Unveiling the Physics of the Mutual Interactions in Paramagnets

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 (Bloc). 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 (Γmag) 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 Bloc; ii) connection between the canonical definition of temperature and Γmag; and iii) possibility of performing adiabatic magnetization by only manipulating the mutual interactions. Our findings unveil unprecedented aspects emerging from the mutual interactions.

Magnetic excitations in solids have been broadly investigated in the past decades, being crucial to the understanding of exotic physical phenomena such as superconductivity 1 and magnetic field induced quantum phase transitions 2 , 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 3,4 and champ moléculaire 5,6 , have been broadly mentioned in the literature [7][8][9][10][11][12][13][14][15][16][17] , 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 [18][19][20][21] , 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 22,23 , as well as for a finite temperature (T) pressure-induced critical point [24][25][26] , 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 27 the analogous physical quantity to the Grüneisen ratio is the so-called magnetic Grüneisen parameter, hereafter Γ mag 22,28,29 . 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 22,23 , 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 30,31 , which in turn enables to change the temperature of a system upon varying adiabatically the external applied magnetic field 22,28,29 . The fingerprints of a genuine magnetic field-induced QCP, besides the gradual suppression of an order parameter (or energy scale) at the QCP [ Fig. 1a)], 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 where B is the external applied magnetic field and  the scaling exponent 23,32 . Particular attention has been paid to the so-called zero-field quantum criticality [ Fig. 1b)], 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 2 Ge 4 33 , β-YbAlB 4 34 , and Au-Al-Yb 35 . In the case of β-YbAlB 4 , in particular, the understanding of possible zero-field quantum criticality 34,36,37 and the emergence of superconductivity at = T 80 temperatures, cf. Fig. 2(A) of ref. 34 . Because of such exotic behavior the system is considered as a "strange metal" 23 .
Based on a scaling analysis of the magnetization, the authors of ref. 34 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 29 . 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 42 , 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 43 , when adiabatic magnetization was employed to produce cooling using paramagnetic salts with ∂ ∂ > S B ( / ) 0 T , 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 29 . 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 11 . 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. 1b)]. In quantitative terms, considering that in an ideal paramagnet neighboring magnetic moments μ → are separated by a distance r 44 , the magnetic dipolar energy is roughly given by μ μ π T·m/A is the vacuum magnetic permeability. Since classically the magnetic energy is given by , being φ the angle between μ → and → B loc , the local dipolar magnetic field is roughly given by . Hence, for r = 5 Å, 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 0 01 loc T can be estimated 11 . 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 45 . 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 .
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 15 : Hypothetical zero-field quantum critical point (yellow bullet). The yellow gradient shadow represents the role played by the mutual interactions between neighboring magnetic moments, which are responsible for the emergence of an effective local magnetic field B loc . We consider that B loc makes an angle θ  90° with B, so that the resultant magnetic field (inset) is given by r l oc 2 2 (cyan arrow). The dotted line represents the transition from quantum paramagnet to a magnetic ordered phase. Figure created with use of Adobe Illustrator (version CC 2017) 62 . Details in the main text. 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 15 . The first term of this simple Hamiltonian (Eq. 1) 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. 1) show a quite similar behavior, being that in this case for B = 0 T the magnetic energy is = U A mag . Analogously, if we consider two neighboring magnetic moments, we can treat the mutual interactions between them employing the Hamiltonian (Eq. 1), 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 μ = − → ⋅ → U B mag l oc and μ = + → ⋅ → U B mag l oc 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 μ → . The modulus of the total angular momentum vector → J is given in terms of J, J describes a particular orientation of the magnetic moment and its respective magnetic energy, since μ μ = g m B J , where g is the gyromagnetic factor 46 . Here it is worth emphasizing that in our analysis we consider μ → oriented along the → z direction. In the frame of Quantum Mechanics, the magnetic energy is written as 46 . 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 ±μB loc as expected classically, and thus we can infer that φ will never be exactly zero or π 46 . In fact, the absence of a perfect alignment between μ → and → B loc can be observed in the results shown in Fig. 2 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 34 , using the effective Yb magnetic moment μ .  1 94 μ B and the Yb-Yb separation of .  r 3 5 Å 47 , results in B loc  0.04 T. Even considering the fact that the system undergoes a superconducting transition at T c = 80 mK 38,39 , intrinsic magnetic moments survive in β-YbAlB 4 at very low-T 34 , so that the Yb valence fluctuations 48 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 49 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
