Unveiling the Physics of the Mutual Interactions in Paramagnets
Lucas Squillante Isys F. Mello Gabriel O. Gomes A. C. Seridonio R. E. Lagos-Monaco H. Eugene Stanley Mariano de Souza
cond-mat.str-el cond-mat.mtrl-sci cond-mat.other cond-mat.stat-mech








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1]Lucas Squillante
[1]Departamento de Física, São Paulo State University (UNESP), IGCE, Rio Claro, 13506-900 SP, Brazil 
1]Isys F. Mello
2]Gabriel O. Gomes
[2]Department of Astronomy, University of São Paulo, 05508-090 SP, Brazil
3]A. C. Seridonio
[3]São Paulo State University (Unesp), Department of Physics and Chemistry, Ilha Solteira - SP, Brazil
1]R. E. Lagos-Monaco
4]H. Eugene Stanley
[4]Boston University, Department of Physics, Boston, 02215, USA
1∗]Mariano de Souza
[^∗]To whom correspondence should be addressed. E-mail: mariano.souza@unesp.br



Unveiling the Physics of the  Mutual Interactions in Paramagnets
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In real paramagnets, there is always a subtle many-body contribution to the system's energy, which can be regarded as a small effective local magnetic field (B_loc). Usually, it is neglected, since it is very small when compared with thermal fluctuations and/or external magnetic fields (B). Nevertheless, as both the temperature (T) → 0 K and B → 0 T, such many-body contributions become ubiquitous. Here, employing the magnetic Grüneisen parameter (Γ_mag) and entropy arguments,  we report on the pivotal role played by the mutual interactions in the regime of ultra-low-T and vanishing B. Our key results are:  i) absence of a genuine zero-field quantum phase transition due to the presence of B_loc; ii) connection between the canonical definition of temperature and Γ_mag; and iii) possibility of performing adiabatic magnetization by only manipulating the mutual interactions. Our findings unveil unprecedented aspects emerging from the mutual interactions.






§ INTRODUCTION




Magnetic excitations in solids have been broadly investigated in the past decades, being crucial to the understanding of exotic physical phenomena such as superconductivity <cit.> and magnetic field induced quantum phase transitions <cit.>, just to mention a few examples. It is well-known that the behavior of paramagnetic metals and insulators are nicely described, respectively, by the Pauli paramagnetism and Brillouin-like model. Such approaches are based on a spin gas scheme, i.e., interactions between magnetic moments are not taken into account and thus the system is treated as an ideal paramagnet. However, in real paramagnets  the magnetic dipolar interactions between adjacent spins are always present, being usually neglected.  Although the mutual interactions, also called zero-field splitting <cit.> and champ moléculaire <cit.>, have been broadly mentioned in the literature  <cit.>, a detailed discussion about their role in the magnetic properties of solids in the regime of ultra low-temperatures and vanishing external magnetic field is still lacking.

