Hole-Induced Anomaly Temperature

• Hole-induced anomaly temperature is defined as the crossover (T_A) where thermal excitations access the hole-like Lieb II branch, producing a distinctive specific-heat peak in 1D Bose gases.
• In black-hole thermodynamics, anomaly corrections adjust the local temperature profile to remain finite and vanish at the horizon, resolving the divergence of the conventional Tolman law.
• In solid-state systems, temperature scales linked to hole degrees of freedom drive transport reorganizations, Lifshitz transitions, and shifts in qubit frequencies.

Searching arXiv for the cited papers and closely related work on hole-induced thermal anomalies. Hole-induced anomaly temperature most precisely denotes the crossover temperature $T_A$ of the one-dimensional repulsive Bose gas at which the specific heat exhibits a pronounced peak because thermal excitations begin to access states up to the top of the hole-like Lieb II branch. In that setting, the anomaly is a thermodynamic crossover rather than a true phase transition, and its characteristic scale is of the same order as the maximal hole energy $\Delta(\gamma)$ (Rosi et al., 2021). Closely related literature uses analogous temperature concepts in technically distinct settings: anomaly-corrected local temperatures near black-hole horizons, transport anomalies controlled by hot holes or hole pockets in solids, and angle-tunable thermal susceptibilities of hole-spin qubits. The common feature is that a hole degree of freedom—or, in black-hole thermodynamics, the horizon together with the associated anomaly structure—sets the scale at which naive thermal behavior fails.

1. Scope and principal usages

In the surveyed literature, the phrase labels several distinct phenomena rather than a single universal observable. In the Lieb-Liniger problem it refers to a sharply defined crossover temperature $T_A$; in black-hole thermodynamics it refers to an anomaly-corrected local temperature profile; and in condensed-matter systems it denotes characteristic temperatures at which hole branches, hole pockets, or hole carriers reorganize transport or spectroscopy.

Context	Anomalous thermal signature	Characteristic scale
1D Bose gas (Rosi et al., 2021)	Peak in specific heat from thermal access to the Lieb II hole branch	$k_B T_A(\gamma)\sim \Delta(\gamma)$
1D Bose gas correlations (Rosi et al., 2023)	Specific-heat peak, chemical-potential maximum, and fading of the $1/k^4$ Tan tail	Above $T_A$
Generalized Tolman temperature (Gim et al., 2015)	Conventional Tolman temperature diverges, anomaly-corrected temperature is finite and vanishes at the horizon	Horizon of a static black hole
Effective Tolman temperature in 4D Schwarzschild (Eune et al., 2015)	Finite temperature everywhere outside the horizon, with $T\to 0$ at $r=2M$	Schwarzschild exterior
Hot holes in silicon (Catherall et al., 2023)	Extra drift-velocity saturation regime and non-monotonic microwave noise	Below $\sim 40$ K
WTe$_2$ (Wu et al., 2015)	Disappearance of hole pockets and temperature-induced Lifshitz transition	$\Delta(\gamma)$0 K

This range of usages does not collapse to a single formula. Instead, each case ties an anomalous temperature scale to the onset of hole-controlled excitations, hole-controlled scattering, or anomaly-controlled corrections to standard thermodynamic relations.

2. Thermodynamic definition in the one-dimensional Bose gas

For the repulsive Lieb-Liniger gas, the anomaly temperature is defined through the exact temperature dependence of the specific heat,

$\Delta(\gamma)$1

computed within the thermal Bethe-Ansatz framework for the Hamiltonian

$\Delta(\gamma)$2

The anomaly is the peak in $\Delta(\gamma)$3, present for any finite interaction strength $\Delta(\gamma)$4, and it shifts to higher temperature and becomes more pronounced as $\Delta(\gamma)$5 increases (Rosi et al., 2021).

The physical interpretation is spectral. The excitation spectrum contains a lower Lieb II branch, identified as the hole branch. Its maximum energy $\Delta(\gamma)$6, located at the Fermi wave number $\Delta(\gamma)$7, acts as a gap-like scale because there is a region of unpopulated states below that maximal hole energy at $\Delta(\gamma)$8. When thermal fluctuations reach that scale, the system can store energy much more efficiently, producing the specific-heat peak. The paper summarizes the relation as

$\Delta(\gamma)$9

and in the Tonks-Girardeau regime the approximation tightens to $T_A$0 (Rosi et al., 2021).

This anomaly is explicitly not a phase transition. One-dimensional finite-temperature long-range order is absent, and the paper treats the phenomenon as a Schottky-like crossover analogue rather than critical symmetry breaking. At sufficiently low temperature, the specific heat reduces to the Luttinger-liquid form

$T_A$1

with interaction-dependent corrections in the Bogoliubov and hard-core limits. The anomaly then marks the breakdown of that low-$T_A$2 regime as hole-branch states become thermally accessible (Rosi et al., 2021).

