Bethe Ansatz Results Research Articles

The large-N limit of 4d superconformal indices for general BPS charges

We study the superconformal index of mathcal{N} = 1 quiver theories at large-N for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic extension method. We are particularly interested in the case of unequal angular momenta, J1 ≠ J2, which has only been partially considered in the literature. We revisit the previous computation with the Bethe Ansatz formulation with generic angular momenta and extend it to encompass a large class of competing exponential terms. In the process, we also provide a simplified derivation of the original result. We consider the newly-developed elliptic extension method as well; we apply it to the J1 ≠ J2 case, finding a good match with the Bethe Ansatz results. We also investigate the relation between the two different approaches, finding in particular that for every saddle of the elliptic action there are corresponding terms in the Bethe Ansatz formula that match it at large-N.

Kubo's response theory and bosonization with a background gauge field and irrelevant perturbations

Using conformal field theory calculations of the energy spectrum, within the XXZ model we investigate effects of the flux insertion and the Umklapp term. We discuss two approaches to the evaluation of the Drude weight, the first corresponding to the linear response theory and the second corresponding to the twisted boson theory with the Umklapp term. Divergences obtained in the context of the former contradict the Bethe ansatz results, with the two approaches coinciding for the free fermion point only. The origin of this discrepancy is in the different order in which the Umklapp term and the flux insertion are treated, where the marginal perturbation should be considered before the irrelevant perturbation. We calculate the scaling of the conductivity with system size and temperature in the long-wave limit.

Repulsive Polarons in One-Dimensional Fermi Gases**Support by the National Natural Science Foundation of China under Grant Nos. 11374331, 11304364, and 11534014

Using the variational method, we study the properties of a spin-down impurity immersed in a one-dimensional (1D) spin-up Fermi sea. With repulsive interactions between them, the impurity is dressed up by surrounding particles in Fermi sea and forms a polaron. We clearly calculate the binding energy, effective mass, momentum distribution, Tan contact, and pair correlation. Even in strong repulsive regimes, the results can agree with the exact Bethe Ansatz results. The repulsive polaron energy E+ is below Fermi energy EF and no negative effective masses are found in whole interaction regimes, unequal masses polarons are also calculated. We show a clear momentum distribution and calculate the Tan contact from three different aspects. Furthermore, we explore the particle-hole excitation and find that the hole terms in Fermi sea have a great influence on the polaron energy and contact in repulsive regime. These results show that the variational method can still be used effectively in 1D repulsive polaron system.

Elaborating the phase diagram of spin-1 anyonic chains

We revisit the phase diagram of spin-1 su(2)_ksu(2)k anyonic chains, originally studied by Gils et. al. . These chains possess several integrable points, which were overlooked (or only briefly considered) so far.Exploiting integrability through a combination of algebraic techniques and exact Bethe ansatz results, we establish in particular the presence of new first order phase transitions, a new critical point described by a Z_kZk parafermionic CFT, and of even more phases than originally conjectured. Our results leave room for yet more progress in the understanding of spin-1 anyonic chains.

Topology of a dissipative spin: Dynamical Chern number, bath-induced nonadiabaticity, and a quantum dynamo effect

We analyze the topological deformations of a spin-1/2 in an effective magnetic field induced by an ohmic quantum dissipative environment at zero temperature. From Bethe Ansatz results and a variational approach, we confirm that the Chern number is preserved in the delocalized phase for $\alpha<1$. We report a divergence of the Berry curvature at the equator when $\alpha_c=1$ that appears at the localization Kosterlitz-Thouless quantum phase transition in this model. Recent experiments in quantum circuits have engineered non-equilibrium protocols in time to access topological properties at equilibrium from the measure of the (quasi-)adiabatic out-of-equilibrium spin expectation values. Applying a numerically exact stochastic Schr\"{o}dinger equation we find that, for a fixed sweep velocity, the bath induces a crossover from (quasi-)adiabatic to non-adiabatic dynamical behavior when the spin bath coupling increases. We also investigate the particular regime $H/\omega_c \ll v/H \ll 1$, where the dynamical Chern number observable built from out-of-equilibrium spin expectation values vanishes at $\alpha=1/2$. In this regime, the mapping to an interacting resonance level model enables us to characterize the evolution of the dynamical Chern number in the vicinity of $\alpha=1/2$. Then, we provide an intuitive physical explanation of the breakdown of adiabaticity in analogy to the Faraday effect in electromagnetism. We demonstrate that the driving of the spin leads to the production of a large number of bosonic excitations in the bath, which in return strongly affect the spin dynamics. Finally, we quantify the spin-bath entanglement and build an analogy with an effective model at thermal equilibrium.

