Discontinuity In Specific Heat Research Articles

Heat capacity double transitions in time-reversal symmetry broken superconductors

Standard superconductors display a ubiquitous discontinuous jump in the electronic specific heat at the critical superconducting transition temperature. In a growing class of unconventional superconductors, however, a second order parameter component may get stabilized and produce a second heat capacity jump at a lower temperature, typically associated with the spontaneous breaking of time-reversal symmetry. The splitting of the two specific heat discontinuities can be controlled by external perturbations such as chemical substitution, hydrostatic pressure, or uniaxial strain. We develop a theoretical quantitative multi-band framework to determine the ratio of the heat capacity jumps, given the band structure and the order parameter momentum structure. We discuss the conditions of the gap profile which determine the amplitude of the second jump. We apply our formalism to the case of Sr$_2$RuO$_4$, and using the gap functions from a microscopic random phase approximation calculation, we show that recently-proposed accidentally degenerate order parameters may exhibit a strongly suppressed second heat capacity jump. We discuss the origin of this result and consider also the role of spatial inhomogeneity on the specific heat. Our results provide a possible explanation of why a second heat capacity jump has so far evaded experimental detection in Sr$_2$RuO$_4$.

Thermal properties of polymer with intrinsic microporosity membranes

The thermal behavior of a polymer with intrinsic microporosity (PIM-1) membrane was investigated via thermomechanical analysis and differential scanning calorimetry. The coefficient of thermal expansion (CTE) was almost constant at 70*106 K−1 below 110 °C and inflection points were observed over the range 120–150 °C, regardless of the fabrication conditions. The CTE after the inflection point strongly depended on the membrane formation conditions and the thermal history. Although the CTE increased with a methanol treatment, the CTE of other membranes decreased because of shrinkage attributed to internal stress. This behavior indicated cooperative molecular motion below 200 °C. In the calorimetry measurements, a specific-heat discontinuity from the glass transition was not observed. However, an extremely broad endothermic peak, which may have been caused by enthalpy relaxation, was observed for a membrane that exhibited advanced physical aging.

Noncommutative inspired black holes in regularized 4D Einstein–Gauss–Bonnet theory

Abstract Low energy limits of string theory indicated that the standard gravity action should be modified to include higher-order curvature terms, in the form of dimensionally continued Gauss–Bonnet densities. If one includes only quadratic curvature terms then the resulting theory is Einstein–Gauss–Bonnet (EGB) gravity valid only in D > 4 dimensions. Recently there has been a surge of interest in regularizing, a D → 4 limit, of EGB gravity, and the resulting regularized 4 D EGB gravities have nontrivial gravitational dynamics. We obtain a static spherically symmetric noncommutative (NC) geometry inspired black hole solution with Gaussian mass distribution as a source in the regularized 4 D EGB and also analyze their properties. The metric of the NC inspired 4 D EGB black hole smoothly interpolates between a de Sitter core around the origin and 4 D EGB metric as r ∕ θ → ∞ . Owing to the NC and GB term corrected black hole, the thermodynamic quantities have also been altered. The phase transitions for the local thermodynamic stability, in the theory, is outlined by a discontinuity of specific heat C + at a critical radius r + = r C , and C + changes from infinitely negative to infinitely positive and then down to a finite positive for the smaller r + . The thermal evaporation process leads to a thermodynamic stable extremal black hole with vanishing temperature.

Thermodynamics of Eliashberg theory in the weak-coupling limit

The weak-coupling limits of the gap and critical temperature computed within Eliashberg theory surprisingly deviate from the BCS theory predictions by a factor of $1/\sqrt{e}$. Interestingly, however, the ratio of these two quantities agrees for both theories. Motivated by this result, here we investigate the weak-coupling thermodynamics of Eliashberg theory, with a central focus on the free energy, specific heat, and the critical magnetic field. In particular, we numerically calculate the difference between the superconducting and normal-state specific heats, and we find that this quantity differs from its BCS counterpart by a factor of $1/\sqrt{e}$, for all temperatures below $T_{c}$. We find that the dimensionless ratio of the specific-heat discontinuity to the normal-state specific heat reduces to the BCS prediction given by $\Delta C_{V}(T_{c})/C_{V,n}(T_c)\approx1.43$. This gives further evidence to the expectation that all dimensionless ratios tend to their "universal values" in the weak-coupling limit.