3. Correlation and excitation signatures above $T_A$3

A later analysis recasts the same temperature scale $T_A$4 as a reference temperature for correlation physics. In that treatment, the hole anomaly is observed as a peak in the specific heat and a maximum in the chemical potential, while the internal energy remains nearly flat for $T_A$5 and rises strongly for $T_A$6 because spectral excitation states become thermally occupied (Rosi et al., 2023).

The central consequence is that high-momentum and short-distance correlations change character above the anomaly threshold. The conventional Tan tail,

$T_A$7

is no longer the dominant asymptotic feature, because it is screened by a $T_A$8 contribution tied to the thermal increase of the internal energy. On the short-distance side, the one-body density matrix

$T_A$9

admits the expansion

$k_B T_A(\gamma)\sim \Delta(\gamma)$0

where $k_B T_A(\gamma)\sim \Delta(\gamma)$1 and

$k_B T_A(\gamma)\sim \Delta(\gamma)$2

Because $k_B T_A(\gamma)\sim \Delta(\gamma)$3 depends on the internal energy, the rise of $k_B T_A(\gamma)\sim \Delta(\gamma)$4 above $k_B T_A(\gamma)\sim \Delta(\gamma)$5 enhances the $k_B T_A(\gamma)\sim \Delta(\gamma)$6 sector relative to the contact-controlled $k_B T_A(\gamma)\sim \Delta(\gamma)$7 sector (Rosi et al., 2023).

The same temperature also governs excitation coherence. Path Integral Monte Carlo calculations of the dynamic structure factor show a sharp quasiparticle-like peak for $k_B T_A(\gamma)\sim \Delta(\gamma)$8, strong broadening near $k_B T_A(\gamma)\sim \Delta(\gamma)$9, and a broad incoherent response for $1/k^4$0. The paper interprets this as a breakdown of the quasiparticle description at the anomaly threshold for any interaction strength, even though no true phase transition occurs (Rosi et al., 2021). Taken together, these results make $1/k^4$1 a boundary between a low-$1/k^4$2 quantum-contact-dominated regime and a higher-$1/k^4$3 thermally screened regime (Rosi et al., 2023).

4. Anomaly-corrected temperatures at black-hole horizons

In black-hole thermodynamics, the relevant anomaly temperature is not a crossover scale but a corrected local temperature that replaces the divergent Tolman law. The conventional static relation,

$1/k^4$4

diverges at a Schwarzschild horizon because $1/k^4$5, even though the renormalized stress tensor in the Hartle-Hawking state remains finite there. The cited works identify the source of this conflict as the assumption of a traceless stress tensor in Tolman’s original derivation, an assumption incompatible with the trace anomaly responsible for Hawking radiation (Eune et al., 2015).

In the two-dimensional formulation, the stress tensor is written as a perfect fluid with trace relation $1/k^4$6, and thermodynamics together with the temperature independence of the trace anomaly yields the generalized Stefan-Boltzmann law

$1/k^4$7

The resulting generalized local temperature is finite everywhere and, in the exactly solvable 2D Schwarzschild example, approaches the Hawking temperature at infinity while vanishing at the horizon (Gim et al., 2015).

The four-dimensional Schwarzschild treatment is anisotropic and uses

$1/k^4$8

with trace relation $1/k^4$9. The anomaly-corrected Stefan-Boltzmann law becomes

$T_A$0

where $T_A$1 for a conformal scalar field. For Ricci-flat Schwarzschild,

$T_A$2

and the explicit effective Tolman temperature factorizes so that the problematic redshift factor is canceled. The temperature is finite everywhere outside the horizon and satisfies $T_A$3 as $T_A$4 (Eune et al., 2015).

A related $T_A$5-dimensional anomaly framework incorporates gravitational anomalies into Tolman-Ehrenfest and Luttinger relations. There the corrected temperature obeys

$T_A$6

with curvature-dependent quantum scales $T_A$7 and $T_A$8. In the black-hole application, these anomaly terms cancel the classical horizon divergence and yield a horizon-vanishing corrected temperature, while in flat-space analogues they appear in nonlinear thermal response, thermal quenches, and Floquet heating (Bermond et al., 2022).

Across these formulations, the consistent point is that the naive Tolman temperature is not the physically relevant local temperature when anomaly terms are present. The corrected temperature aligns the thermal description with a finite renormalized stress tensor and with the statement that a freely falling observer should not encounter an infinite local temperature at the horizon.

5. Hole-carrier and hole-pocket anomaly temperatures in solids

Several condensed-matter systems exhibit temperature scales at which hole carriers or hole pockets qualitatively change transport. In cryogenic $T_A$9-Si, hot holes display an extra, lower-field drift-velocity saturation regime below about $T\to 0$0 K, over roughly $T\to 0$1 to $T\to 0$2, in addition to the usual high-field saturation. The first-principles explanation is that at such temperatures acoustic phonon occupation becomes negligible, so acoustic phonon absorption tends to zero while emission approaches a constant; as the field heats carriers into the $T\to 0$3–30 meV range, scattering rises faster than band velocity and produces a saturation shoulder. The same work attributes a microwave current-noise peak not to convective noise but to the field dependence of the momentum relaxation time, with a Drude-like fit

$T\to 0$4

and a peak near $T\to 0$5 (Catherall et al., 2023).

