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arXiv:1201.5576 · Physical Review A

Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices

Elena V. Gorelik, D. Rost, Thereza Paiva, Richard T. Scalettar, Andreas Klümper, Nils Blümer

Published Jun 7, 2012• 37 citations
arXivHTMLPDF

Abstract

We determine antiferromagnetic (AF) signatures in the half-filled Hubbard model at strong coupling on a cubic lattice and in lower dimensions. Upon cooling, the transition from the charge-excitation regime to the AF Heisenberg regime is signaled by a universal minimum of the double occupancy at entropy $s\ensuremath{\equiv}S/(N{k}_{\text{B}})={s}^{*}\ensuremath{\approx}\mathrm{ln}(2)$ per particle and a linear increase of the next-nearest-neighbor (NNN) spin correlation function for $s<{s}^{*}$. This crossover, driven by a gain in kinetic exchange energy, appears as the essential AF physics relevant for current cold-atom experiments. The onset of long-range AF order (at low $s$ on cubic lattices) is hardly visible in nearest-neighbor spin correlations versus $s$, but could be detected in spin correlations at or beyond NNN distances.

DOI
10.1103/physreva.85.061602

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We determine antiferromagnetic (AF) signatures in the half-filled Hubbard model at strong coupling on a cubic lattice and in lower dimensions. Upon cooling, the transition from the charge-excitation regime to the AF Heisenberg regime is signaled by a universal minimum of the double occupancy at entropy s\ensuremath≡S/(NkB)=s∗\ensuremath≈ln(2)s\ensuremath{\equiv}S/(N{k}_{\text{B}})={s}^{*}\ensuremath{\approx}\mathrm{ln}(2)s\ensuremath≡S/(NkB​)=s∗\ensuremath≈ln(2) per particle and a linear increase of the next-nearest-neighbor (NNN) spin correlation function for s<s∗s<{s}^{*}s<s∗. This crossover, driven by a gain in kinetic exchange energy, appears as the essential AF physics relevant for current cold-atom experiments. The onset of long-range AF order (at low sss on cubic lattices) is hardly visible in nearest-neighbor spin correlations versus sss, but could be detected in spin correlations at or beyond NNN distances.
We verify signatures of antiferromagnetic (AF) correlations in the double occupancy D [Gorelik et al., PRL 105, 065301 (2010)] and study their dimensional dependence using direct quantum Monte Carlo in dimensions d=2,3 and Bethe Ansatz in d=1. We find quantitative agreement with dynamical mean-field theory (DMFT) in the cubic case and qualitative agreement down to d=1. As a function of entropy s=S/(N k_B), D is nearly universal with respect to d; the minimum in D(s) approaches s=log(2) at strong coupling, as predicted by DMFT. Long-range order appears hardly relevant for the current search of AF signatures in cold fermions. Thus, experimentalists need not achieve s
One of the major challenges in realizing antiferromagnetic and superfluid phases in optical lattices is the ability to cool fermions. We determine constraints on the entropy for observing these phases in two-dimensional Hubbard models using determinantal quantum Monte Carlo simulations. We find that an entropy per particle approximately = ln2 is sufficient to observe the insulating gap in the repulsive Hubbard model at half-filling, or the pairing pseudogap in the attractive case. Observing antiferromagnetic correlations or superfluidity in 2D systems requires a further reduction in entropy by a factor of 3 or more. In contrast with higher dimensions, we find that adiabatic cooling is not useful to achieve the required low temperatures. We also show that double-occupancy measurements are useful for thermometry for temperatures greater than the nearest-neighbor hopping energy.
Abstract The interaction-driven crossover from a Slater insulator to a Mott insulator in the Néel-ordered ground state of the Hubbard model is a fundamental paradigm of strongly correlated electrons, yet its quantitative characterization has remained elusive. Here we establish a clear and experimentally accessible thermal criterion for this crossover via the sign change of the temperature derivative of double occupancy, ( ∂D / ∂T ) U , near zero temperature. In the Slater regime, ( ∂D / ∂T ) U > 0 reflects the major role of charge fluctuations; in the Mott regime, the anomalous ( ∂D / ∂T ) U < 0, a manifestation of the Pomeranchuk effect, signals the dominance of low-energy spin superexchange physics. Using exact diagonalization and numerically exact quantum Monte Carlo simulations, we demonstrate that this criterion determines the crossover boundary at U cross / t = 4.0(2) for the half-filled two-dimensional Hubbard model. Furthermore, we obtain a consistent boundary independently from the maximum in the antiferromagnetic correlation length, which also arises from the superexchange physics. These two thermal signatures are theoretically unified through the local minimum of thermal entropy versus interaction U at low temperatures. Our results offer a direct, measurable, and physically intuitive framework to identify the SlaterMott crossover in optical lattice experiments.
The repulsive Hubbard Hamiltonian is one of the foundational models describing strongly correlated electrons and is believed to capture essential aspects of high-temperature superconductivity. Ultracold fermions in optical lattices allow for the simulation of the Hubbard Hamiltonian with control over kinetic energy, interactions, and doping. A great challenge is to reach the required low entropy and to observe antiferromagnetic spin correlations beyond nearest neighbors, for which quantum gas microscopes are ideal. Here, we report on the direct, single-site resolved detection of antiferromagnetic correlations extending up to three sites in spin-1/2 Hubbard chains, which requires entropies per particle well below s* = ln(2). The simultaneous detection of spin and density opens the route toward the study of the interplay between magnetic ordering and doping in various dimensions.
Various topics at the interface between condensed matter physics and the physics of ultra-cold fermionic atoms in optical lattices are discussed. The lectures start with basic considerations on energy scales, and on the regimes in which a description by an effective Hubbard model is valid. Qualitative ideas about the Mott transition are then presented, both for bosons and fermions, as well as mean-field theories of this phenomenon. Antiferromagnetism of the fermionic Hubbard model at half-filling is briefly reviewed. The possibility that interaction effects facilitate adiabatic cooling is discussed, and the importance of using entropy as a thermometer is emphasized. Geometrical frustration of the lattice, by suppressing spin long-range order, helps revealing genuine Mott physics and exploring unconventional quantum magnetism. The importance of measurement techniques to probe quasiparticle excitations in cold fermionic systems is emphasized, and a recent proposal based on stimulated Raman scattering briefly reviewed. The unconventional nature of these excitations in cuprate superconductors is emphasized.
The 2D Hubbard model with nearest-neighbor hopping on the square lattice and an average of one electron per site is known to undergo an extended crossover from metallic to insulating behavior driven by proliferating antiferromagnetic correlations. We study signatures of this crossover in spin and charge correlation functions and present results obtained with controlled accuracy using the diagrammatic Monte Carlo approach in the range of parameters amenable to experimental verification with ultracold atoms in optical lattices. The qualitative changes in charge and spin correlations associated with the crossover are observed at well-separated temperature scales, which encase the intermediary regime of non-Fermi-liquid character, where local magnetic moments are formed and nonlocal fluctuations in both channels are essential.