Third Law of Thermodynamics and The Shape of the Phase Diagram for Systems With a First-Order Quantum Phase Transition

Abstract

The third law of thermodynamics constrains the phase diagram of systems with a first-order quantum phase transition. For zero conjugate field, the coexistence curve has an infinite slope at T = 0. If a tricritical point exists at T > 0, then the associated tricritical wings are perpendicular to the T = 0 plane, but not to the zero-field plane. These results are based on the third law and basic thermodynamics only, and are completely general. As an explicit example we consider the ferromagnetic quantum phase transition in clean metals, where a first-order quantum phase transition is commonly observed.First-order phase transitions are ubiquitous in nature, the solid-to-liquid and liquid-to-gas transitions being the most commonly observed ones. Another common example of a first-order transition is the ferromagnetic transition below the Curie temperature as a function of an external magnetic field. First-order transitions are characterized by a coexistence curve in the phase diagram along which the two phases coexist in thermodynamic equilibrium. (The coexistence curve may be the projection of a higher-dimensional coexistence manifold into a particular plane in the phase diagram.)It has long been known that the curvature of the coexistence curve is determined by the discontinuities of certain observables across it. The Clapeyron-Clausius (CC) equation relates the slope of the coexistence curve in the pressure-temperature (p -T ) plane to the discontinuities of the entropy and the volume [1]:where ∆s = s 1 − s 2 and ∆v = v 1 − v 2 with s 1,2 and v 1,2 the specific entropy and volume per particle, respectively, in the two phases. For definiteness, let 1 and 2 label the ordered and disordered phases, respectively, and for later reference we indicate that an appropriate external field H, if any, is held constant in taking the derivative. The CC equation (1) and its analogs in different planes of the phase diagram are very general, as they rely only on basic thermodynamic arguments. In this Letter we show that for quantum phase transitions, when combined with the third law of thermodynamics, they provide interesting constraints on the shape of the phase diagram. We will consider a pressure-driven transition at T = 0 that is first order, remains first order at low T , and turns second order at higher T via a tricritical point (TCP). The schematic phase diagram in the space spanned by T , p, and H, where H is the field conjugate to the order parameter, is shown in Fig. 1. As we will see, the detailed shape of this phase diagram at low T is constrained by thermodynamics. Our arguments leading to this conclusion are completely general; however, as an explicit example we will discuss the quantum ferromagnetic transition in clean metals [2]. Another example of a first-order quantum phase transition with a TCP in the phase diagram is the Ising antiferromagnet dysprosium arXiv:1504.00644v3 [cond-mat.stat-mech] 9 Jul 2015

The quantum ferromagnetic transition in a clean Kondo lattice is discontinuous

“…It follows from thermodynamic considerations alone, namely, from various Clapeyron-Clausius relations, that the tricritical wings shown in Fig. 1 are perpendicular to the T = 0 plane, but not perpendicular to the p-axis and point in the direction of the paramagnetic phase [38]. These features, as well as the overall structure of the phase diagram, are in excellent agreement with experimental observations [4].…”

Restoration of quantum critical behavior by disorder in pressure-tuned (Mn,Fe)Si

“…8,9 This behavior matches the results in pure MnSi, except for the effect of helical spin correlations, which adds a weakly first-order character to the thermal transition before the TCP is reached, and phase separation at p* < p, which cannot be expected for ideally thermodynamic behavior in systems without disorder. 36,37 In their recent studies, 36,37 quenched disorder was suggested as a possible origin for the phase separation observed in pure MnSi. This situation is illustrated in Fig.…”

“…8,9,[34][35][36][37] Considering the effect of a negative m 3 ln(m) term of the magnetization m (order parameter) on the free energy in itinerant ferromagnets due to soft modes and particle-hole excitations, they proposed that the second-order thermal transition in ferromagnets is replaced by a first-order transition at a tricritical point (TCP) when approaching the quantum critical point (QCP), as illustrated in Fig. 1b.…”

Wing structure in the phase diagram of the Ising ferromagnet URhGe close to its tricritical point investigated by angle-resolved magnetization measurements

“…If T C is decreased by some tuning parameter such as pressure, the nature of the transition changes from second order to first order at a tricritical point TCP, and by application of a small field parallel to the spontaneous moment surfaces or "wings" of firstorder transition emerge [18,19]. The edges of the wing planes are second-order transition lines, terminating at T = 0 in quantum wing critical points (QWCPs) [20]. This type of "T -p-H phase diagram" with pressure p as a tuning parameter has been studied in itinerant FM compounds, such as UGe 2 [21][22][23], ZrZn 2 [24], URhAl [25], UCoGa [26], and an itinerant metamagnet UCoAl [27].…”