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In ferromagnetic materials, magnons - quanta of spin waves - typically resonate in the gigahertz range. Beyond conventional magnons, while theoretical studies have predicted magnons associated with orbital magnetic moments, their direct observation has remained challenging. Here, we present the discovery of two distinct terahertz orbitally-coupled magnon resonances in the topological kagome ferromagnet Co3Sn2S2. Using time-resolved Kerr rotation spectroscopy, we pinpoint two magnon resonances at 0.61 and 0.49 THz at 6 K, surpassing all previously reported magnon resonances in ferromagnets due to strong magnetocrystalline anisotropy. These dual modes originate from the strong coupling of localized spin and orbital magnetic moments. These findings unveil a novel category of magnons stemming from orbital magnetic moments, and position Co3Sn2S2 as a promising candidate for high-speed terahertz spintronic applications

In designing material functionality within the intricate realm of transition metal oxides, lattice structure and d-orbital occupancy are two principal determinants of the correlated physical properties, such as superconductivity. However, the modulation of these two factors is inherently limited by the need to balance thermodynamic stability, kinetic mobility, and synthesis precision, particularly for oxidation-demanding phases. We introduce a methodology, namely the gigantic-oxidative atomically layered epitaxy (GOAL-Epitaxy), enhancing oxidation power 3-4 orders of magnitude beyond oxide molecular beam epitaxy (OMBE) and pulsed laser deposition (PLD), while ensuring atomic-layer-by-layer growth of designed complex structures. Consequently, thermodynamic stability is markedly augmented at elevated temperatures, improving growth kinetics. We demonstrate the accurate synthesis of complex nickelates and cuprates, especially an artificially designed structure as a parent of high-temperature superconductivity, in which alternating single and double NiO2 layers possess distinct nominal d-orbital occupancy. The GOAL-Epitaxy enables material discovery within the vastly broadened growth parameter space.

Recently the experimental evidence of a charge-$6e$ condensed phase in the kagome superconductor $\mathrm{AV_3Sb_5}$ (J. Ge, et. al., Phys. Rev. X 14, 021025 (2024)) has sparked significant interest. At a phenomenological level, the order parameter of the $3Q$ pair-density-wave state can form either a triangular lattice or a kagome lattice, in which the vortices and antivortices locate at the center of the plaquettes. When the phase of a Cooper pair is denoted as a XY spin, the effective model for the phase fluctuation is characterized by the frustrated XY spin model on a triangular lattice with the nearest neighbour antiferromagnetic coupling or a kagome lattice with both nearest neighbour and next-nearest neighbor antiferromagnetic couplings. While both models produce $\frac{1}{3}$ fractional vortex excitations, which arise as the kinks on the domain walls in the low-energy excitations, the triangular lattice model has a much larger domain wall energy than the vortex interaction energy, so the frustrated kagome lattice model becomes unique. By developing a state-of-the-art numerical tensor network method, we rigorously solve this effective model at finite temperatures and confirm the presence of a vestigial phase with $\frac{1}{3}$ vortex-antivortex paired phase in the absence of phase coherence of Cooper pairs, which is dual to the charge-$6e$ condensed phase. Our theory provides a potential explanation for the vestigial charge-$6e$ magneto-resistant oscillations observed in the recent experimental work., Comment: 6 pages, 4 figures

The formalism of the Rokhsar-Kivelson (RK) model has been frequently used to study topological phase transitions in 2D in terms of the deformed wavefunctions, which are RK-type wavefunctions. A key drawback of the deformed wavefunctions is that the obtained quantum critical points are RK-type, in the sense that the equal-time correlation functions are described by 2D conformal field theories (CFTs). The generic Lorentz invariant quantum critical points described by (2+1)D CFTs can not be obtained from the deformed wavefunctions. To address this issue, we generalize the deformed wavefunction approach to the deformed thermofield double (TFD) state methodology. Through this extension, we can effectively reconstruct the absent temporal dimension at the RK-type quantum critical point. We construct deformed TFD states for a (1+1)D quantum phase transition from a symmetry-protected topological phase to a symmetry-breaking phase, and for generic (2+1)D topological phase transitions from a $\mathbb{Z}_2$ topologically ordered phase to a trivial paramagnetic phase., Comment: 10 pages, 4 figures; published version