Chemical potential-functional-theory about the properties of one-dimensional Hubbard model at finite temperature

In this paper, we numerically solve the thermodynamic Bethe-ansatz coupled equations for a one-dimensional Hubbard model at finite temperature and obtain the second order thermodynamics properties, such as the specific heat, compressibility, and susceptibility. We find that these three quantities could embody the phase transitions of the system, from the vacuum state to the metallic state, from the metallic state to the Mott-insulating phase, from the Mott-insulating phase to the metallic state, and from the metallic state to the band-insulating phase. With the increase of temperature, the thermal fluctuation overwhelms the quantum fluctuations and the phase transition points disappear due to the destruction of the Mott-insulating phase. But in the case of the strong interaction strength, the Mott-insulating phase is robust, embodying the compressibility. Furthermore, we study the thermodynamic properties of the inhomogeneous Hubbard model with trapping potential. Making use of the Bethe-ansatz results from the homogeneous Hubbard model, we construct the chemical potential-functional theory (-BALDA) for the inhomogeneous Hubbard model instead of the commonly used density-functional theory, in order to solve the in-convergence problem of the Kohn-Sham equation in the case of the divergence appearing in the exchange-correlation potential. We further point out a multi-dimensional bisection method which changes the Kohn-Shan equation into a problem of finding the fixed points. Through -BALDA we numerically solve the one-dimensional homogeneous Hubbard model of trapping potential. The density profile and the local compressibility are obtained. We find that at a given interaction strength, the metallic phase and the Mott-insulating phase are destroyed and the density profile becomes a Guassian distribution with increasing temperature. To the metallic phase, Friedel oscillation caused by quantum fluctuations is still visible at low temperature. With increasing temperature, Friedel oscillation will disappear. This situation reflects the fact that the thermal fluctuation overwhelms the quantum fluctuations. For the Mott-insulating phase, the Mott-insulating plateau is robust at a certain temperature and only the boundary of the Mott-insulating plateau is destroyed. With increasing temperature, the Mott insulating plateau will be destroyed. And the change of the local compressibility provides the information about such a change. So we conclude that the thermal fluctuation destroys the original quantum phase. Through our analysis, we find that the -BALDA can be used to study the finite temperature properties for the system of the exchange-correlation potential divergence with high efficiency.

Local moment approach as a quantum impurity solver for the Hubbard model

The local moment approach (LMA) has presented itself as a powerful semi-analytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we attempt to complete the phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum-rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. We also show that, in the metallic phase, the low-energy scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong-coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. Finally we study optical properties and find universal features such as absorption peak position governed by the low-energy scale and a doping independent crossing point, often dubbed as the "isosbestic point" in experiments.

Self‐consistent hybridization expansions for static properties of the Anderson impurity model

AbstractBuilding an effective Hamiltonian utilizing a projector‐operator procedure, we derive an approximation based on a self‐consistent hybridization expansion to study the ground state properties of the Anderson impurity model. We applied the approximation to the general case of finite Coulomb repulsion U, extending previous work with the same formalism in the infinite‐U case. The ground state energy and their related zero temperature properties are accurately obtained in the case in which U is large enough, but still finite, as compared with the rest of energy scales involved in the model. The results for the valence of the impurity are compared with exact results that we obtain from equations derived using the Bethe ansatz and with a perturbative approach. The magnetization and magnetic susceptibility is also compared with Bethe ansatz results. In order to do this comparison, we also show how to regularize the Bethe ansatz integral equations necessary to calculate the impurity valence, for arbitrary values of the parameters.

Finite-temperature scaling of spin correlations in a partially magnetized HeisenbergS=12chain

Inelastic neutron scattering is employed to study transverse spin correlations of a Heisenberg $S=1/2$ chain compound in a magnetic field of 7.5 T. The target compound is the antiferromagnetic Heisenberg $S=1/2$ chain material 2(1,4-Dioxane)$\cdot$2(H$_2$O)$\cdot$CuCl$_2$, or CuDCl for short. The validity and the limitations of the scaling relation for the transverse dynamic structure factor are tested, discussed and compared to the Tomonaga-Luttinger spin liquid theory and to Bethe-ansatz results for the Heisenberg model.

