Hot and dense QCD equation of state from the complex Langevin equation

Michael Mandl; Dénes Sexty; Daniel Unterhuber

AI summary · arXiv: 2610.38466v1 · hep-lat

Hot and dense QCD equation of state from the complex Langevin equation

Michael Mandl, Dénes Sexty, Daniel Unterhuber

Original paper Submitted to arXiv on 28 Oct. 2026 4 pages, 4 figures; Proceedings of the 22nd International Conference on QCD in Extreme Conditions (xQCD 2026), 14-17 July 2026, Busan, Korea License: CC BY 4.0 Also in: fr

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The essentials

The paper presents continuum-extrapolated QCD thermodynamic densities and specific heat at physical pion mass, reaching baryon chemical-potential-to-temperature ratios up to 10 using complex Langevin simulations.

Question

Can complex Langevin simulations provide equation-of-state observables for QCD at physical pion mass and baryon densities beyond the range accessible to conventional lattice methods? The study computes energy, entropy, and enthalpy densities and specific heat above the QCD crossover.

Method

The authors use complex Langevin simulations of strangeness- and charm-neutral, isospin-symmetric 2+1+1-flavor QCD, with staggered fermions, a tree-level improved Symanzik gauge action, and four steps of stout smearing. They simulate lattices of sizes 20³×10, 24³×12, and 32³×16 along a line of constant physics, perform continuum extrapolations, and report results up to μB/T=10.

Findings

Result The energy density increases with baryon chemical potential at all studied temperatures and increases with temperature at least for small chemical potentials.
See the passage in the paper
One observes an increase with $\mu_{B}$ for all temperatures as well as an increase with $T$, at least for small chemical potentials.
Authors’ interpretation The authors report that pressure contributes more to the energy density than the trace anomaly.
See the passage in the paper
By comparing with [6], it becomes clear that the contribution of the pressure to the energy density dominates over the trace anomaly.
Result The entropy and enthalpy densities show qualitative behavior similar to the energy density.
See the passage in the paper
The qualitative behavior is very similar to that of the energy density in Fig. 1.
Result The specific heat appears to increase with baryon chemical potential at low temperatures, while its trend at higher temperatures is unclear because of large uncertainties.
See the passage in the paper
The data suggests an increase of $\hat{c}_{V}$ with $\mu_{B}$ for low temperatures. For higher temperatures, on the other hand, the uncertainties become very large and it becomes difficult to determine the monotonicity adequately.
Result The calculations extend continuum-extrapolated physical-point results to μB/T ratios up to 10.
See the passage in the paper
For our continuum-extrapolated lattice results, we have employed the first ever complex Langevin simulations at the physical point.

Limitations

The results cover temperatures well above the QCD crossover, so they do not establish the equation of state in the crossover region.
See the passage in the paper
We note, however, that our results are currently limited to temperatures well above the QCD crossover.
At the highest studied temperature, uncertainties in the observables become sizable, in part because of systematic uncertainty in fitting their temperature dependence.
See the passage in the paper
For our highest temperature, i.e., $T=600\,\mathrm{MeV}$, the (combined statistical and systematic, see [5]) uncertainties become sizable. This is also true for the other observables studied in this work and can be attributed to systematic uncertainties arising in our fitting procedure for the temperature-dependence.
The specific-heat derivative is not evaluated at strictly constant spatial volume, although the authors expect the resulting error to be negligible.
See the passage in the paper
In our calculations, however, we do not enforce this condition. Nonetheless, we expect the ensuing error to be negligible since we consider intensive quantities exclusively and finite-size effects are close to zero.

Why it matters

The authors argue that their continuum-extrapolated, physical-point complex Langevin results substantially extend lattice-QCD equation-of-state calculations to higher baryon chemical potentials than currently achievable with HMC-based approaches. They provide additional thermodynamic observables for the high-temperature, high-density region.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Quantum chromodynamics Complex Langevin Equation of state Baryon density Continuum extrapolation

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Comprehensive Summary
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Original abstract

This paper calculates several thermodynamic quantities in quantum chromodynamics (QCD) at high temperatures and nonzero baryon density, where standard lattice methods face the sign problem. The authors investigate whether complex Langevin simulations can produce continuum-extrapolated equation-of-state results in this regime, including at chemical potentials beyond the range commonly accessible to conventional hybrid Monte Carlo methods. The study focuses on temperatures above the QCD crossover and on matter with strangeness and charm neutrality and isospin symmetry. Complex Langevin dynamics evolves gauge links in the non-compact group SL(3,C), extending them from SU(3). Adaptive time steps and gauge cooling help stabilize the simulations. The calculations use 2+1+1 quark flavors at the physical pion mass, with staggered fermions, a tree-level improved Symanzik gauge action, and four steps of stout smearing. Results from lattices of sizes 20³×10, 24³×12, and 32³×16 are extrapolated to the continuum along a line of constant physics. Cross-checks on larger volumes indicate that finite-size effects are controlled, despite the spatial-to-temporal lattice aspect ratio of 2. The paper reports energy density, entropy density, enthalpy density, and specific heat capacity, derived from baryon density, pressure, and the trace anomaly calculated in the authors’ earlier work. Energy density is the trace anomaly plus three times the pressure. Entropy density is obtained from the energy density, pressure, and chemical-potential contribution to baryon density, while enthalpy density is the sum of energy density and pressure. Specific heat is calculated by differentiating energy density with respect to temperature, using a numerical derivative of a fit. Because the simulations directly determine changes relative to zero chemical potential, the full quantities also incorporate zero-chemical-potential results from prior work. The calculations reach baryon chemical-potential-to-temperature ratios of up to 10, substantially exceeding the approximately 3.5 range reported for current conventional lattice approaches. Normalized energy density rises with baryon chemical potential at all temperatures studied and increases with temperature, at least at small chemical potential. The pressure contribution dominates over the trace anomaly. Entropy and enthalpy show qualitatively similar behavior. At lower temperatures, normalized specific heat appears to rise with chemical potential; at higher temperatures, uncertainties are too large to establish a reliable trend. The paper presents the observables’ dependences in figures rather than reporting single numerical values in the text. The authors describe their results as the first complex Langevin equation-of-state calculations at the physical point and as a continuum-extrapolated extension of lattice QCD thermodynamics to high baryon density. The conclusions apply to temperatures well above the crossover and do not establish the equation of state near or below it. Uncertainties become sizable at the highest reported temperature, 600 MeV, especially for quantities whose temperature dependence relies on fitting. The authors attribute these uncertainties mainly to systematic effects in the fitting procedure. In addition, the specific-heat derivative is not evaluated at strictly fixed spatial volume, although the authors expect the resulting error to be negligible for intensive quantities given the small finite-size effects.