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 Brillouinlike model 11 , 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: : T refers to a characteristic temperature and g = 1.94 34 . Using Eq. 3, it is straightforward to calculate the quantum critical entropy, Considering that B loc makes an angle θ° 90 with B, we can write the resultant magnetic field 11 , as depicted in the inset of Fig. 1b). It turns out that when using B r directly in the partition function for the Brillouin paramagnet 45 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. 4) 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 Fig. 2, 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 11 . Hence, the magnetic entropy is released and the third law of Thermodynamics is obeyed 14 . 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 22 : 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. 2 and 4, so that: mag loc 2 2 for the Brillouin paramagnet 29 , while for β-YbAlB 4 , Γ mag reads: The results of Γ mag for the Brillouin paramagnet and β-YbAlB 4 (Eqs. 6 and 7, respectively) are shown in Figs Fig. 4). Hence, the consequence of taking ≠ B 0 loc 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 Scientific RepoRtS | (2020) 10:7981 | https://doi.org/10.1038/s41598-020-64632-x www.nature.com/scientificreports www.nature.com/scientificreports/ 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. 6. The situation is quite different, for instance, for the one-dimensional Ising model under transverse magnetic field 51 , 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 51 . When analyzing Eqs. 6 and 7, respectively, for the Brillouin paramagnet and β-YbAlB 4 , considering Γ mag ≠ 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 loc B B 2 2 2 2 2 1/2 , i.e., when T = 0 K the maximum is located at B = B loc , cf. Fig. 3. The maxima were obtained by the simple optimization of the Γ mag functions (Eqs. 6 and 7), i.e., making Γ ∂ ∂ ( ) B / mag 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 12,14 , to be discussed into more details in the following. For β-YbAlB 4 , under the condition 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 , being M the magnetization, it is tempting to say that the results of Fig. 2(A) of ref. 34 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. 34 , 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 23 . In ref. 52 , the authors discuss that the free energy scaling for the systems CeCu 6−x Au 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. 34 , see Eq. 3. 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 29,53 , 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 29 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 29 . Hence, considering the similar role played by J′ and B loc , it becomes evident that Γ mag only diverges for vanishing www.nature.com/scientificreports www.nature.com/scientificreports/ 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 5 , 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 3 . Hence, in the same way as the Curie temperature 12 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.
Connection between the canonical definition of temperature and Γ mag . Considering the definition of temperature 1/T = ∂ ∂ S E ( / ) B 11,14 , absolute zero temperature can be inferred when ∂ ∂ → ∞ S E ( / ) B (upper panel of Fig. 5), where E refers to the average magnetic energy, being the partition function for the Brillouin paramagnet 11 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. 5, 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 ln (2) B . From the expression for the average magnetic energy, we can easily write T E B ( , ) as:  , respectively. It turns out that both derivatives are related to each other by the simple form: . 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 μ → and → B 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. 5), upon approaching T = 0 K, ∂ ∂ → ∞ S B ( / ) T and thus absolute zero temperature can be indirectly inferred. However, when ≠ B 0 loc T (red and green circles in the lower panel of Fig. 5

), 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 4 , 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. 5, 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 ln(2) B is nicely recovered for any ≠ T 0 K 4 . This is not the case when B loc is taken into account, being that for B = 0 T the entropy depends on T 4 , as depicted in the lower panel of Fig. 5.
The entropy as a function of the average energy (upper panel of Fig. 5) can be easily obtained through Eq. 9 employing the arbitrarily fixed values of B = 1 and 2 T. Figure 6 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 42 . Interestingly enough, when considering the equilibrium spin populations in a two-level system, well known from classical textbooks 3,4,11 , namely: 2 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. 6 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 50 . 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 11 : B B B Figure 6. The ground-state of a paramagnet. Schematic representation of the intrinsic magnetic moments (black arrows) of the electrons (red spheres), separated by a distance r, for the Brillouin paramagnet. Upon lowering the temperature T 1 > T 2 > T = 0 K in the absence of external magnetic field, all magnetic moments are naturally aligned along the same direction and a magnetically ordered phase takes place, since an effective local magnetic field B loc is intrinsically present due to the mutual interactions between adjacent magnetic moments 3,4,11 . Note the lack of perfect alignment of the magnetic moments in respect to B loc , cf. discussion in the main text. Figure  along the external magnetic field, as follows: 14) From the expression for μ it is possible to write the external magnetic field in respect to μ : 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  50 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. 23 and references cited therein. Upon analyzing the scaling behavior for the Brillouin paramagnet, shown in Fig. 7, 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. 7 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. 34 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 22 : where c B is the specific heat at constant external magnetic field. Upon employing Eq. 19, it can be directly inferred that when if c B is non-singular. Thus, we make use of Eq. 19 in order to infer the divergence of Γ mag for the scaling analysis of the Brillouin paramagnet, as shown in Fig. 7. It is worth mentioning that since Γ mag for the Brillouin paramagnet is temperature independent 29 , 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 51 . We observe from Fig. 7 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. 