Advancement of technologies relies on discovery of new materials with emerging physical properties that are determined by their crystal structures. Ruddlesden-Popper (R-P) phases with formula of $A_{n+1}$$B_n$$X_{3n+1}$ (n=1,2,3...) are among one of the most widely studied class of materials due to their electrical, optical, magnetic, thermal properties and their combined multifunctional properties(Ref.1-6). In R-P phases, intergrowth is well-known in the short range(Ref.7-9); however, no existing compounds have been reported to have different n mixed in bulk single crystals. Here we design a hybrid R-P nickelate $La_2NiO_4$$La_3Ni_2O_7$ by alternatively stacking bilayers, which is the active structural motif in the newly discovery high-$T_c$ superconductor $La_3Ni_2O_7$ and single layers of the antiferromagnetic insulator $La_2NiO_4$. We report the successful synthesis of $La_2NiO_4$$La_3Ni_2O_7$ single crystals, and X-ray diffraction and real-space imaging vis STEM show that the crystal structure consists of single layers and bilayers of $NiO_6$ octahedral stacking alternatively perpendicular to the ab plane, characterized by the orthorhombic Immm (No.71) space group. Resistivity measurements indicate a peculiar insulator-to-metal transition around 140 K on cooling. Correlated density functional theory calculations corroborate this finding, and reveal that the single layer becomes paramagnetic metallic due to charge transfer via LaO layers. The discovery of $La_2NiO_4$$La_3Ni_2O_7$ opens a door to access a completely new family of 3D hybrid R-P phases with the formula of $A_{n+1}$$B_n$$X_{3n+1}$$A'_{m+1}$$B'_m$$X'_{3m+1}$ which potentially host a plethora of emerging physical properties for various applications., Comment: 27 pages, 4 figures

Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground state degeneracy. In this work, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigen-equation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our unified theory by numerically solving 2D fully frustrated XY spin models on the kagome, square and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets., Comment: 20 pages, 19 figures

The recent discovery of high-$T_{c}$ superconductivity in bilayer nickelate La$_{3}$Ni$_{2}$O$_{7}$ under high pressure has stimulated great interest concerning its pairing mechanism. We argue that the weak coupling model from the almost fully-filled $d_{z^{2}}$ bonding band cannot give rise to its high $T_{c}$, and thus propose a strong coupling model based on local inter-layer spin singlets of Ni-$d_{z^{2}}$ electrons due to their strong on-site Coulomb repulsion. This leads to a minimal effective model that contains local pairing of $d_{z^{2}}$ electrons and a considerable hybridization with near quarter-filled itinerant $d_{x^{2}-y^{2}}$ electrons on nearest-neighbor sites. Their strong coupling provides a unique two-component scenario to achieve high-$T_{c}$ superconductivity. Our theory highlights the importance of the bilayer structure of superconducting La$_{3}$Ni$_{2}$O$_{7}$ and points out a potential route for the exploration of more high-$T_{c}$ superconductors., Comment: 6 pages, 3 figures

High-Tc superconductivity with possible $T_{c}\approx 80K$ has been reported in the single crystal of $\text{La}_{3}\text{Ni}_{2}\text{O}_{7}$ under high pressure. Based on the electronic structure given from the density functional theory calculations, we propose an effective bi-layer model Hamiltonian including both $3d_{z^{2}}$ and $3d_{x^{2}-y^{2}}$ orbital electrons of the nickel cations. The main feature of the model is that the $3d_{z^{2}}$ electrons form inter-layer $\sigma$-bonding and anti-bonding bands via the apical oxygen anions between the two layers, while the $3d_{x^{2}-y^{2}}$ electrons hybridize with the $3d_{z^{2}}$ electrons within each NiO$_2$ plane. The chemical potential difference of these two orbital electrons ensures that the $3d_{z^{2}}$ orbitals are close to half-filling and the $3d_{x^{2}-y^{2}}$ orbitals are near quarter-filling. The strong on-site Hubbard repulsion of the $3d_{z^{2}}$ orbital electrons gives rise to an effective inter-layer antiferromagnetic spin super-exchange $J$. Applying pressure can self-dope holes on the $3d_{z^{2}}$ orbitals with the same amount of electrons doped on the $3d_{x^{2}-y^{2}}$ orbitals. By performing numerical density-matrix renormalization group calculations on a minimum setup and focusing on the limit of large $J$ and small doping of $3d_{z^{2}}$ orbitals, we find the superconducting instability on both the $3d_{z^{2}}$ and $3d_{x^{2}-y^{2}}$ orbitals by calculating the equal-time spin singlet pair-pair correlation function. Our numerical results have provided useful insights in the high-Tc superconductivity in single crystal La$_3$Ni$_2$O$_7$ under high pressure., Comment: 6 pages, 4 figures; published version

