Thermodynamical Topology in Complex Systems

• Thermodynamical topology is a framework that uses topological invariants to classify equilibrium states, phase transitions, and defects in diverse systems.
• It integrates defect-based analysis, Morse theory, and contact topology to capture critical behavior in black holes, thermal networks, and spin systems.
• Its applications span assessing black-hole stability, enhancing thermometric precision in networks, and controlling transport mechanisms via topologically encoded data.

Searching arXiv for recent and foundational papers on thermodynamical topology across black holes, thermal networks, and statistical systems. Tool unavailable in this interface, so I will ground the article strictly in the supplied arXiv records and cite them directly. Thermodynamical topology denotes a family of topological constructions used to classify thermodynamic structure through invariants attached to equilibrium states, critical points, relaxation trajectories, or localized thermal modes. In black-hole thermodynamics, the dominant formulation treats on-shell black hole states as topological defects of vector fields built from generalized off-shell free energies, so that winding numbers, Brouwer degrees, or related charges distinguish stable and unstable branches and organize phase structure globally (Gogoi et al., 2023). The supplied literature also uses the term for topology of diffusion operators in thermal networks, topology-controlled thermometric precision, Morse–Euler descriptions of microcanonical spin systems, contact-topological formulations of relaxation, and thermally generated topological charge in chiral magnets. Taken together, this suggests that “thermodynamical topology” is not a single formalism but a research program in which thermodynamic behavior is encoded by topological data rather than only by local response coefficients.

1. Major meanings and conceptual scope

The supplied literature suggests at least four distinct but related meanings of thermodynamical topology. The first, and currently the most developed on the black-hole side, identifies zeros of vector fields derived from generalized free energies or spinodal functions with thermodynamic defects, and classifies the resulting phase structure by local indices and global topological numbers. The second appears in thermal diffusion, where dimerized diffusion operators inherit band-topological invariants and produce edge, hinge, and corner temperature profiles with spectrally isolated diffusive rates. The third is a microcanonical Morse-theoretic program, where topology of energy sublevel sets and Euler characteristics encode thermodynamic observables. The fourth is a non-equilibrium geometric program in contact topology, where equilibrium manifolds are Legendrian and relaxation is realized by contact Hamiltonian flow trajectories (Wu et al., 3 Aug 2025, Chen et al., 2023, Santos et al., 2016, Entov et al., 2021).

Setting	Topological object	Representative invariant
Black-hole thermodynamics	Defects of off-shell or spinodal vector fields	Winding number, Brouwer degree, $W$
Thermal diffusive networks	Boundary-localized diffusive modes	Winding, Zak phase, hierarchy of order
Microcanonical spin systems	Critical points of a Morse function	Euler characteristic $\chi(E)$, Euler entropy
Non-equilibrium thermodynamics	Legendrian submanifolds and contact flows	Interlinking, existence of semi-infinite trajectories

This plurality is substantive rather than terminological. In the black-hole literature, topology is attached to phase branches, spinodal curves, and critical points. In thermal networks and chiral magnets, topology organizes transport or localized thermal textures. In microcanonical and contact-topological settings, topology enters through the geometry of configuration or phase space itself. A common thread is that thermodynamic behavior is represented by globally stable integers or homotopy-type data rather than only by equations of state.

2. Defect-based black-hole formalism

In the defect-based black-hole framework, one introduces an auxiliary inverse temperature $\tau$ and defines a generalized off-shell free energy

$\mathcal{F}=E-\frac{S}{\tau},$

with $E=M$ in extended black-hole thermodynamics. The corresponding vector field is typically taken on a two-dimensional space $(r_+,\Theta)$ or $(S,\Theta)$,

$$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$

so that zeros occur at $\Theta=\pi/2$ and $\partial_{r_+}\mathcal{F}=0$. These zeros are interpreted as thermodynamic defects, and their local winding numbers are computed either from the deflection of the normalized field $\chi(E)$0 around a small contour or from a residue construction after complexifying the horizon radius (Gogoi et al., 2023).

The local and global invariants are

$\chi(E)$1

with stable branches carrying $\chi(E)$2 and unstable branches $\chi(E)$3 in the implementations discussed for Euler–Heisenberg–AdS black holes. The same formalism uses $\chi(E)$4 to generate defect curves $\chi(E)$5, whose intersections with constant $\chi(E)$6 slices enumerate coexisting black-hole branches (Gogoi et al., 2023).

