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arXiv:2006.15868 · Physical Review Research

Structure of spin correlations in high-temperature SU(N) quantum magnets

Christian Romen, Andreas Martin Läuchli

Published Oct 1, 2020• 22 citations
arXivHTMLPDF

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 sizable regime in temperature, starting at T = ∞ down to intermediate temperatures and for all N 2, 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 × 10 9 states.

DOI
10.1103/physrevresearch.2.043009

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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 sizable regime in temperature, starting at T = ∞ down to intermediate temperatures and for all N 2, 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 × 10 9 states.
Ultracold fermionic alkaline earth atoms confined in optical lattices realize Hubbard models with internal SU(N) symmetries, where N can be as large as ten. Such systems are expected to harbor exotic magnetic physics at temperatures below the superexchange energy scale. Employing quantum Monte Carlo simulations to access the low-temperature regime of one-dimensional chains, we show that after adiabatically loading a weakly interacting gas into the strongly interacting regime of an optical lattice, the final temperature decreases with increasing N. Furthermore, we estimate the temperature scale required to probe correlations associated with low-temperature SU(N) magnetism. Our findings are encouraging for the exploration of exotic large-N magnetic states in ongoing experiments.
Motivated by the possibility to load multicolor fermionic atoms in optical lattices, we study the entropy dependence of the properties of the one-dimensional antiferromagnetic SU(N) Heisenberg model, the effective model of the SU(N) Hubbard model with one particle per site (filling 1/N) in the large U/t limit. Using continuous-time world-line Monte Carlo simulations for N=2-5, we show that characteristic short-range correlations develop at low temperature as a precursor of the ground state algebraic correlations. We also calculate the entropy as a function of temperature, and we show that the first sign of short-range order appears at an entropy per particle that increases with N and already reaches 0.8k(B) at N=4, in the range of experimentally accessible values.
The thermal evolution of equal-time spin correlations in the antiferromagnetic isotropic spin-1/2 Heisenberg chain is examined by full exact diagonalization of a ten-site periodic cluster. The calculation retains the complete spectrum and evaluates canonical ensemble averages without decoupling approximations. Correlations through the maximum independent separation are reported over 0.10 ≤ kBT/J ≤ 5. The low-temperature state displays the expected alternating antiferromagnetic pattern and appreciable correlations across the finite ring. Heating suppresses the magnitude of every nonlocal correlator, with longer-range terms disappearing first. The nearest-neighbour correlation approaches the high-temperature limit continuously, while an effective finite-cluster correlation length falls rapidly around kBT/J of order unity. Low- and high-temperature consistency checks, SU (2) symmetry, and comparison with Bethe-ansatz expectations are used to validate the computation. The results provide a compact reproducible reference for interpreting thermal loss of short-range order, while clearly separating finite-size numerical evidence from thermodynamic-limit conclusions.
We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as T=0.067T=0.067T=0.067 and interactions up to U=7U=7U=7, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.
This thesis reports on the microscopic investigation of antiferromagnetic order in Hubbard chains, realized with ultracold repulsively interacting fermions in an optical lattice. Extending a quantum gas microscope on spin resolution opened the possibility to Access the full charge and spin statistics of a strongly correlated fermionic many body system. Using this technique, we measured antiferromagnetic correlations over distances up to three lattice sites. Starting with a repulsive degenerate two component mixture of 6Li atoms trapped in a single plane of a vertical lattice, we loaded the Atoms into one-dimensional tubes of a transverse optical superlattice. By adiabatically ramping up the lattice potential along the tubes, we isentropically entered the strongly correlated regime of the one-dimensional Fermi-Hubbard Hamiltonian. We spatially separated the two spin states into opposite sites of local double wells orthogonal to the direction of the lattice before single atom and site sensitive imaging. In this way we extracted the spin information of every single lattice site and were able to evaluate spin-spin correlations. The two component degenerate gases were produced with sub-Poissonian atom number fluctuations by magnetically driven evaporative cooling in vicinity of a Feshbach resonance, used to set the scattering length. The dependence of antiferromagnetic correlations on the driving superexchange coupling was experimentally investigated by varying the interparticle scattering length before loading the atoms into the optical lattice. Taking advantage of the underlying trapping potential of the optical lattice, which shaped the filling and the entropy of the gases, the influence of density excitations and different entropies on spin correlations was observed. By post-selecting the Hubbard chains based on their average filling and local atom number fluctuations, spin correlations corresponding to 58% of the zero temperature predictions in the Heisenberg regime were measured. By comparing the obtained antiferromagnetic correlations to Quantum-Monte-Carlo predictions for our System the lowest measured entropy per particle could be stated as s = 0.51(4)kB, clearly below the critical value of s = ln(2)kB, which is required to form longer ranged correlations. Below this critical entropy, in contrast to local density fluctuations, these spin correlations strongly depend on the entropy of the system. In the future, such a spin thermometer can be used to benchmark novel cooling techniques, which are required to enter the temperature regime of d-wave superconducting states. The access to the full particle and spin statistics will enable us to read out multi-point correlation functions, which characterize such exotic states of matter.
We study kinetic magnetism for the Fermi-Hubbard models in triangular type lattices, including a zigzag ladder, four- and six-legged triangular cylinders and a full two-dimensional triangular lattice. We focus on the regime of strong interactions, U≫tU\gg tU≫t and filling factors around one electron per site. For temperatures well above the hopping strength, the Curie-Weiss form of the magnetic susceptibility suggests effective antiferromagnetic correlations for systems that are hole doped with respect to ν=1ν=1ν=1, and ferromagnetic correlations for systems with electron dopings. We show that these correlations arise from magnetic polaron dressing of charge carrier propagating in a spin incoherent Mott insulator. Effective interactions corresponding to these correlations can strongly exceed the magnetic super-exchange energy. In the case of hole doping, antiferromagnetic polarons originate from kinetic frustration of individual holes in a triangular lattice. In the case of electron doping, Nagaoka type ferromagnetic correlations are induced by propagating doublons. These results provide a theoretical explanation of recent experimental results in moire TMDC materials. To understand many-body states arising from antiferromagentic polarons at low temperatures, we study hole doped systems in finite magnetic fields. At low dopings and intermediate magnetic fields we find a magnetic polaron phase, separated from the fully polarized state by a metamagnetic transition. With decreasing magnetic field the system shows a tendency to phase separate, with hole rich regions forming antiferromagnetic spinbags. We demonstrate that direct observations of magnetic polarons in triangular lattices can be achieved in experiments with ultracold atoms, which allow measurements of three point hole-spin-spin correlations.