7, 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. 7 demonstrate in another way the non-divergence of Γ mag when ≠ B 0 loc T for the Brillouin paramagnet. Also, the maximum in Γ mag shown in both panels of Fig. 3 when ≠ B 0 loc T is due to the Schottky anomaly 12,14 , as previously mentioned. As well-known from textbooks, such maximum takes place 12,13 . 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. 6, and it is in line with our analysis for the spin populations, cf. Eqs. 11 and 12. The Schottky anomaly can also be captured in the scaling analysis for the Brillouin model (Fig. 7) where a maximum takes place at T/B r ≈ 0.5602. Now, we use again the definition of the magneto-caloric effect (Eq. 16) 22 . Knowing that Γ mag = 1/B for the Brillouin paramagnet 29  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. 11 , 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 11 : 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. 20 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 11 . 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 11 . Following similar arguments, adiabatic magnetization of the mutual interactions can be achieved. As depicted in Fig. 8, 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 B r l oc and, in order to hold S (Eq. 2) 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 E mag of the magnetic energy, which reads: 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 54,55 : , and r nn is the distance between nearest-neighbor spins. The second term of the Hamiltonian (Eq. 22) 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. 21 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 22. Upon analyzing Eq. 21, it is clear that ΔE is minimized when φ → cos 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 Δt should be maximized while ΔE is minimized. However, still considering Eq. 21, for B loc = 0 ⇒ ΔE = 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 4 . 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 φ cos is increased. At this point, we recall the magnetic energy, namely μ = − + Figure 8. Adiabatic demagnetization of the mutual interactions. Main panel: entropy as a function of temperature for = B 0 T (solid cyan curve) and for ΔB loc = 0.0032 T (red dashed line). The yellow gradient indicates the region of relevance of B loc in the regime of ultra-low temperatures. Inset: schematic representation of the adiabatic magnetization employing the mutual interactions. The steps are as follows: i) the system is cooled down to a sufficiently low temperature where the magnetic energy associated with B loc overcomes the thermal fluctuations (path A to B); ii) the temperature is adiabatically increased from B (T 1 ) to C (T 2 ) and, in order to hold the entropy constant, the system is magnetized adiabatically, i.e., a finite value of ΔB loc emerges into the system and thus, the entropy changes from the solid cyan to the fictitious red dashed line, which emulates S versus T for the corresponding value of (B loc + ΔB loc ); iii) upon increasing the temperature from C, the entropy will vary and, as a consequence, the system will be demagnetized at T = 4.35 mK (indicated by the solid black arrow), going back to its original configuration of ΔB loc = 0 T. Such process can be restarted by cooling the system again from T = 4.35 mK to point A. Figure created with use of OriginPro (version 2020) 63 and Wolfram Mathematica (version 11) 64 . (2020) 10:7981 | https://doi.org/10.1038/s41598-020-64632-x www.nature.com/scientificreports www.nature.com/scientificreports/ magnetization process (Fig. 8) 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: where ΔB loc is the magnetic field increment that will emerge into the system to compensate the adiabatic increase of temperature. It is clear that such ΔB loc refers to the adiabatic magnetization employing only the mutual interactions of the system. Then, ΔB loc can be easily determined by: 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 0 01 loc 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 ΔB loc . Thus, the entropy is recovered to its original configuration, i.e., ΔB 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. 8, 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.
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 56 , 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 57,58 , 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 56 . Yet, in dipolar spin-ice systems 55 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 55 . The "spin-spin" interactions in such systems can be nicely tuned via doping, which in turn enables to change the distance between magnetic moments 59 . 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 22. 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 60 , our findings suggest that the mutual interactions can be mimicked by tuning the Hubbard on-site Coulomb repulsion U, as reported in ref. 61 . 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 1 and 6 were created from scratch using the software Adobe Illustrator Version CC 2017. Figures 2, 3, 5, 7 and 8 were plotted using the software OriginPro Version 2020 based on data set generated employing the software Wolfram Mathematica Version 11. Figure 4 was plotted using the 3D plot function (2020) 10:7981 | https://doi.org/10.1038/s41598-020-64632-x www.nature.com/scientificreports www.nature.com/scientificreports/ of the software Wolfram Mathematica Version 11. In Figs. 2, 3 (lower panel) and 5, = 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. 5.
Derivation of the observables considering B loc in the Brillouin paramagnet. The partition function Z for the Brillouin paramagnet 12 reads:  Equation 31 is exactly the same as Eq. 2 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 B loc 2 2 , which is 1 when = B 0 loc . Interestingly enough, the calculation of Γ mag is be affected by this aspect. Considering ∂ ∂ S B ( / ) T in Eq. 5, Γ mag reads: which is also Γ mag = 1/B when = B 0 loc 29 . The very same argument can be used when dealing with the calculations of Γ mag for β-YbAlB 4 34 taking B loc into account.