A closely related but technically distinct approach assigns topology directly to thermodynamic critical points through Duan’s $\chi(E)$7-mapping current. There the zeros of an entropy-based vector field correspond to critical points satisfying

$\chi(E)$8

or, equivalently, the usual van der Waals conditions in $\chi(E)$9 variables. The resulting topological charges divide critical points into conventional and novel types; only the conventional critical point serves as the endpoint of a first-order transition line in the examples studied (Wei et al., 2021).

The spinodal-curve variant replaces the off-shell free-energy field by a scalar function $\tau$0, where $\tau$1 is obtained by eliminating pressure along the spinodal locus. In one dimension the Brouwer degree is

$\tau$2

which makes the topology computable directly from asymptotics without solving the criticality equations explicitly (Bai et al., 2022).

More recent work has removed the auxiliary $\tau$3 variable altogether in grand-canonical constructions. There the vector field is the gradient of an off-shell grand free energy, for example

$\tau$4

and the topological charge is the sum of the indices of all zeros in the physical thermodynamic variables themselves. In that formulation, the charge is conserved under variations of the ensemble parameters and is interpreted simultaneously as a thermodynamic and spacetime invariant (Nam, 28 Oct 2025).

3. Spinodals, higher-order defects, and topological classes

Thermodynamical topology has developed from isolated winding numbers into a hierarchical classification scheme. A unified formulation introduces $\tau$5-th order vector fields $\tau$6 built from successive derivatives of a generalized free energy $\tau$7. In this hierarchy, zeros of $\tau$8 encode phase branches, zeros of $\tau$9 encode spinodal or Davies-type structures, zeros of $\mathcal{F}=E-\frac{S}{\tau},$0 encode critical points, and higher $\mathcal{F}=E-\frac{S}{\tau},$1 resolve multicritical coalescences. The same framework gives

$\mathcal{F}=E-\frac{S}{\tau},$2

and, for even $\mathcal{F}=E-\frac{S}{\tau},$3, associates a higher-order zero with critical exponents

$\mathcal{F}=E-\frac{S}{\tau},$4

For the 7-dimensional Lovelock example, the reported sequence is $\mathcal{F}=E-\frac{S}{\tau},$5, $\mathcal{F}=E-\frac{S}{\tau},$6, $\mathcal{F}=E-\frac{S}{\tau},$7, $\mathcal{F}=E-\frac{S}{\tau},$8, with the higher-order critical behavior effectively resolved by a $\mathcal{F}=E-\frac{S}{\tau},$9 zero (Wu et al., 3 Aug 2025).

The original critical-point classification already exhibited two local types. For 4D RN–AdS, the single critical point carries charge $E=M$0, whereas in 4D Born–Infeld–AdS a representative parameter choice yields one novel critical point with $E=M$1 and one conventional critical point with $E=M$2, so the total topological charge vanishes. This construction is precisely what underlies the statement that the presence of a novel critical point does not, by itself, indicate a nearby first-order transition (Wei et al., 2021).

Spinodal topology in Lovelock gravity shows that the total Brouwer degree constrains the possible phase diagrams. Charged spherical Lovelock AdS black holes in arbitrary $E=M$3 fall into a single topology class with $E=M$4. Uncharged spherical Lovelock black holes split: $E=M$5 again has $E=M$6, while $E=M$7 has $E=M$8. This difference is used to interpret the appearance of reentrant phase transitions and multi-critical behavior in higher-dimensional uncharged cases (Bai et al., 2022).

Einstein–Maxwell–Dilaton theories extend this classification by introducing a temperature-dependent interpolation class $E=M$9. Across $(r_+,\Theta)$0, including the Gubser–Rocha model at the critical dilaton coupling $(r_+,\Theta)$1, the global invariant remains $(r_+,\Theta)$2, while pairs of stable and unstable defects are created and annihilated as $(r_+,\Theta)$3 varies. The terminology $(r_+,\Theta)$4 signals that the same system passes through regimes locally resembling different endpoint classes without changing its global topological number (Babaei-Aghbolagh et al., 20 Aug 2025).

Ultraspinning Kerr–AdS black holes furnish a dimension-dependent but globally simple pattern. Most configurations lie in the standard class $(r_+,\Theta)$5. A distinct subclass $(r_+,\Theta)$6 appears only for odd-dimensional black holes with the maximal number of rotation parameters. Both have $(r_+,\Theta)$7, but $(r_+,\Theta)$8 differs by having an unstable innermost branch and an extra annihilation point in the defect structure (Tian et al., 5 Feb 2026).

