Abstract
One-dimensional atomic mixtures of fermions can effectively realize spin chains and thus constitute a clean and controllable platform to study quantum magnetism. Such strongly correlated quantum systems are also of sustained interest to quantum simulation and quantum computation due to their computational complexity. In this article, we exploit spectral graph theory to completely characterize the symmetry properties of one-dimensional fermionic mixtures in the strong interaction limit. We also develop a powerful method to obtain the so-called Tan contacts associated with certain symmetry classes. In particular, compared to brute force diagonalization that is already virtually impossible for a moderate number of fermions, our analysis enables us to make unprecedented efficient predictions about the energy gap of complex spin mixtures. Our theoretical results are not only of direct experimental interest, but also provide important guidance for the design of adiabatic control protocols in strongly correlated fermion mixtures.
Highlights
From quantum magnetism to the highly debated hightemperature superconductivity, many spectacular phenomena observed in condensed-matter physics emerge from strong interactions among particles with a spin degree of freedom [1]
Our results suggest that strongly confined 1D SU (κ = N ) systems may become promising candidates for quantum adiabatic computing, allowing (i) complex encoding due to the large number of spins, (ii) an adiabatic tuning of the exchange constants of the effective spin chain through the external potential, and (iii) a well-controlled energy gap that can be computed efficiently with our method
This paves the way to further theoretical and experimental investigations that may eventually lead to technological applications [56]
Summary
From quantum magnetism to the highly debated hightemperature superconductivity, many spectacular phenomena observed in condensed-matter physics emerge from strong interactions among particles with a spin degree of freedom [1]. We claim that the spin-chain model associated with 1D strongly repulsive fermionic mixtures has a natural interpretation in terms of spectral graph theory This hitherto unobserved connection to a well-studied mathematical branch [20,21,22,23,24] provides a general and rigorous framework, which enables us to completely elucidate the symmetry structure of the spectrum for arbitrary external potentials and numbers of particles. This has strong implications: For example, it allows us to prove a generalized form of the LiebMattis theorem, which implies in particular that the ground state of the system is unmagnetized [25]. This enables us in particular to compute the energy gap with polynomial efficiency instead of exponential, providing a simple answer to a critical outstanding problem for adiabatic quantum computing [26,27]
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