Spontaneous Breaking Of Rotational Symmetry Research Articles

Nematic Ising superconductivity with hidden magnetism in few-layer 6R-TaS2

In van der Waals heterostructures (vdWHs), the manipulation of interlayer stacking/coupling allows for the construction of customizable quantum systems exhibiting exotic physics. An illustrative example is the diverse range of states of matter achieved through varying the proximity coupling between two-dimensional (2D) quantum spin liquid (QSL) and superconductors within the TaS2 family. This study presents a demonstration of the intertwined physics of spontaneous rotational symmetry breaking, hidden magnetism, and Ising superconductivity (SC) in the three-fold rotationally symmetric, non-magnetic natural vdWHs 6R-TaS2. A distinctive phase emerges in 6R-TaS2 below a characteristic temperature (T*) of approximately 30 K, which is characterized by a remarkable set of features, including a giant extrinsic anomalous Hall effect (AHE), Kondo screening, magnetic field-tunable thermal hysteresis, and nematic magneto-resistance. At lower temperatures, a coexistence of nematicity and Kondo screening with Ising superconductivity is observed, providing compelling evidence of hidden magnetism within a superconductor. This research not only sheds light on unexpected emergent physics resulting from the coupling of itinerant electrons and localized/correlated electrons in natural vdWHs but also emphasizes the potential for tailoring exotic quantum states through the manipulation of interlayer interactions.

Angle-resolved transport non-reciprocity and spontaneous symmetry breaking in twisted trilayer graphene.

The identification and characterization of spontaneous symmetry breaking is central to our understanding of strongly correlated two-dimensional materials. In this work, we utilize the angle-resolved measurements of transport non-reciprocity to investigate spontaneous symmetry breaking in twisted trilayer graphene. By analysing the angular dependence of non-reciprocity in both longitudinal and transverse channels, we are able to identify the symmetry axis associated with the underlying electronic order. We report that a hysteretic rotation in the mirror axis can be induced by thermal cycles and a large current bias, supporting the spontaneous breaking of rotational symmetry. Moreover, the onset of non-reciprocity with decreasing temperature coincides with the emergence of orbital ferromagnetism. Combined with the angular dependence of the superconducting diode effect, our findings uncover a direct link between rotational and time-reversal symmetry breaking. These symmetry requirements point towards exchange-driven instabilities in momentum space as a possible origin for transport non-reciprocity in twisted trilayer graphene.

Spontaneous rotational symmetry breaking in KTaO3 heterointerface superconductors

Broken symmetries play a fundamental role in superconductivity and influence many of its properties in a profound way. Understanding these symmetry breaking states is essential to elucidate the various exotic quantum behaviors in non-trivial superconductors. Here, we report an experimental observation of spontaneous rotational symmetry breaking of superconductivity at the heterointerface of amorphous (a)-YAlO3/KTaO3(111) with a superconducting transition temperature of 1.86 K. Both the magnetoresistance and superconducting critical field in an in-plane field manifest striking twofold symmetric oscillations deep inside the superconducting state, whereas the anisotropy vanishes in the normal state, demonstrating that it is an intrinsic property of the superconducting phase. We attribute this behavior to the mixed-parity superconducting state, which is an admixture of s-wave and p-wave pairing components induced by strong spin-orbit coupling inherent to inversion symmetry breaking at the heterointerface of a-YAlO3/KTaO3. Our work suggests an unconventional nature of the underlying pairing interaction in the KTaO3 heterointerface superconductors, and brings a new broad of perspective on understanding non-trivial superconducting properties at the artificial heterointerfaces.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible -wave symmetry, in place of the conventional -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Enhanced Superconducting Pairing Strength near a Pure Nematic Quantum Critical Point

The quest for high-temperature superconductivity at ambient pressure is a central issue in physics. In this regard, the relationship between unconventional superconductivity and the quantum critical point (QCP) associated with the suppression of some form of symmetry-breaking order to zero temperature has received particular attention. The key question is how the strength of the electron pairs changes near the QCP, and this can be verified by high-field experiments. However, such studies are limited mainly to superconductors with magnetic QCPs, and the possibility of unconventional mechanisms by which nonmagnetic QCP promotes strong pairing remains a nontrivial issue. Here, we report systematic measurements of the upper critical field $H_{{\rm c2}}$ in nonmagnetic FeSe$_{1-x}$Te$_{x}$ superconductors, which exhibit a QCP of electronic nematicity characterized by spontaneous rotational-symmetry breaking. As the magnetic field increases, the superconducting phase of FeSe$_{1-x}$Te$_{x}$ shrinks to a narrower dome surrounding the nematic QCP. The analysis of $H_{{\rm c2}}$ reveals that the Pauli-limiting field is enhanced toward the QCP, implying that the pairing interaction is significantly strengthened via nematic fluctuations emanated from the QCP. Remarkably, this nonmagnetic nematic QCP is not accompanied by a divergent effective mass, distinct from the magnetically mediated pairing. Our observation opens up a nonmagnetic route to high-temperature superconductivity.