In a silicon MOS quantum dot hosting a single hole spin, the anomaly appears as a temperature-dependent Larmor shift that is neither monotonic nor orientation independent. The temperature-induced shift

$T\to 0$6

has a finite Larmor thermal susceptibility

$T\to 0$7

with values as large as $T\to 0$8. The sign and magnitude depend strongly on magnetic-field angle $T\to 0$9: the largest shift, about $r=2M$0, occurs near $r=2M$1, while at $r=2M$2 the measured shift is essentially zero. The paper traces the effect to thermally activated electric dipoles in the oxide that “unfreeze” with temperature and alter the local electrostatic environment, which the hole spin converts into a frequency shift through spin-orbit coupling. In the simplified fit, the thermal sweet spot occurs when

$r=2M$3

and the phenomenological fit yields $r=2M$4 and $r=2M$5 (Champain et al., 19 Sep 2025).

In semimetallic materials, hole pockets often define the relevant temperature scale. In WTe$r=2M$6, ultra-high-resolution ARPES shows that the hole pockets shrink with increasing temperature and vanish above $r=2M$7 because the top of the hole band sinks below the chemical potential. Transport tracks the same reconstruction: the thermoelectric power shows a change of slope at $r=2M$8, and Kohler’s rule breaks down over $r=2M$9. The work identifies this as a temperature-induced Lifshitz transition without structural or magnetic symmetry breaking (Wu et al., 2015).

In HgTe-based semimetal quantum wells, low-temperature resistivity anomalies are likewise hole-controlled but take a different form. Between $\sim 40$0 and $\sim 40$1 K, the resistance can grow either linearly or quadratically with temperature depending on top-gate voltage in the semimetal regime. The theory attributes the distinction to electron-hole Coulomb scattering together with the crossover of the hole subsystem from diffusive motion, $\sim 40$2, to quasiballistic motion, $\sim 40$3. In the diffusive hole regime the dominant correction is linear in $\sim 40$4, while in the quasiballistic regime it is quadratic in $\sim 40$5 (Snegirev et al., 2024).

A further example is Cd$\sim 40$6As$\sim 40$7 nanoplate transport, where the Hall anomaly is the temperature scale at which thermally activated electrons begin to compete strongly with holes. The Hall response is nonlinear, the dominant transport changes from hole-dominated at low temperature to electron-dominated at high temperature, and the anomaly is most significant in the intermediate-to-high temperature range near $\sim 40$8–200 K. The magnetoresistance reaches about $\sim 40$9 at $_2$0 K, consistent with a nearly compensated two-carrier state. The paper explains the observations with a standard two-carrier model and proposes temperature-dependent spin-orbit coupling as a plausible microscopic scenario, while noting that the mechanism is not fully settled (Li et al., 2016).

6. Conceptual interpretation and recurrent misconceptions

Several distinctions are necessary to avoid conflating unlike anomalies. In the one-dimensional Bose gas, $_2$1 is not a critical temperature and not the onset of long-range order; it is a crossover temperature identified by a specific-heat peak and controlled by the maximum of the hole branch. The repeated comparison with a superfluid transition is therefore only analogical and concerns the sharpening of thermodynamic and dynamical signatures, not actual symmetry breaking (Rosi et al., 2021).

In black-hole thermodynamics, the divergent Tolman temperature is not interpreted as a physical horizon bath once the trace or gravitational anomaly is included. The anomaly-corrected formulations instead imply that the local temperature is finite and, in the explicit constructions cited here, vanishes at the horizon. This removes the infinite blueshift and is presented as restoring the equivalence principle for freely falling observers in thermal equilibrium (Gim et al., 2015).

In solid-state hole systems, “anomaly temperature” is system dependent. For hole-spin qubits, the central distinction is between a thermal sweet spot, where static thermal shifts cancel algebraically, and a noise sweet spot, where electric susceptibilities are minimized in quadrature; the two need not coincide (Champain et al., 19 Sep 2025). In hot-hole silicon transport, the noise peak is assigned to the field dependence of $_2$2, not to earlier convective-noise or simple carrier-heating explanations (Catherall et al., 2023). In Cd$_2$3As$_2$4, the essential empirical fact is two-carrier transport with a temperature-driven carrier-type crossover, whereas temperature-dependent spin-orbit coupling remains a proposed microscopic explanation rather than a closed issue (Li et al., 2016).

Taken together, these results show that hole-induced anomaly temperature is best understood as a family of temperature scales or temperature laws that become singularly informative when holes, hole branches, hole pockets, or horizon anomalies dominate the thermal response. In the Bose gas it serves as an exact thermodynamic marker $_2$5; in black-hole physics it defines an anomaly-compatible local temperature; and in electronic materials it identifies the temperature window where hole-mediated scattering, compensation, or electrostatic susceptibility qualitatively reorganize observable behavior.