ABSTRACT We study melting in two‐dimensional classical particles with Gaussian‐core interactions in both pure and disordered environments. The pure system exhibits conventional two‐step melting described by Berezinskii‐Kosterlitz–Thouless–Halperin–Nelson–Young (BKTHNY) theory, with a hexatic phase separating solid and liquid. Disorder alters this scenario remarkably: random pinning stabilizes a hexatic‐like phase down to zero temperature, which melts directly into a liquid at , whereas commensurate pinning anchors a solid which undergoes a single‐step transition to the liquid at , removing any intervening hexaticity. Thus, in both cases, the two‐step melting of pure systems turns into a one‐step transition. Beyond elucidating the equilibrium phase behavior, with and without disorder, we uncover dynamical signatures of cooperative string‐like motion of particles at very low temperatures. Remarkably, such correlated dynamics, usually associated with structural glasses and supercooled liquids, emerge here within equilibrium phases, in both pure and disordered systems. We further show that impurities, in the form of pinning, enhance such cooperative motions, leading to slow relaxation and departures from diffusive dynamics. Our results highlight the roles of topological defects, impurities, and cooperative dynamics in governing two‐dimensional melting.

Motivated by the unique characteristics of the phase diagram of 1T-TiSe2, we investigated the complex interplay of excitonic charge density wave (CDW) and superconductivity (SC) on a two-dimensional triangular lattice, each site accommodating two orbitals. In response to various tuning parameters, such as intercalation, pressure, substitution, and gating, these materials exhibit a generic and gradual waning of CDW, followed by the emergence of SC. Intriguingly, the nature of CDW changes from commensurate to incommensurate with the appearance of SC. Setting up a minimal model based on observed excitonic CDW and electron and hole pockets in the Fermi surface, and analyzing it within a simple mean-field framework, we can comprehend some salient experimental features. We also found that the qualitative phase diagram remains insensitive to details of the band structure. These results offer crucial insights into the nature of the interplay between CDW and SC in transition-metal dichalcogenides.

Emerging granularity in superconducting films by tuning disorder is a well-studied topic, both theoretically and experimentally. However, the orbital magnetic field generates a vortex lattice and contributes to the formation of periodic inhomogeneities. Here, we study superconducting films in the simultaneous presence of disorder and a magnetic field, examining how inhomogeneities in various superconducting correlations evolve under these two perturbations. By performing scanning tunneling spectroscopy (STS) on thin films of Sr2VO_3-xFeAs layer structures under both zero and finite orbital magnetic fields, we report impressive similarities between our theoretical results and the experimental findings. Our results have strong implications for identifying the nature of vortices in disordered superconductors, demonstrating a crossover from Abrikosov to Josephson character with increasing disorder, and provide predictive guidance for interpreting STS and current mapping data in complex superconductors.

We investigate the charge density wave phase in the strongly correlated Hubbard model without any other broken symmetry phase. Starting from the atomic Hamiltonian with no hopping, we generate quasiparticle operators corresponding to holons and doublons in the strongly correlated limit of the repulsive Hubbard model. We develop a real-space composite operator formalism using the equation of motion technique to include the intersite hopping perturbatively. Our fully self-consistent calculation stabilizes multiple unidirectional translation symmetry broken states within the doping range delta = 0.07-0.2. The charge-ordered states become increasingly unfavorable with hole doping. The unidirectional density waves manifest as periodic modulations of half-filled Mott regions separated by hole-rich regions. Notably, density wave solutions with periods of 3-8 lattice spacing remain energetically higher than those with larger periods. Quenched disorder on the charge-ordered states induces the merging of the Mott regions and, consequently, forms short-ranged charge modulations. The density of states shows signatures of strongly correlated Mott regions, potentially relevant to the physics of underdoped cuprates.

We use the Bogoliubov-de Gennes (BDG) formalism to undertake a microscopic investigation of a vortex lattice in a strongly correlated, type-II, d-wave superconductor (SC) treating strong correlation within Gutzwiller formalism. We demonstrate that in the underdoped region, the presence of subdominant charge and bond modulation introduces low-energy core states in the local density of states (LDOS) at the vortex core instead of U-shaped (i.e., with a hard gap) Mott core in a strongly correlated d-wave superconductor. Intriguingly, the LDOS at the vortex core is distinct from a metallic core with a Caroli-de Gennes-Matricon (CdGM) peak in the case of weakly coupled superconductors. We have investigated that such subdominant order changes the structure and spectrum of the d-wave vortex in the underdoped region. We have demonstrated the formation of charge and bond modulation at the vortex center by decreasing the doping and reaching an underdoped zone.

When an orbital magnetic field suppresses superconductivity, forming periodic vortices in type-II superconductors, subdominant orders can emerge in the vortex cores. Rather than competing with superconductivity, we find that the emergent charge order within the halo of a vortex makes superconductivity more robust by enhancing the upper critical field. We establish that charge modulations nucleate in and around the vortex core for model parameters dictated by the underlying non-superconducting state. We further show that the spectral signatures from the Caroli-de Gennes-Matricon (CdGM) bound states in vortex cores track the charge modulation. The CdGM-like peak is found to shift toward the gap edge and oscillate from particle-to-hole bias from site to site, signaling charge modulation.

