc-
curs which is separated from the AF phase by the curve
TN(B) of phase transition. In accordance with experi-
mental facts we assume that at relatively high temper-
atures T/TN(B) ≃ 1 the AF phase transition is of the
second order [1, 32]. In that case, the entropy and the
other thermodynamic functions are continuous functions
at the curve of the phase transitions TN(B). This means
that the entropy of the AF phase SAF(T) coincides with
the entropy S(T) of the NFL state
At elevated tem-
SAF(T → TN(B)) = S(T → TN(B)).(9)
Since the AF phase demonstrates the LFL behavior, that
is SAF(T → 0) → 0, Eq. (9) cannot be satisfied at di-
minishing temperatures T ≤ Tcrdue to the temperature-
independent term S0given by Eq. (5). Thus, in the NFL
region formed by FCQPT the second order AF phase
transition inevitably becomes the first order one at the
tricritical point with T = Tcr, as it is shown in Fig. 2.
At T = 0, the critical field Bc0 is determined by the

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4
FIG. 2: Schematic T − B phase diagram for YbRh2Si2. The
solid and dash TN(B) curves separate AF and non-Fermi liq-
uid (NFL) states representing the field dependence of the N´ eel
temperature. The black dot at T = Tcr shown by the arrow
in the dash curve is the tricritical point, at which the curve
of second order AF phase transitions shown by the solid line
passes into the curve of the first ones. At T < Tcr, the dash
line represents the field dependence of the N´ eel temperature
when the AF phase transition is of the first order. The NFL
state is characterized by the entropy S0given by Eq. (5). The
dash-dot line separating the NFL state and the weakly polar-
ized LFL is represented by T∗(B/Bc0) given by Eq. (8). The
horizontal solid arrow represents the direction along which
the system transits from the NFL behavior to the LFL one at
elevated magnetic field and fixed temperature. The vertical
solid arrow represents the direction along which the system
transits from the LFL behavior to the NFL one at elevated
temperature and fixed magnetic field. The shadowed circle
depict the transition temperature T∗from the NFL to LFL
behavior.
condition that the ground state energy of the AF phase
coincides with the ground state energy of the weakly po-
larized LFL, and the ground state of YbRh2Si2becomes
degenerated at B = Bc0. Therefore, the N´ eel tempera-
ture TN(B → Bc0) → 0.
Upon comparing the phase diagram of YbRh2Si2 de-
picted in Fig. 2 with that of CeCoIn5shown in Fig. 3,
it is possible to conclude that they are similar in many
respects. Indeed, the line of the second order supercon-
ducting phase transitions changes to the line of the first
ones at the tricritical point shown by the the square in
Fig. 3. This transition takes place under the applica-
tion of magnetic field B > Bc2≥ Bc0[9, 24], where Bc2
is the critical field destroying the superconducting state,
and Bc0is the critical field at which the magnetic field
induced QCP takes place [33, 34]. We note that the su-
perconducting boundary line Bc2(T) at lowering temper-
atures acquires the tricritical point due to Eq. (9) that
cannot be satisfied at diminishing temperatures T ≤ Tcr,
i.e. the corresponding phase transition becomes first or-
der [9, 24, 33]. This permits us to conclude that at lower-
ing temperatures, in the NFL region formed by FCQPT
FIG. 3: B − T phase diagram of the CeCoIn5 heavy fermion
metal. The interface between the superconducting and nor-
mal phases is shown by the solid and dash lines. At T < T0,
the curve of the second order superconducting phase transi-
tions passes into a curve of the first order ones at the tricriti-
cal point shown by the square [33]. The interface between the
superconducting and normal phases is shown by the dashed
line. The solid straight line with the experimental points [34]
shown by squares is the interface between the Landau Fermi
liquid (LFL) and non-Fermi liquid (NFL) states [9, 33, 34].
the curve of any second order phase transition passes into
the curve of the first order one at the tricritical point.
