Electron states and magnetic phase diagrams of strongly correlated systems

TL;DR

This paper analyzes various auxiliary-particle methods for electron correlations in many-electron models, applying them to copper-oxide systems and exploring magnetic phase diagrams within Hubbard and Kondo models.

Contribution

It compares auxiliary-particle approaches and employs a generalized Hartree-Fock method to analyze magnetic phases and phase separation in correlated electron systems.

Findings

Correct estimation of doubly occupied states improves phase energy calculations

Magnetic phase diagrams depend on band filling and interaction parameters

Slave-boson approach effectively captures correlations in Hubbard model

Abstract

Various auxiliary-particle approaches to treat electron correlations in many-electron models are analyzed. Applications to copper-oxide layered systems are discussed. The ground-state magnetic phase diagrams are considered within the Hubbard and $s$-$d$ exchange (Kondo) models for square and simple cubic lattices vs. band filling and interaction parameter. A generalized Hartree-Fock approximation is employed to treat commensurate ferro-, antiferromagnetic, and incommensurate (spiral) magnetic phases, and also magnetic phase separation. The correlations are taken into account within the Hubbard model by using the slave-boson approach. The main advantage of this approach is correct estimating the contribution of doubly occupied states number and therefore the paramagnetic phase energy.

Equations

$${\mathcal{H}}_{\mathrm{H}}=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{†}{c}_{j\sigma }+U\sum _i{n}_{i\uparrow }{n}_{i\downarrow },$$

$${\mathcal{H}}_{t_J}=-\sum _{ij\sigma }{t}_{ij}{X}_i(0\sigma )X_j(\sigma 0)+\sum _{ij}{J}_{ij}{\mathbf{S}}_i{\mathbf{S}}_j.$$

$$c_{i\sigma }^{†}=X_i(\sigma ,0)+\sigma X_i(2,-\sigma )\to f_{i\sigma }^{†}{e}_i+\sigma d_i^{†}{f}_{i-\sigma }.$$

$$\mathbf{S}_{i}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}a_{i\sigma}^{\dagger}\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}}a_{i\sigma^{\prime}},\,$$

$$c_{i\sigma }^{†}\to f_{i\sigma }^{†}{z}_{i\sigma }^{†}$$

$$\begin{gathered} z_{i\sigma }^{†}=(1-e_i^{†}{e}_i-p_{i-\sigma }^{†}{p}_{i-\sigma }{)}^{-1/2}(p_{i\sigma }^{†}{e}_i+d_i^{†}{p}_{i-\sigma }) \\ \times (1-d_i^{†}{d}_i-p_{i\sigma }^{†}{p}_{i\sigma }{)}^{-1/2}. \end{gathered}$$

$$c_{i\sigma }^{†}=\sum _{{\sigma }^{\prime }}{f}_{i\sigma }^{†}{z}_{i\sigma {\sigma }^{\prime }}^{†},{\hat{z}}_i=e_i^{†}{\hat{L}}_i{M}_i{\hat{R}}_i{\hat{p}}_i+{\hat{\tilde{p}}}_i^{†}{\hat{L}}_i{M}_i{\hat{R}}_{\hat{i}}{d}_i$$

$$c_{i\sigma }^{†}=f_{i\sigma }^{†}\exp (i{\theta }_i)$$

$$\begin{gathered} {\mathcal{H}}_{\mathrm{A}}=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{†}{c}_{j\sigma }+{ϵ}_d\sum _{i\sigma }{d}_{i\sigma }^{†}{d}_{i\sigma } \\ +V\sum _{i\sigma }(c_{i\sigma }^{†}{d}_{i\sigma }+d_{i\sigma }^{†}{c}_{i\sigma })+U\sum _i{d}_{i\uparrow }^{†}{d}_{i\uparrow }{d}_{i\downarrow }^{†}{d}_{i\downarrow }, \end{gathered}$$

$$\mathcal{H}_{s-d}=\sum_{\mathbf{k}\sigma}t_{\mathbf{k}}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}-I\sum_{i\sigma\sigma^{\prime}}(\mathbf{S}_{i}\cdot\mbox{\boldmath$\sigma$}_{\sigma\sigma^{\prime}})c^{\dagger}_{i\sigma}c_{i\sigma^{\prime}},$$

$$I=V^2[1/({ϵ}_d-{ϵ}_{\mathrm{F}})-1/(U+{ϵ}_d-{ϵ}_{\mathrm{F}})],$$

$$\mathcal{H}$$

$$X{{}_{i}(0,-\sigma)}=-\frac{\sigma}{2\sqrt{2}}\sum_{\sigma^{\prime}}d_{i\sigma^{\prime}}^{\dagger}(1-n_{i,-\sigma^{\prime}})[\delta_{\sigma\sigma^{\prime}}-2(\mathbf{S}_{i}\mbox{\boldmath$\sigma$}_{\sigma^{\prime}\sigma})].$$