We uncover the dynamics of particles with Gaussian core interactions across melting in pure and disordered two-dimensional (2D) systems. Intriguing signatures of cooperative motion of particles in string-like paths are found at low temperatures. Such a motion, while common to glasses and supercooled liquids, are realized here in traditional equilibrium phases, including in pure systems. We explore the interplay of such motion and impurities and report their repercussions on spatiotemporal correlations. In particular, cooperative motion seems to cause a departure from the diffusive dynamics, causing slow relaxation.

We carry out a microscopic study of a vortex lattice in a strongly correlated, type-II, d-wave superconductor(SC) using Bogoliubov de Gennes (BDG) formalism. In weak-coupling theory, commonly accepted truism is that a vortex binds to impurity. We demonstrate that in unconventional SCs, the binding of vortex to an impurity depends on relevant parameters. In particular, we illustrate such dependency on the sign of impurity, i.e. attractive or repulsive, as well as doping. We emphasize that this seemingly unanticipated behavior arises from strong correlation effects and is absent in weak coupling descriptions.

We study melting in a two-dimensional system of classical particles with Gaussian-core interactions in disordered environments. The pure system validates the conventional two-step melting with a hexatic phase intervening between the solid and the liquid. This picture is modified in the presence of pinning impurities. A random distribution of pinning centers forces a hexaticlike low-temperature phase that transits into a liquid at a single melting temperature T_{m}^{RP}. In contrast, pinning centers located at randomly chosen sites of a perfect crystal anchor a solid at low temperatures which undergoes a direct transition to the liquid at T_{m}^{CP}. Thus, the two-step melting is lost in either case of disorder. We discuss the characteristics of melting depending on the nature of the impurities.

We show that while an orbital magnetic field and disorder, acting individually, weaken superconductivity, acting together they produce an intriguing evolution of a two-dimensional type-II s-wave superconductor. For weak disorder, the critical field Hc at which the superfluid density collapses is coincident with the field at which the superconducting energy gap gets suppressed. However, with increasing disorder these two fields diverge from each other, creating a pseudogap region with insulating vortex cores. Our results naturally explain two outstanding puzzles: the gigantic magnetoresistance peak observed as a function of magnetic field in thin disordered superconducting films and the disappearance of the celebrated zero-bias Caroli-de Gennes-Matricon peak in the local density of states at the vortex core in disordered superconductors.

We show that strong electronic repulsion transforms a vortex core from a metallic type in the overdoped regime to a Mott insulator at underdoping of a strongly correlated d-wave superconductor. This changeover is accompanied by an accumulation of electron density at the vortex core toward local half filling in the underdoped region, which in turn facilitates the formation of the Mott-insulating core. We find that the size of vortices evolves nonmonotonically with doping. A similar nonmonotonicity of critical field H-c2, as extracted from superfluid stiffness, is also found. Our results explain some recent experimental puzzles of cuprate superconductors.

We comprehend the role of imperfections arising from different origins on the universal features in materials consisting of interacting particles. Specifically, we report the static and dynamic responses in a cluster of Coulomb particles in two dimensions. Properties associated with melting in confined systems with pinned impurities are studied, and results are compared with those from irregularly trapped system of particles. While the disorder of first type leads to diffusive single-particle dynamics, the motion of a particle in an irregular trap remains chaotic but ballistic. The many-particle system does not differentiate between these two models of disorder insofar as their qualitative properties are concerned—particularly for describing the thermal melting of the underlying Coulomb-“solid”. However, quantitative differences persist and could be observed—the relaxation time scales differ significantly by tuning impurity concentration. Graphic abstract

The coexistence of multiple quasi-degenerate orders is the hallmark of the strongly correlated materials. Experiments often reveal several spatially modulated orders in the underdoped cuprates. This has come to the forefront with the possible detection of the pair density wave states. However, microscopic calculations often struggle to stabilize such spatially modulating orders as the ground state in the strong correlation limit. This work uses the t − t (cid:48) − J -model with an additional nearest-neighbor repulsion to stabilize spatially oscillating charge, bond, and pairing orders in the underdoped regime. We employ the standard Gutzwiller approach while treating the inhomogeneity for the spatial orders using the self-consistent Hartree-Fock-Bogoliubov methodol-ogy. Our calculations reveal that unidirectional bond density states coexisting with charge and pairing modulations can have lower energy than the uniform superconducting state over an extensive doping range. These modulating states vanish monotonically as the modulation wavevector becomes shorter with increased dopings. The finite momentum orders give way to a vestigial nematic phase upon increasing doping which only breaks the rotational symmetry of the system. The nematic order vanishes upon further increasing doping, and only uniform superconductivity survives. The spatial features of the ground state at each doping reveal multiple wavevectors, which potentially drives the incommensuration of charge orders. Interestingly, the spatially modulating states are absent when the strong correlations criteria are relaxed, suggesting that the removal of double occupancy aids the stabilization of density wave orders.