IV.THE TRICRITICAL POINT IN THE B − T
PHASE DIAGRAM OF YbRh2Si2
The Landau theory of the second order phase transi-
tions is applicable as the tricritical point is approached,
T ≃ Tcr, since the fluctuation theory can lead only to fur-
ther logarithmic corrections to the values of the critical
indices. Moreover, near the tricritical point, the differ-
ence TN(B) − Tcris a second order small quantity when
entering the term defining the divergence of the specific
heat [2]. As a result, upon using the Landau theory we
obtain that the Sommerfeld coefficient γ0= C/T varies
as γ0∝ |t− 1|−αwhere t = T/TN(B) with the exponent
is α ≃ 0.5 as the tricritical point is approached at fixed
magnetic field [2]. We will see that α = 0.5 gives good
description of the facts collected in measurements of the
specific heat on YbRh2Si2. Taking into account that the
specific heat increases in going from the symmetrical to
the asymmetrical AF phase [2], we obtain
γ0(t) = A +
B
?|t − 1|. (10)
Here, B = B±are the proportionality factors which are
different for the two sides of the phase transition, the
parameters A = A±related to the corresponding specific

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5
heat (C/T)±are also different for the two sides, and “+”
stands for t > 1, “−” stands for t < 1.
?
?
?
FIG. 4: The temperature dependence of the normalized Som-
merfeld coefficient γ0/A+as a function of the normalized tem-
perature t = T/TN(B = 0) given by the formula (10) is shown
by the solid line. The normalized Sommerfeld coefficient is ex-
tracted from the facts obtained in measurements on YbRh2Si2
at the AF phase transition [1] and shown by the triangles.
The attempt to fit the available experimental data for
γ0= C(T)/T in YbRh2Si2at the AF phase transition in
zero magnetic fields [1] by the function (10) is reported
in Fig. 4. We show there the normalized Sommerfeld
coefficient γ0/A+ as a function of the normalized tem-
perature T/TN(B = 0). It is seen that the normalized
Sommerfeld coefficient γ0/A+extracted from C/T mea-
surements on YbRh2Si2[1] is well described in the entire
temperature range around the AF phase transition by
the formula (10) with A+= 1.
?
?
FIG. 5: The temperature dependence of the ratios (γ0−A)/B
for t < 1 and t > 1 versus |1 − t| given by the formula (11)
is shown by the solid line. The ratios are extracted from the
facts obtained in measurements of γ0 on YbRh2Si2 at the AF
phase transition [1] and depicted by the triangles as shown in
the legend.
Now transform Eq. (10) to the form
γ0(t) − A
B
=
1
?|t − 1|. (11)
It follows from Eq. (11) that the ratios (γ0− A)/B for
t < 1 and t > 1 versus |1 − t| are to collapse into a
single line in logarithmic-logarithmic plot. The extracted
from experimental facts [1] ratios are depicted in Fig. 5,
coefficients A and B are taken from the fitting γ0shown
in Fig. 4. It is seen from Fig. 5 that the ratios (γ0−A)/B
shown by the upward and downwards triangles for t < 1
and t > 1 respectively do collapse into the single line
shown by the solid straight line.
A few remarks are in order here.
shown in Figs.4 and 5 of the experimental facts by
the functions (10) and (11) with the critical exponent
α = 1/2 allows us to conclude that the specific-heat mea-
surements on YbRh2Si2[1] are taken near the tricritical
point and to predict that the second order AF phase tran-
sition in YbRh2Si2 changes to the first order under the
application of magnetic field as it is shown by the arrow
in Fig. 2. It is seen from Fig. 4 that at t ≃ 1 the peak is
sharp, while one would expect that anomalies in the spe-
cific heat associated with the onset of magnetic order are
broad [1, 35, 36]. Such a behavior presents fingerprints
that the phase transition is to be changed to the first or-
der one at the tricritical point, as it is shown in Fig. 2.
As seen form Fig. 4, the Sommerfeld coefficient is larger
below the phase transition than above it. This fact is in
accord with the Landau theory stating that the specific
heat is increased when passing from t > 1 to t < 1 [2].
The good fitting
V.ENTROPY IN YbRh2Si2 AT LOW
TEMPERATURES
It is instructive to analyze the evolution of mag-
netic entropy in YbRh2Si2 at low temperatures.
start with considering the derivative of magnetic entropy
dS(B,T)/dB as a function of magnetic field B at fixed
temperature Tf when the system transits from the NFL
behavior to the LFL one as shown by the horizontal solid
arrow in Fig. 2. Such a behavior is of great importance
since exciting experimental facts [37] on measurements
of the magnetic entropy in YbRh2Si2allow us to analyze
reliability of the employed theory and to study the scal-
ing behavior of the entropy when the system in its NFL,
transition and LFL states.