Nowadays, exotic manifestations of matter, like non-Fermi-liquid behavior and unconventional superconductivity <cit.>, emerging in the immediate vicinity of a quantum critical point (QCP), have been attracting high interest of the community. It is well-established that for a pressure-induced QCP <cit.>, as well as for a finite temperature (T) pressure-induced critical point <cit.>, the Grüneisen ratio, i.e., the ratio between thermal expansivity and specific heat at constant pressure, is enhanced upon approaching the critical values of the tuning parameter and it diverges right at the critical point. For a magnetic field-induced QCP <cit.> the analogous physical quantity to the Grüneisen ratio is the so-called magnetic Grüneisen parameter, hereafter  Γ_mag  <cit.>. The enhancement of both the Grüneisen ratio and Γ_mag in the immediate vicinity of a magnetic field-induced QCP is merely a direct consequence of the high entropy accumulation in that region  <cit.>, which in turn is related to the fluctuations of the order parameter. Also, it is well-known that Γ_mag  quantifies the magneto-caloric effect <cit.>, which in turn enables to change the temperature of a system upon varying adiabatically the external applied magnetic field <cit.>. The fingerprints of a genuine magnetic field-induced QCP, besides the gradual suppression of an order parameter (or energy scale) at the QCP [Fig. <ref> a)], are: i) the divergence of Γ_mag for T → 0 K at the critical magnetic field B_c; ii) the sign-change of Γ_mag upon crossing B_c, and iii) its typical scaling behavior in the form T/(B - B_c)^ϵ, where B is the external applied magnetic field and ϵ the scaling exponent <cit.>.
Particular attention has been paid to the so-called zero-field quantum criticality [Fig. <ref> b)], i.e., the system is inherently quantum critical (B_c = 0 T) and thus no field-sweep is required for achieving the QCP.  Examples include several materials, such as YbCo_2Ge_4 <cit.>, β-YbAlB_4 <cit.>, and Au-Al-Yb <cit.>. In the case of β-YbAlB_4, in particular, the understanding of possible zero-field quantum criticality <cit.> and the emergence of superconductivity at T_c = 80 mK <cit.> remains elusive <cit.>. Interestingly, although β-YbAlB_4 is metallic <cit.>,  its magnetic susceptibility displays a typical Curie-Weiss-like behavior <cit.> and it is thus evident that resultant local magnetic moments are still present into the system at very low temperatures, cf. Fig. 2(A) of Ref. <cit.>. Because of such exotic behavior the system is considered as a “strange metal” <cit.>. Based on a scaling analysis of the magnetization,  the authors of Ref. <cit.> argue on the crossover of β-YbAlB_4 from a non Fermi liquid to a Fermi liquid behavior.
Recently, some of us reported on a surprisingly divergent behavior of Γ_mag  for model systems, including the one-dimensional Ising model under longitudinal B and Brillouin-like paramagnets <cit.>.
Here, we report on the  absence of zero-field quantum criticality for any paramagnetic insulator with non-zero effective local magnetic field (B_loc) and discuss the intricate role played by mutual interactions in the regime of T → 0 K and B → 0 T.  We demonstrate the validity of our analysis for the textbook Brillouin-like paramagnet and for the proposed zero-field quantum critical system  β-YbAlB_4.
At some extent, our approach is reminiscent of the famous Mermin-Wagner theorem <cit.>, since at finite temperatures the mutual interactions give rise to long-range intrinsic magnetic fluctuations.
Also, the connection between Γ_mag and the canonical definition of temperature is reported.  Yet, we propose the possibility of carrying out adiabatic magnetization by only manipulating the mutual interactions. It is to be noted that a discussion in the literature concerning on adiabatic magnetization was reported about seventy years ago  <cit.>, when adiabatic magnetization was employed to produce cooling using paramagnetic salts with (∂ S/∂ B)_T > 0, where S is the entropy.
When dealing with possible zero-field quantum criticality in paramagnets, we need to be very careful by analyzing the divergence of Γ_mag for B → 0 T experimentally, since an enhancement of Γ_mag solely does not suffice to assign genuine zero-field quantum criticality <cit.>.
It turns out that due to the mutual interactions a spontaneous magnetically ordered phase emerges, which prevents Γ_mag to diverge.
For the sake of completeness, it is worth recalling that paramagnetic systems have been used to achieve low-temperatures in the range of μK (electronic spins) and nK (nuclear spins) employing the adiabatic demagnetization method <cit.>. Nevertheless, the achievement of exactly 0 K using this process is limited by the increase of the mutual interactions' relevance upon approaching the ground-state [Fig. <ref> b)].
In quantitative terms, considering that in an ideal paramagnet neighboring magnetic moments μ⃗ are separated by a distance r <cit.>, the magnetic dipolar energy is roughly given by U_dip≃ (μ_0μ^2)/(4π r^3)<cit.>, where μ_0 = (4π×10^-7) T·m/A is the vacuum magnetic permeability. Since classically the magnetic energy is given by U_mag = -μ B_loc cosϕ, being ϕ the angle between μ⃗ and B⃗_loc, the local dipolar magnetic field is roughly given by B_loc≃μ_0μ/4π r^3. Hence, for r = 5, a typical distance between neighboring spins in a paramagnet, and considering μ = μ_B = (9.27×10^-24) J/T, an intrinsic effective local magnetic field B_loc≃ 0.01 T can be estimated <cit.>.
It is worth emphasizing that although we have considered that B_loc emerges purely from the magnetic dipolar interactions between neighboring magnetic moments, it is clear that the electrostatic energy is also present into the system. Nevertheless, the electrostatic energy overcomes the magnetic energy only in the regime of relatively high temperatures <cit.>. Considering that we are interested in the physical properties of paramagnets in the ultra-low T (<< 1 K) regime our analysis of B_loc for real systems remains appropriate, when only the magnetic dipolar interactions are considered.
If we consider the Hydrogen atom, for instance, we must take into account the interaction between electronic and nuclear spins, i.e. , the hyperfine coupling, so that when the distance between electron and nucleus is the Bohr radius, the electron perceives a local magnetic field from the nuclear spin B_loc≃ 0.0063 T. The latter is roughly one order of magnitude lower than typical values of  B_loc in real paramagnets as expected, since the nuclear magnetic moment is roughly 1000 times lower than the electronic one.
In the frame of Quantum Mechanics, the Eigenenergies of the Hydrogen for the Zeeman splitting are obtained through the following Hamiltonian <cit.>:


    H = A(σ^e·σ^p)-μ_e σ^e·B-μ_p σ^p·B,


where the indexes e and p refers to electron and proton, respectively; σ is the spin operator and A is a constant related to the magnetic interaction between electron and proton <cit.>. The first term of this simple Hamiltonian (Eq. <ref>) is independent of the external magnetic field and, therefore, it is connected to the zero-field splitting. Indeed, the Zeeman splitting starts at U_mag = 0 J and the magnetic energy difference between the energy levels enhances as the external magnetic field is increased. It turns out that the Eigenenergies of the Hamiltonian (Eq. <ref>) show a quite similar behavior, being that in this case for B = 0 T the magnetic energy is U_mag = A. Analogously, if we consider two neighboring magnetic moments, we can treat the mutual interactions between them employing the Hamiltonian (Eq. <ref>), being necessary only to associate them to the indexes e and p.  As we have mentioned before, given its relatively low strength, B_loc is relevant only for vanishing external applied magnetic fields and in the regime of ultra low-temperatures. In other words,  B_loc begins to be important only when the thermal and magnetic energies are comparable, recalling that U_mag = -μ⃗·B⃗_loc and U_mag = + μ⃗·B⃗_loc represent, respectively, the minimum and maximum magnetic energies.
The magnetic energy can also be expressed in terms of the total angular momentum quantum number J,  as well as in terms of the magnetic quantum number m_J, which represents the number of allowed orientations of μ⃗. The modulus of the total angular momentum vector J⃗ is given in terms of J, namely |J⃗| = ħ√(J(J+1)), where ħ is Planck's constant divided by 2π. Each value of m_J = -J, -J+1, ..., J-1, J describes a particular orientation of the magnetic moment and its respective magnetic energy, since μ = gμ_B m_J, where g is  the gyromagnetic factor <cit.>.  Here it is worth emphasizing that in our analysis we consider μ⃗ oriented along the z⃗ direction.
In the frame of Quantum Mechanics, the magnetic energy is written as U_mag = -[m_J/√(J(J+1)) ]μ B_loc <cit.>.
At this point, we stress that in our analysis B_loc is the effective magnetic field generated by the magnetic dipolar interaction between  neighboring magnetic moments. Since √(J(J+1)) is always greater than |m_J|, the magnetic energy is not exactly ± μ B_loc as expected classically, and thus we can infer that ϕ will never be exactly zero or  π <cit.>.
In fact, the absence of a perfect alignment between μ⃗ and B⃗_loc can be observed in the results shown in Fig. <ref>  for the Brillouin paramagnet (upper panel) and β-YbAlB_4 (lower panel). Note that the entropy is lowered upon increasing B, but it will never be exactly zero for any finite value of B.
Yet, in quantitative terms, in the case of β-YbAlB_4 <cit.>, using the effective Yb magnetic moment  μ≃ 1.94μ_B and the Yb-Yb separation of r ≃ 3.5  <cit.>, results in B_loc ≃ 0.04 T.
Even considering the fact that the system undergoes a superconducting transition at T_c = 80 mK <cit.>, intrinsic magnetic moments survive in β-YbAlB_4 at very low-T <cit.>, so that the Yb valence fluctuations <cit.> do not affect our analysis. We consider that such resultant magnetic moment is responsible for the emergence of B_loc and thus, even into the superconducting dome, B_loc is relevant and prevents that zero-field quantum criticality takes place.
Interestingly enough, in the case of  α-YbAlB_4 <cit.> the magnetic moments are fully screened and thus it is not possible to infer resultant magnetic dipolar interactions. After this Introduction, we present and discuss our findings.



