Research Interest: Prof. Dr. Christian Hainzl

Physical Context

The homogeneous electron gas at high density is a paradigmatic model in quantum many-body theory. While the Hartree–Fock approximation captures mean-field exchange, it fails to incorporate many-body correlation effects such as screening and collective excitations. These contributions dominate the correlation energy, which is predicted to scale as c₁ ρ log ρ + c₂ ρ. This expansion, known as the Gell-Mann–Brueckner formula, reflects fundamental physics of long-range Coulomb interactions and collective plasmon modes.

Mathematical Contributions

In a series of works, Christiansen–Hainzl–Nam developed a novel and rigorous approach to derive the leading-order correlation energy for high-density Fermi gases in the mean-field approximation on finite boxes. Their method is based on approximate bosonization of particle–hole excitations and a Bogoliubov diagonalization of a quadratic effective Hamiltonian. This framework allowed for a precise derivation of the correlation energy, including both the direct plasmon contribution and the subtle exchange correction.

Their results include the first rigorous proof of the full Gell-Mann–Brueckner formula for the electron gas, even for singular interactions such as the Coulomb potential. The sequence of works culminates in matching upper and lower bounds that validate the predicted high-density energy expansion. Although the derivation is confined to the mean-field scaling on finite boxes, it represents a significant advance toward solving the full problem in the thermodynamic limit.

Earlier, in collaboration with Gontier and Lewin, I analyzed the Hartree–Fock approximation at high density. We rigorously confirmed that symmetry-breaking instabilities, while allowed in Hartree–Fock, produce only exponentially small energy gains compared to the uniform Fermi sea. This result justifies the uniform free Fermi gas as a suitable reference state for perturbative analysis.

Selected Publications

• Christiansen, Hainzl, Nam (2024): , arXiv:2405.01386.
• Christiansen, Hainzl, Nam (2023): , Commun. Math. Phys. 401 (2023).
• Christiansen, Hainzl, Nam (2023): , Forum Math. Pi 11, e32.
• Hainzl, Porta, Rexze (2020): , Commun. Math. Phys. 374.
• Gontier, Hainzl, Lewin (2019): , Phys. Rev. A 99, 052501.

Complementary Work

A complementary line of research by Benedikter, Nam, Porta, Schlein, and Seiringer introduced a patch decomposition method to approximate particle–hole excitations as bosons. Their work was the first to rigorously derive the leading correlation energy using this technique. However, their application is limited to sufficiently regular potentials and does not cover the Coulomb interaction. In contrast, the approach developed by Christiansen–Hainzl–Nam is robust enough to include singular interactions and yields the full high-density expansion.

The low-density Fermi gas is a central model in many-body physics. In the spin-½ case, the ground-state energy admits an expansion in the density ρ, known as the Huang–Yang formula. It includes the free energy term, a second-order interaction term proportional to the scattering length a, and a universal third-order correction arising from many-body correlations.

The expansion reads:

e(ρ) = e_free(ρ) + 2πa ρ² + (4(11 - 2 log 2) / 35π²)(6π²)^{4/3} a² ρ^{7/3} + o(ρ^{7/3})

In collaboration with E. Giacomelli, P. T. Nam, R. Seiringer, M. Falconi, and M. Porta, I have rigorously confirmed the Huang–Yang expansion up to third order. Our results provide both matching upper and lower bounds for the energy and demonstrate the universality of the expansion for a broad class of short-range repulsive interactions.

In “The Huang–Yang Conjecture for the Low-Density Fermi Gas” (), we established the full asymptotic expansion with a matching lower bound. This extends our earlier work “The Huang–Yang Formula for the Low-Density Fermi Gas: Upper Bound” (), where we proved the optimal upper bound and identified the third-order term explicitly.

An earlier result using Bogoliubov theory is given in “The Dilute Fermi Gas via Bogoliubov Theory” with M. Falconi, E. Giacomelli, and M. Porta, published in Ann. Henri Poincaré 22 (2021), where we reproduced the first two terms and introduced a new method for fermionic trial states.

Our work builds upon and complements the foundational contribution by Lieb, Seiringer, and Solovej (2005), who rigorously derived the second-order term (the mean-field energy) and established its universality even for hard-core potentials. While their approach does not access the third-order term, it remains a cornerstone of the rigorous theory of dilute Fermi gases.

