Davies Phase-Transition Curves

• Davies phase-transition curves are loci defined by conditions like (∂/∂S)(1/T)=0, signaling divergent heat capacity and critical behavior in black holes.
• They illuminate ensemble-specific critical loci and reveal distinct thermodynamic properties, including topological charges and dynamical signatures in quasinormal modes.
• Methodologies such as Euclidean path integrals, tunneling radiation, and fractional derivative analyses provide insights into black-hole stability and phase-transition order.

Searching arXiv for recent and foundational papers on Davies phase-transition curves and Davies points. Davies phase-transition curves are loci in black-hole thermodynamic parameter space at which a response function—most commonly the heat capacity at fixed generalized displacement—diverges, separating thermodynamic branches and marking what the standard Ehrenfest scheme treats as second-order criticality. In the literature considered here, the term is most precise for charged and rotating black holes, where the defining condition can be written either as $C_{X_i}\to\infty$ or, assuming $T\neq 0$, $$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$. Subsequent work has shown that these curves admit finer fractional classifications, topological interpretations, Euclidean canonical-ensemble derivations, and dynamical signatures in quasinormal modes and tunneling radiation (Wang et al., 2022, Bhattacharya et al., 2024, Fernandes et al., 2024).

1. Thermodynamic definition and ensemble dependence

The generic thermodynamic setup is the black-hole first law

$TdS=dM-\sum_i Y_i\,dX_i,$

with $X_i$ denoting generalized displacements such as electric charge $Q$ or angular momentum $J$, and $Y_i$ the corresponding potentials. For Davies-type transitions one studies the specific heat at fixed $X_i$,

$$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$

which can be rewritten as

$T\neq 0$0

Hence, for nonzero temperature, the Davies critical condition is

$T\neq 0$1

This criterion identifies the Davies phase-transition curve or point in a form that is independent of any one spacetime example (Bhattacharya et al., 2024).

In concrete ensembles, the same condition is often expressed more directly through the horizon radius. For the Reissner–Nordström–AdS black hole in the canonical ensemble with fixed $T\neq 0$2 and non-extended thermodynamics, the variables are $T\neq 0$3, the first law is $T\neq 0$4, the Helmholtz free energy is $T\neq 0$5, and the heat capacity diverges precisely when $T\neq 0$6. This makes the Davies set the locus where the temperature curve $T\neq 0$7 has either an extremum or an inflection-type critical point, depending on the parameters (Wang et al., 2022).

A plausible implication is that Davies curves are best understood not as a single universal geometric object, but as ensemble-specific critical loci whose explicit form depends on which extensive variables are held fixed.

2. Explicit realizations in charged black holes

For charged Reissner–Nordström families, the Davies curve is the divergence set of the fixed-charge heat capacity, and the same condition can be represented in several coordinate systems: $T\neq 0$8, $T\neq 0$9, $$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$0, or $$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$1. In asymptotically flat four-dimensional Reissner–Nordström spacetime, the heat capacity is

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$2

so the Davies point satisfies $$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$3, equivalently

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$4

For the de Sitter case,

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$5

and the divergence condition becomes

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$6

In higher-dimensional asymptotically flat Reissner–Nordström black holes, the same role is played by

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$7

For canonical Reissner–Nordström–AdS,

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$8

so the divergence condition is

$$\left(\frac{\partial}{\partial S}\frac{1}{T}\right)_{X_i}=0$$9

with Davies radii

$TdS=dM-\sum_i Y_i\,dX_i,$0

These are the Davies phase-transition curves or points in that setting (Wei et al., 2019, Wang et al., 2022, Fernandes et al., 2024).