§ RESULTS AND DISCUSSION




i) Absence of a genuine zero-field quantum phase transition due to the presence of B_loc.

We start our analysis based on entropy arguments.
In the case of an ideal paramagnet described by the textbook Brillouin-like model <cit.>, the entropy can be derived from the probabilities of the spins (we consider J = 1/2) to align parallel or anti-parallel to B and it is given by:

    S(B,T) = -B μ_B/Ttanh(B μ_B/k_B T) + k_B[2cosh(B μ_B/k_B T)],

where k_B is the Boltzmann constant (1.38×10^-23) J/K.
The entropy S (Eq. <ref>) as a function of B at T = 5 mK (chosen arbitrarily), assuming B_loc = 0  (blue asterisks) and B_loc =  0.01 T (red circles), is plotted in the upper panel of Fig. <ref>. Now, we recall the proposed quantum critical free-energy F_QC for β-YbAlB_4 <cit.>:

    F_QC(B,T) = -1/(k_B T̃)^1/2[(gμ_B B)^2+(k_B T)^2]^3/4,

where k_BT̃ ≈ 6.6 eV ≈ (1.06×10^-18) J, T̃ refers to a characteristic temperature and g = 1.94 <cit.>. Using Eq. <ref>, it is straightforward to calculate the quantum critical entropy, S_QC = -(∂ F_QC/∂ T)_B, namely:

    S_QC(B,T) = 3k_B^2 T/2(k_B T̃)^1/2(B^2 g^2μ_B^2 + k_B^2 T^2)^1/4.

Considering that B_loc makes an angle θ≃ 90^∘ with B, we can write the resultant magnetic field  B_r ≃ √(B^2 + B_loc^2) <cit.>, as depicted in the inset of Fig. <ref> b).
It turns out that when using B_r directly in the partition function for the Brillouin paramagnet <cit.> instead of B, the derived physical quantities will naturally have B replaced by B_r in their respective mathematical expressions. Thus, a key point of our analysis is that we have  replaced B by B_r in the expression of the entropy for the Brillouin paramagnet following the latter approach.


The obtained entropy S_QC (Eq. <ref>) as a function of B for  β-YbAlB_4  considering arbitrarily chosen T = 5 mK, B_loc = 0 (green asterisks) and B_loc  =  0.04 T (orange circles), is plotted in the lower panel of Fig. <ref>. Essentially, in our analysis of the entropy we have used B_r = B (B_loc = 0 T) and B_r = √(B^2 + B_loc^2) in  Eqs. <ref> and <ref>.
As depicted in Fig. <ref>, for both the Brillouin-like paramagnet (upper panel) and β-YbAlB_4 (lower panel) the entropy is expressively lowered at zero external magnetic field when B_loc  is taken into account, since B_loc favours long-range magnetic order <cit.>.
Hence, the magnetic entropy is released and the third law of Thermodynamics is obeyed <cit.>. These results suggest that, upon considering B_loc, the entropy accumulation when B → 0 T is lowered and as a consequence Γ_mag is only enhanced, but it does not diverge.




We focus now on the analysis of Γ_mag, which can be calculated employing the well-known relation <cit.>:

    Γ_mag = -1/T(∂ S/∂ B)_T/(∂ S/∂ T)_B,

taking into account B_loc. Hence, in the following we consider in addition to the external magnetic field (B) the effects of B_loc on Γ_mag. We thus replace B by B_r in Eqs. <ref> and <ref>, so that:

    Γ_mag = B/B^2 + B_loc^2,

for the Brillouin paramagnet <cit.>, while for β-YbAlB_4, Γ_mag reads:

    Γ_mag = B g^2μ_B^2/k_B^2T^2 + 2g^2μ_B^2(B^2+B_loc^2).

