Three-site Terms Research Articles

Floquet topological superconductivity induced by chiral many-body interaction

Non-equilibrium engineering is becoming a seminal way for realising novel quantum phases that are unimaginable in equilibrium. In particular, Floquet theory applied to quantum mechanics revealed that we can even control the band topology in semimetallic/insulating systems, while the straightforward application to topological superconductivity fails for typical superconductors because the supercondicting gap function does not couple to the electromagnetic field in a direct manner. Here we show that we can overcome this difficulty by taking account of correlation effects. Namely, we study how a d-wave superconductivity is changed when illuminated by circularly-polarised light (CPL) in the repulsive Hubbard model in the strong-coupling regime. We adopt the Floquet formalism for the Gutzwiller-projected effective Hamiltonian with the time-periodic Schrieffer-Wolff transformation. We find that CPL induces a topological superconductivity with a d + id pairing, which arises from the chiral spin coupling and the three-site term generated by the CPL. The latter term remains significant even for low frequencies and low intensities of the CPL. This is clearly reflected in the obtained phase diagram against the laser intensity and temperature for various frequencies red-detuned from the Hubbard U, with the transient dynamics also examined. The phenomenon revealed here can open a novel, dynamical way to induce a topological superconductivity.

SU(4) Chiral Spin Liquid, Exciton Supersolid, and Electric Detection in Moiré Bilayers.

We propose a moiré bilayer as a platform where exotic quantum phases can be stabilized and electrically detected. Moiré bilayers consist of two separate moiré superlattice layers coupled through the interlayer Coulomb repulsion. In the small distance limit, an SU(4) spin can be formed by combining layer pseudospin and the real spin. As a concrete example, we study an SU(4) spin model on triangular lattice in the fundamental representation. By tuning a three-site ring exchange term K∼(t^{3}/U^{2}), we find the SU(4) symmetric crystallized phase and an SU(4)_{1} chiral spin liquid at the balanced filling. We also predict two different exciton supersolid phases with interlayer coherence at imbalanced filling under displacement field. Especially, the system can simulate an SU(2) Bose-Einstein condensation by injecting interlayer excitons into the magnetically ordered Mott insulator at the layer polarized limit. Smoking gun evidences of these phases can be obtained by measuring the pseudospin transport in the counterflow channel.

Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians

We show that the combination of charge and dipole conservation---characteristic of fracton systems---leads to an extensive fragmentation of the Hilbert space, which in turn can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total $S^z$) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature auto-correlation saturates to a finite value---showcasing non-thermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local $S^z$ basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: the system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits non-thermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment conserving random unitary circuit models for which we reach identical conclusions.

Phase diagrams of Heisenberg chains with different cell spins: role of the tree-site exchange interactions

Additional three-site exchange interactions, naturally arising, e.g., in the strong-coupling expansion of the two-band Hubbard model and in various effective spin Hamiltonians, may stabilize various exotic spin phases which are not typical for the pure Heisenberg models on frustrated lattices and/or with extra biquadratic exchange terms. The reason is hidden in some specific features of the isotropic three-site exchange interactions, such as the promotion of collinear spin configurations as well as the reinforcing tendency towards clustering of the quantum spins on the shortest exchange bonds. As a result, systems with integer and half­integer composite spins in the unit cell might be expected to support different quantum phase diagrams. In this work, we study this phenomenon by comparing the quantum phase diagrams of two Heisenberg chains with extra three-site exchange terms and two different alternating pairs of site spins S and σ [(S, σ) = and (], resulting in half-integer and integer cell spins S + σ, respectively.

Ground-state phase diagram of the one-dimensional t−J model with pair hopping terms

The $t$-$J$ model is a standard model of strongly correlated electrons, often studied in the context of high-$T_c$ superconductivity. However, most studies of this model neglect three-site terms, which appear at the same order as the superexchange $J$. As these terms correspond to pair-hopping, they are expected to play an important role in the physics of superconductivity when doped sufficiently far from half-filling. In this paper we present a density matrix renormalisation group study of the one-dimensional $t$-$J$ model with the pair hopping terms included. We demonstrate that that these additional terms radically change the one-dimensional ground state phase diagram, extending the superconducting region at low fillings, while at larger fillings, superconductivity is completely suppressed. We explain this effect by introducing a simplified effective model of repulsive hardcore bosons.