System	Divergence condition	Representative Davies locus
4D RN	$TdS=dM-\sum_i Y_i\,dX_i,$1	$TdS=dM-\sum_i Y_i\,dX_i,$2
4D RN-dS	$TdS=dM-\sum_i Y_i\,dX_i,$3	$TdS=dM-\sum_i Y_i\,dX_i,$4
$TdS=dM-\sum_i Y_i\,dX_i,$5-dimensional RN	heat-capacity denominator $TdS=dM-\sum_i Y_i\,dX_i,$6	$TdS=dM-\sum_i Y_i\,dX_i,$7
RN-AdS canonical	$TdS=dM-\sum_i Y_i\,dX_i,$8	$TdS=dM-\sum_i Y_i\,dX_i,$9

The four-dimensional canonical Euclidean treatment makes this curve particularly explicit. There the Hawking-temperature saddle occurs at

$X_i$0

and this is identified as the Davies point. The lower-radius branch $X_i$1 is stable, while the upper-radius branch $X_i$2 is unstable (Fernandes et al., 2024).

3. Order of transition and local critical structure

The standard interpretation treats a Davies point as a second-order phase transition because the heat capacity diverges there. In the Euclidean canonical treatment of charged black holes, this statement is made precise by the fact that the Helmholtz free energy and its first derivatives remain finite while the second derivatives are discontinuous. The same language is used in conventional analyses of Reissner–Nordström–AdS thermodynamics (Wang et al., 2022, Fernandes et al., 2024).

A more refined picture emerges in the fractional-derivative generalization of the Ehrenfest classification. For canonical Reissner–Nordström–AdS black holes there are two distinct types of Davies points. When $X_i$3, the temperature curve $X_i$4 has two distinct extrema satisfying

$X_i$5

These are the first type of Davies points. Near such a point, the free energy contains a noninteger contribution of the form

$X_i$6

and the first singular fractional derivative is of order $X_i$7. These points are therefore classified as $X_i$8-order phase transitions rather than ordinary second-order transitions (Wang et al., 2022).

When $X_i$9, the two roots merge into a critical inflection point satisfying

$Q$0

This is the second type of Davies point. For generic directions of approach in the $Q$1 plane, the free energy contains an $Q$2 term,

$Q$3

so the transition is classified as $Q$4-order. The paper also identifies a special direction,

$Q$5

along which the fractional-power terms disappear and the free energy becomes analytic. Along that line the behavior is consistent with an ordinary second-order-type boundary in Hilfer’s language (Wang et al., 2022).

This refines a common misconception: not all Davies points are thermodynamically equivalent merely because they all make $Q$6 diverge. The local structure of $Q$7 matters, and the generalized Ehrenfest framework resolves that difference.

4. Dynamical and radiative diagnostics

Davies curves have been connected to black-hole dynamics through the eikonal quasinormal-mode correspondence. For static, spherically symmetric charged black holes, the eikonal spectrum is

$Q$8

where $Q$9 is the angular velocity and $J$0 the Lyapunov exponent of the unstable photon orbit. In both asymptotically flat and de Sitter Reissner–Nordström spacetimes, spiral-like structures appear in the complex quasinormal-mode plane, but the starting point of those spirals does not coincide with the Davies point. In four-dimensional asymptotically flat Reissner–Nordström, for example, the spiral begins at

$J$1

whereas the Davies point is

$J$2

The exact relation found instead is that the Davies point coincides with the maximum of the temperature in the $J$3-$J$4 and $J$5-$J$6 planes; that statement remains valid even in higher-dimensional asymptotically flat cases where the spiral-like structure disappears (Wei et al., 2019).

A complementary interpretation comes from tunneling radiation. In the Parikh–Wilczek framework for Reissner–Nordström black holes, the tunneling rate can be written as

$J$7

with

$J$8

The leading nonthermal correction is therefore proportional to $J$9. At the Davies critical point, where $Y_i$0, the nonthermal contribution vanishes and the radiation becomes purely thermal. On the $Y_i$1 side, the correction enhances emission; on the $Y_i$2 side, it suppresses emission. The same phase-separating role persists for Kerr–Newman black holes, even when emissions of energy, charge, and angular momentum are included (La, 2010).

Taken together, these results show that Davies curves are not only thermodynamic singular loci. They also organize dynamical behavior, although different dynamical observables encode the curve in different ways.

