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Intertwined Order in Strongly Correlated Quantum Systems

Milan Rakic

Published Feb 24, 2025• 0 citations

Abstract

Supersolids are a fascinating quantum phase of matter that have both superfluid and solid properties, and are in some ways an extension of a superfluid, in that it possesses an additional rich and dynamic set of phenomenological features. Supersolids are an example of strongly correlated intertwined order, which is the notion that systems can form phases of matter that are more than the sum of its parts. Put simply, a supersolid is not a superfluid + solid. This intertwined order generates new and exciting physics which looks a little like its constituent components, but also not at all. In this thesis we develop a homogenisation technique, which begins with a microscopic theory of a supersolid and ends with a phenomenological Lagrangian. By doing this we marry two opposite ends of supersolid theory, providing the phenomenological toolset to study supersolids, but rooted in the foundation of microscopic theory. We derive the mathematical framework necessary to employ homogenisation, apply it to the Gross-Pitaevskii Lagrangian, elucidate crucial contributions from the thermodynamic ensemble of a supersolid, and obtain an effective theory of the Goldstone modes. We conduct numerics on a soft-core bosonic system, applying homogenisation to our simulated supersolids, and verify our results with Bogoliubov theory. We find an unexpected result that the supersolid has two distinct values for bulk compressibility, which takes it outside of the class of materials for which this thermodynamic relation holds. To the best of our knowledge, this is the first such example of this particular physical feature in any material. We then extend homogenisation to include beyond-meanfield effects, in order to apply it to a dipolar BEC. The dipolar BEC has a considerably richer phase diagram, in which we find several exciting results.

DOI: 10.25560/117861

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Supersolids are a fascinating quantum phase of matter that have both superfluid and solid properties, and are in some ways an extension of a superfluid, in that it possesses an additional rich and dynamic set of phenomenological features. Supersolids are an example of strongly correlated intertwined order, which is the notion that systems can form phases of matter that are more than the sum of its parts. Put simply, a supersolid is not a superfluid + solid. This intertwined order generates new and exciting physics which looks a little like its constituent components, but also not at all. In this thesis we develop a homogenisation technique, which begins with a microscopic theory of a supersolid and ends with a phenomenological Lagrangian. By doing this we marry two opposite ends of supersolid theory, providing the phenomenological toolset to study supersolids, but rooted in the foundation of microscopic theory. We derive the mathematical framework necessary to employ homogenisation, apply it to the Gross-Pitaevskii Lagrangian, elucidate crucial contributions from the thermodynamic ensemble of a supersolid, and obtain an effective theory of the Goldstone modes. We conduct numerics on a soft-core bosonic system, applying homogenisation to our simulated supersolids, and verify our results with Bogoliubov theory. We find an unexpected result that the supersolid has two distinct values for bulk compressibility, which takes it outside of the class of materials for which this thermodynamic relation holds. To the best of our knowledge, this is the first such example of this particular physical feature in any material. We then extend homogenisation to include beyond-meanfield effects, in order to apply it to a dipolar BEC. The dipolar BEC has a considerably richer phase diagram, in which we find several exciting results.

We study a supersolid in the context of a Gross-Pitaevskii theory with a nonlocal effective potential. We employ a homogenization technique which allows us to calculate the elastic moduli, supersolid fraction, and other state variables of the system. Our methodology is verified against numerical simulations of elastic deformations. We can also verify that the long-wavelength Goldstone modes that emerge from this technique agree with Bogoliubov theory. We find a thermodynamic anomaly that the supersolid does not obey the thermodynamic relation ∂ P / ∂ V | N = − n ( ∂ P / ∂ N | V ) , which we claim is a feature unique to supersolids. Published by the American Physical Society 2024