4. Ensemble dependence, matter couplings, and representative black-hole results

A recurring conclusion in the recent literature is that thermodynamic topology is often ensemble dependent. For 4D Euler–Heisenberg–AdS black holes, the canonical ensemble at fixed $(r_+,\Theta)$9 gives $(S,\Theta)$0 for $(S,\Theta)$1 and $(S,\Theta)$2 for $(S,\Theta)$3, independent of $(S,\Theta)$4 and $(S,\Theta)$5. Adding higher-order QED corrections removes that sign dependence and enforces $(S,\Theta)$6 for all $(S,\Theta)$7. In the grand canonical ensemble at fixed $(S,\Theta)$8, both the Euler–Heisenberg and higher-order corrected systems instead give $(S,\Theta)$9, independent of $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$0, $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$1, and $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$2. The same model therefore exhibits explicit ensemble dependence of its thermodynamic topological class (Gogoi et al., 2023).

Hořava–Lifshitz black holes in $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$3 display a related structure in two ensembles. In the fixed $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$4 ensemble, spherical horizons $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$5 support a $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$6 line of continuous second-order phase transitions. The reported critical values are $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$7 for $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$8 and $$\phi=\left(\partial_{r_+}\mathcal{F},-\cot\Theta\,\csc\Theta\right),$$9 for $\Theta=\pi/2$0. Across that line the number of defects changes, but the total topological number remains $\Theta=\pi/2$1 before the GR limit is reached. In the fixed $\Theta=\pi/2$2 ensemble, the $\Theta=\pi/2$3 line disappears and the total topological number depends on $\Theta=\pi/2$4 for several horizon topologies, showing again that a Legendre transform can reshuffle the defect structure while preserving only ensemble-specific invariants (Hazarika et al., 2023).

Dyonic AdS black holes with quasitopological electromagnetism exhibit multiple defect curves. Depending on the chosen pressure and temperature, there can be one, three, or five black-hole states. Representative cases explicitly yield alternating local indices such as $\Theta=\pi/2$5 or $\Theta=\pi/2$6, yet the total topological number remains $\Theta=\pi/2$7. In that setting, the appearance of two separated first-order coexistence curves in the phase diagram corresponds topologically to the emergence of multiple defect curves in the $\Theta=\pi/2$8 plane (Chen et al., 2024).

The comparison across bulk–boundary thermodynamics, restricted phase space, and extended phase space shows both stability and nontrivial variation. AdS RN and AdS EGB black holes are reported to share $\Theta=\pi/2$9 in both bulk–boundary and restricted phase space descriptions. AdS EPYM black holes behave differently: the local charges are $\partial_{r_+}\mathcal{F}=0$0, and the total topological number is $\partial_{r_+}\mathcal{F}=0$1 in bulk–boundary thermodynamics, restricted phase space, and extended phase space alike. This three-way equivalence is singled out as a distinctive consequence of the nonlinear YM charge sector (Sadeghi et al., 2023).

Regular black holes generated from pure gravity by an infinite tower of higher-curvature corrections provide a different universality statement. The small-horizon behavior of the Hawking temperature forces at least one zero in the physical interval, and the analyzed family falls into the class $\partial_{r_+}\mathcal{F}=0$2 with global $\partial_{r_+}\mathcal{F}=0$3: a stable small black hole with $\partial_{r_+}\mathcal{F}=0$4 and an unstable large black hole with $\partial_{r_+}\mathcal{F}=0$5 form a universal pair for the pure-gravity constructions discussed in $\partial_{r_+}\mathcal{F}=0$6 (Wang et al., 2024).

A still more ambitious extension argues that, in grand-canonical variables, black holes sharing the same background geometry carry the same topological charge. In the formulation based on the gradient of the off-shell grand free energy, the paper reports $\partial_{r_+}\mathcal{F}=0$7 for Minkowski-flat and de Sitter classes, and $\partial_{r_+}\mathcal{F}=0$8 for AdS. The proposed implication is that the thermodynamic topological charge can be treated as an invariant of spacetime rather than only of a specific black-hole solution (Nam, 28 Oct 2025).

5. Thermal networks, thermometry, and topology-controlled transport

Outside gravity, thermodynamical topology also refers to topological organization of thermal diffusion itself. In thermal diffusive networks built as generalized $\partial_{r_+}\mathcal{F}=0$9-dimensional SSH analogues, the diffusion operator can be written in a chiral-symmetric block-off-diagonal form

$\chi(E)$00

The central relation reported there is that, in an $\chi(E)$01-dimensional thermal SSH network, a $\chi(E)$02-th-order topological state exhibits confined temperature profiles of dimension $\chi(E)$03 with constant diffusive rates. The work demonstrates edge, hinge, and corner states up to $\chi(E)$04, including an intermediate-order 3D phase with hinge states but no corner states, and near-zero-decay “thermal still” modes generated by chiral-symmetry engineering (Chen et al., 2023).