Infinite critical boson non-Fermi liquid

We study a distinct type of non-Fermi liquid where there exists an infinite number of critical bosonic modes instead of finite number of bosonic modes for the conventional ones. We consider itinerant magnets with both conduction electrons and fluctuating magnetic moments in three dimensions. With Dzyaloshinskii–Moriya interaction, the moments fluctuate near a boson surface in the reciprocal space at low energies when the system approaches an ordering transition. The infinite number of critical modes on the boson surface strongly scatter the gapless electrons on the Fermi surface and convert the metallic sector into a non-Fermi liquid. We explain the physical properties of this non-Fermi liquid. On the ordered side, a conventional non-Fermi liquid emerges due to the scattering by the gapless Goldstone mode from the spontaneous breaking of the global rotational symmetry. We discuss the general structure of the phase diagram in the vicinity of the quantum phase transition and clarify various crossover behaviors.

Rhombic skyrmion lattice coupled with orthorhombic structural distortion in EuAl4

The centrosymmetric tetragonal itinerant magnet EuAl$_{4}$ exhibits an intricate magnetic phase diagram including rhombic and square skyrmion-lattice (SkL) phases in the external magnetic field. Here, we report a multi-axis dilatometric investigation of EuAl$_{4}$ by means of a newly designed fiber-Bragg-grating technique complemented by a resonant x-ray scattering experiment, revealing anisotropic magnetostriction and magnetovolume effect associated with successive phase transitions. The rhombic and square SkL phases are found to possess $\sim$0.10% and $\sim$0.03% orthorhombic structural distortion within the $ab$ plane, respectively. We propose that the coupling between the spin system and the lattice deformation should be essential for the structural instability in EuAl$_{4}$, yielding a rich variety of topological spin textures with spontaneous rotational-symmetry breaking as well as a potential controllability of the SkL phases by uniaxial stress or pressure.

In-plane electronic anisotropy revealed by interlayer resistivity measurements on the iron-based superconductor parent compound CaFeAsF

Both cuprates and iron-based superconductors demonstrate nematicity, defined as the spontaneous breaking of rotational symmetry in electron systems. The nematic state can play a role in the high-transition-temperature superconductivity of these compounds. However, the microscopic mechanism responsible for the transport anisotropy in iron-based compounds remains debatable. Here, we investigate the electronic anisotropy of CaFeAsF by measuring its interlayer resistivity under magnetic fields with varying field directions. Counterintuitively, the interlayer resistivity was larger in the longitudinal configuration ($B \parallel I \parallel c$) than in the transverse one ($B \perp I \parallel c$). The interlayer resistivity exhibited a so-called coherence peak under in-plane fields and was highly anisotropic with respect to the in-plane field direction. At $T$ = 4 K and $B$ = 14 T, the magnetoresistance $\Delta\rho/\rho_0$ was seven times larger in the $B \parallel b_o$ than in the $B \parallel a_o$ configuration. Our theoretical calculations of the conductivity based on the first-principles electronic band structure qualitatively reproduced the above observations but underestimated the magnitudes of the observed features. The proposed methodology can be a powerful tool for probing the nematic electronic state in various materials.

Robust superconductivity in magic-angle multilayer graphene family.

The discovery of correlated states and superconductivity in magic-angle twisted bilayer graphene (MATBG) established a new platform to explore interaction-driven and topological phenomena. However, despite multitudes of correlated phases observed in moiré systems, robust superconductivity appears the least common, found only in MATBG and more recently in magic-angle twisted trilayer graphene. Here we report the experimental realization of superconducting magic-angle twisted four-layer and five-layer graphene, hence establishing alternating twist magic-angle multilayer graphene as a robust family of moiré superconductors. This finding suggests that the flat bands shared by the members play a central role in the superconductivity. Our measurements in parallel magnetic fields, in particular the investigation of Pauli limit violation and spontaneous rotational symmetry breaking, reveal a clear distinction between the N = 2 and N > 2-layer structures, consistent with the difference between their orbital responses to magnetic fields. Our results expand the emergent family of moiré superconductors, providing new insight with potential implications for design of new superconducting materials platforms.

