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Quantitative relationship between structural orthorhombicity, shear modulus, and heat capacity anomaly of the nematic transition in iron-based superconductors
Electronic nematicity in iron pnictide materials has been extensively studied by various experimental techniques, yet its heat capacity anomaly at the phase transition has not been examined quantitatively. In this paper, we review the thermodynamic description of nematicity in Ba(Fe 1- x Cox) 2 As 2 using the Landau free energy, which defines the behavior of three thermodynamic quantities: the structural orthorhombicity that develops below the nematic transition, the softening shear modulus above the transition, and the discontinuous heat capacity at the transition. We derive a quantitative relationship between these three quantities, which is found to hold for a range of dopings. We report this result shows that the nematic transition is exceedingly well described by a mean-field model in the underdoped regime of the phase diagram.
Elastocaloric evidence for a multicomponent superconductor stabilized within the nematic state in Ba(Fe 1− x Co x ) 2 As 2
The iron-based high-$T_c$ superconductors (SCs) exhibit rich phase diagrams with intertwined phases, including magnetism, nematicity, and superconductivity. The superconducting $T_c$ in many of these materials is maximized in the regime of strong nematic fluctuations, making the role of nematicity in influencing the superconductivity a topic of intense research. Here, in this study, we use the AC elastocaloric effect (ECE) to map out the phase diagram of Ba(Fe 1− x Co x ) 2 As 2 near optimal doping. The ECE signature at $T_c$ on the overdoped side, where superconductivity condenses without any nematic order, is quantitatively consistent with other thermodynamic probes that indicate a single-component superconducting state. In contrast, on the slightly underdoped side, where superconductivity condenses within the nematic phase, ECE reveals a second thermodynamic transition proximate to and below $T_c$. We rule out magnetism and reentrant tetragonality as the origin of this transition and find that our observations strongly suggest a phase transition into a multicomponent superconducting state. This implies the existence of a subdominant pairing instability that competes strongly with the dominant $s^±$ instability. Our results highlight the significant role of nematic order in determining the pairing symmetry close to optimal doping in this extensively studied iron-based SC, while also demonstrating the power of ECE in uncovering strain-tuned phase diagrams of quantum materials.
Multiple Charge Density Waves and Superconductivity Nucleation at Antiphase Domain Walls in the Nematic Pnictide Ba 1– x Sr x Ni 2 As 2
How superconductivity interacts with charge or nematic order is one of the great unresolved issues at the center of research in quantum materials. Ba 1– x Sr x Ni 2 As 2 (BSNA) is a charge ordered pnictide superconductor recently shown to exhibit a sixfold enhancement of superconductivity due to nematic fluctuations near a quantum phase transition (at $x_c$ = 0.7). The superconductivity is, however, anomalous, with the resistive transition for $0.4 < x < x_c$ occurring at a higher temperature than the specific heat anomaly. Here, using x-ray scattering, we discovered a new charge density wave (CDW) in BSNA in this composition range. The CDWis commensurate with a period of two lattice parameters, and is distinct from the two CDWs previously reported in this material. We argue that the anomalous transport behavior arises from heterogeneous superconductivity nucleating at antiphase domain walls in this CDW. We also present new data on the incommensurate CDW, previously identified as being unidirectional, showing that it is a rotationally symmetric “4$\textit{Q}$” state with $C_4$ symmetry. Our study establishes BSNA as a rare material containing three distinct CDWs, and an exciting test bed for studying coupling between CDW, nematic, and SC orders.
Growth of Nanocrystal Superlattices from Liquid Crystals
The growth of superlattices (SLs) made from self-assembled nanocrystals (NCs) is a powerful method for creating new materials and gaining insight into fundamental molecular dynamics. Previous explorations of NCSL syntheses have mostly compared them to crystallization. However, NCSL synthesis has not broadly shown cooling crystallization from saturated solutions as a reversible crystallization–dissolution process. We demonstrate the reversible growth of NCSLs by dispersing NCs in liquid crystal (LC) “smart solvents,” and harnessing the transitions between the isotropic and nematic phases of the LCs. The growth mode and morphology can be tuned. Furthernore, this process is a model platform for studying crystallization and demonstrates great potential in manufacturing NCSLs as colloidal crystals through liquid-phase epitaxy or colloidal synthesis.