The results of Γ_mag for the Brillouin paramagnet and β-YbAlB_4 (Eqs. <ref> and <ref>, respectively) are shown in Figs. <ref> and <ref>.
A careful analysis of  Γ_mag for the Brillouin paramagnet (upper panel of Fig. <ref>) enables us to relate our findings with the seminal experiment proposed by Purcell and Pound <cit.> regarding the achievement of negative temperatures in laboratory <cit.>, as well as with the well-known impossibility of achieving absolute zero temperature for B_loc≠ 0 T, to be discussed in the next subsection. In the case of β-YbAlB_4, Γ_mag→∞  for  T → 0 K and B → 0 T when B_loc = 0 T (upper panel of Fig. <ref>), but it does not do so for B_loc≠ 0 T (lower panel of Fig. <ref>).
Hence, the  consequence of taking B_loc≠ 0 T into account is that Γ_mag will never diverge for T→ 0 K and B → 0 T, since B_r will never be zero and thus we cannot infer a genuine zero-field quantum critical point. This is one of the main results of our work. We stress that we are not dealing with a simple shift in the position of the maximum value of Γ_mag, cf. Eq. <ref>. The situation is quite different, for instance, for the one-dimensional Ising model under transverse magnetic field <cit.>, where the critical field for the divergence of Γ_mag is shifted when the ratio of the critical field to coupling constant between nearest neighbor is changed <cit.>.
When analyzing Eqs. <ref> and <ref>, respectively, for the Brillouin paramagnet and β-YbAlB_4, considering B_loc ≠ 0 T, we observe that the maximum of Γ_mag is centered at B = B_loc for the Brillouin paramagnet, while for β-YbAlB_4 it is centered at  B = [B_loc^2 + (k_B^2 T^2)/(2g^2 μ_B^2)]^1/2, i.e., when T = 0 K the maximum is located at B = B_loc, cf. Fig. <ref>. The maxima were obtained by the simple optimization of the Γ_mag functions (Eqs. <ref> and <ref>), i.e., making (∂Γ_mag/∂ B)_T = 0 in both cases.
Yet, it is worth mentioning that the position of the maximum value of  Γ_mag is related to the well-known Schottky anomaly <cit.>, to be discussed into more details in the following.
For β-YbAlB_4, under the condition k_B T  = μ_B B_loc, considering B_loc = 0.04 T, we obtain T = μ_B B_loc/k_B ≃ 24 mK. The latter indicates the temperature onset of B_loc relevance  for this system. It is worth mentioning that the lowest temperature of the experiments reported in Fig. 2(A) of Ref. <cit.> for β-YbAlB_4 was T ≃ 10 mK and the lowest external magnetic field was 0.31 mT. In terms of Maxwell-relations, namely (∂ M/∂ T)_B = (∂ S/∂ B)_T, being M the magnetization, it is tempting to say that the results of Fig. 2(A) of Ref. <cit.> are at odds with a diverging Γ_mag.
Yet, considering β-YbAlB_4, our analysis is corroborated by the results presented in Fig. S2 of Ref.<cit.>, namely  Γ_mag versus external magnetic field for various temperatures. There, a  clear decrease of Γ_mag was observed experimentally upon decreasing B. However, in the frame of zero-field quantum criticality, we would expect an enhancement of Γ_mag for T → 0 K and B → 0 T <cit.>.  In Ref.<cit.>, the authors discuss that the free energy scaling for the systems CeCu_6-xAu_x and Au-Al-Yb, approximant and quasicrystal, respectively, follow the same scaling behavior as  β-YbAlB_4. These systems are also considered to be zero-field quantum critical. Then, our analysis can be extended to all systems that follow the same scaling behavior of the free energy reported in Ref. <cit.>, see Eq. <ref>. The main result of this subsection is that zero-field quantum criticality will not hold in any system where resultant magnetic moments are non-negligible.
A corresponding situation is also observed for the one-dimensional (1D) Ising model under longitudinal magnetic field <cit.>, where the magnetic coupling constant J' (we make use of J' to avoid confusion with the momentum quantum number J) plays the role analogously to B_loc, i.e.,  Γ_mag only diverges for J' → 0. In fact, the 1D Ising model under longitudinal field <cit.> is equivalent to the Brillouin paramagnet in two distinct cases, namely: i) at the ferromagnetic ground state where J' can be associated with a local magnetic field, which in turn acts as B_loc and; ii) in the limit T →∞ for finite B, where due to the increase of the thermal energy the ferromagnetic ordering is suppressed giving rise to a paramagnetic phase <cit.>. Hence, considering the similar role played by J' and B_loc, it becomes evident that Γ_mag only diverges for vanishing values of J', analogously to the case for the Brillouin paramagnet when B_loc = 0 T, as discussed previously. In general terms, any system with finite mutual interactions will not show a diverging Γ_mag and thus zero field quantum criticality cannot take place. This analysis reinforces the universal character of the mutual interactions and their role in the field of quantum criticality.
In the frame of the original work reported by Weiss <cit.>, the molecular field is associated with the mutual interactions, which in turn leads to the ordering of the magnetic moments within the regime of relevance of such interactions <cit.>. Hence, in the same way as the Curie temperature <cit.> represents the critical temperature for ferromagnets, analogously for the mutual interactions we can infer a pseudo critical temperature T_c, which defines their regime of relevance.



ii) Connection between the canonical definition of temperature and Γ_mag.

Considering the definition of temperature 1/T = (∂ S/∂ E)_B <cit.>, absolute zero temperature can be inferred when (∂ S/∂ E)_B →∞ (upper panel of Fig. <ref>), where E refers to the average magnetic energy, being E = k_B T^2 ∂ln Z/∂ T = -μ_B B Ntanh(μ_B B/k_B T), with Z = 2cosh(μ_B B/k_B T) the partition function for the Brillouin paramagnet <cit.> and N refers to the number of particles. Upon analyzing the behavior of the entropy S as a function of the external magnetic field at various temperatures for the Brillouin paramagnet, depicted in the lower panel of Fig. <ref>, we observe that there is an intrinsic entropy accumulation when B → 0 T. This is naturally expected, since for B = 0 T the entropy achieves its maximum value, namely S = k_Bln(2).
From the expression for the average magnetic energy, we can easily write T(E,B) as:

    T(E,B) = μ_B B/k_B (-E/μ_B B N),

and then plug it in the expression for the entropy (Eq. <ref>) obtaining thus S(E,B):

    S(E,B) = -Ek_B (E/μ_B B N)/B μ_B N + k_Bln2/√(1 - E^2/B^2 μ_B^2 N^2).

Note that when E = 0 J in Eq. <ref>, S = k_Bln(2) is nicely recovered, cf. upper panel of Fig. <ref>.
Then, employing Eq. <ref>, we compute separately the magnetic field and energy derivatives of S, namely (∂ S/∂ B)_T and (∂ S/∂ E)_B, respectively. It turns out that both derivatives are related to each other by the simple form:

    (∂ S/∂ B)_T = E/B(-∂ S/∂ E)_B .

