##### Light-Induced Quantum Droplet Phases of Lattice Bosons in Multimode Cavities

P. Karpov, F. Piazza

Physical Review Letters 128, 103201 (2022).

Multimode optical cavities can be used to implement interatomic interactions which are highly tunable in strength and range. For bosonic atoms trapped in an optical lattice we show that, for any finite range of the cavity-mediated interaction, quantum self-bound droplets dominate the ground state phase diagram. Their size and in turn density is not externally fixed but rather emerges from the competition between local repulsion and finite-range cavity-mediated attraction. We identify two different regimes of the phase diagram. In the strongly glued regime, the interaction range exceeds the droplet size and the physics resembles the one of the standard Bose-Hubbard model in a (self-consistent) external potential, where in the phase diagram two incompressible droplet phases with different filling are separated by one with a superfluid core. In the opposite weakly glued regime, we find instead direct first order transitions between the two incompressible phases, as well as pronounced metastability. The cavity field leaking out of the mirrors can be measured to distinguish between the various types of droplets.

##### Machine-learned phase diagrams of generalized Kitaev honeycomb magnets

N. Rao, K. Liu, M. Machaczek, L. Pollet

Physical Revies Research 3, 033223 (2021).

We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine, to investigate the low-temperature classical phase diagram of a generalized HeisenbergKitaev-Gamma (J-K-F) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated S-3 x Z(3) phase, which emerges through the competition between the Kitaev and Gamma spin liquids, against Heisenberg interactions. The results imply that, in the restricted parameter space spanned by the three primary exchange interactions-J, K, and Gamma, the representative Kitaev material alpha-RuCl3 lies close to the boundaries of several phases, including a simple ferromagnet, the unconventional S-3 x Z(3), and nested zigzag-stripy magnets. A zigzag order is stabilized by a finite Gamma' and/or J(3) term, whereas the four magnetic orders may compete in particular if Gamma' is antiferromagnetic.

##### Inferring hidden symmetries of exotic magnets from detecting explicit order parameters

N. Rao, K. Liu, L. Pollet

Physical Review E 104, 015311 (2021).

An unconventional magnet may be mapped onto a simple ferromagnet by the existence of a high-symmetry point. Knowledge of conventional ferromagnetic systems may then be carried over to provide insight into more complex orders. Here we demonstrate how an unsupervised and interpretable machine-learning approach can be used to search for potential high-symmetry points in unconventional magnets without any prior knowledge of the system. The method is applied to the classical Heisenberg-Kitaev model on a honeycomb lattice, where our machine learns the transformations that manifest its hidden O(3) symmetry, without using data of these high-symmetry points. Moreover, we clarify that, in contrast to the stripy and zigzag orders, a set of D2 and D2h ordering matrices provides a more complete description of the magnetization in the Heisenberg-Kitaev model. In addition, our machine also learns the local constraints at the phase boundaries, which manifest a subdimensional symmetry. This paper highlights the importance of explicit order parameters to many-body spin systems and the property of interpretability for the physical application of machine-learning techniques.

##### Pairing and the spin susceptibility of the polarized unitary Fermi gas in the normal phase

L.Physical Review A Rammelmüller, Y. Hou, J.E. Drut, J. Braun

Physical Review A 103, 043330 (2021).

We theoretically study the pairing behavior of the unitary Fermi gas in the normal phase. Our analysis is based on the static spin susceptibility, which characterizes the response to an external magnetic field. We obtain this quantity by means of the complex Langevin approach and compare our calculations to available literature data in the spin-balanced case. Furthermore, we present results for polarized systems, where we complement and expand our analysis at high temperature with high-order virial expansion results. The implications of our findings for the phase diagram of the spin-polarized unitary Fermi gas are discussed in the context of the state of the art.

##### Revealing the phase diagram of Kitaev materials by Machine Learning: Cooperation and Competition between spin Liquids

K. Liu, N. Sadoune, N. Rao, J. Greitemann, L. Pollet

Physical Review Research 3, 023016 (2021).

Kitaev materials are promising materials for hosting quantum spin liquids and investigating the interplay of topological and symmetry-breaking phases. We use an unsupervised and interpretable machine-learning method, the tensorial-kernel support vector machine, to study the honeycomb Kitaev-Γ model in a magnetic field. Our machine learns the global classical phase diagram and the associated analytical order parameters, including several distinct spin liquids, two exotic S3 magnets, and two modulated S3×Z3 magnets. We find that the extension of Kitaev spin liquids and a field-induced suppression of magnetic order already occur in the large-S limit, implying that critical parts of the physics of Kitaev materials can be understood at the classical level. Moreover, the two S3×Z3 orders are induced by competition between Kitaev and Γ spin liquids and feature a different type of spin-lattice entangled modulation, which requires a matrix description instead of scalar phase factors. Our work provides a direct instance of a machine detecting new phases and paves the way towards the development of automated tools to explore unsolved problems in many-body physics.

##### The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet

J. Greitemann, K. Liu, L. Pollet

Journal of Physics-Condensed Matter 33 (5), 054002 (2021).

We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial D-3h spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading 3x3<i correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.

##### Z(2) Parton Phases in the Mixed-Dimensional t - J(z) Model

F. Grusdt, L. Pollet

Physical Review Letters 125 (25), 256401 (2020).

