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A firmer grip on the Hubbard model

Nature volume 545, pages 414415 (25 May 2017) | Download Citation

The Hubbard model describes the behaviour of interacting quantum particles, but many of its properties remain unknown. A system of ultracold atoms could provide the key to determining the model's underlying physics. See Letter p.462

To understand the properties of many materials, we must solve the exceedingly difficult problems that are associated with systems of interacting quantum particles. Among the theoretical descriptions of these systems, the Hubbard model1 can explain phenomena such as magnetic order (the ordering of magnetic moments) and metal–insulator transitions. However, despite its apparent simplicity, the model has proved extremely challenging to analyse. On page 462, Mazurenko et al.2 report a realization of the Hubbard model using a system of ultracold atoms (at temperatures of tens of nanokelvin). The authors demonstrate the emergence and disappearance of magnetic order in their system when it is cooled and when its density is reduced, respectively. These observations suggest that the authors' system could potentially reach the regime in which previously unknown physics is expected to exist in this model.

The Hubbard model1 was introduced in 1963 as being simple, yet also having the main ingredients necessary for describing interacting fermions (elementary particles such as electrons) moving in a solid. These ingredients are: the existence of an atomic lattice in which the fermions can move only between lattice sites; the kinetic energy of the fermions; and a simplified picture in which two fermions interact with one another only if they occupy the same lattice site (Fig. 1a). Remove one of these ingredients, and most of the non-trivial physics is lost. The Hubbard model does not provide the most microscopically accurate description of real materials, but its usefulness is rather like that of the fruit fly Drosophila in genetic studies or the Ising model3 in phase transitions, in that understanding its properties is a key step forward.

Figure 1: Key properties of the Hubbard model.
Figure 1

a, The Hubbard model describes the behaviour of interacting quantum particles called fermions in an atomic lattice. A fermion can be represented by its position in the lattice and its magnetic moment, the latter being in one of two states: 'up' (blue arrow) or 'down' (red arrow). In the Hubbard model, fermions can hop between lattice sites (indicated by purple dashed arrows), and if two particles occupy the same site, they interact with one another. b, At low temperatures and when there is exactly one fermion per lattice site, the magnetic moments are ordered (there is a regular arrangement of up and down magnetic moments). The system is an insulator with well-understood properties. c, However, for an average density lower than one particle per site, the existence of holes (the absence of particles) in the lattice rapidly scrambles the magnetic order. In this regime, the physics of the Hubbard model is largely unknown, with the possibility that it has exotic properties such as superconductivity — the transport of electric current without energy dissipation. Mazurenko et al.2 report an experiment that probes this largely unexplored regime.

Unfortunately, we lack good analytical methods for tackling the Hubbard model, and because the model deals with fermions that obey the bizarre rules of quantum mechanics, it does not easily lend itself to numerical simulations on a classical (non-quantum) computer. So far, we understand the physics of the model in one spatial dimension4 and in an infinite number of dimensions5. But in 2D — the most relevant case for exotic materials such as high-temperature superconductors6 (in which electric current can be transferred without energy dissipation) — the model stubbornly resists giving up its secrets.

In the late 1990s, help came unexpectedly from a different field of physics, quantum optics. Trapping and cooling matter using lasers (in research that was awarded the 1997 and 2001 physics Nobel prizes) led to the development of artificial solids7, in which the lattice is made by light and atoms are the mobile particles. Such systems offered many advantages in terms of controlling the lattice and the particle interactions, and resulted in interactions on the lattice sites becoming exact experimental features instead of a simplification of reality. This allowed the construction of analog quantum computers that could directly access the physics of the Hubbard model — as was spectacularly demonstrated for the counterpart of the model8, in which the lattice contains particles called bosons, rather than fermions.

Since then, the quest to study the fermionic Hubbard model using ultracold matter has been actively pursued9, with remarkable progress. However, the ultimate goal seems to be blocked by two main obstacles. First, although the temperature of these ultracold atomic systems is low in absolute terms (a few tens of nanokelvin), it is high compared with other energy scales of the system, such as the kinetic energy of the fermions. So far, ultracold-atom experiments have probed the Hubbard model at temperatures corresponding to about 1,000 K in a solid, whereas new physics is expected to arise6 at temperatures of several hundred kelvin.

The second obstacle is that the physics of the Hubbard model is highly dependent on the density of fermions in the lattice. If there is exactly one particle per lattice site, the model offers no surprises — the system has magnetic order and is an insulator whose properties are well known in all spatial dimensions (Fig. 1b). But if the density of fermions in the system is reduced, in a process called doping, the magnetic order is lost and the model enters unknown territory (Fig. 1c). Unfortunately, ultracold atomic systems are usually confined in such a way that their density becomes highly spatially inhomogeneous, causing these different physical properties to be scrambled. Despite intensive efforts and partial solutions to these two problems, it has been unclear how the physics of the 'doped' Hubbard model could be observed in a realistic experiment.

Mazurenko and colleagues' study provides a key step in this direction. The authors use a device called a fermionic microscope, which allows fermions to be accessed and controlled at the level of individual lattice sites. The experimental set-up consists of two parts: a system of particles whose properties are measured and a surrounding sea of particles that acts as a coolant. The authors' system attains temperatures that are substantially lower (at about 12 nK) than observed in previous experiments2. These temperatures allow them to achieve magnetic order for their whole system — a first for these types of experiment.

Even more importantly, Mazurenko et al. can precisely control the density of particles in their system. They show that the magnetic order disappears if the system is doped, in agreement with theoretical expectations of the Hubbard model. Although the authors do not observe new physics, their work is a spectacular advance that puts ultracold-atom experiments firmly in a position to crack the physics of the 2D Hubbard model — in regimes for which the simulations performed in classical computers are becoming almost intractable.

Clearly, Mazurenko and colleagues' experiment will stimulate researchers to use quantum microscopes or other techniques to push the analysis of the Hubbard model further. There is therefore little doubt that rapid progress will be made, and that the doped regime will be explored in a controlled way. The Hubbard model could very well reveal all of its secrets soon, opening the door to an understanding of the materials of the twenty-first century.

Notes

References

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  1. Thierry Giamarchi is in the Department of Quantum Matter Physics, University of Geneva, CH1211 Geneva 4, Switzerland.

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Correspondence to Thierry Giamarchi.

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