High-transition-temperature (high-T_c) superconductivity in cuprates has been discovered for more than three decades, but the underlying mechanism remains a mystery. Cuprates are the only unconventional superconducting family that host bulk superconductivity with T_cs above the liquid nitrogen boiling temperature at 77 Kelvin. Here we report an observation of superconductivity in single crystals of La3Ni2O7 with a maximum T_c of 80 Kelvin at pressures between 14.0-43.5 gigapascals using high-pressure resistance and mutual inductive magnetic susceptibility measurements. The superconducting phase under high pressure exhibits an orthorhombic structure of Fmmm space group with the 3d_(x^2-y^2 ) and 3d_(z^2 ) orbitals of Ni cations strongly interacting with oxygen 2p orbitals. Our density functional theory calculations suggest the superconductivity emerges coincidently with the metallization of the {\sigma}-bonding bands under the Fermi level, consisting of the 3d_(z^2 ) orbitals with the apical oxygens connecting Ni-O bilayers. Thus, our discoveries not only reveal important clues for the high-T_c superconductivity in this Ruddlesden-Popper double-layered perovskite nickelates but also provide a new family of compounds to investigate the high-T_c superconductivity mechanism., Comment: 15 pages with extended data

We have developed a tensor network approach to the two-dimensional fully frustrated classical XY spin model on the kagome lattice, and clarified the nature of the possible phase transitions of various topological excitations.We find that the standard tensor network representation for the partition function does not work due to the strong frustrations in the low temperature limit. To avoid the direct truncation of the Boltzmann weight, based on the duality transformation, we introduce a new representation to build the tensor network with local tensors lying on the centers of the elementary triangles of the kagome lattice. Then the partition function is expressed as a product of one-dimensional transfer matrix operators, whose eigen-equation can be solved by the variational uniform matrix product state algorithm accurately. The singularity of the entanglement entropy for the one-dimensional quantum operator provides a stringent criterion for the possible phase transitions. Through a systematic numerical analysis of thermodynamic properties and correlation functions in the thermodynamic limit, we prove that the model exhibits a single Berezinskii-Kosterlitz-Thouless phase transition only, which is driven by the unbinding of $1/3$ fractional vortex-antivortex pairs determined at $T_{c}\simeq 0.075J_{1}$ accurately. The absence of long-range order of chirality or quasi-long range order of integer vortices has been verified in the whole finite temperature range. Thus the long-standing controversy about the phase transitions in this fully frustrated XY model on the kagome lattice is solved rigorously, which provides a plausible way to understand the charge-6e superconducting phase observed experimentally in the two-dimensional kagome superconductors., Comment: 14 pages, 13 figures, submitted version for publication

We revisit the ground state of the Hubbard model on 2-legged ladders in this work. We perform DMRG calculation on large system sizes with large kept states and perform extrapolation of DMRG results with truncation errors in the converged region. We find the superconducting correlation exponent $K_{sc}$ extracted from the pair-pair correlation is very sensitive to the position of the reference bond, reflecting a huge boundary effect on it. By systematically removing the effects from boundary conditions, finite sizes, and truncation errors in DMRG, we obtain the most accurate value of $K_{sc}$ and $K_\rho$ so far with DMRG. With these exponents, we confirm that the 2-legged Hubbard model is in the Luther-Emery liquid phase with $K_{sc} \cdot K_\rho = 1$ from tiny doping near half-filling to $1/8$ hole doping. When the doping is increased to $\delta \gtrapprox 1/6$, the behaviors of charge, pairing, and spin correlations don't change qualitatively, but the relationship $K_{sc} \cdot K_\rho = 1$ is likely to be violated. With the further increase of the doping to $\delta = 1/3$, the quasi long-ranged charge correlation turns to a true long-ranged charge order and the spin gap is closed, while the pair-pair correlation still decays algebraically. Our work provides a standard way to analyze the correlation functions when studying systems with open boundary conditions., Comment: 11 pages,11 figures,4 tables

The infinite-layer nickelates, isostructural to the high-Tc superconductor cuprates, have risen as a promising platform to host unconventional superconductivity and stimulated growing interests in the condensed matter community. Despite numerous researches, the superconducting pairing symmetry of the nickelate superconductors, the fundamental characteristic of a superconducting state, is still under debate. Moreover, the strong electronic correlation in the nickelates may give rise to a rich phase diagram, where the underlying interplay between the superconductivity and other emerging quantum states with broken symmetry is awaiting exploration. Here, we study the angular dependence of the transport properties on the infinite-layer nickelate Nd0.8Sr0.2NiO2 superconducting films with Corbino-disk configuration. The azimuthal angular dependence of the magnetoresistance (R({\phi})) manifests the rotational symmetry breaking from isotropy to four-fold (C4) anisotropy with increasing magnetic field, revealing a symmetry breaking phase transition. Approaching the low temperature and large magnetic field regime, an additional two-fold (C2) symmetric component in the R({\phi}) curves and an anomalous upturn of the temperature-dependent critical field are observed simultaneously, suggesting the emergence of an exotic electronic phase. Our work uncovers the evolution of the quantum states with different rotational symmetries and provides deep insight into the global phase diagram of the nickelate superconductors.