Eigenstate thermalization hypothesis and integrability in quantum spin chains

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg $(XXX)$ chain. We provide numerical evidence that the ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as ${L}^{\ensuremath{-}1/2}$, with $L$ the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the $XXX$ chain. We numerically verify that typical midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).

Spin-wave theory applied to the low-temperature properties of the spin-[formula omitted] ferromagnetic chain with the y-direction exchange anisotropy

In this paper, we apply the spin-wave theory within the simple self-consistent method, and provide a simple way to investigate the spin-1/2 ferromagnetic chain with the y-direction exchange anisotropy. It is found that the anisotropy has an influence on the ground-state and low-temperature properties of the system. Anisotropy dependences of both the susceptibility maximum and specific heat maximum fit well to the exponential laws. The linear decrease is shown for the position of the susceptibility maximum. The position of the specific heat maximum displays three kinds of the anisotropy dependence, i.e., the power law, constant law and exponential law. The specific heat coefficient is calculated to be a monotonically increasing function of the anisotropy in the zero-temperature limit. At some low temperature, the Heisenberg behavior is observed to cross over the XY behavior, which agrees with the experimental result of the quasi-one-dimensional compound (C6H11NH3)CuBr3. In the isotropic and anisotropic cases, our results are in good accord with the exact Bethe-ansatz results and quantum Monte Carlo estimates, and have some advantages over the findings of Green-function method and the renormalization group method at low temperatures.

Transport through nanostructures: Finite time versus finite size

Numerical simulations and experiments on nanostructures out of equilibrium usually exhibit strong finite size and finite measuring time ${t}_{\mathrm{m}}$ effects. We discuss how these affect the determination of the full counting statistics for a general quantum impurity problem. We find that, while there are many methods available to improve upon finite-size effects, any real-time simulation or experiment will still be subject to finite-time effects: In short size matters, but time is limiting. We show that the leading correction to the cumulant generating function (CGF) at zero temperature for single-channel quantum impurity problems is proportional to $ln{t}_{\mathrm{m}}$, where the constant of proportionality is universally related to the steady state CGF itself for non-interacting systems; universal in this context means independent of details of the quench procedure, i.e., independent of the switching on of both voltage and counting field. We give detailed numerical evidence for the case of the self-dual interacting resonant level model that this relation survives the addition of interactions. This allows the extrapolation of finite measuring time in our numerics to the long-time limit, in excellent agreement with Bethe-ansatz results.

Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z -axis anisotropy λ and magnetic field B . To demonstrate its accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first seven lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z -axis anisotropy and the magnetic field, the 2- and 4-spin correlation functions are calculated, and the results are compared with those obtained by bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λ space is determined by Roomany–Wyld renormalization group (RG) finite size scaling. The results are in good agreement with those obtained by the level-spectroscopy method.

Full density-matrix numerical renormalization group calculation of impurity susceptibility and specific heat of the Anderson impurity model

Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models (see A. Weichselbaum and J. von Delft) by using the completeness of the eliminated states introduced by F. B. Anders and A. Schiller (2005). While these developments prove particularly useful in the calculation of transient response and finite temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to the Anderson impurity model. We compare the results for the susceptibility and specific heat to both the conventional approach within NRG and to exact Bethe ansatz results. We also point out a subtlety in the calculation of the susceptibility (in a uniform field) within the FDM approach. Finally, we show numerically that for the Anderson model, the susceptibilities in response to a local and a uniform magnetic field coincide in the wide-band limit, in accordance with the Clogston-Anderson compensation theorem.

Estimating quasi-long-range order via Rényi entropies

We show how entanglement entropies allow for the estimation of quasi-long-range order in one-dimensional systems whose low-energy physics is well captured by the Tomonaga-Luttinger liquid universality class. First, we check our procedure in the exactly solvable XXZ spin-$1/2$ chain in its entire critical region, finding very good agreement with Bethe ansatz results. Then, we show how phase transitions between different dominant orders may be efficiently estimated by considering the superfluid-charge density wave transition in a system of dipolar bosons. Finally, we discuss the application of this method to multispecies systems such as the one-dimensional Hubbard model. Our work represents the first proof of a direct relationship between the Luttinger parameter and R\'enyi entropies in both bosonic and fermionic lattice models.

