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arXiv:2605.00137

Demonstration of a fermion Quadrupling Condensate via Quantum Monte Carlo Simulation

Alexandru Golic, Egor S. Babaev, Johan Carlström

Published Apr 30, 2026• 0 citations
arXivHTMLPDF

Abstract

Fermionic condensation typically occurs via pairing. In recent decades, however, a fundamental question has emerged: whether alternative forms of order exist, such as condensates of fermion quadruplets. These states--including ``charge-4e" superconductors and ``charge-0" counterflow condensates--lie beyond the standard Bardeen-Cooper-Schrieffer framework, and require strong fluctuations and correlation effects that invalidate the BCS mean-field description. This makes the problem notoriously difficult to study numerically at a microscopic level, as it involves both strong interactions and the fermionic sign problem. Here, we present a microscopic fermionic model featuring correlated hopping that significantly mitigates the sign problem, enabling rigorous Monte-Carlo-based analysis. Using large-scale simulations, we demonstrate the existence of a fermion-quadrupling condensate with a transition temperature comparable to the hopping energy scale. These results provide direct numerical evidence for quartic fermionic order in a microscopic system and suggest that these exotic states are also experimentally accessible in ultracold atomic gases.

DOI
10.48550/arxiv.2605.00137

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Fermionic condensation typically occurs via pairing. In recent decades, however, a fundamental question has emerged: whether alternative forms of order exist, such as condensates of fermion quadruplets. These states--including charge-4e" superconductors and charge-0" counterflow condensates--lie beyond the standard Bardeen-Cooper-Schrieffer framework, and require strong fluctuations and correlation effects that invalidate the BCS mean-field description. This makes the problem notoriously difficult to study numerically at a microscopic level, as it involves both strong interactions and the fermionic sign problem. Here, we present a microscopic fermionic model featuring correlated hopping that significantly mitigates the sign problem, enabling rigorous Monte-Carlo-based analysis. Using large-scale simulations, we demonstrate the existence of a fermion-quadrupling condensate with a transition temperature comparable to the hopping energy scale. These results provide direct numerical evidence for quartic fermionic order in a microscopic system and suggest that these exotic states are also experimentally accessible in ultracold atomic gases.
The condensation of electron quartets, known as charge-4e superconductivity (SC), represents a novel quantum state of matter beyond the standard paradigm of Cooper pairing. However, concrete microscopic models realizing this phase in two dimensions remain a central challenge. Here, we introduce a non-engineered and sign-problem-free model, unambiguously demonstrating the emergence of a robust and high-temperature charge-4e SC phase using unbiased quantum Monte Carlo simulations. At zero temperature, the phase diagram reveals that charge-4e SC is the primary ground state in the strong-coupling regime. At finite temperature in the absence of charge-2e SC, we identify charge-4e SC through a Berezinskii-Kosterlitz-Thouless transition, marked by a universal jump in the superfluid stiffness consistent with a condensate of charge 4e. Remarkably, the transition temperature Tc increases nearly linearly with interaction strength, providing a robust mechanism for high-Tc quartet superconductivity. Furthermore, spectral analysis reveals a prominent pseudogap above Tc arising from strong phase fluctuations. Our results establish a canonical and numerically exact model system for charge-4e superconductivity, offering crucial guidance for its realization in experimental platforms such as moiré materials and ultracold atomic systems.