Based on Eq. <ref> we discuss in the next the connection between (∂ S/∂ E)_B and (∂ S/∂ B)_T.
Employing the expression for the average magnetic energy, discussed previously, and upon analyzing Fig. <ref> (upper panel), it is clear that negative values of E correspond to positive values of T, since the employed values of B = 1 and 2 T  in our calculations are positive and constant making thus T the quantity that rules the sign of E. Hence, upon analyzing the results shown in the lower panel of Fig. <ref>, we can infer directly neither negative, infinite nor absolute zero temperature in the same way as upon analyzing the upper panel of Fig. <ref>, since a fixed positive value of T was employed. However, for finite values of B  when the limit T → ∞ is considered in the lower panel of Fig. <ref>, the entropy saturates (S = k_Bln2) and thus (∂ S/∂ B)_T → 0. In the same way, when T → 0 K, (∂ S/∂ B)_T → ∞. This is in agreement with both a diverging (zero temperature) and vanishing (infinite temperature) of (∂ S/∂ E)_B. However, strictly speaking (∂ S/∂ E)_B will never diverge since μ⃗ and B⃗ cannot be perfectly aligned, as discussed previously, and thus the achievement of absolute zero temperature is prevented. Yet, when the local magnetic field  B_loc = 0 T (blue asterisks and open circles in the lower panel of Fig. <ref>), upon approaching T = 0 K, (∂ S/∂ B)_T →∞ and thus absolute zero temperature can be indirectly inferred. However, when B_loc≠ 0 T (red and green circles in the lower panel of Fig. <ref>), for vanishing T and B, such an entropy accumulation [(∂ S/∂ B)_T → ∞] is suppressed and, as a consequence, the achievement of absolute zero temperature is also prevented. Evidently, such a divergence of the entropy for vanishing T and B would violate the third law of Thermodynamics <cit.>, strengthening thus our argument that a genuine zero-field QCP cannot take place, cf. discussions in the previous subsection. Thus, we have demonstrated in two different ways that the achievement of absolute zero temperature is not possible when considering finite B_loc.
Also, in the lower panel of Fig. <ref>, when B_loc = 0 T, the entropy for B = 0 T is exactly the same for both data set (blue asterisks and open circles), since for B = 0 T the resultant entropy S = k_B ln(2) is nicely recovered for any T ≠ 0 K <cit.>. This is not the case when B_loc is taken into account, being that for B = 0 T the entropy  depends on T <cit.>, as depicted in the lower panel of Fig. <ref>.



The entropy as a function of the average energy (upper panel of Fig. <ref>) can be easily obtained through Eq. <ref> employing the arbitrarily fixed values of B = 1 and 2 T. Figure <ref> depicts schematically the relevance of B_loc for T → 0 K and B → 0 T, since a ferromagnetic ground state only emerges due to the presence of finite mutual interactions in the system. Recalling that the absence of long-range magnetic ordering in low-dimensional systems at finite temperature  is a direct consequence of the Mermin-Wagner theorem <cit.>.   Interestingly enough, when considering the equilibrium spin populations in a two-level system, well known from classical textbooks <cit.>, namely:


    N_1/N_T = exp(μ B/k_B T)/exp(μ B/k_B T) + exp(-μ B/k_B T)

and

    N_2/N_T = exp(-μ B/k_B T)/exp(μ B/k_B T) + exp(-μ B/k_B T),


being N_1 and N_2 the spin populations regarding the lower and upper energy levels, respectively, and N_T the total number of spins. It turns out that for vanishing T and positive values of B, the ratio N_1/N_T becomes 1 and N_2/N_T is zero, which means that all spins point roughly along the same direction, i.e., a ferromagnetic ordering takes place, as schematically shown in Fig. <ref> for T = 0 K. In the following we make a link between Γ_mag, the definition of temperature and the Purcell and Pound's experiment <cit.>. Before starting the discussions, we stress that we are not proposing a new definition of temperature.
Recalling that the average magnetic energy is given by <cit.>:

    E = -Nμ_B Btanh(μ_B B/k_BT),

it is possible to rewrite E as a function of the average magnetic moment  ⟨μ⟩ = N μ_B tanh(μ_B B/ k_B T) along the external magnetic field,  as follows:

    E = -⟨μ⟩ B.

From the expression for ⟨μ⟩ it is possible to write the external magnetic field in respect to ⟨μ⟩:

    B(T, ⟨μ⟩) = k_B T/μ_B(⟨μ⟩/Nμ_B).

Equation <ref> indicates that at a certain temperature T, the value of an external magnetic field B is associated with a spin configuration that corresponds to a specific average magnetic moment ⟨μ⟩. Recalling that the magneto-caloric effect can be quantified by Γ_mag <cit.>:

    Γ_mag=1/T(∂ T/∂ B)_S,

we can compute the temperature derivative of B in Eq. <ref> and replace it into Eq. <ref>, resulting:

    Γ_mag = μ_B/Tk_B(⟨μ⟩/Nμ_B).

Since Γ_mag = 1/B for the Brillouin paramagnet <cit.>, we write T as a function of B and ⟨μ⟩:

    T = B μ_B/k_B (⟨μ⟩/Nμ_B).

Equation <ref> connects in an unprecedent way the Purcell and Pound experiment <cit.> and Γ_mag itself. When B and ⟨μ⟩ are ∥, positive temperatures are inferred. However, when B and ⟨μ⟩ are anti-∥, then negative temperatures can be associated.