Abstract Supersolid states simultaneously feature properties typically associated with a solid and with a superfluid. Like a solid, they possess crystalline order, manifesting as a periodic modulation of the particle density; but unlike a typical solid, they also have superfluid properties, resulting from coherent particle delocalization across the system. Such states were initially envisioned in the context of bulk solid helium, as a possible answer to the question of whether a solid could have superfluid properties1–5. Although supersolidity has not been observed in solid helium (despite much effort)6, ultracold atomic gases provide an alternative approach, recently enabling the observation and study of supersolids with dipolar atoms7–16. However, unlike the proposed phenomena in helium, these gaseous systems have so far only shown supersolidity along a single direction. Here we demonstrate the extension of supersolid properties into two dimensions by preparing a supersolid quantum gas of dysprosium atoms on both sides of a structural phase transition similar to those occurring in ionic chains17–20, quantum wires21, 22 and theoretically in chains of individual dipolar particles23, 24. This opens the possibility of studying rich excitation properties25–28, including vortex formation29–31, and ground-state phases with varied geometrical structure7, 32 in a highly flexible and controllable system.

Supersolidity, combining superfluid and crystalline orders, has been realized in dipolar Bose-Einstein condensates by tuning interatomic interactions. Here we show that supersolidity can also emerge from mode coupling at a superfluid-solid interface, without modifying bulk interactions and for a broad class of superfluids. Using an analytical and numerical treatment of the coupled superfluid and phonon fields, we derive the criterion for a density-modulation instability driven by interfacial coupling and dependent on dimensionality. In superfluid helium, the instability first appears at the roton mode, while in a Bose-Einstein condensate with contact interactions it occurs at the lowest accessible wave vector set by the system size. Beyond the threshold, the ground state acquires an interfacial density modulation while the bulk remains superfluid, forming a hybrid superfluid-supersolid phase. Our results identify interfacial mode coupling as a promising route to supersolidity, enabling the simultaneous exploitation of interfacial supersolid and bulk superfluid quantum properties.

In one dimension, quantum fluctuations prevent the appearance of long-range order in a supersolid, and only quasi-long-range order can survive. We derive this quantum critical behavior and study its influence on the superfluid response and properties of the solid. The analysis is based on an effective low-energy description accounting for the two coupled Goldstone modes. We find that the quantum phase transition from the superfluid to the supersolid is shifted by quantum fluctuations from the position where the local formation of a solid structure takes place. For current experimental parameters with dipolar atomic gases, this shift is extremely small and cannot be resolved yet, i.e., current observations in experiments are expected to be in agreement with predictions from mean-field theory based on the extended Gross-Pitaevskii formalism.

The supersolid phase of soft-core bosons in two dimensions is investigated using the self-consistent Hartree-Fock and quantum Monte Carlo methods. An approximate phase diagram at finite temperatures is initially constructed using the mean-field approach, which is subsequently validated through precise path-integral simulations, enabling a microscopic characterization of the various phases. Superfluid and melting/freezing transitions are analyzed through the superfluid density and the long-range behavior of correlation functions associated with positional and orientational order, in accordance with the general picture of Berezinskii-Kosterlitz-Thouless transitions. A broad region at low temperatures is identified where the supersolid phase exists, separating the uniform superfluid phase from the normal quasicrystal phase. Additionally, a potential intermediate hexatic phase with quasi-long-range orientational order is identified in a narrow region between the normal solid and fluid phases. These findings establish self-consistent Hartree-Fock theory beyond the local density approximation as an effective tool, complementary to computationally intensive quantum Monte Carlo simulations, for investigating the melting of the supersolid phase and the possible emergence of the hexatic superfluid phase in bosonic systems with various interaction potentials.

Conclusive experimental evidence of a supersolid phase in any known condensed matter system is presently lacking. On the other hand, a supersolid phase has been recently predicted for a system of spinless bosons in continuous space, interacting via a broad class of soft-core, repulsive potentials. Such an interaction can be engineered in assemblies of ultracold atoms, providing a well-defined pathway to the unambiguous observation of this fascinating phase of matter. In this Letter, we study by first principles computer simulations the elementary excitation spectrum of the supersolid, and show that it features two distinct modes, namely, a solidlike phonon and a softer collective excitation, related to broken translation and gauge symmetry, respectively.