Taking QCD thermodynamics into denser territory

How does matter behave when it is both extremely hot and rich in baryons? Quantum chromodynamics (QCD), the theory of quarks and gluons, can answer this question in principle. In practice, however, calculations at nonzero baryon density run into the notorious “sign problem,” which limits the reach of standard lattice methods.

In “Hot and dense QCD equation of state from the complex Langevin equation,” Michael Mandl, Dénes Sexty, and Daniel Unterhuber explore an alternative: complex Langevin simulations. Their work calculates several thermodynamic properties at temperatures above the QCD crossover and pushes to chemical potentials well beyond the range commonly reached with conventional lattice techniques.

Why high-density calculations are difficult

Lattice QCD represents space-time as a grid, making it possible to study the interactions of quarks and gluons numerically. At zero baryon density, widely used simulation methods work reliably. At nonzero density, the mathematical weights in the calculation become complex rather than behaving like ordinary probabilities. This sign problem makes conventional simulations increasingly difficult as the chemical potential grows.

Complex Langevin takes a different route. Instead of keeping the gauge links—the variables describing the gluon fields—within the SU(3) group used in ordinary QCD, the method extends them into the larger, non-compact group SL(3,C). The simulations use adaptive time steps and a stabilization technique called gauge cooling to help control the resulting dynamics.

A continuum calculation at physical quark masses

The study uses four quark flavors: up, down, strange, and charm, with the physical pion mass. The calculations employ staggered fermions, an improved gauge action, and stout smearing, a procedure that smooths the fields entering the fermion calculation. The matter is constrained to have zero net strangeness and charm, with isospin symmetry.

To estimate results in the continuum rather than on a particular lattice grid, the researchers simulate three lattice sizes: 20³×10, 24³×12, and 32³×16. They then extrapolate along a line of constant physics. Additional checks on larger volumes suggest that finite-size effects are controlled, even though the spatial-to-temporal lattice aspect ratio is 2.

The calculations reach baryon chemical-potential-to-temperature ratios of up to 10. That is a substantial extension beyond the roughly 3.5 range reported for current conventional lattice approaches. The authors describe this as the first complex Langevin equation-of-state calculation at the physical point and a continuum-extrapolated extension of lattice QCD thermodynamics to high baryon density.

What the study measures

The paper reports energy density, entropy density, enthalpy density, and specific heat. These quantities are built from baryon density, pressure, and the trace anomaly, with the latter inputs calculated in the authors’ earlier work. Because the new simulations determine changes relative to zero chemical potential, the full results also incorporate zero-density results from prior research.

Energy density combines the trace anomaly with three times the pressure.
Entropy density is obtained from energy density, pressure, and the chemical-potential contribution associated with baryon density.
Enthalpy density is the sum of energy density and pressure.
Specific heat measures how energy density changes with temperature and is calculated by differentiating a fit to the data.

How the thermodynamic quantities behave

Across the temperatures studied, normalized energy density rises as baryon chemical potential increases. It also rises with temperature, at least at small chemical potential. The pressure contribution is larger than the contribution from the trace anomaly, so pressure plays the dominant role in the energy-density result. Entropy and enthalpy show broadly similar patterns.

Specific heat is more difficult to pin down. At lower temperatures, its normalized value appears to increase with chemical potential. At higher temperatures, the uncertainties are too large to establish a dependable trend. The paper presents these dependencies in figures rather than giving single numerical values for the observables in its text.

What the results do—and do not—tell us

The results provide a promising demonstration that complex Langevin can extend lattice calculations into a high-density regime that is hard to access with standard methods. They concern temperatures well above the QCD crossover, however, and do not establish the equation of state near or below that transition.

Uncertainties also grow at the highest reported temperature, 600 MeV, particularly for quantities that depend on fitting their temperature variation. The authors attribute these larger errors mainly to systematic effects in the fitting procedure. In addition, the specific-heat derivative is not evaluated at strictly fixed spatial volume. The researchers expect this to have a negligible effect on intensive quantities because their finite-size checks indicate small volume effects.

Taken together, the study is a step toward mapping hot, dense QCD with first-principles calculations. Its extended reach is notable, while its temperature limits and fitting uncertainties make clear where further work is needed.

Created on 02 Nov. 2026

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Hot and dense QCD equation of state from the complex Langevin equation

The essentials

Question

Method

Findings

Limitations

Why it matters

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Definitions

Taking QCD thermodynamics into denser territory

Why high-density calculations are difficult

A continuum calculation at physical quark masses

What the study measures

How the thermodynamic quantities behave

What the results do—and do not—tell us

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