We study the interplay of spin and charge degrees of freedom in a doped Ising antiferromagnet, where the motion of charges is restricted to one dimension. The phase diagram of this mixed-dimensional t - J(z) model can be understood in terms of spinless chargons coupled to a Z(2) lattice gauge field. The antiferromagnetic couplings give rise to interactions between Z(2) electric field lines which, in turn, lead to a robust stripe phase at low temperatures. At higher temperatures, a confined meson-gas phase is found for low doping whereas at higher doping values, a robust deconfined chargon-gas phase is seen, which features hidden antiferromagnetic order. We confirm these phases in quantum Monte Carlo simulations. Our model can be implemented and its phases detected with existing technology in ultracold atom experiments. The critical temperature for stripe formation with a sufficiently high hole concentration is around the spin-exchange energy J(z), i.e., well within reach of current experiments.

##### Constrained random phase approximation of the effective Coulomb interaction in lattice models of twisted bilayer graphene

T.I. Vanhala, L. Pollet

Physical Review B 102 (3), 035154 (2020).

Recent experiments on twisted bilayer graphene show the urgent need for establishing a low-energy lattice model for the system. We use the constrained random phase approximation to study the interaction parameters of such models, taking into account screening from the moire bands left outside the model space. Based on an atomic-scale tight-binding model, we numerically compute the polarization function and study its behavior for different twist angles. We discuss an approximation scheme which allows us to compute the screened interaction, in spite of the very large number of atoms in the unit cell. We find that the polarization has three different momentum regimes. For small momenta, the polarization is quadratic, leading to a linear dielectric function expected for a two-dimensional dielectric material. For large momenta, the polarization becomes independent of the twist angle and approaches that of uncoupled graphene layers. In the intermediate-momentum regime, the dependence on the twist angle is strong. Close to the largest magic angle the dielectric function peaks at a momentum of 1/(4 nm), attaining values of 18-25, depending on the exact model, meaning very strong screening at intermediate distances. We also calculate the effective screened Coulomb interaction in real space and give estimates for the on-site and extended interaction terms for the recently developed hexagonal-lattice model. For freestanding twisted bilayer graphene, the effective interaction decays slower than 1/r at intermediate distances r, while it remains essentially unscreened at large enough r.

##### Identification of emergent constraints and hidden order in frustrated magnets using tensorial kernel methods of machine learning

J. Greitemann, K. Liu, L.D.C. Jaubert, H. Yan, N. Shannon, L. Pollet

Physical Review B 100 (17), 174408 (2019).

Machine-learning techniques have proved successful in identifying ordered phases of matter. However, it remains an open question how far they can contribute to the understanding of phases without broken symmetry, such as spin liquids. Here we demonstrate how a machine-learning approach can automatically learn the intricate phase diagram of a classical frustrated spin model. The method we employ is a support vector machine equipped with a tensorial kernel and a spectral graph analysis which admits its applicability in an effectively unsupervised context. Thanks to the interpretability of the machine we are able to infer, in closed form, both order parameter tensors of phases with broken symmetry, and the local constraints which signal an emergent gauge structure, and so characterize classical spin liquids. The method is applied to the classical XXZ model on the pyrochlore lattice where it distinguishes, among others, between a hidden biaxial spin-nematic phase and several different classical spin liquids. The results are in full agreement with a previous analysis by Taillefumier et al. [Phys. Rev. X 7, 041057 (2017)], but go further by providing a systematic hierarchy between disordered regimes, and establishing the physical relevance of the susceptibilities associated with the local constraints. Our work paves the way for the search of new orders and spin liquids in generic frustrated magnets.

##### Magnetization, d-wave superconductivity, and non-Fermi-liquid behavior in a crossover from dispersive to flat bands

P. Kumar, P. Torma, T.I. Vanhala

Physical Review B 100 (12), 125141 (2019).

We explore the effect of inhomogeneity on electronic properties of the two-dimensional Hubbard model on a square lattice using dynamical mean-field theory (DMFT). The inhomogeneity is introduced via modulated lattice hopping such that in the extreme inhomogeneous limit the resulting geometry is a Lieb lattice, which exhibits a flat-band dispersion. The crossover can be observed in the uniform sublattice magnetization which is zero in the homogeneous case and increases with the inhomogeneity. Studying the spatially resolved frequency-dependent local self-energy, we find a crossover from Fermi-liquid to non-Fermi-liquid behavior happening at a moderate value of the inhomogeneity. This emergence of a non-Fermi liquid is concomitant of a quasiflat band. For finite doping the system with small inhomogeneity displays d-wave superconductivity coexisting with incommensurate spin-density order, inferred from the presence of oscillatory DMFT solutions. The d-wave superconductivity gets suppressed for moderate to large inhomogeneity for any finite doping while the incommensurate spin-density order still exists.

##### Learning multiple order parameters with interpretable machines

K. Liu, J. Greitemann, L. Pollet.

Physical Review B 99, 104410 (2019).

Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.

##### Probing hidden spin order with interpretable machine learning

J. Greitemann, K. Liu, L. Pollet.

Physical Review B 99, 060404(R) (2019).

The search of unconventional magnetic and nonmagnetic states is a major topic in the study of frustrated magnetism. Canonical examples of those states include various spin liquids and spin nematics. However, discerning their existence and the correct characterization is usually challenging. Here we introduce a machine-learning protocol that can identify general nematic order and their order parameter from seemingly featureless spin configurations, thus providing comprehensive insight on the presence or absence of hidden orders. We demonstrate the capabilities of our method by extracting the analytical form of nematic order parameter tensors up to rank 6. This may prove useful in the search for novel spin states and for ruling out spurious spin liquid candidates.