Two-spin Correlation Function Research Articles

Floquet prethermal phase protected by U(1) symmetry on a superconducting quantum processor

Periodically driven systems, or Floquet systems, exhibit many novel dynamics and interesting out-of-equilibrium phases of matter. Those phases arising with the quantum systems' symmetries, such as global U(1) symmetry, can even show dynamical stability with symmetry protection. Here we experimentally demonstrate a U(1) symmetry-protected prethermal phase, via performing a digital-analog quantum simulation on a superconducting quantum processor. The dynamical stability of this phase is revealed by its robustness against external perturbations. We also find that the site-averaged square of the two-spin correlation function in this phase is stabilized by the interaction between the spins. Our work reveals a promising prospect in discovering emergent quantum dynamical phases with digital-analog quantum simulators.

Spin excitations and thermodynamics of the t-J model on the honeycomb lattice

We present a spin-rotation-invariant Green-function theory for the dynamic spin susceptibility in the spin-1/2 antiferromagnetic t-J Heisenberg model on the honeycomb lattice. Employing a generalized mean-field approximation for arbitrary temperatures and hole dopings, the electronic spectrum of excitations, the spin-excitation spectrum and thermodynamic quantities (two-spin correlation functions, staggered magnetization, magnetic susceptibility, correlation length) are calculated by solving a coupled system of self-consistency equations for the correlation functions. The temperature and doping dependence of the magnetic (uniform static) susceptibility is ascribed to antiferromagnetic short-range order. Our results on the doping dependencies of the magnetization and susceptibility are analyzed in comparison with previous results for the t-J model on the square lattice.

Spin excitations and thermodynamics of the antiferromagnetic Heisenberg model on the layered honeycomb lattice

We present a spin-rotation-invariant Green-function theory for the dynamic spin susceptibility in the spin-1/2 antiferromagnetic Heisenberg model on a stacked honeycomb lattice. Employing a generalized mean-field approximation for arbitrary temperatures, the thermodynamic quantities (two-spin correlation functions, internal energy, magnetic susceptibility, staggered magnetization, Néel temperature, correlation length) and the spin-excitation spectrum are calculated by solving a coupled system of self-consistency equations for the correlation functions. The temperature dependence of the magnetic (uniform static) susceptibility is ascribed to antiferromagnetic short-range order. The Néel temperature is calculated for arbitrary interlayer couplings. Our results are in a good agreement with numerical computations for finite clusters and with available experimental data on the β-Cu2V2O2 compound.

One-dimensional Ising model with multispin interactions

We study the spin-1/2 Ising chain with multispin interactions K involving the product of m successive spins, for general values of m. Using a change of spin variables the zero-field partition function of a finite chain is obtained for free and periodic boundary conditions and we calculate the two-spin correlation function. When placed in an external field H the system is shown to be self-dual. Using another change of spin variables the one-dimensional Ising model with multispin interactions in a field is mapped onto a zero-field rectangular Ising model with first-neighbour interactions K and H. The 2D system, with size m × N/m, has the topology of a cylinder with helical BC. In the thermodynamic limit , , a 2D critical singularity develops on the self-duality line, .

Proton NMR study of spin dynamics in the magnetic organic chainsM(hfac)3NITEt(M=Eu3+,Gd3+)