On the Theory of Plasmon Dispersion in Electron-Doped Cuprates

An explicit expression for the dynamic charge susceptibility for electron-doped cuprates has been derived. This expression accurately reproduces the wave vector dependence of the plasmon frequency observed in inelastic X-ray scattering experiments for Nd2 – xCexCuO4. The imaginary part of the charge susceptibility along the triangular path in the Brillouin zone is plotted. It is demonstrated that the spectral weight of the plasmon mode near q = 0 is negligibly low. The calculated frequencies of the plasmon mode for all wave vectors in the Brillouin zone turn out to lie outside the range of damping related to electron−hole excitations. A formula for the charge susceptibility is derived within the t−t′−t″−J model supplemented by the Coulomb interaction operator and three-site terms. The derivation is performed by the Green’s function technique employing the formalism of composite Hubbard operators and the Mori projection method, which have proved themselves in the analysis of collective spin excitations. The used Fourier transform of the Coulomb interaction corresponds to the monolayer model with a spatially periodic structure, which is embedded in a three-dimensional crystal lattice.

Signatures of strong correlation effects in resonant inelastic x-ray scattering studies on cuprates

Recently, spin excitations in doped cuprates are measured using the resonant inelastic X-ray scattering (RIXS). The paramagnon dispersions show the large hardening effect in the electron-doped systems and seemingly doping-independence in the hole-doped systems, with the energy scales comparable to that of the antiferromagnetic magnons. This anomalous hardening effect was partially explained by using the strong coupling t-J model but with a three-site term(Nature communications 5, 3314 (2014)). However we show that hardening effect is a signature of strong coupling physics even without including this extra term. By considering the t-t'-t"-J model and using the Slave-Boson (SB) mean field theory, we obtain, via the spin-spin susceptibility, the spin excitations in qualitative agreement with the experiments. These anomalies is mainly due to the doping-dependent bandwidth. We further discuss the interplay between particle-hole-like and paramagnon-like excitations in the RIXS measurements.

Band-renormalization Effect on Relationship between Superconductivity and Antiferromagnetism in t-J Model

Having cuprate superconductors in mind, we study a band-renormalization effect on a coexisting state of antiferromagnetism (AF) and superconductivity (SC) in a t-J model on a square lattice with a diagonal transfer t′ and three-site hopping terms, using a variational Monte Carlo method. It is found that this t-J model with t′/t = –0.3 exhibits coexistence of AF and SC orders in lightly doped regime and robust SC for a wide range of doping rate δ. This result contrasts with the previous results of a large-U/t Hubbard model, in which the two orders are mutually exclusive and a coexisting state does not appear; the SC state appears only in heavily doped regime (δ≳ 0.2), after the pure AF state vanishes at δ ∼ 0.2.

Realization of higher Wess-Zumino-Witten models in spin chains

Building on the generalization of the exactly dimerized Majumdar-Ghosh ground state to arbitrary spin $S$ for the Heisenberg chain with a three-site term $({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\text{H.c.}$, we use density-matrix renormalization group simulations and exact diagonalizations to determine the nature of the dimerization transition for $S=1$, $3/2$, and 2. The resulting central charge and critical exponent are in good agreement with the $\text{SU}{(2)}_{k=2S}$ Wess-Zumino-Witten values $c=3k/(2+k)$ and $\ensuremath{\eta}=3/(2+k)$. Since the three-site term that induces dimerization appears naturally if exchange interactions are calculated beyond second order, these results suggest that $\text{SU}{(2)}_{k>1}$ Wess-Zumino-Witten models might finally be realized in actual spin chains.

Repulsive fermions in optical lattices: Phase separation versus coexistence of antiferromagnetism andd-wave superfluidity

We investigate a system of fermions on a two-dimensional optical square lattice in the strongly repulsive coupling regime. In this case, the interactions can be controlled by laser intensity as well as by Feshbach resonance. We compare the energetics of states with resonating valence bond $d$-wave superfluidity, antiferromagnetic long-range order, and a homogeneous state with coexistence of superfluidity and antiferromagnetism. Using a variational formalism, we show that the energy density of a hole ${e}_{\mathrm{hole}}(x)$ has a minimum at doping $x={x}_{c}$ that signals phase separation between the antiferromagnetic and $d$-wave paired superfluid phases. The energy of the phase-separated ground state is, however, found to be very close to that of a homogeneous state with coexisting antiferromagnetic and superfluid orders. We explore the dependence of the energy on the interaction strength and on the three-site hopping terms and compare with the nearest-neighbor hopping t-J model.