Crossover from multiple- to single-gap superconductivity in Nb5Ir3−xPtxO alloys

By using mostly the muon-spin rotation/relaxation ($\mu$SR) technique, we investigate the superconductivity (SC) of Nb$_5$Ir$_{3-x}$Pt$_x$O ($x = 0$ and 1.6) alloys, with $T_c = 10.5$ K and 9.1 K, respectively. At a macroscopic level, their superconductivity was studied by electrical resistivity, magnetization, and specific-heat measurements. In both compounds, the electronic specific heat and the low-temperature superfluid density data suggest a nodeless SC. The superconducting gap value and the specific heat discontinuity at $T_c$ are larger than that expected from the Bardeen-Cooper-Schrieffer theory in the weak-coupling regime, indicating strong-coupling superconductivity in the Nb$_5$Ir$_{3-x}$Pt$_x$O family. In Nb$_5$Ir$_3$O, multigap SC is evidenced by the field dependence of the electronic specific heat coefficient and the superconducting Gaussian relaxation rate, as well as by the temperature dependence of the upper critical field. Pt substitution suppresses one of the gaps, and Nb$_5$Ir$_{1.4}$Pt$_{1.6}$O becomes a single-gap superconductor. By combining our extensive experimental results, we provide evidence for a multiple- to single-gap SC crossover in the Nb$_5$Ir$_{3-x}$Pt$_x$O family.

Broken time-reversal symmetry in superconducting Pr1−xLaxPt4Ge12

The superconducting state of the filled skutterudite alloy series Pr$_{1-x}$La$_{x}$Pt$_{4}$Ge$_{12}$ has been systematically studied by specific heat, zero-field muon spin relaxation ($\mu$SR), and superconducting critical field measurements. An additional inhomogeneous local magnetic field, indicative of broken time-reversal symmetry (TRS), is observed in the superconducting states of the alloys. For $x \lesssim 0.5$ the broken-TRS phase sets in below a temperature $T_m$ distinctly lower than the superconducting transition temperature $T_c$. For $x \gtrsim 0.5$ $T_m \approx T_c$. The local field strength decreases as $x \to 1$, where LaPt$_{4}$Ge$_{12}$ is characterized by conventional pairing. The lower critical field $H_{c1}(T)$ of PrPt$_{4}$Ge$_{12}$ shows the onset of a second quadratic temperature region below $T_q \sim T_m$. Upper critical field $H_{c2}(T)$ measurements suggest multiband superconductivity, and point gap nodes are consistent with the specific heat data. In Pr$_{1-x}$La$_{x}$Pt$_{4}$Ge$_{12}$ only a single specific heat discontinuity is observed at $T_c$, in contrast to the second jump seen in PrOs$_{4}$Sb$_{12}$ below $T_c$. These results suggest that superconductivity in PrPt$_{4}$Ge$_{12}$ is characterized by a complex order parameter.

Structure and superconductivity in the binary Re1−xMox alloys

The binary Re$_{1-x}$Mo$_x$ alloys, known to cover the full range of solid solutions, were successfully synthesized and their crystal structures and physical properties investigated via powder x-ray diffraction, electrical resistivity, magnetic susceptibility, and heat capacity. By varying the Re/Mo ratio we explore the full Re$_{1-x}$Mo$_x$ binary phase diagram, in all its four different solid phases: hcp-Mg ($P6_3/mmc$), $\alpha$-Mn ($I\overline{4}3m$), $\beta$-CrFe ($P4_2/mnm$), and bcc-W ($Im\overline{3}m$), of which the second is non-centrosymmetric with the rest being centrosymmetric. All Re$_{1-x}$Mo$_x$ alloys are superconductors, whose critical temperatures exhibit a peculiar phase diagram, characterized by three different superconducting regions. In most alloys the $T_c$ is almost an order of magnitude higher than in pure Re and Mo. Low-temperature electronic specific-heat data evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling across the series. Considering that several $\alpha$-Mn-type Re$T$ alloys ($T$ = transition metal) show time-reversal symmetry breaking (TRSB) in the superconducting state, while TRS is preserved in the isostructural Mg$_{10}$Ir$_{19}$B$_{16}$ or Nb$_{0.5}$Os$_{0.5}$, the Re$_{1-x}$Mo$_x$ alloys represent another suitable system for studying the interplay of space-inversion, gauge, and time-reversal symmetries in future experiments expected to probe TRSB in the Re$T$ family.

