&Bullet; physics 14, 88

A new model suggests that disorder can be a critical component in generating non-Fermi-liquid behavior in a system of interacting fermions.

In 1956, the physicist Lev Landau developed a theory that describes the low-temperature properties of helium-3 as those of a liquid of interacting fermions. Landau’s Fermi fluid theory proved to be broadly applicable. For example, it successfully describes the low temperature properties of most metals. However, there are numerous fermionic systems that are not Fermi-liquids, including one-dimensional “Luttinger” liquids, Mott insulators, and heavy Fermion materials. Understanding these exotic systems is an important direction of research in modern condensed matter physics, but describing them theoretically is devilishly difficult. For simplified cases, there are only a handful of solvable non-Fermi-liquid models. Based on one of these models, the so-called Sachdev-Ye-Kitaev (SYK) model [1] , a team led by Subir Sachdev from Harvard University has developed a new theory that answers an important, unsolved question: How does a non-Fermi-liquid behavior arise in a system of condensed matter – for example one in which Fermions (like electrons) are coupled to massless bosons (like phonons)? The team’s results show that non-Fermi fluid behavior can occur in a 2D system if the coupling between the bosons and the fermions is sufficiently disordered [2] .

In order to understand what could destroy the behavior of Fermi fluids, it is helpful to remember an important property of Fermi fluids – they allow a purely local description in momentum space. That means that their stimuli are “quasiparticles” with well-defined impulses. As a result of the uncertainty principle, this localization in the momentum space corresponds to a local delocalization – Fermi liquids show extensive entanglement in real space. This delocalization makes Fermi liquids robust against local disturbances such as short-range repulsive Coulomb interactions. However, changing the sign of the interactions can destroy the Fermi-liquid behavior. In conventional superconductors, for example, the coupling of electrons to phonons creates an attractive interaction – the Cooper pairing interaction. This change in sign creates the superconducting state, which can be viewed as the instability of a Fermi liquid.

In the absence of an attractive interaction, one possible strategy for finding non-Fermi fluids is to introduce some type of long-range or non-local interaction. Indeed, an early proposal took advantage of the far-reaching correlations emerging at a critical point in a quantum phase transition [3] . However, such an idea has led to a theory that is not “well behaved”.[4] Violation of a recipe for the construction of a low-energy theory by the physicist Kenneth Wilson [5] .

In the early 1990s, researchers made advances by developing two precisely solvable models that use the non-local interaction pathway to generate non-Fermi-fluid behavior. Both models evoke all-to-all interactions – each particle interacts with all other particles. The Hatsugai-Kohmoto (HK) model involves the hopping of electrons with a given kinetic energy between neighboring positions in a 2D lattice and involves a constant two-body interaction between all electrons that meet a center of mass constraint [6] . The result is an exactly solvable model that creates a Mott isolation phase when the interaction exceeds the width of the electronic bands [6, 7] . To date, this approach is the only model for Mott insulators that can be solved independently of the grid dimensions. The second approach is the SYK model, which contains two-body all-to-all interaction terms that, in contrast to those in the HK model, are not constant, but are distributed randomly. This model is precisely solvable if the number of “flavors” of fermions (fermions belonging to different bands can be viewed as different flavors) is large. The SYK model provides significant deviations from the Fermi liquid behavior, such as an anomalous power law scaling of the decay of the temporal correlations between particles. The original SYK model is zero-dimensional (it contains time but no space coordinates), but extensions to higher dimensions have been suggested [8] .

Both models show that non-local interactions, be they constant (HK) or random (SYK), are sufficient to break the long-range real space entanglement on which the quasiparticle image of a Fermi liquid is based. Researchers have attempted to extend the SYK model to the case of a Fermi fluid coupled to a massless boson, but these attempts had the side effect of producing non-zero unphysical entropy at a temperature of zero [9] .

The theory developed by Sachdev’s team avoids the entropy problem. The trick behind the researchers’ result lies in a special coupling of the bosons to the fermions. In earlier work this coupling was assumed to be constant. The innovation is that the coupling strength is a random variable that is tied to the boson and fermion flavors and is described by a Gaussian distribution. The best way to think about flavor space randomness is to look at a model in which fermions in multiple bands are coupled to phonons via band-dependent phonon coupling that is randomly distributed. Such a randomness can be thought of as a disorder in the system [10] . The first step in the researchers’ analysis is to average over the random distribution. Assuming a large number of flavors, the team comes up with a non-Fermi liquid theory that shows strong coupling between the bosons and fermions, but also captures the overall nature of the perturbation through certain correlation functions. The researchers also find that the resulting theory has a space-time-like symmetry that leads to a disappearance of the soft mode at the critical Fermi surface (the Fermi surface at the quantum critical point). It is precisely the absence of this soft mode that makes the theory well educated [2] .

The overall picture that paints this work is that the right theory is all about introducing the right type of disorder. SYK alone is not enough to get such a theory. Adding a second element – fermion-coupled bosons – to SYK is also insufficient [8] . However, adding clutter in the fermion-boson coupling might do the job. Since there are experimental proposals to test the SYK physics with experiments with cold atoms, the mechanism described by Sachdev’s team may also motivate experimental realizations. Finally, a thought-provoking idea is to examine the analogies between the breakdown of the quasiparticle image of Fermi fluid and the breakdown of the particle image adopted by a theory known as “undarticle physics”. [11] . Could one achieve a reformulation of the boson-fermion coupling that resembles the continuous mass formulation of non-particle physics?

## References

1. S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet”, Phys. Rev. Lett.70, 3339 (1993).
2. I. Esterlis et al., “Large-No Theory of the critical Fermi surfaces “, Phys. Rev. B103, 235129 (2021).
3. JA Hertz, “Quantum-Critical Phenomena”, Phys. Rev. B14th, 1165 (1976).
4. S.-S. Lee, “Low-energy effective theory of the Fermi surface coupled with U (1) -Eichfeld in
$2+1$

Dimensions,” Phys. Rev. B80, 165102 (2009).

5. Wilson showed that proper low-energy theory requires integration of the high-energy degrees of freedom, but the early proposals did just the opposite. In practice, the integration of low energy modes leads to an infinite number of quantum corrections.
6. Y. Hatsugai and M. Kohmoto, “Exactly solvable model of correlated lattice electrons in any dimensions”, J.Phys. Social Jpn61, 2056 (1992).
7. PW Phillips et al., “Exact theory for superconductivity in a doped Mott insulator”, Nat. Phys.16, 1175 (2020).
8. Y. Gu et al., “Local criticality, dissemination and chaos in generalized Sachdev-Ye-Kitaev models”, J. High Energy Phys.2017, 125 (2017).
9. AA Patel et al., “Magnetotransport in a model of a disordered strange metal”, Phys. Rev. X8th, 021049 (2018).
10. Y. Wang, “Solvable strongly coupling quantum dot model with a non-Fermi-liquid pairing transition”, Phys. Rev. Lett.124, 017002 (2020); EE Aldape et al., “Solvable Theory of a Strange Metal in the Collapse of a Heavy Fermi Fluid”, arXiv: 2012.00763.
11. NG Deshpande and X.-G. He, “Realization of subparticles through continuous mass-scale invariant theories”, Phys. Rev. D78, 055006 (2008).

## About the author

Philip W. Phillips is a condensed matter theorist who studies highly correlated electron matter, mainly high temperature superconductivity and strange metals. He approaches such problems by applying ideas at the interface between condensed matter and high energy physics, such as the gauge-gravitational duality.

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