In this work, we present a nuclear magnetic resonance (NMR) study of the spin dynamics in the rare-earth-based low-dimensional molecular magnetic chains $\mathrm{Eu}{(\mathrm{hfac})}_{3}\mathrm{NITEt}$ and $\mathrm{Gd}{(\mathrm{hfac})}_{3}\mathrm{NITEt}$ (in short, Eu-Et and Gd-Et). Although both samples are based on the same chemical building block, [(${\mathrm{hfac})}_{3}\mathrm{NITEt}]$, their magnetic properties change dramatically when the ${\mathrm{Eu}}^{3+}$ ion, which is nonmagnetic at low temperatures, is substituted by the magnetic ${\mathrm{Gd}}^{3+}$ ion. The present proton NMR investigation shows that, down to the lowest investigated temperature ($T=1.5$ K for Gd-Et and $T=3$ K for Eu-Et), the Eu-Et chain behaves as a one-dimensional Heisenberg model with antiferromagnetic exchange coupling ($J=\ensuremath{-}20$ K) between $s=1/2$ organic radicals, and has a $T$-independent exchange frequency (${\ensuremath{\omega}}_{e}=2.6\ifmmode\times\else\texttimes\fi{}{10}^{12}$ rad/s). In the Gd-Et chain, in contrast, a competition arises between nearest-neighbor ferromagnetic coupling and next-nearest-neighbor antiferromagnetic coupling; moreover, two phase transitions have previously been found, in agreement with Villain's conjecture: a first transition, at ${T}_{0}=2.2$ K, from a high temperature paramagnetic phase to a chiral spin liquid phase, and a second transition, at ${T}_{N}=1.9$ K, to a three-dimensional helical spin solid phase. Contrary to the Eu-Et chain (whose three-dimensional ordering temperature is estimated to insurge at very low, ${T}_{N}\ensuremath{\approx}0.3$ K), critical spin dynamics effects have been measured in the Gd-Et chain on approaching ${T}_{N}=1.9$ K: namely, a divergence of the proton nuclear spin-lattice relaxation rate $1/{T}_{1}$, which in turn produces a sudden wipe-out of the NMR signal in a very narrow ($\mathrm{\ensuremath{\Delta}}T\ensuremath{\sim}0.04$ K) temperature range above ${T}_{N}$. Below ${T}_{N}$, an inhomogeneous broadening of the NMR line indicates a complete spin freezing. At ${T}_{0}=2.2$ K, instead, such critical effects are not observed because NMR measurements probe the two-spin correlation function, while the chiral spin liquid phase transition is associated with a divergence of the four-spin correlation function.

High temperature spin dynamics in linear magnetic chains, molecular rings, and segments by nuclear magnetic resonance

We present the room temperature proton nuclear magnetic resonance (NMR) nuclear spin-lattice relaxation rate (NSLR) results in two 1D spin chains: the Heisenberg antiferromagnetic (AFM) Eu(hfac)3NITEt and the magnetically frustrated Gd(hfac)3NITEt. The NSLR as a function of external magnetic field can be interpreted very well in terms of high temperature spin dynamics dominated by a long time persistence of the decay of the two-spin correlation function due to the conservation of the total spin value for isotropic Heisenberg chains. The high temperature spin dynamics are also investigated in Heisenberg AFM molecular rings. In both Cr8 closed ring and in Cr7Cd and Cr8Zn open rings, i.e., model systems for a finite spin segment, an enhancement of the low frequency spectral density is found consistent with spin diffusion but the high cut-off frequency due to intermolecular anisotropic interactions prevents a detailed analysis of the spin diffusion regime.

Magnetic Properties of Mg x Cu1−x Cr2O4 Spinels are Studied by Different Theoretical Methods

The external magnetic field induced the reorientation of magnetization of a ferromagnetic (or antiferromagnetic) treated within the framework of many-body Green’s function theory by considering all components of the magnetization. We present a new method for the calculation of expectation values in terms of the eigenvalues and eigenvectors of the equations of motion matrix for the set of Green’s functions. Magnetization and magnetic susceptibility are investigated when an external magnetic field is applied in (x-z)-plane. The mean field theory is applied to calculate the nearest neighbour and the next-neighbour super-exchange J 1(Cr−Cr) and J 2(Cr−(Mg(Cu)−O)−Cr), respectively, for the Mg x Cu1−x Cr2O4 in the range of 0 ≤ x ≤ 1. The intra-planar and the inter-planar interactions are deduced. The high-temperature series expansions (HTSEs) are derived for the magnetic susceptibility and two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random-phase approximation (RPA). The magnetic-phase diagram is deduced. A spin-glass phase is predicted for intermediate range of concentration. The spin glass is obtained. The obtained results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) have been deduced. The obtained values are comparable to those of 3D Heisenberg model.

Magnetic susceptibility and short-range order in iron pnictides: Anisotropic J1-J2 Heisenberg model

A spin-rotation-invariant Green-function theory for the dynamic spin susceptibility in the anisotropic non-frustrated J 1-J 2 model for the iron pnictides AFe2As2 (A = Ca, Sr, Ba) is presented. The two-spin correlation functions, the spin-excitation spectrum, and the staggered magnetization are calculated self-consistently as functions of temperature and spin. The temperature dependence of the magnetic (uniform static) susceptibility, in particular the linear increase with T in a wide temperature region that is in good quantitative agreement with experiments, is ascribed to a strong in-plane antiferromagnetic short-range order. The values of the Neel temperature resulting from the localized spin model with realistic out-of-plane couplings are found to be much higher than the measured values. From this an extension of the model by including itinerant degrees of freedom is suggested.