5. Topological and Euclidean reformulations

Two recent lines of work recast Davies-type transitions in geometrical or topological language. In one formulation, Duan’s $Y_i$3-mapping topological current theory is applied directly to thermodynamic phase space by introducing

$Y_i$4

This gives

$Y_i$5

The zero of $Y_i$6 occurs at

$Y_i$7

so the Davies point is identified as a zero of the vector field carrying nonzero winding number. In the explicit phantom RN-(A)dS example with $Y_i$8, the enclosing contour has topological charge $Y_i$9, while a contour not enclosing the critical point has $X_i$0 (Bhattacharya et al., 2024).

A related construction uses a single common vector field for both Davies-type and Hawking–Page transitions,

$X_i$1

Because

$X_i$2

the zeros arise either from $X_i$3 or from $X_i$4. The former gives the Hawking–Page transition, while the latter gives the Davies point. Across Schwarzschild AdS, grand-canonical RN-AdS, and grand-canonical Kerr-AdS examples, the topological charge is $X_i$5 for the Davies point and $X_i$6 for the Hawking–Page point (Hazarika et al., 2024).

The Euclidean canonical-ensemble derivation supplies a complementary structural interpretation. Starting from the Gibbons–Hawking path integral with fixed temperature $X_i$7 and fixed electric charge $X_i$8 at infinity, the zero-loop approximation yields two black-hole solutions below a critical temperature: a lower-radius branch $X_i$9, which is stable, and an upper-radius branch $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$0, which is unstable. The saddle point at $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$1 is exactly where the heat capacity changes sign and diverges, so it is the Davies curve in the canonical ensemble. In four dimensions this reproduces Davies’ thermodynamic theory exactly. The same analysis also distinguishes this second-order branch transition from the first-order competition with hot flat space, which has $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$2 while the stable black-hole branch has positive free energy (Fernandes et al., 2024).

6. Scope of the term and related usages

Outside black-hole thermodynamics, the phrase “Davies phase-transition curves” can denote conceptually related but technically distinct objects. In open quantum systems, the relevant “Davies” object is the Davies generator: the weak-coupling Markovian Lindbladian that relaxes to a Gibbs state and satisfies the KMS condition

$$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$3

A generalized framework extends this structure to arbitrary full-rank stationary states $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$4. In that setting, the paper’s explicit critical curve is not a Davies thermodynamic curve, but a measurement-induced phase transition at

$$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$5

where $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$6 correlators exhibit critical fluctuations and power-law-like scaling with distance, consistent with the Ising critical point (Guo et al., 2024).

Orthographically similar terminology can also be misleading. The tetragonal$$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$7cubic boundary in davemaoite is a pressure–temperature free-energy equality curve,

$$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$8

separating $$C_{X_i}=T\left(\frac{\partial S}{\partial T}\right)_{X_i},$$9 and $T\neq 0$00 structures. That is a mineral phase boundary, not a Davies curve in the black-hole sense (Wu et al., 2024).

Some condensed-matter and finite-system papers invoke only an analogy. In BaFe$T\neq 0$01(As$T\neq 0$02P$T\neq 0$03)$T\neq 0$04, a line built from the anomalous $T\neq 0$05 and $T\neq 0$06 points is interpreted as a vortex-lattice transition and fitted by

$T\neq 0$07

which the authors describe as being in the same broader spirit as Davies-type phase-transition constructions. In SLE and in constrained caloric curves for hot nuclei, the resemblance is even looser: the comparison is only to sharp thresholds or anomalous curvature, not to Davies criticality as defined by heat-capacity divergence (Jr. et al., 2015, Beliaev et al., 2020, Borderie et al., 2013).

The resulting terminological boundary is clear. In its strict usage, a Davies phase-transition curve is a black-hole thermodynamic critical locus defined by divergent specific heat or an equivalent condition such as $T\neq 0$08. Broader uses are either analogical extensions or arise because “Davies” refers to a different construct altogether.