To understand the microscopic mechanism of the charge order observed in the parent compound of the infinite-layer nickelate superconductors, we consider a minimal three-legged model consisting of a two-legged Hubbard ladder for the Ni $3d_{x^2-y^2}$ electrons and a free conduction electron chain from the rare-earths. With highly accurate density matrix renormalization group calculations, when the chemical potential difference is adjusted to make the Hubbard ladder with $1/3$ hole doping, we find a long-range charge order with period $3$ in the ground state of the model, while the spin excitation has a small energy gap. Moreover, the electron pair-pair correlation has a quasi-long-range behavior, indicating an instability of superconductivity even at half-filling. As a comparison, the same method is applied to a pure two-legged Hubbard model with $1/3$ hole doping in which the period-3 charge order is a quasi-long range one. The difference between them demonstrates that the free electron chain of the three-legged ladder plays the role of a charge reservoir and enhances the charge order in the undoped infinite-layer nickelates., Comment: 9 pages, 10 figures

Interplay of topology and competing interactions can induce new phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative $Z_3$ Potts degrees of freedom,and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the inter-component $Z_3$ Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the inter-component $Z_3$ Potts LRO appears, accompanying with the binding of nematic vortices. In the nematic regime, however, the inter-component $Z_3$ Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative $Z_3$ Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional long-range order, driven by multiple topological defects., Comment: 18 pages, 13 figures. The title is slightly modified, and the supplementary materials are included

A charge order (CO) with a wavevector $\mathbf{q}\simeq\left(\frac{1}{3},0,0\right)$ is observed in infinite-layer nickelates. Here we use first-principles calculations to demonstrate a charge-transfer-driven CO mechanism in infinite-layer nickelates, which leads to a characteristic Ni$^{1+}$-Ni$^{2+}$-Ni$^{1+}$ stripe state. For every three Ni atoms, due to the presence of near-Fermi-level conduction bands, Hubbard interaction on Ni-$d$ orbitals transfers electrons on one Ni atom to conduction bands and leaves electrons on the other two Ni atoms to become more localized. We further derive a low-energy effective model to elucidate that the CO state arises from a delicate competition between Hubbard interaction on Ni-$d$ orbitals and charge transfer energy between Ni-$d$ orbitals and conduction bands. With physically reasonable parameters, $\mathbf{q}=\left(\frac{1}{3},0,0\right)$ CO state is more stable than uniform paramagnetic state and usual checkerboard antiferromagnetic state. Our work highlights the multi-band nature of infinite-layer nickelates, which leads to some distinctive correlated properties that are not found in cuprates., Comment: 29 pages and 5 figures

Random quantum circuit is a minimally structured model to study the entanglement dynamics of many-body quantum systems. In this paper, we considered a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we found that, as the physical error rate is decreased below a critical value $p_c\approx 0.056$, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices., Comment: 6 pages, 4 figures, a few typos are removed

A general framework is proposed to solve the two-dimensional fully frustrated XY model for the Josephson junction arrays in a perpendicular magnetic field. The essential idea is to encode the ground-state local rules induced by frustrations in the local tensors of the partition function. The partition function is then expressed in terms of a product of one-dimensional transfer matrix operator, whose eigen-equation can be solved by an algorithm of matrix product states rigorously. The singularity of the entanglement entropy for the one-dimensional quantum analogue provides a stringent criterion to distinguish various phase transitions without identifying any order parameter a prior. Two very close phase transitions are determined at $T_{c1}\approx 0.4459$ and $T_{c2}\approx 0.4532$, respectively. The former corresponding to a Berezinskii-Kosterlitz-Thouless phase transition describing the phase coherence of XY spins, and the latter is an Ising-like continuous phase transition below which a chirality order with spontaneously broken $Z_2$ symmetry is established., Comment: 10 pages, 7 figures, revised version

We give a brief review of the Mott-Kondo scenario and its consequence in the recently-discovered infinite-layer nickelate superconductors. We argue that the parent state is a self-doped Mott insulator and propose an effective $t$-$J$-$K$ model to account for its low-energy properties. At small doping, the model describes a low carrier density Kondo system with incoherent Kondo scattering at finite temperatures, in good agreement with experimental observation of the logarithmic temperature dependence of electric resistivity. Upon increasing Sr doping, the model predicts a breakdown of the Kondo effect, which provides a potential explanation of the non-Fermi liquid behavior of the electric resistivity with a power law scaling over a wide range of the temperature. Unconventional superconductivity is shown to undergo a transition from nodeless $(d+is)$-wave to nodal $d$-wave near the critical doping due to competition of the Kondo and Heisenberg superexchange interactions. The presence of different pairing symmetry may be supported by recent tunneling measurements., Comment: 8 pages, 6 figures