We show that while orbital magnetic field and disorder, acting individually weaken superconductivity, acting together they produce an intriguing evolution of a two-dimensional type-II s-wave superconductor. For weak disorder, the critical field H_c at which the superfluid density collapses is coincident with the field at which the superconducting energy gap gets suppressed. However, with increasing disorder these two fields diverge from each other creating a pseudogap region. The nature of vortices also transform from Abrikosov vortices with a metallic core for weak disorder to Josephson vortices with gapped and insulating cores for higher disorder. Our results naturally explain two outstanding puzzles: (1) the gigantic magnetoresistance peak observed as a function of magnetic field in thin disordered superconducting films; and (2) the disappearance of the celebrated zero-bias Caroli-de Gennes-Matricon peak in disordered superconductors.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is an unconventional superconducting state found under the influence of strong Zeeman field. This phase is identified by finite center-of-mass momenta in the Cooper pairs, causing the pairing amplitude to oscillate in real space. Repulsive correlations, on the other hand, smear out spatial inhomogeneities in d-wave superconductors. We investigate the FFLO state in a strongly correlated d-wave superconductor within a consolidated framework of Hartree-Fock-Bogoliubov theory and Gutzwiller approximation. We find that the profound effects of strong correlations lie in shifting the BCS-FFLO phase boundary towards a lower Zeeman field and thereby enlarging the window of the FFLO phase. In the FFLO state, our calculation features a sharp mid-gap peak in the density of states, indicating the formation of strongly localized Andreev bound states. We also find that the signatures of the FFLO phase survive even in the presence of an additional translational symmetry breaking competing order in the ground state. This is demonstrated by considering a broken symmetry ground state with a simultaneous presence of the d-wave superconducting order and a spin-density wave order, often found in unconventional superconductors.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is an unconventional superconducting state found under the influence of a strong Zeeman field. This phase is identified by finite center-of-mass momenta in the Cooper pairs, causing the pairing amplitude to oscillate in real space. Repulsive correlations, on the other hand, smear out spatial inhomogeneities in d-wave superconductors. We investigate the FFLO state in a strongly correlated d-wave superconductor within a consolidated framework of Hartree-Fock-Bogoliubov theory and the Gutzwiller approximation. We find that the crucial effects of strong correlations lie in shifting the BCS-FFLO phase boundary towards a lower Zeeman field and thereby enlarging the window of the FFLO phase. In the FFLO state, our calculation features a sharp midgap peak in the density of states, indicating the formation of strongly localized Andreev bound states. We also find that the signatures of the FFLO phase survive even in the presence of an additional translational-symmetry-breaking competing order in the ground state. This is demonstrated by considering a broken-symmetry ground state with a simultaneous presence of the d-wave superconducting order and a spin-density wave order, often found in unconventional superconductors.

We explore the interplay of a charge density wave (CDW) order and s-wave superconductivity (sSC) in a disordered system. Recent experiments on 1T-TiSe_2, where the pristine sample has a commensurate CDW order and the superconductivity appears upon copper intercalation, motivates our study. Starting with an extended Hubbard model, with parameters which yield a CDW ground state within Hartree-Fock-Bogoliubov formalism in pure systems, we show that the addition of disorder quickly wipes out the global charge order by disrupting periodic modulation of density at some (low) strength of disorder. Along with this, the subdominant superconducting order emerges in regions that spatially anti-correlates with islands of strong local CDW order. The short-range density modulations, however, continue to persist and show discernible effects up to a larger disorder strength. The local CDW puddles reduce in size with increasing disorder and they finally lose their relevance in effecting the properties of the system. Our results have strong implications for the experimental phase diagram of transition metal dichalcogenides.

We study various properties of the vibrational normal modes for Coulomb-interacting particles in two-dimensional irregular confinement using numerical simulations. By analyzing the participation ratio and spectral statistics, we characterize the vibrational modes for Coulomb clusters as localized, quasilocalized, and delocalized. We also study a correlation function to understand the spatial structure of these different kinds of modes and subsequently extract the associated characteristic length scales. We further demonstrate that, at any given temperature, particles exhibiting larger displacement over a time interval comparable to the structural relaxation time are strongly correlated with the low-frequency quasilocalized modes of the inherent structure corresponding to the initial configuration. Establishing this correlation for Coulomb clusters paves the path to identify the particular feature of the initial configuration that determines the previously observed heterogeneous dynamics of the particles at low temperatures in these systems.

Motivated by recent proposals of correlation induced insensitivity of d-wave superconductors to impurities, we develop a simple pairing theory for these systems for up to a moderate strength of disorder. Our description implements the key ideas of Anderson, originally proposed for disordered s-wave superconductors, but in addition takes care of the inherent strong electronic repulsion in these compounds, as well as disorder induced inhomogeneities. We first obtain the self-consistent one-particle states, that capture the effects of disorder exactly, and strong correlations using Gutzwiller approximation. These ‘normal states’, representing the interplay of strong correlations and disorder, when coupled through pairing attractions following the path of BardeenCooper-Schrieffer (BCS), produce results nearly identical to those from a more sophisticated Gutzwiller augmented Bogoliubov-de Gennes analysis.