Magnetic properties of ferromagnetic diluted Zn x Cd1–x Cr2Se4 spinels are studied by Green's functions, mean field theory and high temperature series expansions theories

The field induced reorientation of the magnetization of ferromagnetic (or antiferromagnetic) structure is treated within the framework of many-body Green's function theory by considering all components of the magnetization. The mean field theory is used to calculate the nearest neighbour and the next-neighbour super-exchange J1(Cr–Cr) and J2(Cr–(Zn(Cd)–Se)–Cr), respectively, for the Zn1–x Cd x Cr2Se4 in the range 0 < x < 1. The intraplanar and the interplanar interactions are deduced. The high temperature series expansions (HTSEs) are derived for the magnetic susceptibility and the two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random phase approximation (RPA). The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The obtained results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) have been deduced. The obtained values are comparable to those of 3D Heisenberg model.

Some Upper Bounds for the Critical Temperature for Ising Model with Four-spin Interactions

Upper bounds for the critical temperature of Ising models with different types of four-spin interactions on honeycomb and square lattices, which act only between the nearest-neighbor sites or to the nearest- and next-nearest-neighbor sites in addition to the conventional pair interactions, are obtained, using an exact relation for the two-spin correlation functions and rigorous inequalities for the spin correlation functions.

Spin nematic phase in one-dimensional and quasi-one-dimensional frustrated magnets in a strong magnetic field

We discuss spin-1/2 one-dimensional (1D) and quasi-1D magnets with competing ferromagnetic nearest-neighbor $J_1$ and antiferromagnetic next-nearest-neighbor $J$ exchange interactions in strong magnetic field $H$. It is well known that due to attraction between magnons quantum phase transitions (QPTs) take place at $H=H_s$ from the fully polarized phase to nematic ones if $J>|J_1|/4$. Such a transition at $J>0.368|J_1|$ is characterized by softening of the two-magnon bound-state spectrum. Using a bond operator formalism we propose a bosonic representation of the spin Hamiltonian containing, aside from bosons describing one-magnon spin-1 excitations, a boson describing spin-2 excitations which spectrum coincides at $H\ge H_s$ with the two-magnon bound-state spectrum obtained before. The presence of the bosonic mode in the theory that softens at $H=H_s$ makes substantially standard the QPT consideration. In 1D case at $H<H_s$, we find an expression for the magnetization which describes well existing numerical data. Expressions for spin correlators are obtained which coincide with those derived before either in the limiting case of $J\gg|J_1|$ or using a phenomenological theory. In quasi-1D magnets, we find that the boundary in the $H$--$T$ plane between the fully polarized and the nematic phases is given by $H_s(0)-H_s(T)\propto T^{3/2}$. Simple expressions are obtained in the nematic phase for static spin correlators, spectra of magnons and the soft mode, magnetization and the nematic order parameter. All static two-spin correlation functions are short ranged with the correlation length proportional to $1/\ln(1+|J_1|/J)$. Dynamical spin susceptibilities are discussed and it is shown that the soft mode can be observed experimentally in the longitudinal channel.

Theory ofL-edge resonant inelastic x-ray scattering for magnetic excitations in two-leg spin ladders

We study the magnetic excitation spectra of Cu ${L}_{2,3}$-edge resonant inelastic x-ray scattering (RIXS) from the spin-liquid ground state in two-leg spin ladder cuprates. By applying the projection method developed by the present authors, we have derived the formulas of the magnetic RIXS spectra, which are expressed by one- and two-spin correlation functions in the polarization changing and preserving channels, respectively. The one-spin correlation function includes both one- and two-triplon excitations and they are picked up separately by choosing rung wave vectors $\ensuremath{-}{q}_{a}=\ensuremath{\pi}$ and 0, respectively. An application to Sr${}_{14}$Cu${}_{24}$O${}_{41}$ reveals that the calculated RIXS spectrum captures well the dispersive behavior shown by the lower boundary of two-triplon continuum. An exception is absence of the intensity at zero-momentum transfer where weak but finite intensity was observed in the experiment. By adjusting the geometrical configuration of the measurement, one-triplon dispersion around the zone center could be detectable in the RIXS experiment. The observed weak intensity in the higher-energy region might be attributed to the two-spin correlation function, which could be detected more clearly at the ${M}_{3}$-edge RIXS spectra.