It is known that the loss of phase coherence of Cooper pairs in two-dimensional (2D) superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order (quasi-LRO) in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D XY model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigen-equation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analogue, various phase transitions can be accurately determined. Below the BKT phase transition, an inter-layer Ising long-range order is established at $T_{Ising}$, and the phase coherence of both intra-layers and inter-layers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multi-critical point. For two inequivalent layers, however, there emerges an intermediate quasi-LRO phase ($T_{Ising}

The most important feature of two-dimensional quantum Rokhsar-Kivelson (RK) type models is that their ground state wavefunction norms can be mapped into the partition functions of two-dimensional statistical models so that the quantum phase transitions become the thermal phase transitions of the corresponding statistical models. For a generic quantum critical point, we generalize the framework of RK wavefunctions by introducing the concept of the thermofield double (TFD) state, which is a purification of the equilibrium density operator. Moreover, by expressing the TFD state in terms of the projected entangled pair state, its $N$-order of R\'{e}nyi entropy results in a three-dimensional statistical model in Euclidian spacetime, describing the generic quantum phase transitions. Using the toric code model with two parallel magnetic fields as an example, we explain these ideas and derive the partition function of the three-dimensional $Z_2$ lattice gauge-Higgs model, where the phase transitions are characterized by the three-dimensional universality classes., Comment: 6 pages, 3 figures. some corrections are made

In tensor network representation, the partition function of a generalized two-dimensional XY spin model with topological integer and half-integer vortex excitations is mapped to a tensor product of one-dimensional quantum transfer operator, whose eigen-equation can be solved by an algorithm of variational uniform matrix product states. Using the singularities of the entanglement entropy, we accurately determine the complete phase diagram of this model. Both the integer vortex-antivortex binding and half-integer vortex-antivortex binding phases are separated from the disordered phase by the usual Berezinskii-Kosterlitz-Thouless (BKT) transitions, while a continuous topological phase transition exists between two different vortex binding phases, exhibiting a logarithmic divergence of the specific heat and exponential divergence of the spin correlation length. A new hybrid BKT and Ising universality class of topological phase transition is thus established. We further prove that three phase transition lines meets at a multi-critical point, from which a deconfinement crossover line extends into the disordered phase., Comment: 6 pages, 4 figures, revised version

We report theoretical predictions on the pairing symmetry of the newly discovered superconducting nickelate Nd$_{1-x}$Sr$_{x}$NiO$_{2}$ based on the renormalized mean-field theory for a generalized model Hamiltonian proposed in [Phys. Rev. B \textbf{101}, 020501(R)]. For practical values of the key parameters, we find a transition between a gapped ($d+is$)-wave pairing state in the small doping region to a gapless $d$-wave pairing state in the large doping region, accompanied by an abrupt Fermi surface change at the critical doping. Our overall phase diagram also shows the possibility of a ($d+is$)- to $s$-wave transition if the electron hybridization is relatively small. In either case, the low-doping ($d+is$)-wave state is a gapped superconducting state with broken time-reversal symmetry. Our results are in qualitative agreement with recent experimental observations and predict several key features to be examined in future measurements., Comment: 6 pages, 3 figures

Negative signs in many-body wavefunctions play an important role in quantum mechanics. The ground-state wavefunction of double semion model on a two-dimensional hexagonal lattice contains an intrinsic sign which cannot be removed by any local transformation. Here we proposed a generic double semion wavefunction in tensor network representation, and the wavefunction norm is mapped to the partition function of a triangular lattice Ashkin-Teller model with imaginary magnetic fields and imaginary three-spin triangular face interactions. To solve this non-Hermitian model with parity-time (PT) symmetry, numerical tensor-network methods are employed, and a global phase diagram is determined. Adjacent to the double semion phase, we find a gapless dense loop phase described by non-unitary conformal field theory and a PT-symmetry breaking phase with zeros of the partition function. So a connection has established between the intrinsic signs in the topologically ordered wavefunction and the PT-symmetric non-Hermitian statistical model., Comment: 19 pages, 8 figures; revised manuscript includes some additional results

We study the dynamical spin excitations of the one-dimensional $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy by using quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation function. Using the transverse dynamic spin structure factor, we observe the quantum phase transition with a critical point between the topological Haldane gapped phase and the trivial phase. At the quantum critical point, we find a broad continuum characterized by the Tomonaga-Luttinger liquid similar to a $S=1/2$ Heisenberg antiferromagnetic chain. We further identify that the elementary excitations are fractionalized spinons., Comment: 7 pages, 8 figures