Positional ordering induced by dynamic steric interactions in superparamagnetic rods
The dynamic motion produced by precessing magnetic fields can drive matter into far-from-equilibrium states. We predict 1D periodic ordering in systems of precessing rods when magnetic interactions between rods remain insignificant. The precession angle of the rods is completely determined by the field's precession angle and the ratio of the field's precession frequency and the characteristic response frequency of the rods. Here, we develop a molecular dynamics model that explicitly calculates magnetic interactions between particles, and we also simulate rods in the limit of a strong and fast precessing magnetic field where inter-rod magnetic interactions are negligible, using a purely steric model. Our simulations show how steric interactions drive the rods from a positionally disordered phase (nematic) to a layered (smectic) phase. As the rod precession angle increases, the nematic–smectic transition density significantly decreases. The minimization of unfavorable steric interactions also induces phase separation in binary mixtures of rods of different lengths. This effect is general to any force that produces precession in elongated particles. This work will advance the understanding and control of out-of-equilibrium soft matter systems.
Spontaneous orbital polarization in the nematic phase of FeSe
The origin of nematicity in FeSe remains a critical outstanding question towards understanding unconventional superconductivity in proximity to nematic order. To understand what drives the nematicity, it is essential to determine which electronic degree of freedom admits a spontaneous order parameter independent from the structural distortion. In this work, we use X-ray linear dichroism at the Fe K pre-edge to measure the anisotropy of the 3d orbital occupation as a function of in situ applied stress and temperature across the nematic transition. Along with using X-ray diffraction to precisely quantify the strain state, we reveal a lattice-independent, spontaneously ordered orbital polarization within the nematic phase, as well as an orbital polarizability that diverges as the transition is approached from above. These results provide strong evidence that spontaneous orbital polarization serves as the primary order parameter of the nematic phase.
Phase diagram of generalized XY model using the tensor renormalization group
We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T , and the deformation parameter, Δ , and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024
Elasticity and dynamics of uniaxial nematic liquid crystal with defects: Nemator model
Not Available
Discovery of spontaneous mesoscopic strain waves in nematic domains using dark-field x-ray microscopy
Electronic nematic order arises when correlated electrons spontaneously break the rotational symmetry of a crystal lattice. When electronic nematic order couples bilinearly to symmetry-breaking lattice strain, both appear together at a single ferroelastic phase transition, producing structural twin domains with distinct orientations of the nematic director. While the effects of applied strain on these domains are well established, the intrinsic behavior of spontaneous subdomain strain fields has remained unexplored. Here, we report the discovery of spontaneous mesoscopic strain waves within individual nematic domains of an iron-based superconductor, observed using dark-field x-ray microscopy (DFXM). Using this advanced full-field imaging technique, we visualize subdomain strain modulations emerging concurrently with nematic order. Elastic compatibility relations governing inhomogeneous strains provide a natural mechanism for the strain waves. Our findings reveal a broadly relevant form of strain self-organization and position DFXM as a powerful probe of the local interplay between lattice strain and electronic order.
Hydrostatic and Uniaxial Pressure Tuning of Iron-Based Superconductors: Insights into Superconductivity, Magnetism, Nematicity, and Collapsed Tetragonal Transitions
Iron-based superconductors are well-known for their intriguing phase diagrams, which manifest a complex interplay of electronic, magnetic and structural degrees of freedom. Among the phase transitions observed are superconducting, magnetic, and several types of structural transitions, including a tetragonal-to-orthorhombic and a collapsed-tetragonal transition. In particular, the widely-observed tetragonal-to-orthorhombic transition is believed to be a result of an electronic order that is coupled to the crystalline lattice and is, thus, referred to asnematic transition. Nematicity is therefore a prominent feature of these materials, which signals the importance of the coupling of electronic and lattice properties. Correspondingly, these systems are particularly susceptible to tuning via pressure (hydrostatic, uniaxial, or some combination). We review efforts to probe the phase diagrams of pressure-tuned iron-based superconductors, with a strong focus on our own recent insights into the phase diagrams of several members of this mate-rial class under hydrostatic pressure. These studies on FeSe, Ba(Fe 1-x Co x ) 2 As 2 , Ca(Fe 1-x Co x ) 2 As 2 and CaK(Fe 1-x Ni x ) 4 As 4 were, to a significant extent, made possible by advances of what measurements can be adapted to the use under differing pressure environments. We point out the potential impact of these tools for the study of the wider class of strongly correlated electron systems.