Renormalized mean-fieldt−Jmodel of high-Tcsuperconductivity: Comparison to experiment

Using an advanced version of the renormalized mean-field theory (RMFT) for the t-J model, we examine spin-singlet superconducting (SC) state of $d_{x^2 - y^2}$-symmetry. Overall doping dependence of the SC gap magnitude is in good agreement with experimental results for $\text{Bi}_{2}\text{Sr}_{2}\text{Ca} \text{Cu}_2 \text{O}_{8 + \delta}$ (BSCCO) and $\text{La}_{2-x}\text{Sr}_{x}\text{Cu} \text{O}_{4}$ (LSCO) compounds at the optimal doping and in the overdoped regime. We also calculate the dispersion relation for the Bogoliubov quasiparticles and compare our findings both with the angle resolved photoemission data for the cuprates, as well as with the variational Monte Carlo and other mean-field studies. Within the method proposed by Fukushima [cf. Phys. Rev. B \textbf{78}, 115105 (2008)], we analyze different forms of the t-J Hamiltonian, i.e. modifications caused by the form of exchange interaction, and by the presence of three-site terms. It is shown that although the former has a small influence, the latter suppresses strongly the superconductivity. We also analyze the temperature dependence of the gap magnitude and compare the results with those of the recently introduced finite-temperature renormalized mean-field theory (TRMFT) of Wang et al. [cf. Phys. Rev. B \textbf{82}, 125105 (2010)].

Orbitally induced string formation in the spin-orbital polarons

We study the spectral function of a single hole doped into the (a,b) plane of the Mott insulator LaVO$_3$, with antiferromagnetic (AF) spin order of S=1 spins accompanied by alternating orbital (AO) order of active $\{d_{yz},d_{zx}\}$ orbitals. Starting from the respective t-J model, with spin-orbital superexchange and effective three-site hopping terms, we derive the polaron Hamiltonian and show that a hole couples simultaneously to the collective excitations of the AF/AO phase, magnons and orbitons. Next, we solve this polaron problem using the self-consistent Born approximation and find a stable quasiparticle solution -- a spin-orbital polaron. We show that the spin-orbital polaron resembles the orbital polaron found in $e_g$ systems, as e.g. in K$_2$CuF$_4$ or (to some extent) in LaMnO$_3$, and that the hole may be seen as confined in a string-like potential. However, the spins also play a crucial role in the formation of this polaron -- we explain how the orbital degrees of freedom: (i) confine the spin dynamics acting on the hole as the classical Ising spins, and (ii) generate the string potential which is of the joint spin-orbital character. Finally, we discuss the impact of the results presented here on the understanding of the phase diagrams of the lightly doped cubic vanadates.

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

Abstract It is shown that the spin- 1 2 anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- 1 2 Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang–Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang–Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang–Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang–Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang–Lee and Fisher edge singularity exponents are universal and equal to each other, σ = − 1 2 .

Absence of Hole Confinement in Transition-Metal Oxides with Orbital Degeneracy

We investigate the spectral properties of a hole moving in a two-dimensional Hubbard model for strongly correlated t(2g) electrons. Although superexchange interactions are Ising-like, a quasi-one-dimensional coherent hole motion arises due to effective three-site terms. This mechanism is fundamentally different from the hole motion via quantum fluctuations in the conventional spin model with SU(2) symmetry. The orbital model describes also propagation of a hole in some e(g) compounds, and we argue that orbital degeneracy alone does not lead to hole self-localization.

Competition between spin exchange and correlated hopping

The ground-state phase diagram is numerically studied for an electronic model consisting of the spin exchange term ( J) and the correlated hopping term ( t 3: the three-site term). This model has no single-particle hopping and the ratio of the two terms is controlled by a parameter α≡4 t 3/ J. The case of α=1 corresponds to complete suppression of single-particle hopping in the strong-coupling limit of the Hubbard model. In one dimension, phase separation takes place below a critical value α c=0.36–0.63 which depends on the electron density. Spin gap opens in the whole region except the phase-separated one. For α≳1.2 and low hole densities, charge-density-wave correlations are the most dominant, whereas singlet-pairing correlations are the most dominant in the remaining region. The possibility of superconductivity in the two-dimensional case is also discussed, based on equal-time pairing correlations.