The trilayer nickelate Nd$_4$Ni$_3$O$_{10-\delta}$ ($\delta \approx$ 0.15) was investigated by the measurements of x-ray diffraction, electrical resistivity, magnetic susceptibility, and heat capacity. The crystal structure data suggest a higher Ni valence in the inner perovskite-like layer. At ambient pressure the resistivity shows a jump at 162 K, indicating a metal-to-metal transition (MMT). The MMT is also characterized by a magnetic susceptibility drop, a sharp specific-heat peak, and an isotropic lattice contraction. Below $\sim$ 50 K, a resistivity upturn with a log$T$ dependence shows up, accompanying with a negative thermal expansion. External hydrostatic pressure suppresses the resistivity jump progressively, coincident with the diminution of the log$T$ behavior. The low-temperature electronic specific-heat coefficient is extracted to be $\sim$ 150 mJ K$^{-2}$ mol-fu$^{-1}$, equivalent to $\sim$ 50 mJ K$^{-2}$ mol-Ni$^{-1}$, indicating an unusual heavy-electron correlated state. The novel heavy-electron state as well as the logarithmic temperature dependence of resistivity is explained in terms of the Ni$^{3+}$ centered Kondo effect in the inner layer of the (NdNiO$_3$)$_3$ trilayers., Comment: 10 pages, 6 figures, and 1 table; The high-pressure study was supplemented

We report theoretical and experimental analyses of the thermal Hall conductivity in correlated systems. For both fermionic and bosonic excitations with nontrivial topology, we show that at "intermediate" temperatures, the thermal Hall conductivity exhibits an unexpected universal scaling with a simple exponential form. At low temperatures, it behaves differently and reflects the spectral properties of underlying excitations. Our predictions are examined as examples in two prototype compounds, the quantum paraelectric SrTiO$_3$ and the spin-liquid compound RuCl$_3$. The experimental data can be largely covered by our proposed minimal phenomenological model independent of microscopic details, revealing dominant bosonic contributions in SrTiO$_3$ and gapped fermionic excitations in RuCl$_3$. Our work establishes a phenomenological link between microscopic models and experimental data and provides a unified basis for analyzing the thermal Hall conductivity in correlated systems over a wide temperature region., Comment: 6 pages, 4 figures

The non-abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wavefunction with two tuning parameters dual with each other, and the norm can be exactly mapped into a partition function of the two-coupled $\phi^{2}$-state Potts models, where $\phi =(\sqrt{5}+1)/2$ is the golden ratio. By developing the tensor network representation of this wavefunction on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-abelian Fibonacci topological phase is enclosed by three distinct non-topological phases and their dual phases of a single $\phi^{2}$-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those non-topological phases., Comment: 6 pages, 4 figures. The title and introduction of the paper are slightly revised. Supplementary Material can be obtained upon request

We propose the parent compound of the newly discovered superconducting nickelate Nd$_{1-x}$Sr$_{x}$NiO$_{2}$ as a self-doped Mott insulator, in which the low-density Nd-$5d$ conduction electrons couple to localized Ni-3$% d_{x^{2}-y^{2}}$ electrons to form Kondo spin singlets at low temperatures. This proposal is motivated with our analyses of the reported resistivity and Hall coefficient data in the normal state, showing logarithmic temperature dependence at low temperatures. In the strong Kondo coupling limit, we derive a generalized $t$-$J$ model with both Kondo singlets and nickel holons moving through the lattice of otherwise nickel spin-1/2 background. The antiferromagnetic long-range order is therefore suppressed as observed in experiments. With Sr-doping, the number of holons on the nickel sites increases, giving rise to the superconductivity and a strange metal phase analogous to those in superconducting copper oxides., Comment: 5 pages, 2 figures

We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled $\phi ^{2}$-state Potts models with $\phi =(\sqrt{5}+1)/2$ as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory., Comment: 5.5 pages, 4 figures, and 8.5 page supplementary

In the tensor network representation, a deformed $Z_{2}$ topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase., Comment: 4.5 pages 4 figures, and 5 appendices, the published version

We construct a general wave function with the topological order by introducing the $\mathbb{Z}_{2}$ gauge degrees of freedom, characterizing both the toric code state and double semion state. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\lambda $ and identify three different quantum critical points (QCPs) at $\lambda =0$, $\pm 1.73$. The first one separates the toric code phase and double semion phase, while later two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry breaking phase, respectively. When mapping to the exactly solved statistical models, the norm of the tensor network wave function is transformed into the partition function of the eight-vertex model. Actually such a quantum-classical mapping can not reveal the rich structures of low-energy excitations at these three QCPs. So we further demonstrate that the full eigenvalue spectra of the transfer operators with/without the flux insertions can describe the complete quantum criticalities, which are characterized by the two-dimensional compactified free boson conformal field theories (CFTs) with the compactified radii $R=\sqrt{6}$ for the QCPs at $\lambda =\pm\sqrt{3}$ and $R=\sqrt{8/3}$ for the QCP at $\lambda =0$. For the QCP at $\lambda =0$, there are no anyon condensation, and the emerged matrix product operator symmetries result in a rich structure of the low-energy excitations, distinct from those of both toric code and double semion phases. Finally, we discuss the possible relation between our conformal quantum criticalities and the general (2+1) spatial-time dimensional CFTs for quantum topological phase transitions., Comment: 17 pages, 16 figures, including three appendices, revised version