The smectic Z A phase: Antiferroelectric smectic order as a prelude to the ferroelectric nematic
We have structurally characterized the liquid crystal (LC) phase that can appear as an intermediate state when a dielectric nematic, having polar disorder of its molecular dipoles, transitions to the almost perfectly polar-ordered ferroelectric nematic. This intermediate phase, which fills a 100-y-old void in the taxonomy of smectic LCs and which we term the “smectic Z A ,” is antiferroelectric, with the nematic director and polarization oriented parallel to smectic layer planes, and the polarization alternating in sign from layer to layer with a 180 Å period. A Landau free energy, originally derived from the Ising model of ferromagnetic ordering of spins in the presence of dipole–dipole interactions, and applied to model incommensurate antiferroelectricity in crystals, describes the key features of the nematic–SmZ A –ferroelectric nematic phase sequence.
Cascade of vestigial orders in two-component superconductors: Nematic, ferromagnetic, s -wave charge- 4 e , and d -wave charge- 4 e states
Electronically ordered states that break multiple symmetries can melt in multiple stages, similarly to liquid crystals. In a partially melted phase, known as vestigial phase, a bilinear made out of combinations of the multiple components of the primary order parameter condenses. Multicomponent superconductors are thus natural candidates for vestigial order since they break both the U(1)-gauge and also time-reversal or lattice symmetries. Here, we use group theory to classify all possible real-valued and complex-valued bilinears of a generic two-component superconductor on a tetragonal or hexagonal lattice. While the more widely investigated real-valued bilinears correspond to vestigial nematic or ferromagnetic order, the little explored complex-valued bilinears correspond to a vestigial charge-4e condensate, which itself can have an underlying s-wave, d x 2 –y 2 -wave, or d xy -wave symmetry. To properly describe the fluctuating regime of the superconducting Ginzburg-Landau action and thus access these competing vestigial phases, we employ both a large-N and a variational method. We show that while vestigial order can be understood as a weak-coupling effect in the large-N approach, it is akin to a moderate-coupling effect in the variational method. Despite these distinctions, both methods yield similar results in wide regions of the parameter space spanned by the quartic Landau coefficients. Specifically, we find that the nematic and ferromagnetic phases are the leading vestigial instabilities, whereas the various types of charge-4e order are attractive albeit subleading vestigial channels. The only exception is for the hexagonal case, in which the nematic and s-wave charge-4e vestigial states are degenerate. Finally, we discuss the limitations of our approach, as well as the implications of our results for the realization of exotic charge-4e states in material candidates.
Thermally reconfigurable monoclinic nematic colloidal fluids
Fundamental relationships are believed to exist between the symmetries of building blocks and the condensed matter phases that they form. For example, constituent molecular and colloidal rods and disks impart their uniaxial symmetry onto nematic liquid crystals, such as those used in displays. Low-symmetry organizations could form in mixtures of rods and disks, but entropy tends to phase-separate them at the molecular and colloidal scales, whereas strong elasticity-mediated interactions drive the formation of chains and crystals in nematic colloids. To have a structure with few or no symmetry operations apart from trivial ones has so far been demonstrated to be a property of solids alone, but not of their fully fluid condensed matter counterparts, even though such symmetries have been considered theoretically and observed in magnetic colloids. In this work, we show that dispersing highly anisotropic charged colloidal disks in a nematic host composed of molecular rods provides a platform for observing many low-symmetry phases. Depending on the temperature, concentration and surface charge of the disks, we find nematic, smectic and columnar organizations with symmetries ranging from uniaxial to orthorhombic and monoclinic. With increasing temperature, we observe unusual transitions from less- to more-ordered states and re-entrant phases. Most importantly, we demonstrate the presence of reconfigurable monoclinic colloidal nematic order, as well as the possibility of thermal and magnetic control of low-symmetry self-assembly. Our experimental findings are supported by theoretical modelling of the colloidal interactions between disks in the nematic host and may provide a route towards realizing many low-symmetry condensed matter phases in systems with building blocks of dissimilar shapes and sizes, as well as their technological applications.