Key References

• E. L. Giacomelli, C. Hainzl, P. T. Nam, R. Seiringer, , arXiv:2505.22340 (2025).
• E. L. Giacomelli, C. Hainzl, P. T. Nam, R. Seiringer, , arXiv:2409.17914 (2024).
• M. Falconi, E. L. Giacomelli, C. Hainzl, M. Porta, , Ann. Henri Poincaré 22 (2021), 2283–2353.

Our work on dilute Bose gases focuses on the rigorous derivation of the Lee–Huang–Yang (LHY) correction beyond mean-field theory, both at zero and positive temperature. These corrections account for quantum fluctuations and refine the Gross–Pitaevskii description of Bose–Einstein condensates in the dilute regime.

In the Gross–Pitaevskii limit, together with B. Schlein and A. Triay, we provided a simplified derivation of the ground-state LHY correction based on Bogoliubov theory. This result was published as , Forum Math. Sigma 10 (2022). It provides an alternative route to the thermodynamic limit proof by Fournais and Solovej and confirms that the second-order quantum correction appears already in the GP setting.

In a separate paper, , J. Math. Phys. 62 (2021), we provided a conceptually simple proof of Bose–Einstein condensation for interacting gases, emphasizing Neumann boundary conditions and reduced kinetic energy arguments.

At positive temperature, in joint work with F. Haberberger, P. T. Nam, R. Seiringer, and A. Triay, we established for the first time a rigorous LHY correction for the free energy of dilute Bose gases. This was achieved by combining Bogoliubov transformations with Neumann boundary conditions and large-box thermodynamic estimates.

The free energy per unit volume at low temperature and in the dilute limit satisfies:

T ≪ ρ^{2/3}, in the dilute limit ρ a^3 ≪ 1

F(ρ, T) = F_0(ρ, T) + 2π a ρ² + (128 / 15√π) a^{5/2} ρ^{5/2} + o(ρ^{5/2}),

where \( F_0(ρ, T) \) denotes the free energy of the ideal Bose gas. These results appear in:

• F. Haberberger, C. Hainzl, P. T. Nam, R. Seiringer, A. Triay: The Free Energy of Dilute Bose Gases at Low Temperatures, preprint.
• F. Haberberger, C. Hainzl, B. Schlein, A. Triay: Upper Bound for the Free Energy of Dilute Bose Gases at Low Temperature, preprint.

This line of work builds upon earlier foundational contributions: Lieb and Yngvason rigorously proved the leading-order GP energy, Lieb and Seiringer extended this framework to characterize BEC, and Schlein and collaborators developed the Bogoliubov techniques we refined. Fournais and Solovej later provided the first rigorous proof of the LHY term in the thermodynamic limit. Our results complement these developments and represent the first complete derivation of the LHY correction at positive temperature.

4.1 Translation‑Invariant BCS Functional

We rigorously analyzed the BCS functional for homogeneous fermion systems. With E. Hamza, R. Seiringer, and J. P. Solovej, we proved that a non-trivial superconducting solution exists exactly when a certain linear operator has negative spectrum, establishing a precise criterion for pairing and the onset of superconductivity. (*Comm. Math. Phys. 281, 2008*)

Together with R. L. Frank, S. Naboko, and R. Seiringer, we studied the weak-coupling regime and showed that the critical temperature depends on the smallest eigenvalue of an operator defined by the interaction’s Fourier transform restricted to the Fermi sphere—not simply the scattering length. (*J. Geom. Anal. 17, 2007*)

In collaboration with R. Seiringer, we analyzed the energy gap and critical temperature in broader potentials with negative scattering length, demonstrating pairing even without two-particle bound states. (*Phys. Rev. B 77, 2008*)

With G. Bräunlich and R. Seiringer, we showed that Hartree–Fock corrections only renormalize the chemical potential, leaving the critical temperature unaffected. We also proved that translation invariance remains stable for a wide class of potentials. (*Rev. Math. Phys. 26, 2014*; follow-up in *J. Phys. B* 49, 2016)

