Abstract
Quantum magnets with a large SU($N$) symmetry are a promising playground for the discovery of new forms of exotic quantum matter. Motivated by recent experimental efforts to study SU($N$) quantum magnetism in samples of ultracold fermionic alkaline-earth-like atoms in optical lattices, we study here the temperature dependence of spin correlations in the SU($N$) Heisenberg spin model in a wide range of temperatures. We uncover a sizeable regime in temperature, starting at $T=\infty$ down to intermediate temperatures and for all $N\ge2$, in which the correlations have a common spatial structure on a broad range of lattices, with the sign of the correlations alternating from one Manhattan shell to the next, while the amplitude of the correlations is rapidly decreasing with distance. Focussing on the one-dimensional chain and the two-dimensional square and triangular lattice for certain $N$, we discuss the appearance of a disorder and a Lifshitz temperature, separating the commensurate Manhattan high-$T$ regime from a low-$T$ incommensurate regime. We observe that this temperature window is associated to an approximately $N$-independent entropy reduction from the $\ln(N)$ entropy at infinite temperature. Our results are based on high-temperature series arguments and as well as large-scale numerical full diagonalization results of thermodynamic quantities for SU($3$) and SU($4$) square lattice samples, corresponding to a total Hilbert space of up to $4\times 10^9$ states.
Highlights
More than a decade ago first proposals put forward the use of internal states of ultracold atoms in order to implement various models of SU(N ) quantum magnetism and multiorbital physics [1,2,3,4]
We find a unifying pattern, the Manhattan structure, where spin correlations are organized in shells of equal Manhattan distance, and alternating in sign from one shell to the
For selected cases we have investigated how the Manhattan regime breaks down at low temperature through indicators such as the disorder or Lifshitz temperature
Summary
More than a decade ago first proposals put forward the use of internal states of ultracold atoms in order to implement various models of SU(N ) quantum magnetism and multiorbital physics [1,2,3,4]. While the general phase diagram of this model is to a large extent unknown, and the charting thereof constitutes one of the goals of the experimental investigations, it is clear that at integer fillings, in the limit of strong repulsive interactions U |t| and low temperature T U , Mott insulating phases do occur, where charge fluctuations are suppressed, and the system is insulating, i.e. charge transport is inhibited In this limit, the description of the system can be simplified by projecting out the local occupancies away from the considered integer filling and adopting an SU(N ) symmetric effective spin model. As mentioned in the introduction, our goal is to explore and characterize the structure of spin correlations in a temperature regime where the charge fluctuations can be neglected, i.e., at T U This is an experimentally relevant regime, as some of the currently reported experiments operate at entropies per particle around or somewhat below ln N [25] in the Mott regime. Our work is based on simple high-temperature series considerations [98], complete numerical exact diagonalization (ED) of periodic finite size clusters using a basis of SU(N ) Young tableaux [77] and numerical linked cluster expansion [99] results [ in the SU(N ) Young tableaux basis], and aims to explore the structure and temperature behavior of spin correlations in the currently experimentally accessible temperature or entropy regime in SU(N ) Mott insulators across a variety of (mostly two-dimensional) lattices
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