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Water from a faucet boils quickly, but in the last moment before the boiling point, everything seems to freeze — bubbles hesitate, the temperature stalls. So it is in quantum systems: when a material approaches a critical point, its internal clock stretches to infinity. Modeling this directly is like trying to count every drop in a waterfall: computational costs explode.

A group of physicists found a way to compress this "frozen" spacetime. They packed time into compact influence matrices (semigroup ones, to be precise) and space into matrix product states, resembling a chain of linked train cars. The result was a neat tensor network — not a bulky four-dimensional array, but a flexible structure that can be computed step by step.

The method was tested on the two-impurity Anderson model with four fermionic "flavors" of particles. The picture was detailed: from the Kondo resonance, where guest electrons dance around a magnetic impurity, through a non-Fermi-liquid critical point, a state unlike an ordinary electron liquid, and to the gapped singlet phase, where particles freeze into pairs. The authors computed spectral functions (the system's "fingerprints," so to speak) and extracted the universal exponent of the two-channel Kondo effect. The number matched predictions, and the calculation took a reasonable time.

Thus, multi-bath influence matrices become a working tool for real-time dynamics in strongly correlated multi-orbital systems. From now on, physicists can not only peek into equilibrium but also watch how a quantum material lives its history — from a sudden perturbation to a smooth transition.