Magnetic excitations inL-edge resonant inelastic x-ray scattering from one-dimensional cuprates

We study the magnetic excitation spectra of L-edge resonant inelastic x-ray scattering (RIXS) from the spin singlet ground state in one-dimensional undoped cuprates. Analyzing the transition amplitudes of the magnetic excitations in the second-order dipole allowed process, we find that the magnetic excitations are brought about not only on the core-hole site but also on the neighboring sites. The RIXS spectra are expressed by the one-spin correlation function in the scattering channel with changing polarization, and the two-spin correlation function in the scattering channel without changing polarization. The latter could not be brought about within the so-called UCL approximation. We calculate these correlation functions on a finite-size ring. An application to the possible RIXS spectra in Sr_2CuO_3 demonstrates that the contribution of the two-spin correlation function could be larger than that of the one-spin correlation function in the \sigma polarization for the momentum transfer around the zone center.

Dynamic spin susceptibility in thet−Jmodel

A relaxation-function theory for the dynamic spin susceptibility in the $t$--$J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperaturs. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on La_{2-\delta}Sr_\delta)CuO_4. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its (\omega/T) scaling behavior are calculated in a reasonable agreement with experimental and ED data.

Diagrammatic representation of the two-spin correlation function for the generalized heisenberg model

We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ(D) = –JΣijS(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites . Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.

XXZ-like phase in the F-AF anisotropic Heisenberg chain

By means of the Density Matrix Renormalization Group technique, we have studied the region where XXZ-like behavior is most likely to emerge within the phase diagram of the F-AF anisotropic extended (J-J’) Heisenberg chain. We have analyzed, in great detail, the equal-time two-spin correlation functions, both in- and out-of- plane, as functions of the distance (and momentum). Then, we have extracted, through an accurate fitting procedure, the exponents of the asymptotic power-law decay of the spatial correlations. We have used the exact solution of XXZ model (J’ = 0) to benchmark our results, which clearly show the expected agreement. A critical value of J’ has been found where the relevant power-law decay exponent is independent of the in-plane nearest-neighbor coupling.

Spin-1 Ising model: Exact damage-spreading relations and numerical simulations

The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.

Two-Step Magnetic Ordering in Quasi-One-Dimensional Helimagnets: Possible Experimental Validation of Villain’s Conjecture about a Chiral Spin Liquid Phase

Low-temperature specific heat, magnetic susceptibility, and zero-field muon spin resonance (microSR) measurements have been performed in the quasi-one-dimensional molecular helimagnetic compound Gd(hfac)3NITEt. The specific heat presents two anomalies at T(0)=2.19+/-0.02 K and T(N)=1.88+/-0.02 K, which both disappear upon the application of a weak magnetic field. Conversely, magnetic susceptibility and muSR data show the divergence of two-spin correlation functions only at T(N)=1.88+/-0.02 K. These results suggest an experimental validation of Villain's conjecture of a two-step magnetic ordering in quasi-one-dimensional XY helimagnets; i.e., the paramagnetic phase and the helical spin solid phase are separated by a chiral spin liquid phase, where translational invariance is broken without violation of rotational invariance.

Quantum critical scaling behavior of deconfined spinons

We perform a renormalization group analysis of some important effective field theoretic models for deconfined spinons. We show that deconfined spinons are critical for an isotropic $\mathrm{SU}(N)$ Heisenberg antiferromagnet, if $N$ is large enough. We argue that nonperturbatively this result should persist down to $N=2$ and provide further evidence for the so-called deconfined quantum criticality scenario. Deconfined spinons are also shown to be critical for the case describing a transition between quantum spin nematic and dimerized phases. On the other hand, the deconfined quantum criticality scenario is shown to fail for a class of easy-plane models. For the cases where deconfined quantum criticality occurs, we calculate the critical exponent $\ensuremath{\eta}$ for the decay of the two-spin correlation function to first-order in $ϵ=4\ensuremath{-}d$. We also note the scaling relation $\ensuremath{\eta}=d+2(1\ensuremath{-}\ensuremath{\varphi}∕\ensuremath{\nu})$ connecting the exponent $\ensuremath{\eta}$ for the decay to the correlation length exponent $\ensuremath{\nu}$ and the crossover exponent $\ensuremath{\varphi}$.

Magnetic phase diagram of diluted spinel Zn 1−xCu xCr 2Se 4 system

By using mean field theory, we have evaluated the nearest-neighbour and the next-neighbour super-exchange J 1( x) and J 2( x), respectively, for Zn 1− x Cu x Cr 2Se 4 in the range 0⩽ x⩽1. The intraplanar and the interplanar interactions are deduced. High-temperature series expansions are derived for the magnetic susceptibility and two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random phase approximation. The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility ( γ) and the correlation lengths ( ν) have been deduced. The values are comparable to those of the 3D Heisenberg model, and are insensitive to the dilution x.