Pure nematic quantum critical point accompanied by a superconducting dome

SignificanceThe notion of the quantum critical point (QCP) is at the core of modern condensed matter physics. Near a QCP of the symmetry-breaking order, associated quantum-mechanical fluctuations are intensified, which can lead to unconventional superconductivity. Indeed, dome-shaped superconducting phases are often observed near the magnetic QCPs, which supports the spin fluctuation-driven superconductivity. However, the fundamental question remains as to whether a nonmagnetic QCP of electronic nematic order characterized by spontaneous rotational symmetry breaking can promote superconductivity in real materials. Here, we provide an experimental demonstration that a pure nematic QCP exists near the center of a superconducting dome in nonmagnetic FeSe[Formula: see text] Tex. This result evidences that nematic fluctuations enhanced around the nematic QCP can boost superconductivity.

Supersolidity in Two-Dimensional Trapped Dipolar Droplet Arrays.

We theoretically investigate the ground states and the spectrum of elementary excitations across the superfluid to droplet crystallization transition of an oblate dipolar Bose-Einstein condensate. We systematically identify regimes where spontaneous rotational symmetry breaking leads to the emergence of a supersolid phase with characteristic collective excitations, such as the Higgs amplitude mode. Furthermore, we study the dynamics across the transition and show how these supersolids can be realized with standard protocols in state-of-the-art experiments.

Nematicity in a cuprate superconductor revealed by angle-resolved photoemission spectroscopy under uniaxial strain

The nature of the pseudogap and its relationship with superconductivity are one of the central issues of cuprate superconductors. Recently, a possible scenario has been proposed that the pseudogap state is a distinct phase characterized by spontaneous rotational symmetry breaking called “nematicity” based on transport and magnetic susceptibility measurements, where the symmetry breaking was observed below the pseudogap temperature T∗. Here, we report a temperature-dependent ARPES study of nematicity in slightly overdoped Bi1.7Pb0.5Sr1.9CaCu2O8+δ triggered by a uniaxial strain applied along one of the Cu–O bond directions. While the nematicity was enhanced in the pseudogap state as in the previous studies, it was suppressed in the superconducting state. These results indicate that the pseudogap state is characterized by spontaneous rotational symmetry breaking and that the nematicity may compete with superconductivity. Relationship between the nematicity and charge-density waves, both of which are observed in the pseudogap state, is discussed.

Unconventional density waves and superconductivities in Fe-based superconductors and other strongly correlated electron systems

To seek high-$T_c$ pairing mechanism, many scientists have focused on the mysterious spontaneous rotational symmetry breaking above Tc, such as nematic order at $q=0$ and smectic order at $q\ne0$. Such exotic correlation-driven symmetry breaking in metals has become a central issue in condensed matter physics. We demonstrate the emergence of the nematic and smectic orders due to orbital polarization ($n_{xz}\ne n_{yz}$) and the symmetry breaking in the correlated intersite hopping (= bond order $\delta t_{i,j}$) in Fe-based and cuprate superconductors. In addition, we discuss exotic spontaneous loop current orders driven by the pure imaginary $\delta t_{i,j}$. These interesting ``unconventional density-waves'' originate from the quantum interference between different spin fluctuations that is described by the vertex correction (VC) in the field theory. In the next stage, we discuss electron-correlation driven superconductivity due to the fluctuations of unconventional density-waves. For this purpose, we suggest the beyond-Migdal-Eliashberg gap equation by including the VCs into the equation. In Fe-based superconductors, high-$T_c$ $s$-wave superconductivity can be mediated by nematic and smectic fluctuations because the pairing interaction is magnified by the VCs. We also discuss the multipolar fluctuation pairing mechanism in heavy fermion systems, owing to the cooperation between the strong spin-orbit interaction and the strong electron correlation. To summarize, we suggest that the quantum interference mechanism described by the VCs is the key ingredients to explain not only various unconventional density-waves, but also exotic superconducting states in many strongly correlated metals. We finally discuss some interesting future issues with respect to the quantum interference mechanism.

Electric-field-tunable electronic nematic order in twisted double-bilayer graphene

Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moiré scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moiré degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moiré systems.

Superconductivity and intra-unit-cell electronic nematic phase in the three-band model of cuprates

The intra-unit-cell nematic phase is studied within the three-band Emery model of the cuprates by using the diagrammatic expansion of the Gutzwiller wave function (DE-GWF). According to our analysis a spontaneous rotational (C4) symmetry breaking of the electronic wave function, leading to the nematic behavior, can appear due to electron correlations induced mainly by the onsite Coulomb repulsion, even in the absence of the corresponding intersite oxygen–oxygen repulsion term. The latter has been considered as the triggering factor of the nematic state formation in a number of previous studies. Also, we show that at the transition to the nematic phase, electron concentration transfer from d- to p-orbitals takes place, apart from the usually discussed px∕py polarization. The nematicity appears in a similar doping range as the paired phase, showing that both phases may have a common origin, even though they compete. As we show a coexistence region of both superconductivity and nematicity appears in a relatively wide doping range. The results are discussed in view of the experimental findings corresponding to the relation between nematicity and pseudogap behavior.Graphical abstract

Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model

The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-N limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding spacetime. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.