Time-Reversal Symmetry Breaking in Re-Based Superconductors.

To trace the origin of time-reversal symmetry breaking (TRSB) in Re-based superconductors, we performed comparative muon-spin rotation and relaxation (μSR) studies of superconducting noncentrosymmetric Re_{0.82}Nb_{0.18} (T_{c}=8.8 K) and centrosymmetric Re (T_{c}=2.7 K). In Re_{0.82}Nb_{0.18}, the low-temperature superfluid density and the electronic specific heat evidence a fully gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling. In both Re_{0.82}Nb_{0.18} and pure Re, the spontaneous magnetic fields revealed by zero-field μSR below T_{c} indicate time-reversal symmetry breaking and thus unconventional superconductivity. The concomitant occurrence of TRSB in centrosymmetric Re and noncentrosymmetric ReT (T=transition metal), yet its preservation in the isostructural noncentrosymmetric superconductors Mg_{10}Ir_{19}B_{16} and Nb_{0.5}Os_{0.5}, strongly suggests that the local electronic structure of Re is crucial for understanding the TRSB superconducting state in Re and ReT. We discuss the superconducting order parameter symmetries that are compatible with the experimental observations.

A Theory of Specific Heat Discontinuity of the Superconducting Crystals by Using the Linear Model for Critical Magnetic Field

A Theory of Specific Heat Discontinuity of the Superconducting Crystals by Using the Linear Model for Critical Magnetic Field

Phase Transitions in Definite Total Spin States of Two-Component Fermi Gases.

Second-order phase transitions have no latent heat and are characterized by a change in symmetry. In addition to the conventional symmetric and antisymmetric states under permutations of bosons and fermions, mathematical group-representation theory allows for non-Abelian permutation symmetry. Such symmetry can be hidden in states with defined total spins of spinor gases, which can be formed in optical cavities. The present work shows that the symmetry reveals itself in spin-independent or coordinate-independent properties of these gases, namely as non-Abelian entropy in thermodynamic properties. In weakly interacting Fermi gases, two phases appear associated with fermionic and non-Abelian symmetry under permutations of particle states, respectively. The second-order transitions between the phases are characterized by discontinuities in specific heat. Unlike other phase transitions, the present ones are not caused by interactions and can appear even in ideal gases. Similar effects in Bose gases and strong interactions are discussed.

Ginzburg–Landau expansion in BCS–BEC crossover region of disordered attractive Hubbard model

We have studied disorder effects on the coefficients of Ginzburg - Landau (GL) expansion for attractive Hubbard model within the generalized DMFT+Sigma approximation for the wide region of the values of attractive potential U - from the weak-coupling limit, where superconductivity is described by BCS model, towards the strong coupling, where superconducting transition is related to Bose - Einstein condensation (BEC) of compact Cooper pairs. For the case of semi-elliptic initial density of states disorder influence on the coefficients A and B before the square and the fourth power of the order parameter is universal for at all values of electronic correlations and is related only to the widening of the initial conduction band (density of states) by disorder. Similar universal behavior is valid for superconducting critical temperature T_c (the generalized Anderson theorem) and specific heat discontinuity at the transition. This universality is absent for the coefficient C before the gradient term, which in accordance with the standard theory of "dirty" superconductors is strongly suppressed by disorder in the weak-coupling region, but can slightly grow in BCS - BEC crossover region, becoming almost independent of disorder in the strong coupling region. This leads to rather weak disorder dependence of the penetration depth and coherence length, as well as the slope of the upper critical magnetic field at T_c, in BCS - BEC crossover and strong coupling regions.