In joint work with A. Deuchert, A. Geisinger, and M. Loss, we proved that translational symmetry is preserved in two-dimensional systems under quite general conditions. (*Ann. Henri Poincaré* 19 (2018))

4.2 Derivation of Ginzburg–Landau Theory

In collaboration with R. Frank, R. Seiringer, and J. P. Solovej, we were the first to rigorously derive the Ginzburg–Landau functional from BCS theory near the critical temperature. This proof shows that the GL order parameter emerges naturally from the microscopic Cooper-pair wave function. (*J. Amer. Math. Soc. 25, 2012*)

We later extended this GL derivation to include non-zero magnetic flux, working with A. Deuchert and M. Maier through systematic treatment of slowly varying external fields. (*Probab. Math. Phys. 4, 2023*; *Calc. Var. PDE 62, 2023*)

4.3 Shift in Critical Temperature

We quantified how external fields affect the superconducting transition temperature. In advanced works with R. Frank, R. Seiringer, and J. P. Solovej, we demonstrated that the shift of the critical temperature is determined by the lowest eigenvalue of a linearized GL operator. (*Comm. Math. Phys. 342, 2016*)

With R. Frank and E. Langmann, we calculated the linear reduction of \(T_c\) in a uniform magnetic field. With A. Deuchert and M. Maier, we extended these results to more general weak external fields. (*J. Spectr. Theory 9, 2019*; *Probab. Math. Phys. 4, 2023*; *Calc. Var. PDE 62, 2023*)

4.4 High‑Temperature Superconductivity

With M. Loss, we developed generalized BCS models allowing non-zero momentum pairing, exploring unconventional phases of superconductivity. (*Eur. Phys. J. B 90, 2017*)

In collaboration with M. Duell and E. Hamza, we proposed a mechanism for enhanced pairing in cuprate-type crystals, combining optical phonon coupling and Jahn–Teller distortions. This work predicts pair-density wave ordering and indicates conditions for elevated critical temperatures. (*Phys. Rev. B 106, 2022*)

Selected Publications

• Hainzl & Seiringer, J. Math. Phys. 57, 021101 (2016)
• Hainzl, Hamza, Seiringer, Solovej, Comm. Math. Phys. 281 (2008)
• Frank, Hainzl, Naboko, Seiringer, J. Geom. Anal. 17 (2007)
• Hainzl & Seiringer, Phys. Rev. B 77, 184517 (2008)
• Bräunlich, Hainzl, Seiringer, Rev. Math. Phys. 26 (2014); J. Phys. B 49 (2016)
• Deuchert, A.; Geisinger, A.; Hainzl, C.; Loss, M., Persistence of translational symmetry in the BCS model with radial pair interaction, Ann. Henri Poincaré 19 (2018), no. 5, 1507–1527.
• Frank, Hainzl, Seiringer, Solovej, J. Amer. Math. Soc. 25 (2012)
• Frank, Hainzl, Seiringer, Solovej, Comm. Math. Phys. 342 (2016)
• Frank, Hainzl, Langmann, J. Spectr. Theory 9 (2019)
• Deuchert, Hainzl, Maier, Probab. Math. Phys. 4 (2023); Calc. Var. PDE 62 (2023)
• Hainzl & Loss, Eur. Phys. J. B 90 (2017)
• Duell, Hainzl, Hamza, Phys. Rev. B 106 (2022)

In fermionic systems with attractive interactions, the mechanism of superfluidity evolves from loosely bound Cooper pairs (as described by BCS theory) to tightly bound bosonic molecules that undergo Bose–Einstein condensation (BEC) in the strong-coupling or low-density regime. We rigorously studied this crossover within the BCS framework and confirmed that the theory smoothly transitions from the BCS to the BEC side.

In collaboration with Robert Seiringer, we showed that in the low-density limit, the BCS energy functional reduces to an effective Gross–Pitaevskii theory for composite bosons formed by fermion pairs. This result provides a precise derivation of BEC of tightly bound pairs from first principles.

In joint work with Benjamin Schlein, we extended this result to the time-dependent setting. We demonstrated that the dynamics of the BCS functional in the low-density regime is governed, in leading order, by a nonlinear Schrödinger equation describing the evolution of the condensate wave function.