Emergent tetragonality in a fundamentally orthorhombic material
Symmetry plays a key role in determining the physical properties of materials. By Neumann’s principle, the properties of a material remain invariant under the symmetry operations of the space group to which the material belongs. Continuous phase transitions are associated with a spontaneous reduction in symmetry. Less common are examples where proximity to a continuous phase transition leads to an increase in symmetry. We find signatures of an emergent tetragonal symmetry close to a charge density wave (CDW) bicritical point in a fundamentally orthorhombic material, ErTe 3 , for which the two distinct CDW phase transitions are tuned via anisotropic strain. We first establish that tension along the a axis favors an abrupt rotation of the CDW wave vector from the c to a axis and infer the presence of a bicritical point where the two continuous phase transitions meet. We then observe a divergence of the nematic elastoresistivity approaching this putative bicritical point, indicating an emergent tetragonality in the critical behavior.
Half-quantum vortices in nematic and chiral phases of 3 He
In this work, we report theoretical results for the stability of half-quantum vortices (HQVs) in the superfluid phases of 3 He confined in highly anisotropic Nafen aerogel. Superfluidity of 3 He confined in Nafen is the realization of a “nematic superfluid” with Cooper pairs condensed into a single p -wave orbital aligned along the anisotropy axis of the Nafen aerogel. In addition to the nematic phase, we predict a second “chiral” phase that onsets at a lower transition temperature. This chiral phase spontaneously breaks time-reversal symmetry and is a topological superfluid. Both superfluid phases are equal-spin pairing condensates that host arrays of HQVs as equilibrium states of rotating superfluid 3 He. We present results for the structure of HQVs, including magnetic and topological signatures of HQVs in both the nematic and chiral phases of 3 He-Nafen.
Role of Nematicity in Controlling Spin Fluctuations and Superconducting Tc in Bulk FeSe
FeSe undergoes a transition from a tetragonal to a slightly orthorhombic phase at 90 K and becomes a superconductor below 8 K. The orthorhombic phase is sometimes called a nematic phase because quantum oscillation, neutron, and other measurements detect a significant asymmetry in x and y. How nematicity affects superconductivity has recently become a matter of intense speculation. Here, we employ an advanced ab initio Green's function description of superconductivity and show that bulk tetragonal FeSe would, in principle, superconduct with almost the same critical temperature Tc as the nematic phase. The mechanism driving the observed nematicity is not yet understood. Since the present theory underestimates it, we simulate the full nematic asymmetry by artificially enhancing the orthorhombic distortion. For benchmarking, we compare theoretical spin susceptibilities against experimentally observed data over all energies and relevant momenta. When the orthorhombic distortion is adjusted to correlate with observed nematicity in spin susceptibility, the enhanced nematicity causes spectral weight redistribution in the Fe-3dxz and Fe-dyz orbitals, but it leads to at most a 10-15% increment in Tc. This is because the dxy orbital always remains the most strongly correlated and provides most of the source of the superconducting glue. Nematicity suppresses the density of states at the Fermi level; nevertheless, Tc increases, in contradiction to both BCS theory and the theory of Bose-Einstein condensation. We show how the increase is connected to the structure of the particle-particle vertex. Our results suggest that while nematicity may be an intrinsic property of bulk FeSe, it is not the primary force driving the superconducting pairing.
Strain-Induced Quantum Phase Transitions in Magic-Angle Graphene
In this work, we investigate the effect of uniaxial heterostrain on the interacting phase diagram of magic-angle twisted bilayer graphene. Using both self-consistent Hartree-Fock and density-matrix renormalization group calculations, we find that small strain values (ε~0.1%-0.2%) drive a zero-temperature phase transition between the symmetry-broken "Kramers intervalley-coherent" insulator and a nematic semimetal. The critical strain lies within the range of experimentally observed strain values, and we therefore predict that strain is at least partly responsible for the sample-dependent experimental observations.