It is remarkable that the definition of temperature is encoded in Γ_mag and vice-versa. Next, we focus on the scaling analysis of the magnetization. When investigating the behavior of Γ_mag, an important consideration is its scaling behavior, see, e.g. Ref. <cit.> and references cited therein.
Upon analyzing the scaling behavior for the Brillouin paramagnet, shown in Fig. <ref>, we observe that all data collapse in the same line only for external magnetic field values close to B_loc. By considering much higher or lower values of external magnetic field when compared with B_loc, a breakdown of the scaling behavior of the magnetization is observed. As a matter of fact, the scaling analysis reflects the behavior of Γ_mag as a function of T, so that the maxima depicted in Fig. <ref> are associated with the Schottky anomaly, which in turn is only observed in the presence of mutual interactions.
In other words, due to the intrinsic mutual interactions present into the system, the validity of the scaling in the Brillouin paramagnet is limited only to external magnetic field values close to B_loc. Thus, our scaling analysis also demonstrates the relevance of B_loc in the limit of both B → 0 T and T → 0 K. In Fig. S3 of Ref. <cit.> the authors show a scaling behavior of β-YbAlB_4 upon analyzing the critical contribution of the magnetization M_c. Interestingly enough, there is a breakdown of the scaling behavior for applied magnetic fields higher than 0.5 T, which resembles the scaling behavior of the Brillouin paramagnet discussed here, and the scaling is only valid for low values of B for β-YbAlB_4.
At this point, it is worth recalling that Γ_mag can also be written as follows <cit.>:

    Γ_mag = - (dM/dT)_B/c_B,

where c_B is the specific heat at constant external magnetic field.
Upon employing Eq. <ref>, it can be directly inferred that when (- dM/dT)_B →∞⇒Γ_mag→∞ if c_B is non-singular. Thus, we make use of Eq. <ref> in order to infer the divergence of Γ_mag for the scaling analysis of the Brillouin paramagnet, as shown in Fig. <ref>. It is worth mentioning that since Γ_mag for the Brillouin paramagnet is temperature independent <cit.>, we thus make use of the magnetization scaling in our analysis. Interestingly, also for the one-dimensional Ising model under transverse magnetic field  Γ_mag also independs on T <cit.>.
We observe from Fig. <ref> that when B_loc = 0 T,  for arbitrarily chosen B = 0.001 T,   the scaling consists in a straight line. Since a logarithm scale was used in both axes of Fig. <ref>, the straight line indicates a divergence of Γ_mag for  B → 0 T and T → 0 K. However, when B_loc = 0.01 T the divergence of Γ_mag is suppressed and such linear behavior is no longer observed, opening the way for the appearance of a Schottky-like maximum. Hence, the scaling plots depicted in Fig. <ref> demonstrate in another way the non-divergence of Γ_mag when B_loc≠ 0 T for the Brillouin paramagnet.
Also, the maximum in Γ_mag shown in both panels of Fig. <ref> when B_loc≠ 0 T  is due to the Schottky anomaly <cit.>, as previously mentioned. As well-known from textbooks, such maximum takes place when (k_B T)/(μ_B B) ≃ 0.834 <cit.>. Upon continuously decreasing the temperature of the system, the spins occupy preferably the lower energy level and, as a consequence, all spins will occupy the same energy level in the ground state. This can be visualized in the spins scheme depicted in Fig. <ref>, and it is in line with our analysis for the spin populations, cf. Eqs. <ref> and <ref>. The Schottky anomaly can also be captured in the scaling analysis for the Brillouin model (Fig. <ref>) where a maximum takes place at T/B_r ≈ 0.5602. Now, we use again the definition of the magneto-caloric effect (Eq. <ref>) <cit.>. Knowing that Γ_mag = 1/B for the Brillouin paramagnet <cit.>, it is straightforward to write T/B = (∂ T/∂ B)_S. Hence, this simple analysis indicates that temperature and magnetic field are interconnected. Indeed, in the adiabatic demagnetization the temperature is decreased upon removing the external applied magnetic field, in order to keep the ratio (μ_B B_r)/(k_B T) constant.



iii) Possibility of performing adiabatic magnetization by only manipulating the mutual interactions.

Our proposal is distinct from the adiabatic demagnetization method itself, since no external magnetic field is required.
As pointed out in Ref. <cit.>, although counter-intuitive it is possible to increase the temperature adiabatically. The idea behind is based on the connection between the uncertainty principle and the entropy. The entropy of the system can be written as follows <cit.>:

    S = -k_B ∑_j P(Ψ_j)ln(Ψ_j),

where P(Ψ_j) is the probability of the system to be at the energy Eigenstate Ψ_j and j is the label of the corresponding Eigenstate.

It is clear that the entropy increases with the uncertainty associated with the particle according with Eq. <ref> and thus, in order to hold the entropy constant, the uncertainty must also be held constant.
In an adiabatic expansion, the increase of the volume causes an enhancement in the spatial uncertainty <cit.>. However, in this process, the gas particles that do work will lose energy and then they will have a decrease in their linear momentum, i.e., the momentum uncertainty is lowered as well. Thus, there is a compensation of such uncertainties and, as a consequence, the entropy remains constant during this process <cit.>. Following similar arguments, adiabatic magnetization of the mutual interactions can be achieved. As depicted in Fig. <ref>, upon increasing the temperature adiabatically, a spontaneous magnetization takes place as a direct consequence of the constrain of holding the entropy constant. Since there is no external applied magnetic field we have B_r = B_loc and, in order to hold S (Eq. <ref>) constant, the magnetic energy should also be changed in this process, so that adiabatic magnetization is achieved. This can also be easily understood in terms of the uncertainty principle.  It is straightforward to calculate the uncertainty Δ E = U_mag - E of the magnetic energy, which reads:
 
    Δ E = μ_B B_loc[N tanh(μ_B B_loc/k_B T)-cosϕ].