Together with Gerhard Bräunlich and Robert Seiringer, we developed a general framework for the Bogoliubov–Hartree–Fock theory in the strong-coupling limit. We proved that in this limit, the BCS ground state converges to a Bose–Einstein condensate of bound pairs, and we derived the effective interactions between the bosons.

These works rigorously confirm the long-standing physical prediction that fermionic superfluidity connects smoothly to molecular Bose–Einstein condensation when interactions are strong or densities are low. They also show that the BCS functional, when extended and analyzed carefully, captures both ends of the crossover — from weakly paired fermions to tightly bound bosons.

Selected Publications

• Hainzl, C.; Seiringer, R., Low density limit of BCS theory and Bose–Einstein condensation of fermion pairs, Lett. Math. Phys. 100 (2012), 119–138.
• Hainzl, C.; Schlein, B., Dynamics of Bose–Einstein condensates of fermion pairs in the low density limit of BCS theory, J. Funct. Anal. 265 (2013), no. 3, 399–423.
• Bräunlich, G.; Hainzl, C.; Seiringer, R., Bogolubov–Hartree–Fock theory for strongly interacting fermions in the low density limit, Math. Phys. Anal. Geom. 19 (2016), no. 2, Art. 13, 27 pp.

One of the central challenges in mathematical physics is to rigorously understand the thermodynamic limit of quantum Coulomb systems — that is, the behavior of large collections of charged particles as the system size tends to infinity. This problem is closely tied to the fundamental concept of the stability of matter and the long-range nature of the Coulomb interaction.

Together with Mathieu Lewin and Jan Philip Solovej, we developed a general framework that provides abstract conditions under which a thermodynamic limit exists for a wide class of quantum systems. Our approach applies to classical and quantum particles, to systems with or without magnetic fields, and includes new models not covered by previous theories.

A key idea in our work is to identify a minimal set of physically meaningful properties — such as translation invariance, strong subadditivity, and screening — that are sufficient to guarantee the existence of a thermodynamic limit. In particular, we relied on the Graf–Schenker inequality to quantify the notion of electrostatic screening in quantum Coulomb systems. This technique allows one to rigorously control the energy contributions of subsystems and plays a crucial role in extending thermodynamic results beyond special geometries or particle configurations.

Using this method, we were able to recover and generalize classic results by Lieb and Lebowitz (1972) and Fefferman (1985), as well as extend the theory to systems with periodic external fields or optimized nuclear configurations. Our method also provided simpler and more transparent proofs of known results, replacing technically demanding estimates with a unified variational strategy.

This work offers a powerful and flexible variational toolbox for the study of large Coulomb systems, confirming that under suitable screening and locality assumptions, the free energy per unit volume becomes independent of boundary conditions and domain shapes in the limit of large system size.

Selected Publications

• Hainzl, C.; Lewin, M.; Solovej, J. P., The thermodynamic limit of quantum Coulomb systems, Duke Math. J. 162 (2013), no. 3, 367–415.
• Hainzl, C.; Lewin, M.; Solovej, J. P., General conditions for the thermodynamic limit of Coulomb systems, Comm. Math. Phys. 347 (2016), no. 2, 349–388.

In relativistic quantum electrodynamics (QED), the Dirac operator’s negative spectrum poses a serious challenge to stability. We addressed this problem by constructing a mathematically rigorous mean-field approximation to QED without photons, where the Dirac sea is modeled as a self-consistent Hartree–Fock state.

Together with Mathieu Lewin and Jan Philip Solovej, we introduced the Bogoliubov–Dirac–Fock (BDF) model, which defines the vacuum through a variational principle involving infinite-rank projectors. This model incorporates real particles, vacuum polarization, and external fields while ensuring boundedness of energy. A key result was the proof of the existence and uniqueness of the vacuum state, described by a self-consistent projector.

In further joint work with Mathieu Lewin and Éric Séré, we analyzed the existence of atoms and molecules in this framework. We proved that bound states of electrons exist in the presence of nuclei and that the BDF energy converges to the Hartree–Fock energy in the non-relativistic limit. We also studied charge renormalization and demonstrated that the physical coupling constant remains finite in the limit of large ultraviolet cut-off.