Spectral function of one hole in several one-dimensional spin arrangements

The spectral function of one hole in different magnetic states of the one-dimensional $t\ensuremath{-}J$ model including three-site term and frustration ${J}^{\ensuremath{'}}$ is studied. In the strong-coupling limit $\stackrel{\ensuremath{\rightarrow}}{J}0$ (corresponding to $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ of the Hubbard model) a set of eigenoperators of the Liouvillian is found which allows us to derive an exact expression for the one-particle Green's function that is also applicable at finite temperature and in an arbitrary magnetic state. The spinon dispersion of the pure $t\ensuremath{-}J$ model with the ground state of the Heisenberg model can be obtained by treating the corrections due to a small exchange term by means of the projection method. The spectral function for the special frustration ${J}^{\ensuremath{'}}=J/2$ with the Majumdar-Ghosh wave function is discussed in detail. Besides the projection method, a variational ansatz with the set of eigenoperators of the t term is used. We find a symmetric spinon dispersion around the momentum $k=\ensuremath{\pi}/(2a)$ and a strong damping of the holon branch. Below the continuum a bound state is obtained with finite spectral weight and a very small separation from the continuum. Furthermore, the spectral function of the ideal paramagnetic case at a temperature ${k}_{B}T\ensuremath{\gg}J$ is discussed.

Atomic transport coefficients in a binary random alloy

The frequency-dependent phenomenological coefficients for the atomic transport by the vacancy mechanism in a simple cubic binary random alloy with a very small vacancy content have been calculated through the kinetic equation approach. The contribution of the three-site terms (second order in fluctuations of the atomic occupancy variables) has been obtained by neglecting terms of the third order in these fluctuations in the kinetic equation. The results for the generalized transport coefficients are compared with the computer simulation results and a self-consistent approximation used earlier for the three-site terms. Compared with the self-consistent approximation our results are less accurate for the tracer and non-tracer correlation factors but show better agreement for the frequency-dependent conductivity when compared with the simulation results for the highest-to-lowest-jump-frequency ratio of ten.

Stability of the striped phase in the two-dimensional t– J model: A variational Monte Carlo study

By using a variational Monte Carlo technique, stability of the striped phases is studied for the two-dimensional t– J model with the three-site term near 1/8 hole doping. It is found that both diagonally and vertically striped phases are comparable in energy with the d x 2− y 2 -wave superconducting state in the small- J/ t region. Hole-`filled’ domain walls prefer diagonal stripes, whereas `half-filled’ domain walls vertical ones, in accordance with the situation observed in nickelates and cuprates.

Spectral function and quasiparticle weight in the generalized t– J model

We extend to the spectral function an approach which allowed us to calculate the quasiparticle weight for destruction of a real electron Z cσ ( k ) (in contrast to that of creation of a spinless holon Z h ( k )) in a generalized t– J model, using the self-consistent Born approximation (SCBA). We compare our results with those obtained using the alternative approach of Sushkov et al. [O.P. Sushkov, G.A. Sawatzky, R. Eder, H. Eskes, Phys. Rev. B. 56 (1997) 11769], which also uses the SCBA. The results for Z cσ ( k ) are also compared with the results obtained using the string picture and with exact diagonalizations of a 32-site square cluster. While on a qualitative level, all results look similar, our SCBA approach seems to compare better with the ED one. The effect of hopping beyond nearest neighbors, and that of the three-site term are discussed.

Phase separation and three-site hopping in the two-dimensional t- J model

We study the t- J model with the inclusion of the so-called three-site term which is due to the t U → 0 expansion of the Hubbard model. We find that this singlet pair hopping term has no qualitative effect on the structure of the pure mean field phase diagram for non-magnetic states. In accordance with experimental data on high- T c materials and some numerical studies, we also find wide regions of phase coexistence whenever the coupling J is greater than a critical value J c. We show that J c varies linearly with the temperature T, going to zero at T = 0.