Pauli limit: assumptions, measurements and apparent violations

Scope

The Pauli paramagnetic limit compares the energy gained by a spin-polarized normal state with the condensation energy of a specified superconducting state. It is a model-dependent reference, not a universal upper bound on every superconductor. Exceeding a weak-coupling, g = 2 estimate signals that its assumptions need examination. It does not by itself establish triplet pairing, a spatially modulated state or a topological phase. Chandrasekhar and Clogston's 1962 publication identities are checked here; the following explicit calculation is independent rather than a claimed reproduction of their complete original papers. R1 R2

Suppressing orbital depairing in a very thin film can expose spin depairing that would otherwise be hidden. Geometry alone does not remove the spin-energy cost. Spin–orbit coupling, an altered gap scale, anisotropic response, exchange compensation and multiband physics require distinct models. Keep the applied field, internal exchange field and spin–orbit energy scale separate when interpreting an apparently large critical field. R3 R4 R7

Mathematical assumptions

Use $B$ in tesla and dimensionless SI volume susceptibilities. With positive zero-field condensation-energy density $U_c$, constant spin susceptibilities and no orbital contribution, the free-energy difference is $F_s-F_n=-U_c+(\chi_n-\chi_s)B^2/(2\mu_0)$. For $\chi_n\gt\chi_s$, its crossing occurs at $B_P=\sqrt{2\mu_0U_c/(\chi_n-\chi_s)}$. If the susceptibilities vary strongly with field, this quadratic expression must be replaced by the appropriate magnetization integral. Equal susceptibilities remove this particular quadratic penalty, not every physical limit on superconductivity. R8

For a simple uniform, fully gapped spin-singlet state at zero temperature, set $\chi_s=0$. Let $n$ be the single-spin density of states per unit energy and volume. Taking $U_c=n\Delta_0^2/2$ and $\chi_n=\mu_0n(g\mu_B)^2/2$ yields $B_P=\sqrt{2}\Delta_0/(g\mu_B)$. At $g=2$ and $\Delta_0=1.764k_BT_c$, the numerical coefficient is about 1.857 T/K. The familiar approximate 1.84–1.86 T/K benchmark is meaningful only with its gap and response assumptions. It is a zero-temperature reference, not a measured transition curve at arbitrary temperature. R1 R2 R8

An independent calculation checks eighteen combinations of transition temperature, g factor and gap ratio, and verifies the free-energy sign on both sides of each crossing. Holding the other assumptions fixed, halving g doubles the reference field; increasing the gap ratio from 1.764 to 2.2 multiplies it by about 1.247. A synthetic $T_c=3$ K, g = 2 example gives approximately 5.571 T with the first gap ratio. These parameter changes demonstrate sensitivity, not fitted explanations for a named material. R1 R2

A further controlled example fixes $U_c$ and $\chi_n$ while assigning $r=\chi_s/\chi_n$. The crossing grows by $1/\sqrt{1-r}$: r = 0, 0.25 and 0.75 give factors 1, approximately 1.155 and 2. This scaling cannot be carried unchanged into a system where spin–orbit coupling also changes the normal susceptibility or condensation energy. The finite checks verify the stated energy comparison; they do not solve a self-consistent gap equation or select the stable pairing state. R8

Experimental criteria

For Xi and colleagues' monolayer NbSe2 study, the approximately 35 T zero-temperature scale comes from fitting a spin–orbit-protected pair-breaking model. An independent contact-conductance measurement extends to 31.5 T and finds a strongly diminished Andreev-related feature. The author text gives 0.3 K, while its Figure 4 caption specifies 0.36 K for the inset; those labels are preserved here. The fitted effective spin–orbit scale near 660 T is not an applied laboratory field. Field-transition criteria and two-terminal versus four-terminal devices must also remain distinct. R7

De la Barrera and colleagues report that monolayer TaS2's parallel-field transition remains above their 34.5 T measurement ceiling below 2 K. Their quoted 65.6 T is a near-transition square-root fit parameter, not a directly measured critical field. They define phase-boundary points using half the normal-state resistance, which is distinct from zero resistance. Alignment is optimized at fixed temperature and field by minimizing resistance; maintaining alignment during a temperature sweep is harder. Their 25 mK/12 T and 300 mK/34.5 T apparatus limits must not be combined into one unperformed measurement. R8

Calorimetry provides another kind of evidence. Bianchi and colleagues observe a high-field, low-temperature anomaly in CeCoIn5 and interpret it as a possible finite-momentum superconducting state. The final publication explicitly retains “Possible” in its title. Subsequent neutron evidence establishes coupled magnetic order in the high-field superconducting phase. An extra phase boundary therefore requires assessment of competing or coexisting order; it is not an automatic image of a spatially modulated superconducting gap. The older calorimetry and later magnetic observations have different review depths here. R5 R6

Limits and disputes

The orbital field in a Maki ratio must come from an orbital model or suitable independent estimate: $\alpha_M=\sqrt{2}B_{c2}^{\mathrm{orb}}/B_P$. Inserting an already Pauli-limited measured field into its numerator changes the meaning. The dirty, single-band orbital estimate involving 0.693 times $T_c$ and the magnitude of the near-transition slope has restricted assumptions. Unequal band diffusivities can change the extrapolation. A large ratio motivates further investigation; it does not certify a finite-momentum state or determine a universal onset threshold. R3 R4

Complete scientific acceptance remains open. Selected TMD methods and figures have been checked, but raw transport, full supplementary calibration, complete final-version comparisons and material-specific susceptibility reconstructions remain incomplete. A fitted spin–orbit parameter, a resistance midpoint and a bulk thermodynamic boundary are different quantities. Even when a spin-sensitive probe agrees with a proposed interpretation, compare alternative states under the same field direction, temperature and experimental conditions before assigning pairing symmetry. R5 R6 R7 R8

Discovery implications

Record the measured field range and any censoring or extrapolation, the superconducting criterion, field direction, sample thickness and transition temperature used in the reference ratio. Report the chosen g factor, gap ratio, susceptibility assumptions and orbital model separately. Use departures from the simple benchmark to design discriminating measurements, not to rank triplet content by the size of a critical-field ratio. A future model must explain both the enhanced direction and the directions in which superconductivity is more easily destroyed. R3 R4 R7

Related topics

Magnetic field, singlet and triplet pairing, exchange compensation, and field-induced superconductivity.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.