To address the ultraviolet behavior and extend the model to magnetic fields, we developed a Pauli–Villars regularization technique. In collaboration with Philippe Gravejat, Lewin, and Séré, we showed that the polarized vacuum remains stable under general electromagnetic fields. This work rigorously captured vacuum polarization as a nonlinear response and verified consistency with known results from QED at the one-loop level.

These results were consolidated in a comprehensive review article with Lewin, Séré, and Solovej, which summarizes the mathematical framework, the key ideas of the variational formulation, and the physical consequences for relativistic quantum theory.

Selected Publications

• Hainzl & Siedentop, Comm. Math. Phys. 243 (2003)
• Hainzl, Lewin & Séré, Comm. Math. Phys. 257 (2005)
• Hainzl, Lewin & Séré, J. Phys. A 38 (2005)
• Hainzl, Lewin & Solovej, Comm. Pure Appl. Math. 60 (2007)
• Hainzl, Lewin & Séré, Arch. Ration. Mech. Anal. 192 (2009)
• Gravejat, Hainzl, Lewin & Séré, Arch. Ration. Mech. Anal. 208 (2013)
• Hainzl, Lewin, Séré & Solovej, Oberwolfach Report 39/2010

In non-relativistic QED, quantum particles (such as electrons) are coupled to the quantized electromagnetic field, capturing the interaction with photons. This framework introduces rich physical effects, including radiative corrections to energy levels, enhanced binding due to field coupling, and the Lamb shift. We have contributed to several foundational aspects of this theory, particularly concerning the influence of photon interactions on binding mechanisms and energy shifts.

In collaboration with Vitali Vougalter and Semyon Vugalter, we proved that a potential which is too weak to bind an electron in the absence of field coupling can nevertheless produce a bound state once the electron is coupled to the quantized electromagnetic field. This phenomenon, known as enhanced binding, was the first rigorous demonstration of how coupling to the field can effectively increase the particle’s mass and promote localization.

Together with Isabelle Catto, we developed the first mathematically rigorous expansion of the electron’s self-energy in powers of the fine-structure constant. We showed that Kato’s perturbation theory fails in this setting due to the lack of a spectral gap, and introduced an iterative scheme as a substitute. In a joint paper with Catto and Pavel Exner, we extended the enhanced binding result to more singular potentials.

In further work with Robert Seiringer, we studied the Lamb shift in a one-electron atom. By carefully analyzing the renormalized binding energy and its dependence on the coupling strength and ultraviolet cutoff, we were able to recover Bethe’s original formula for the Lamb shift. Our approach also highlighted the role of the electron mass renormalization in non-relativistic QED.

We also investigated infrared and ultraviolet divergences, the role of spin, and the structure of the effective mass, including a comparison between Pauli-Fierz and Nelson-type models. In particular, our results on the Nelson Hamiltonian allowed for uniform estimates in the ultraviolet cutoff, which is typically not achievable in the Pauli-Fierz setting.

These works collectively reveal how coupling to the quantized field modifies binding energies and ground state properties, and they provide a rigorous foundation for several key aspects of quantum electrodynamics at low energies.

Selected Publications

• Catto, I.; Hainzl, C., Self-energy of one electron in non-relativistic QED, J. Funct. Anal. 207 (2004), 68–110.
• Catto, I.; Exner, P.; Hainzl, C., Enhanced binding revisited for a spinless particle in nonrelativistic QED, J. Math. Phys. 45 (2004), 4174–4185.
• Hainzl, C.; Seiringer, R., The Lamb shift revisited, J. Phys. A: Math. Theor. 42 (2009), 425206.
• Hainzl, C.; Vougalter, V.; Vugalter, S., Binding of a particle to a field in non-relativistic QED, Comm. Math. Phys. 233 (2003), 13–26.
• Hainzl, C.; Hirokawa, M.; Spohn, H., Binding energy for hydrogen-like atoms in the Nelson model without cut-off, Ann. Henri Poincaré 4 (2003), 1007–1025.

We have investigated mathematical models that describe the interplay between quantum mechanics and gravity, with a particular focus on the collapse dynamics of self-gravitating fermionic systems, and time-dependent singularities in nonlinear quantum evolution equations.