Dynamic susceptibilities of the single-impurity Anderson model within an enhanced noncrossing approximation

The single-impurity Anderson model is studied within an enhanced noncrossing approximation (ENCA). This method is extended to the calculation of susceptibilities and thoroughly tested, also in order to prepare applications as a building block for the calculation of susceptibilities and phase transitions in correlated lattice systems. A wide range of model parameters, such as impurity occupancy, temperature, local Coulomb repulsion, and hybridization strength, are studied. Results for the spin and charge susceptibilities are presented. By comparing the static quantities to exact Bethe ansatz results, it is shown that the description of the magnetic excitations of the impurity within the ENCA is excellent, even in situations with large valence fluctuations or vanishing Coulomb repulsion. The description of the charge susceptibility is quite accurate in situations where the singly occupied ionic configuration is the unperturbed ground state; however, it seems to overestimate charge fluctuations in the asymmetric model at too low temperatures. The dynamic spin-excitation spectrum is dominated by the Kondo screening of the impurity spin through the conduction band, i.e., the formation of the local Kondo singlet. A finite local Coulomb interaction $U$ leads to a drastic reduction in the charge response as processes involving the doubly occupied impurity state are suppressed. In the asymmetric model, the charge susceptibility is enhanced for excitation energies smaller than the Kondo scale ${T}_{K}$ due to the influence of valence fluctuations.

Spin-charge decoupling and the photoemission line-shape in one-dimensional insulators

The recent advances in angle resolved photoemission techniques allowed the unambiguous experimental confirmation of spin charge decoupling in quasi one dimensional (1D) Mott insulators. This opportunity stimulates a quantitative analysis of the spectral function $A(k,\omega)$ of prototypical one dimensional correlated models. Here we combine Bethe Ansatz results, Lanczos diagonalizations and field theoretical approaches to obtain $A(k,\omega)$ for the 1D Hubbard model as a function of the interaction strength. By introducing a {\it single spinon approximation}, an analytic expression is obtained, which shows the location of the singularities and allows, when supplemented by numerical calculations, to obtain an accurate estimate of the spectral weight distribution in the $(k,\omega)$ plane. Several experimental puzzles on the observed intensities and line-shapes in quasi 1D compounds, like ${\rm SrCuO_2}$, find a natural explanation in this theoretical framework.

Solutions ofPT-symmetric tight-binding chain and its equivalent Hermitian counterpart

We study the Non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an $N$-site tight-binding chain with two conjugated imaginary potentials $\pm i\gamma $ at two end sites, which Hamiltonian has parity-time symmetry ($\mathcal{PT}$ symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases, an unbroken symmetry phase with a purely real energy spectrum in the region $\gamma \prec \gamma_{c}$ and a spontaneously-broken symmetry phase with $N-2$ real and 2 imaginary eigenvalues in the region $\gamma \succ \gamma_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of $\gamma_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.

Field-dependent specific heat ofYb4As3: Agreement between a spin-12model and experiment

We report the low-temperature specific-heat measurements on polydomain ${\text{Yb}}_{4}{\text{As}}_{3}$ at magnetic fields up to 20 T. Taking into account the Bethe ansatz results, the zero-field data have been used for the estimation of the lattice specific heat, resulting in a value of the exchange integral for the Heisenberg model of the antiferromagnetic spin $S=\frac{1}{2}$ chain of $J/{k}_{B}=\ensuremath{-}28\text{ }\text{K}$. A quantitative agreement has been achieved between the experimental magnetic specific-heat data in magnetic field and the numerical results obtained by the quantum transfer-matrix (QTM) simulation technique. The finite-size QTM approximants have been analyzed and an extrapolation procedure recovering the known density matrix renormalization group (DMRG) results down to very low temperature has been proposed. On the basis of the data in magnetic field and using the earlier DMRG results, the energy-gap size $\ensuremath{\Delta}$ has been analyzed as a function of the applied magnetic field $B$, leading to an experimental verification of the scaling law $\ensuremath{\Delta}\ensuremath{\propto}{B}^{2/3}$ following from the sine-Gordon model.

Entanglement perturbation theory for the elementary excitation in one dimension

The entanglement perturbation theory is developed to calculate the excitation spectrum in one dimension. Applied to the spin- 1 2 antiferromagnetic Heisenberg model, it reproduces the des Cloiseaux–Pearson Bethe ansatz result. As for spin-1, the spin-triplet magnon spectrum has been determined for the first time for the entire Brillouin zone, including the Haldane gap at k = π .