According to the well-known thermodynamic equality
dM/dT = dS/dB, and ∆M/∆T ≃ dS/dB. To carry
out a quantitative analysis of the scaling behavior of
dS(B,T)/dB, we calculate the entropy S as a function
of B at fixed temperature Tf within the model of ho-
mogeneous electron liquid taking into account that the
electronic system of YbRh2Si2is located at FCQPT [23].
Fig. 6 reports the normalized (dS/dB)Nas a function of
the normalized magnetic field. The normalized function
We

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6
??
?
?
?
????
????????
?
?
?
?
?
?
?
FIG.
(dS/dB)N ≃ (∆M/∆T)N versus normalized magnetic field
when the system transits from the NFL region to the LFL
one along the horizontal solid line shown in Fig. 2. Normal-
ized magnetization difference divided by temperature incre-
ment (∆M/∆T)N versus normalized magnetic field at fixed
temperatures listed in the legend is extracted from the facts
collected on YbRh2Si2 [37]. Our calculation of the normal-
ized derivative (dS/dB)N versus normalized magnetic field is
shown by the solid line.
6:Thebehavior of thenormalizedderivative
(dS/dB)N is obtained by normalizing (−dS/dB) by its
maximum taking place at BM, and the field B is scaled
by BM. The measurements of −∆M/∆T are normalized
in the same way and depicted in Fig. 6 as (∆M/∆T)N
versus normalized field. It is seen from Fig. 6 that our
calculations shown by the solid line are in good agreement
with the facts and the scaled functions (∆M/∆T)N ex-
tracted from the facts show the scaling behavior in wide
range variation of the normalized magnetic field B/BM.
Now we are in position to evaluate the entropy S at
temperatures T ? T∗in YbRh2Si2.
system in its LFL state, the effective mass is independent
of T, and is a function of magnetic field B [8, 32]
At T < T∗the
M
M∗(B)= a2
?
B
Bc0
− 1,(12)
where a2 is a parameter. In the LFL state at T < T∗
when the system moves along the vertical arrow shown in
Fig. 2, the entropy is given by the well-known relation,
S = M∗Tπ2/p2
(8) and (12) we obtain that at T ≃ T∗the entropy is
independent of both magnetic field and temperature,
S(T∗) ≃ γ0T∗≃ S0 ≃ a1MTNπ2/a2p2
the data [32], we obtain that for fields applied along
the hard magnetic direction S0(Bc0?c) ∼ 0.03Rln2,
and for fields applied along the easy magnetic direction
S0(Bc0⊥c) ∼ 0.005Rln2.
facts collected on YbRh2Si2 [32] we conclude that the
entropy contains the temperature-independent part S0
[8, 27] which gives rise to the tricritical point.
F= γ0T [17]. Taking into account Eqs.
F. Upon using
Thus, in accordance with
VI.CONCLUSIONS
We have predicted that the curve of the second order
AF phase transitions in YbRh2Si2passes into the curve
of the first order ones at the tricritical point under the
application of magnetic field. Moreover, we have shown
that in the NFL region formed by FCQPT the curve of
any second order phase transition passes into a curve
of the first order one at the tricritical point leading to
the violation of the critical universality of the fluctuation
theory. This change is generated by the temperature-
independent entropy S0 formed behind FCQPT. Near
the tricritical point the Landau theory of second order
phase transitions is applicable and gives the critical in-
dex α ≃ 1/2. Bearing in mind that a theory is an im-
portant input in understanding of what we observe, we
demonstrate that this value of α is in good agreement
with the specific-heat measurements on YbRh2Si2[1] and
conclude that the critical universality of the fluctuation
theory is violated at the AF phase transition [1] since the
second order phase transition is about to change to the
first order one making α → 1/2.
VII.ACKNOWLEDGEMENTS
This work was supported in part by the grants: RFBR
No. 09-02-00056 and the Hebrew University Intramu-
ral Funds. V.R.S. is grateful to the Lady Davis Foun-
dation and thanks the generosity of the Forchheimer
Fund for supporting his visit to the Hebrew University
of Jerusalem.
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