In collaboration with Benjamin Schlein, we analyzed the gravitational collapse of white dwarfs. We derived and studied a dynamical model based on the pseudo-relativistic Hartree–Fock equation coupled with gravitational interaction. This work rigorously demonstrated both static and dynamical collapse phenomena and provided insight into Chandrasekhar's limit within a time-dependent framework.

Together with Enno Lenzmann, Mathieu Lewin, and Schlein, we studied blow-up solutions in generalized time-dependent Hartree and Hartree–Fock equations. These models describe the evolution of large bosonic or fermionic systems with mean-field interactions, and our work revealed critical mass thresholds and mechanisms leading to finite-time singularities.

In joint work with Felix Finster, we explored the interplay of quantum fields and general relativity in cosmological settings. We studied a model of the universe described by an Einstein–Dirac system under spatial homogeneity and isotropy assumptions. Remarkably, we proved that quantum oscillations in the Dirac field can prevent the big bang singularity, replacing it with a bounce.

These contributions combine mathematical analysis, quantum many-body theory, and general relativity to provide insight into the behavior of matter under extreme conditions, such as in astrophysical and early-universe scenarios.

Selected Publications

• Hainzl, C.; Schlein, B., Dynamics of Bose–Einstein condensates of self-gravitating particles, Comm. Math. Phys. 287 (2009), 705–717.
• Hainzl, C., On the static and dynamical collapse of white dwarfs, Contemp. Math. 529 (2010), 189–202.
• Hainzl, C.; Lenzmann, E.; Lewin, M.; Schlein, B., On blow-up for time-dependent generalized Hartree–Fock equations, Ann. Henri Poincaré 11 (2010), 1023–1052.
• Finster, F.; Hainzl, C., A spatially homogeneous and isotropic Einstein–Dirac cosmology, J. Math. Phys. 52 (2011), 042501.
• Finster, F.; Hainzl, C., Quantum oscillations can prevent the big bang singularity in an Einstein–Dirac cosmology, Found. Phys. 40 (2010), 116–124.

Beyond the primary themes of my research, I have worked on several topics that contribute to the mathematical understanding of quantum many-body systems and related fields in mathematical physics.

In collaboration with Thomas Chen, Nataša Pavlović, and Robert Seiringer, we studied the Gross–Pitaevskii hierarchy, which arises as a limiting description of Bose–Einstein condensates in the mean-field regime. Using tools from the quantum de Finetti theorem, we provided new results on the well-posedness, unconditional uniqueness, and scattering theory for this infinite hierarchy of equations. These results helped clarify the link between microscopic quantum dynamics and effective nonlinear models.

In another direction, I explored the electronic structure of graphene through Hartree–Fock theory. Together with Mathieu Lewin and Christof Sparber, we analyzed the ground-state properties of graphene sheets, incorporating the relativistic dispersion relation of Dirac-type fermions and long-range Coulomb interactions. This work offered a rigorous perspective on the emergence of charge-density modulations and interaction-induced effects in graphene-like materials.

In an earlier collaboration with Robert Seiringer, we derived a general decomposition of radial functions in ℝⁿ, which has proven useful in the analysis of rotationally symmetric many-body quantum systems. This mathematical tool plays a role in simplifying interaction terms in high-dimensional quantum problems.

Selected Publications

• Chen, T.; Hainzl, C.; Pavlović, N.; Seiringer, R., Unconditional uniqueness for the cubic Gross–Pitaevskii hierarchy via quantum de Finetti, Comm. Pure Appl. Math. 68 (2015), 1845–1884.
• Chen, T.; Hainzl, C.; Pavlović, N.; Seiringer, R., On the well-posedness and scattering for the Gross–Pitaevskii hierarchy via quantum de Finetti, Lett. Math. Phys. 104 (2014), 871–891.
• Hainzl, C.; Lewin, M.; Sparber, C., Ground state properties of graphene in Hartree–Fock theory, J. Math. Phys. 53 (2012), 095220.
• Hainzl, C.; Seiringer, R., General decomposition of radial functions on ℝⁿ and applications to N-body quantum systems, Lett. Math. Phys. 61 (2002), 75–84.