Strictly speaking, the treatment of the mutual interactions would require a many body approach.
In fact, in a many-body picture the mutual interactions can be described by the Hamiltonian <cit.>:

    H=-J'∑_(ij)S⃗_i^z_i·S⃗_j^z_j+Dr_nn^3 ∑_i>jS⃗_i^z_i·S⃗_j^z_j/|r⃗_ij|^3-3(S⃗_⃗i⃗^z_i·r⃗_ij)(S⃗_⃗j⃗^z_j·r⃗_ij)/|r⃗_ij|^5,

where S⃗ is the spin vector oriented along the local z_i Ising <111> axis, i and j refer to the two sites of the lattice, r⃗ is the position vector, D=(μ_0μ^2)/(4π r_nn^3), and r_nn is the distance between nearest-neighbor spins. The second term of the Hamiltonian (Eq. <ref>) embodies the magnetic energy associated with the interaction between a single magnetic moment and its nearest neighbors, since the dipolar interaction is short range. Hence, in our analysis of Eq. <ref> we consider that the local magnetic field perceived by a certain magnetic moment is associated with the resultant magnetic field generated by the various neighboring magnetic moments in its immediate surrounding, cf. Hamiltonian <ref>.
Upon analyzing Eq. <ref>, it is clear that Δ E is minimized when cosϕ→ 1. Such a condition would indicate a maximum alignment of the magnetic moments because of the local magnetic field. Thus, in order to  increase the temperature of the system adiabatically, a quasi-static process is required, i.e., the temporal uncertainty Δ t should be maximized while Δ E is minimized. However, still considering Eq. <ref>, for B_loc = 0 ⇒Δ E = 0, indicating that in the absence of mutual interactions the uncertainty principle would be violated. In the same way,  the absence of the mutual interactions would imply in the violation of the third law of Thermodynamics, as pointed out previously <cit.>.  The only way to reduce the magnetic energy adiabatically is, however, varying the total angular momentum projection. Such a behavior is naturally expected in the adiabatic magnetization, since cosϕ is increased. At this point, we recall the magnetic energy, namely U_mag=-m_J/√(J(J+1))μ B_loc. In the adiabatic magnetization process, the projection of J⃗ on the z-axis is increased, while its projections on the x and y-axes are reduced and, as a consequence, the magnetic energy decreases, being thus the magnetization of the system increased, since m_J = -J, -J+1, ..., J-1, J.
This is the principle of work behind the adiabatic magnetization of the mutual interactions here proposed. The magnetic field increment  Δ B_loc  of the mutual interaction that emerges in the adiabatic magnetization is associated with the increased projection of J⃗ along the z-axis. In general terms,
in the adiabatic magnetization process (Fig. <ref>) of the mutual interactions, the temperature is adiabatically increased from T_1 to T_2 and, in order to do so, the magnetic energy needs to compensate the temperature variation in order to hold the entropy constant. Thus, we can write:

    μ_B B_loc/k_B T_1 = μ_B √(B_loc^2 + Δ B_loc^2)/k_B T_2,

where Δ B_loc is the magnetic field increment that will emerge into the system to compensate the adiabatic increase of temperature. It is clear that such Δ B_loc refers to the adiabatic magnetization employing only the mutual interactions of the system. Then, Δ B_loc can be easily determined by:

    Δ B_loc = B_loc√(T_2^2/T_1^2 - 1).

It is worth emphasizing that such adiabatic magnetization process can only be performed in the temperature range where the magnetic energy associated with the mutual interactions are not overcome by the thermal fluctuations. In the case of real paramagnets (B_loc≃  0.01 T), as discussed previously, the typical temperature onset of the mutual interactions relevance is T ≤ 6 mK. After magnetizing the system adiabatically, if the temperature is further increased such magnetization will be released when the thermal energy equals the corresponding magnetic energy associated with Δ B_loc. Thus, the entropy is recovered to its original configuration, i.e., Δ B_loc = 0 T, making this process a single shot closed-cycle, which can be restarted. Although we have provided a particular numerical example in Fig. <ref>, namely T_1 (T_2) = 2.0 (2.1) mK, such process can be carried out  between any T_1 and T_2 within the temperature range of relevance of B_loc. The fascinating aspect behind this process is that no external magnetic field is required in order to perform the adiabatic magnetization, being only the mutual interactions of the system employed.



iv) On the role of the mutual interactions in other physical systems.

An analogous situation to the mutual interactions in paramagnets can also be found for the Bose-Einstein condensation (BEC). Strictly speaking, BEC should occur at T = 0 K, but due to the presence of finite interactions between particles BEC takes place at T ≠ 0 K <cit.>, being the energy of the system equivalent to the chemical potential itself and all particles occupy the same energy level. The fact that interactions between particles prevent an ideal BEC corresponds to a similar physical situation of how the mutual interactions in insulating paramagnets prevent the existence of ideal paramagnets. Also, in the frame of BEC in magnetic insulators <cit.>, where dimers formed by pairs of spin S = 1/2 with a ground state S = 0 (spin-singlet) and S = 1 (triplet) bosonic excitations, the interdimer interaction plays the role analogous to the mutual interaction
in insulating paramagnets. Indeed, the interdimer interaction is the key ingredient for the stabilization of a long-range magnetic ordering <cit.>. Yet, in dipolar spin-ice systems <cit.> the dipole-dipole interactions are relevant, so that we consider an “effective nearest-neighbor energy scale” composed by two distinct contributions, namely nearest-neighbor exchange energy between <111> Ising magnetic moments and a nearest neighbor dipolar energy scale <cit.>. The “spin-spin” interactions in such systems can be nicely tuned via doping, which in turn enables to change the distance between magnetic moments <cit.>. For spins arranged in a pyrochlore structure of corner-sharing tetrahedra spin-ice, we have a residual entropy originating from the degeneracy of the “disordered” ground-state.