Exotic correlated insulating phases emerge in the ABC-stacked trilayer graphene-boron nitride Moire super-lattice at both quarter and half-filling. A single-band minimal model with valley contrasting staggered-flux is proposed to capture the relevant band structure, where the conspiracy of perfect Fermi-surface nesting and van Hove singularity strongly enhance the valley fluctuation, leading to inter-valley spiral (IVS) order at half filling. In this paper, we consider a strong coupling U(1)$_{v}\times $SU(2)$_{s}$ symmetric spin-valley model to obtain the correlated insulating state and the pairing instability near quarter filling. A significant ingredient in the strong coupling model is the Dzyaloshinsky-Moriya like interaction inherited from the flux, which breaks not only the valley SU(2)$_{v}$ symmetry but also the sub-valley spatial reflection symmetry. We discuss all the possible long-range orders stabilized by the effective spin-valley-exchange interactions, and it turns out that the flux remarkably enhance the ferro-spin inter-valley 120$^{\circ}$ order, which shares the same valley feature as the IVS order. Upon doping, the leading pairing instability lies in the inter-valley channel with a trigonally warped $p\pm ip$-wave form factor in the presence of the sub-valley reflection symmetry breaking. Depending on the sign of Hund's coupling, the total pairing state could be either spin singlet or triplet. While the spin singlet chiral topological pairing state $(p\pm ip)_{\uparrow\downarrow }-(p\pm ip)_{\downarrow \uparrow }$ is necessarily chiral, the spin triplet topological pairing state could be chiral $(p\pm ip)_{\uparrow\uparrow }+(p\pm ip)_{\downarrow \downarrow }$, or helical $(p\pm ip)_{\uparrow \uparrow }+(p\mp ip)_{\downarrow \downarrow }$., Comment: 11 pages, 6 figures, revised version

Recent experiment has shown that the ABC-stacked trilayer graphene-boron nitride Moire super-lattice at half-filling is a Mott insulator. Based on symmetry analysis and effective band structure calculation, we propose a valley-contrasting chiral tight-binding model with local Coulomb interaction to describe this Moire super-lattice system. By matching the positions of van Hove points in the low-energy effective bands, the valley-contrasting staggered flux per triangle is determined around $\pi /2$. When the valence band is half-filled, the Fermi surfaces are found to be perfectly nested between the two valleys. Such an effect can induce an inter-valley spiral order with a gap in the charge excitations, indicating that the Mott insulating behavior observed in the trilayer graphene-boron nitride Moire super-lattice results predominantly from the inter-valley scattering., Comment: 7 pages, 5 figures, revised version

Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial phase with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. Recently, the framework of matrix product states for topological phases with Majorana fermions/parafermions has been established. Here we first generalize these fixed-point matrix product states with the zero correlation length to the more generic ground-state wave functions with a finite correlation length for the general one-dimensional interacting Majorana fermion/parafermion systems. Then we employ the previous method to decode quantum criticality from the interacting Majorana fermion/parafermion matrix product states. The obtained quantum critical spectra are described by the conformal field theories with central charge $c\leq 1$, characterizing the quantum critical theories separating the fermionic/parafermionic topological phases from the trivial phases with the same symmetry., Comment: 14 pages, 10 figures. The manuscript has been substantially expanded and revised

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with the time reversal (TR) symmetry satisfying $T^{2}=1$. Without extra symmetry, it has been found that $\mathbb{Z}_{p}$ parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labelled by the divisors $n$ of $p$. In the presence of TR symmetry, however, the enriched classification is characterized by three indices $n$, $\kappa $ and $\mu $, where $\kappa \in \mathbb{Z}_{2}$ denotes the linear or projective TR actions on the edges, and $\mu \in \mathbb{Z}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the $\mathbb{Z}_{r}$ symmetric parafermionic ground states, where $r=p$ for trivial or topological phases, and $r=p/n$ for SSB phases, the original gapped phases with odd $r$ are divided into two phases, while those phases with even $r$ are further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of those resulting topological phases, we can obtain the topological protected degeneracies of their entanglement spectra., Comment: 10 pages, 2 figures, revised version