Graph topology also enters thermometry in a quantitatively different way. For a finite system modeled by a graph Laplacian Hamiltonian, the Landau fluctuation bound

$\chi(E)$05

coincides with the optimal Cramér–Rao bound because the quantum Fisher information for a thermal state satisfies

$\chi(E)$06

The graph topology fixes the spectrum of $\chi(E)$07, and thereby the temperature dependence of $\chi(E)$08 and $\chi(E)$09. The reported design rule is that low connectivity is advantageous for precise low-temperature thermometry, whereas high connectivity is advantageous at high temperature. Complete graphs, cycles, stars, paths, and several 2D lattices are worked out explicitly (Candeloro et al., 2021).

In multilevel absorption machines, the steady-state heat currents can be decomposed into circuit contributions associated with cycles of the master-equation graph. Each circuit is thermodynamically consistent, and the total current is a sum over cycle currents. The paper proves that the overall performance of a multilevel machine is smaller or equal than the performance of the largest circuit contribution, while the magnitude of the heat currents is controlled by a purely topological parameter that generally increases with graph connectivity. For a fixed number of levels, the preferred construction is therefore the most connected graph compatible with equal circuit performance (1712.06368).

A related but distinct network result concerns interacting unicyclic machines on different interaction topologies. Exact solutions for all-to-all and star-like minimal topologies, together with Gillespie simulations on homogeneous and heterogeneous graphs, show that topology strongly affects performance when individual energies are small, partly because of first-order phase transitions. As individual energies increase, the topology becomes less important and the behavior approaches that of the all-to-all case (Mamede et al., 2023).

6. Microcanonical, contact-topological, and thermally driven perspectives

In classical microcanonical spin systems, thermodynamical topology takes the form of a Morse–Euler program. The potential energy is treated as a Morse function on configuration space, and thermodynamic information is extracted from the Euler characteristic of energy sublevel sets. The central object is the Euler entropy

$\chi(E)$10

which, for the mean-field and short-range $\chi(E)$11 models studied, reproduces the critical temperature, magnetization, correlation functions, and susceptibility in the thermodynamic limit. The same framework relates loss of regularity of the Morse function to unstable and metastable mean-field branches, so topology is not merely a detector of phase transitions but also an organizer of branch stability (Santos et al., 2016).

In non-equilibrium thermodynamics, contact topology furnishes a different geometrization. Thermodynamic phase space is modeled as a contact manifold $\chi(E)$12, equilibrium sets are Legendrian submanifolds, and relaxation processes are contact Hamiltonian flows. “Hard” contact topology yields interlinking results: under suitable sign conditions on the contact Hamiltonian and robust interlinking between Legendrians, one obtains semi-infinite trajectories that start on one Legendrian and asymptotically converge to another. The paper develops this abstract machinery and illustrates it with a contact embedding of Glauber dynamics for the mean-field Ising model (Entov et al., 2021).

A further usage appears in chiral magnets, where thermal fluctuations themselves generate topology. The local spin texture carries topological charge density

$\chi(E)$13

or, on the lattice, a triangulated solid-angle sum. Monte Carlo and $\chi(E)$14 analysis show a high-field, high-temperature upturn of the thermal average of the topological charge, even outside ordered skyrmion phases. The leading high-temperature scaling is

$\chi(E)$15

with a subleading correction proportional to $\chi(E)$16. Here topology is not a defect of an off-shell free-energy field but an emergent thermally biased flux variable generated by Dzyaloshinskii–Moriya canting and a Zeeman field (Hou et al., 2017).

These non-black-hole developments reinforce a broad interpretation. In one branch of the literature, thermodynamical topology classifies defect structure of thermodynamic potentials. In another, it identifies topological properties of thermal operators, microcanonical level sets, or relaxation geometry itself. The supplied corpus therefore suggests a unifying principle: thermodynamic behavior can often be encoded by topological invariants attached either to the state manifold, the flow generating relaxation, or the operator governing thermal transport. Within that principle, the defect-based black-hole formalism remains the most standardized current usage, but it coexists with distinct microcanonical, contact-geometric, and transport-theoretic lineages rather than replacing them.