A proof of the Biswas–Mitra–Bhattacharyya conjecture for the ideal quantum gas trapped under the generic power law potential in d dimensions

.The well known relation for the ideal classical gas , which does not remain valid for quantum systems, is revisited. A new connection is established between the energy fluctuation and specific heat for quantum gases, which is valid in both the classical limit and the degenerate quantum regime. Most importantly, the proposed Biswas–Mitra–Bhattacharyya conjecture (Biswas et al 2015 J. Stat. Mech. P03013) relating the hump in energy fluctuation and the discontinuity of specific heat is essentially proved and made precise in this manuscript.

Attractive Hubbard model: Homogeneous Ginzburg–Landau expansion and disorder

We derive Ginzburg - Landau (GL) expansion in disordered attractive Hubbard model within the combined Nozieres - Schmitt-Rink and DMFT+Sigma approximation. Restricting ourselves to the case of homogeneous expansion, we analyze disorder dependence of GL expansion coefficients on disorder for the wide range of attractive potentials U, from weak BCS coupling region to the strong coupling limit, where superconductivity is described by Bose - Einstein condensation (BEC) of preformed Cooper pairs. We show, that for the case of semi - elliptic "bare" density of states of conduction band, disorder influence on GL coefficients A and B before quadratic and fourth -- order terms of the order parameter, as well as on the specific heat discontinuity at superconducting transition, is of universal nature at any strength of attractive interaction and is related only to the general widening of the conduction band by disorder. In general, disorder growth increases the values of coefficients A and B, leading either to the suppression of specific heat discontinuity (in the weak coupling limit), or to its significant growth (in the strong coupling region). However, this behavior actually confirms the validity of the generalized Anderson theorem, as disorder dependence of superconducting critical temperature T_c, is also controlled only by disorder widening of conduction band (density of states).

Universal scaling law for the condensation energy across a broad range of superconductor classes

One of the goals in understanding any new class of superconductors is to search for commonalities with other known superconductors. The present work investigates the superconducting condensation energy, U, in the iron based superconductors (IBS), and compares their U with a broad range of other distinct classes of superconductor, including conventional BCS elements and compounds and the unconventional heavy Fermion, Sr2RuO4, Li0.1ZrNCl, kappa-(BEDT-TTF)2Cu(NCS)2 and optimally doped cuprate superconductors. Surprisingly, both the magnitude and Tc dependence (U=0.1Tc**3.4 +- 0.2) of U are, contrary to the previously observed behavior of the specific heat discontinuity at Tc, deltaC, quite similar in the IBS and BCS materials for Tc>1.4 K. In contrast, the heavy Fermion superconductors U vs Tc are strongly (up to a factor of 100) enhanced above the IBS/BCS while the cuprate superconductors U are strongly (factor of 8) reduced. However, scaling of U with the specific heat gamma (or deltaC) brings all the superconductors investigated onto one universal dependence upon Tc. This apparent universal scaling U/gamma = 0.2Tc**2 for all superconductor classes investigated, both weak and strong coupled and both conventional and unconventional, links together extremely disparate behaviors over almost seven orders of magnitude for U and almost three orders of magnitude for Tc. Since U has not yet been explicitly calculated beyond the weak coupling limit, the present results can help direct theoretical efforts into the medium and strong coupling regimes.

On the discontinuity of the specific heat of the Ising model on a scale-free network

We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay $P(K)\sim K^{-\lambda}$. It is well established that the model is characterized by classical mean-field exponents for $\lambda>5$. In this note we show that the specific-heat discontinuity $\delta c_h$ at the critical point remains $\lambda$-dependent even for $\lambda>5$: $\delta c_h=3(\lambda-5)(\lambda-1)/[2(\lambda-3)^2]$ and attains its mean-field value $\delta c_h=3/2$ only in the limit $\lambda\to \infty$. We compare this behaviour with recent measurements of the $d$ dependency of $\delta c_h$ made for the Ising model on lattices with $d>4$ [Lundow P.H., Markstr\"{o}m K., Nucl. Phys. B, 2015, Vol. 895, 305].

Influence of Defects on the Charge Density Wave of ([SnSe](1+δ))1(VSe2)1 Ferecrystals.