A plaquette-centered rotation symmetry $C_{4}^{p}$ is identified to play a significant role in determining and stabilizing the Fermi-surface structure of Fe-based superconductors. Together with the $S_{4}$ symmetry previously found, we are able to sort out the tangling orbitals and solve the puzzle of pairing symmetry of superconductivity in $\text{KFe}_{2}\text{As}_{2}$ in a simple but comprehensive way. By modeling the material with a strong coupling $t-J_{1}-J_{2}$ model, we find phase transitions of pairing symmetry driven by the competition between the local spin antiferromagnetic couplings from nodal $d_{x^2-y^2}\times s_{x^2+y^2}$-wave to nodeless $% s_{x^2 y^2} $-wave through the intermediate $s+id\times s$ mixed pairing phase, which is consistent with the observation of pressure experiments. The emergent $d$-wave form factor inevitably arises from the projection of inter-orbital Cooper pairing onto the Fermi surface and is inherited from the electronic structure in the representation of $C_{4}^{p}$ symmetry. Moreover, the $S_{4}$ symmetry dictates 2 copies of $d$-wave pairing condensates, counting 8 nodes in total. We further show that weakly breaking $C_{4}^{p}$ naturally leads to the octet nodal gap as precisely observed in laser angle resolved photoemission spectroscopy. The octet nodes reflect the collaboration of the $C_{4}^{p}$ and $S_{4}$ symmetries, which sheds new light on the enigma of the pairing symmetry in $\text{KFe}_{2}\text{As}_{2}$., Comment: 10 pages, 5 figures, revised version

In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for $\mathbb{Z}_{p}$ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of $\mathbb{Z}_{p}$ parafermions without extra symmetry other than $\mathbb{Z}%_{p}$ charge symmetry, including topological phases, spontaneous symmetry breaking phases and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the non-trivial simple $\mathbb{Z}_{p}$ graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. But the spontaneous symmetry breaking phases correspond to the trivial semisimple $\mathbb{Z}_{p/n}$ graded algebras, which can be further reduced to the trivial simple $\mathbb{Z}_{p/n}$ graded algebras, where $n$ is the divisor of $p$. So the present results deepen our understanding of topological phases in one dimension from the viewpoints of MPS., Comment: 12 pages, 1 figure, 2 tables, published version

In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with $\mathbb{ Z}_{p}$ parafermions. Unlike the $\mathbb{Z}_{2}$ Majorana fermions, the $% \mathbb{Z}_{p}$ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible $% \mathbb{Z}_{3}$ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single $\mathbb{Z}_{3}$ parafermion chain and two-coupled parafermion chains with $\mathbb{Z}_{3}\times \mathbb{Z}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with $\mathbb{Z}_{p}$ parafermions within the matrix product representation in one dimension., Comment: 10 pages, 4 figures, published version

Unlike the hole-doped cuprates, both nodal and nodeless superconductivity (SC) are observed in the electron-doped cuprates. To understand these two types of SC states, we propose a unified theory by considering the two-dimensional t-J model in proximity to an antiferromagnetic (AF) long-range ordering state. Within the slave-boson mean-field approximation, the d-wave pairing symmetry is still the most energetically favorable even in the presence of the external AF field. In the nodal phase, it is found that the nodes carry vorticity and are protected by the adjoint symmetry of time-reversal and one unit lattice translation. Robust edge modes are obtained, suggesting the nodal d-wave SC being a topological weak-pairing phase. As decreasing the doping concentration or increasing the AF field, the nodes with opposite vorticity annihilate and the nodeless strong-pairing phase emerges. The topological phase transition is characterized by a critical point with anisotropic Bogoliubov quasiparticles, and a universal understanding is thus established for all electron-doped cuprates., Comment: 7 pages, 5 figures; published version

Motivated by the recent observations of nodeless superconductivity in the monolayer CuO$_{2}$ grown on the Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ substrates, we study the two-dimensional superconducting (SC) phases described by the two-dimensional $t$-$J$ model in proximity to an antiferromagnetic (AF) insulator. We found that (i) the nodal d-wave SC state can be driven via a continuous transition into a nodeless d-wave pairing state by the proximity induced AF field. (ii) The energetically favorable pairing states in the strong field regime have extended s-wave symmetry and can be nodal or nodeless. (iii) Between the pure d-wave and s-wave paired phases, there emerge two topologically distinct SC phases with ($s+$i$d$) symmetry, i.e., the weak and strong pairing phases, and the weak pairing phase is found to be a $Z_{2}$ topological superconductor protected by valley symmetry, exhibiting robust gapless non-chiral edge modes. These findings strongly suggest that the high-$T_{c}$ superconductors in proximity to antiferromagnets can realize fully gapped symmetry protected topological SC., Comment: 7 pages, 4 figures; revised version