Fundamentals of fermion-condensation physics are outlined. Fermion condensation is a phase transition occurring in a strongly correlated Fermi system upon a topological reconstruction of the Landau ground state and leading to the formation of a fermion condensate that possesses the dispersionless single-particle spectrum ϵ(p)=0\epsilon({\mathbf{p}})=0ϵ(p)=0 in the momentum-space region adjacent to the Fermi surface and, accordingly, an anomalously enhanced density of single-particle states. A original method is developed for solving the set of nonlinear integral equations of fermion-condensation theory. This method makes it possible to analyze the problem of quantum chaos in strongly interacting multifermion systems. The computational technique used is demonstrated by applying it to superdense quark–gluon plasma, where the structure of exchange quark–quark interaction is well known. In electron systems featuring a fermion condensate, the magnitude of the gap appearing in the single-particle spectrum owing to Cooper pairing is shown to be much larger than that in Bardeen–Cooper–Schrieffer (BCS) theory. This explains both a high superconducting-transition temperature TcT_{c}Tc​ and, with allowance for C4C_{4}C4​ crystal-lattice symmetry, the DDD -wave pairing-gap structure observed in cuprates. It is found that, in addition to the BCS gap Δ\DeltaΔ , the spectrum of single-particle excitations of superconducting systems where a fermion condensate is present develops a nonsuperconducting gap Υ\UpsilonΥ that owes its existence to the interaction of condensate particles with normal quasiparticles residing outside the condensate region in momentum space. The question of whether these results are pertinent to the two-gap structure recently unearthed in the excitation spectrum of cuprates upon analyzing ARPES data is addressed.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
The condensation of paired fermions into superfluid states changes progressively depending on the coupling strength. At the midpoint of the crossover between Bardeen-Cooper-Schrieffer (BCS) weak-coupling and Bose-Einstein condensate (BEC) strong-coupling limits, paired fermions condensate most robustly, thereby leading to the emergence of a pseudogap due to enhanced pairing fluctuations. In the case of electrons in solids, excessively strong interactions often induce competing electronic orders instead of strong-coupling superconductivity, and experimental comprehension of the pseudogap remains incomplete. In this study, we provide experimental evidence demonstrating the opening of a pseudogap, marking the incipient stage of the BCS-BEC crossover in the organic system $\ensuremath{\kappa}\text{\ensuremath{-}}(\mathrm{BEDT}\text{\ensuremath{-}}{\mathrm{TTF})}_{2}X$. By controlling electron correlations, we investigate the thermodynamic properties of the BCS-BEC crossover and pseudogap phase. Since the superconductivity of $\ensuremath{\kappa}\text{\ensuremath{-}}{(\text{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}X$ arises from a simple Fermi liquid that does not exhibit any other electronic orders, our study sheds light on the inherent nature of the BCS-BEC crossover.
The Hubbard model of attractively interacting fermions provides a paradigmatic setting for fermion pairing. It features a crossover between Bose-Einstein condensation of tightly bound pairs and Bardeen-Cooper-Schrieffer superfluidity of long-range Cooper pairs, and a "pseudo-gap" region where pairs form above the superfluid critical temperature. We directly observe the nonlocal nature of fermion pairing in a Hubbard lattice gas, using spin- and density-resolved imaging of [Formula: see text]1000 fermionic potassium-40 atoms under a bilayer microscope. Complete fermion pairing is revealed by the vanishing of global spin fluctuations with increasing attraction. In the strongly correlated regime, the fermion pair size is found to be on the order of the average interparticle spacing. Our study informs theories of pseudo-gap behavior in strongly correlated fermion systems.
Many features of charge-4e4e4e superconductors remain unknown because even the ``mean-field Hamiltonian'' describing them is an interacting model. Here we introduce an interacting model to describe a charge-4e4e4e superconductor (SC) deep in the superconducting phase and explore its properties using quantum Monte Carlo (QMC) simulations. The QMC is sign-problem-free but only when a Majorana representation is employed. As a function of the chemical potential we observe two sharply-distinct behaviors: a ``strong'' quarteting phase in which charge-4e4e4e quartets are tightly bound (like molecules) so that charge-2e2e2e pairing does not occur even in the temperature $T\ensuremath{\rightarrow}0$ limit, and a ``weak'' quarteting phase in which a further transition to a charge-2e2e2e superconducting phase occurs at a lower critical temperature. Analogous issues arise in a putative Z4{Z}_{4}Z4​ spin liquid with a pseudo-Fermi surface and other interacting models with composite order parameters. Under certain circumstances, we also identified a stable T=0T=0T=0 charge-4e4e4e SC phase with gapless nodal quasiparticles. We further discuss possible relevance of our results to various experimental observations in 18\frac{1}{8}81​-doped LBCO.