Spontaneous rotational symmetry breaking in a Kramers two-level system

Here, I develop a model for a two-level system that respects the time-reversal symmetry of the atom Hamiltonian and the Kramers theorem. The two-level system is formed by two Kramers pairs of excited and ground states. It is shown that due to the spin-orbit interaction it is in general impossible to find a basis of atomic states for which the crossed transition dipole moment vanishes. The parametric electric polarizability of the Kramers two-level system for a definite ground-state is generically nonreciprocal. I apply the developed formalism to study Casimir-Polder forces and torques when the two-level system is placed nearby either a reciprocal or a nonreciprocal substrate. In particular, I investigate the stable equilibrium orientation of the two-level system when both the atom and the reciprocal substrate have symmetry of revolution about some axis. Surprisingly, it is found that when chiral-type dipole transitions are dominant the stable ground state is not the one in which the symmetry axes of the atom and substrate are aligned. The reason is that the rotational symmetry may be spontaneously broken by the quantum vacuum fluctuations, so that the ground state has less symmetry than the system itself.

Possible two-component pairings in electron-doped Bi2Se3 based on a tight-binding model

Recent experiments show the spontaneous breaking of rotational symmetry in the superconducting topological insulators M$_{x}$Bi$_{2}$Se$_{3}$ (M represents Cu, Sr, or Nd), suggesting that the pairing belongs to a two-dimensional representation of the $D_{3d}$ symmetry group of the crystal. Motivated by these progresses, we construct an exhaustive list of possible two-component pairings of the M$_{x}$Bi$_{2}$Se$_{3}$ superconductors, both for the odd-parity $E_{u}$ representation and for the even-parity $E_{g}$ representation. Starting from a tight-binding model for the normal phase of Bi$_{2}$Se$_{3}$ and M$_{x}$Bi$_{2}$Se$_{3}$, we firstly construct the pairing channels in the spin-orbital basis, up to second-nearest-neighbor pairing correlations in the basal plane. We then infer the properties of these pairings by transforming them to the band (pseudospin) basis for the conduction band. A comparison with the key experimental consensuses on M$_{x}$Bi$_{2}$Se$_{3}$ superconductors shows that the true pairings should also be multichannel. Besides a nematic and time-reversal symmetric pairing combination, the other pairings that we have identified are chiral and nematic at the same time, which may be nonunitary and have a spontaneous magnetization. A complementary set of experiments are proposed to identify the true pairing symmetries of these superconductors and their evolution with the doping concentration $x$.

Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules.

We report on spontaneous rotational symmetry breaking in a minimal model of complex macromolecules with branches and cycles. The transition takes place as the strength of the self-repulsion is increased. At the transition point, the density distribution transforms from isotropic to anisotropic. We analyze this transition using a variational mean-field theory that combines the Gibbs-Bogolyubov-Feynman inequality with the concept of the Laplacian matrix. The density distribution of the broken symmetry state is shown to be determined by the eigenvalues and eigenvectors of this Laplacian matrix. Physically, this reflects the increasing role of the underlying topological structure in determining the density of the macromolecule when repulsive interactions generate internal tension. Eventually, the variational free energy landscape develops a complex structure with multiple competing minima.

Nematic Superconductivity in Doped Bi2Se3 Topological Superconductors

Nematic superconductivity is a novel class of superconductivity characterized by spontaneous rotational-symmetry breaking in the superconducting gap amplitude and/or Cooper-pair spins with respect to the underlying lattice symmetry. Doped Bi 2 Se 3 superconductors, such as Cu x Bi 2 Se 3 , Sr x Bi 2 Se 3 , and Nb x Bi 2 Se 3 , are considered as candidates for nematic superconductors, in addition to the anticipated topological superconductivity. Recently, various bulk probes, such as nuclear magnetic resonance, specific heat, magnetotransport, magnetic torque, and magnetization, have consistently revealed two-fold symmetric behavior in their in-plane magnetic-field-direction dependence, although the underlying crystal lattice possesses three-fold rotational symmetry. More recently, nematic superconductivity was directly visualized using scanning tunneling microscopy and spectroscopy. In this short review, we summarize the current research on the nematic behavior in superconducting doped Bi 2 Se 3 systems and discuss issues and perspectives.