A series of ferecrystalline compounds ([SnSe]1+δ)1(VSe2)1 with varying Sn/V ratios were synthesized using the modulated elemental reactant technique. Temperature-dependent specific heat data reveal a phase transition at 102 K, where the heat capacity changes abruptly. An abrupt increase in electrical resistivity occurs at the same temperature, correlated with an abrupt increase in the Hall coefficient. Combined with the magnitude and nature of the specific heat discontinuity, this suggests that the transition is similar to the charge density wave transitions in transition metal dichalcogenides. An ordered intergrowth was formed over a surprisingly wide compositional range of Sn/V ratios of 0.89 ≤ 1 + δ ≤ 1.37. X-ray diffraction and transmission electron microscopy reveal the formation of various volume defects in the compounds in response to the nonstoichiometry. The electrical resistivity and Hall coefficient data of samples with different Sn/V ratios show systematic variation in the carrier concentration with the Sn/V ratio. There is no significant change in the onset temperature of the charge density wave transition, only a variation in the carrier densities before and after the transition. Given the sensitivity of the charge density wave transitions of transition metal dichalcogenides to variations in composition, it is very surprising that the charge density wave transition observed at 102 K for ([SnSe]1.15)1(VSe2)1 is barely influenced by the nonstoichiometry and structural defects. This might be a consequence of the two-dimensional nature of the structurally independent VSe2 layers.

The discontinuity of the specific heat for the 5D Ising model

In this paper we investigate the behaviour of the specific heat around the critical point of the Ising model in dimension 5 to 7. We find a specific heat discontinuity, like that for the mean field Ising model, and provide estimates for the left and right hand limits of the specific heat at the critical point. We also estimate the singular exponents, describing how the specific heat approaches those limits. Additionally, we make a smaller scale investigation of the same properties in dimension 6 and 7, and provide strongly improved estimates for the critical temperature Kc in d=5,6,7 which bring the best MC-estimate closer to those obtained by long high temperature series expansions.

Energy fluctuation and discontinuity of specific heat

The specific heat per particle (cv) of an ideal gas, on many occasions, is interpreted as energy fluctuation per particle (▵ϵ2) of the ideal gas through the relation ▵ϵ2 = kT2cv, where k is the Boltzmann constant and T is the temperature. This relationship is true only in the classical limit and deviates significantly in the quantum degenerate regime. We have analytically explored quantum to classical crossover of this relationship, in particular, for 3D free Bose and Fermi gases. We have explored the same for harmonically trapped cases. We have obtained a hump of around its condensation point for 3D harmonically trapped Bose gas. We have discussed the possibility of an occurring phase transition with the discontinuity of heat capacity from the existence of such a hump for other Bose and Fermi systems.

N-soliton statistics and condensate formation in dense quantum gases

Statistics of N quantum density soliton waves [B. M. Mirza, Mod. Phys. Lett. B28 (2014) 1450148] is extended here to the case of systems with symmetric wave function. Since many such systems exhibit condensation phenomena, application is made of the soliton wave statistics to investigate condensation and phase transition in quantum gases such as 4 He and also dense systems such as the alkali atoms. Specific heat discontinuities are used to determine the condensation temperature for dense quantum gases and liquids. For the model case of helium the statistical theory is shown to predict not only the observed superfluid condensation temperature (2.17 ± 0.01 K) correctly but also the normal condensation temperature (4.21 ± 0.02 K), as well as the exact specific heat λ-profile.

Normal and superconducting state specific heat of Mg1 − xAlxB2 (0 < x < 0.2)

A theoretical analysis of specific heat C(T) behaviour of the Mg1 − xAlxB2 (0 < x < 0.2) in the temperature domain 0 ≤ T ≤ 300 K is presented. Calculations of C(T) have been made within the two-component scheme: one is the phonon and the other is the electronic contribution. Phonon specific heat is well estimated from the Debye and Einstein temperature for Mg1 − xAlxB2 obtained following an overlap repulsive potential. The interatomic potential of this model includes contributions from the long-range Coulomb attraction and the short-range overlap repulsion and the van der Waals attraction. The fermion constituent as the electronic specific heat is deduced using a suitable trial function above and below Tc. At the critical temperature, the discontinuity in specific heat is higher than that of other samples and the transition is sharper than for most samples. The theoretical results from boson and fermion terms are then compared with the experimental results.