§ CONCLUSIONS

Our findings suggest the absence of zero-field quantum criticality in the presence of mutual interactions. Interactions will eventually, as T → 0 K and B → 0 T, break down the spin gas scheme, so spontaneous magnetization will emerge at some very low-temperature with B = 0 T.
We have shown that the magnetic Grüneisen parameter for any paramagnet with mutual interactions will never diverge as T → 0 K and B → 0 T. As a consequence, zero-field quantum criticality cannot be inferred. As a matter of fact, we are faced to a many body interactions problem, cf. Hamiltonian <ref>.
Our analysis, based on the magnetic Grüneisen parameter and entropy arguments, although validated by the results obtained for the textbook Brillouin-paramagnet and for the proposed zero-field quantum critical paramagnet β-YbAlB_4, is universal and can be applied to any system with mutual interactions. Also, we have discussed the impossibility of achieving absolute zero temperature due to the presence of finite mutual interactions in real paramagnets regarding the connection between Γ_mag and the canonical definition of temperature. Given the recent advances in the investigation of model systems employing cold atoms <cit.>, our findings suggest that the mutual interactions can be mimicked by tuning the Hubbard on-site Coulomb repulsion U, as reported in Ref.<cit.>. Yet, we have proposed the concept of adiabatic magnetization employing solely the mutual interactions. We anticipate that the proof of concept regarding adiabatic magnetization of the mutual interactions can be achieved, with some technical efforts, in any laboratory where temperatures of a few mK are attainable precisely. The role of the mutual interactions in other physical systems like Bose-Einstein condensates, Bose-Einstein condensation of magnons and spin-ice were also discussed.



§ METHODS



Softwares.
All the calculations presented in this work were performed employing the software Wolfram Mathematica® Version 11.  Figures <ref> and <ref> were created from scratch using the software Adobe Illustrator® Version CC 2017. Figures <ref>, <ref>, <ref>,  <ref>, and <ref> were plotted using the software OriginPro® Version 2018 based on data set generated employing the software Wolfram Mathematica® Version 11. Figure <ref> was plotted using the 3D plot function of the software Wolfram Mathematica® Version 11. In Figs. <ref>, <ref> (lower panel) and <ref>, T = 5 mK was chosen arbitrarily due to the numerical impossibility of computing S and Γ_mag exactly at T = 0 K. The same holds true for T = 1 mK in Fig. <ref>.



Derivation of the observables considering B_loc in the Brillouin paramagnet.
The partition function Z for the Brillouin paramagnet <cit.> reads:

    Z(B,T) = ∑_m_J = -J^J(μ_B g m_J B/k_B T),

converging to the expression:

    Z(B,T) = sinh[(1+1/2J)μ_B J g B/k_B T]/sinh(μ_B J g B/k_B T).

Considering J = 1/2 and g = 2 we have:

    Z(B,T) = sinh(2μ_B B/k_B T)/sinh(μ_B B/k_B T),

which in turn is simplified as:

    Z(B,T) = 2cosh(μ_B B/k_B T).

Considering the existence of a local magnetic field B_loc which can  vectorially be added to the external magnetic field B, the resultant magnetic field is B_r = √(B^2 + B_loc^2), cf. previous discussions. Replacing B by B_r in Eq. <ref>, the partition function is now given by:

    Z(B,T) = 2cosh(μ_B √(B^2+B_loc^2)/k_B T).

Thus, the corresponding Helmholtz free energy reads:

    F(B,T) = -k_B Tln[2cosh(μ_B √(B^2+B_loc^2)/k_B T)].

From Eq. <ref>, the entropy S = -(∂ F/ ∂ T)_B is:

    S(B,T) = -√(B^2+B_loc^2)μ_B/Ttanh(√(B^2+B_loc^2)μ_B/k_B T) + k_B[2cosh(√(B^2+B_loc^2)μ_B/k_B T)].

Equation <ref> is exactly the same as Eq. <ref> when B is replaced by B_r. We thus have shown that only replacing B as B_r in the expressions for the Brillouin paramagnet is consistent, since computing the observables from the partition function considering B_r instead of B provides the same results.
It is worth mentioning that we have performed all the magnetic field derivatives of the entropy with respect to B instead of B_r, since B_loc was considered constant in our analysis. If we compute the magnetic field derivatives of S with respect to B_r instead of B, there would be a difference between such expressions by a factor of B/√(B^2 + B_loc^2), which is 1 when B_loc = 0. Interestingly enough, the calculation of Γ_mag is be affected by this aspect. Considering (∂ S/ ∂ B)_T in Eq. <ref>, Γ_mag reads:

    Γ_mag = B/(B^2 + B_loc^2),

which becomes Γ_mag = 1/B when B_loc = 0.
Yet, when (∂ S/ ∂ B_r)_T is considered in Eq. <ref>, Γ_mag is given by:

    Γ_mag = 1/√(B^2 + B_loc^2),

which is also Γ_mag = 1/B when B_loc = 0 <cit.>. The very same argument can be used when dealing with the calculations of Γ_mag for β-YbAlB_4 <cit.> taking B_loc into account.

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§ ACKNOWLEDGEMENTS

M. de S. acknowledges financial support from the São Paulo Research Foundation – Fapesp (Grants No. 2011/22050-4 and 2017/07845-7), National Council of Technological and Scientific Development – CNPq (Grants No. 302498/2017-6), and T.U.V.S.O.T.E. ACS acknowledges CNPq (Grant No. 305668/2018-8). This work was partially granted by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (Capes) - Finance Code 001 (Ph.D. fellowship of L.S. and I.F.M.). The Boston University Center for Polymer Studies is supported by NSF
Grants PHY-1505000, CMMI-1125290, and CHE-1213217, and by DTRA Grant
HDTRA1-14-1-0017. A.C.S. acknowledges financial support from the National
Council of Technological and Scientific Development – CNPq
(Grant No. 305668/2018-8).



§ AUTHOR CONTRIBUTIONS

L.S., G.O.G., and I.F.M. carried out the calculations and generated the figures. L.S. and M. de S. wrote the paper with contributions from A.C.S., R.E.L., G.O.G., and I.F.M. All authors revised the manuscript. M. de S. conceived and supervised the project.



§ ADDITIONAL INFORMATION

The authors declare no competing interests.