$${\mathcal{H}}_{\mathrm{eff}}=\sum _{ij\sigma {\sigma }^{\prime }}{t}_{ij}^{\sigma {\sigma }^{\prime }}{f}_{i\sigma }^{†}{f}_{j{\sigma }^{\prime }}{z}_{i\sigma }^{†}{z}_{j{\sigma }^{\prime }}+U\sum _i{d}_i^{†}{d}_i,$$

$${\chi }_{\mathbf{Q}}^0=\frac{1}{N}\sum _{\mathbf{k}}\frac{f_{\mathbf{k}+\mathbf{Q}}-f_{\mathbf{k}}}{t_{\mathbf{k}}-t_{\mathbf{k}+\mathbf{Q}}}$$

Full text

Magnetic properties of strongly correlated transition-metal compounds and their relation to doping, lattice geometry, band structure and interaction parameters are still being extensively investigated. In particular, the details of magnetic order in the ground state remain to be examined both theoretically and experimentally. During recent decades, the two-dimensional (2D) case closely related to the problem of high-temperature superconductivity in cuprates and iron arsenides has been intensively investigated theoretically.

The ground state of strongly correlated systems is characterized by a competition of ferromagnetic (FM) and antiferromagnetic (AFM) ordering which results in occurrence of spiral magnetic ordering Igoshev:2010 or the magnetic phase separation Visscher:1973 ; Igoshev:2010 ; Igoshev:2015 . The consideration of these problems is performed within a number of many-electron models. In the present work we discuss theoretical approaches to treat these model within auxiliary-particle representations (Sect.1) and present some results of numerical calculations (Sect.2).

I Theoretical models and slave particle representations

To describe the properties of such systems one uses many-electron models like the Hubbard, $s$-$d$ exchange (Kondo) and Anderson lattice models. These are widely applied, e. g., for high-$T_{c}$ cuprates and rare earth compounds. There exist some relations (mappings) between these models in various parameter regions.

The Hamiltonian of the Hubbard model reads

[TABLE]

where $c^{\dagger}_{i\sigma}$ are electron creation operators. In the limit of large Hubbard parameter $U$ and band filling $n<1$ (hole doping) this is reduced to the $t-J$ model

[TABLE]

where $X_{i}(\Gamma,\Gamma^{\prime})=|\Gamma_{i}\rangle\langle\Gamma^{\prime}_{i}|$ are the Hubbard X-operators acting on the $i$ site local subspace 654 , $J_{ij}=2t^{2}_{ij}/U$.

To proceed with analytical and numerical calculations, it is convenient to use auxiliary (“slave”) boson and fermion representations. In connection with the theory of high-temperature superconductors, Anderson 633a put forward the idea of the separation of the spin and charge degrees of freedom of electron ($\sigma=\pm 1$):

[TABLE]

Here, $f_{i\sigma}^{\dagger}$ are the creation operators for neutral fermions (spinons), and $e_{i}^{\dagger}$, $d_{i}^{\dagger}$ are the creation operators for charged spinless bosons (holons and doublons). For large $U$, we have to retain only holon or doublon degrees of freedom for hole or electron doping, respectively.

In fact, the choice of the Fermi statistics for spinons and Bose one for holons is not unique and can be varied depending on the physical picture (e.g., presence or absence of magnetic ordering, see also Kane ). Thus, the spin operators in Eq. (2) are presented as the bilinear form

[TABLE]

in terms of Schwinger bosons ($a_{i\sigma}^{\dagger}=b_{i\sigma}^{\dagger}$) or fermionic spinons ($a_{i\sigma}^{\dagger}=f_{i\sigma}^{\dagger}$), $\sigma$ being Pauli matrices.

A more complicated representation proposed by Kotliar and Ruckenstein Kotliar86 introduces the slave boson operators $e_{i},\,p_{i\sigma},\,d_{i}$, so that

[TABLE]

where $f_{i\sigma},f_{i\sigma}^{{\dagger}}$ are the slave Fermi operators, and

[TABLE]

There exists also a rotationally invariant version Fresard:1992a

[TABLE]

where the factors $L,M,R$ are similar to those in (6), the scalar and vector slave boson fields $p_{i0}$ and $\mathbf{p}_{i}$ are introduced as \hat{p}_{i}=\frac{1}{2}(p_{i0}\sigma_{0}+\mathbf{p}_{i}\mbox{\boldmath\sigma}) and $\widehat{\tilde{p}}_{i}$ is the time reverse of operator $\hat{p}_{i}$. This version is suitable for magnetically ordered phases to take into account spin fluctuation corrections. In particular, it can describe in a simple way non-quasiparticle states owing electron-magnon scattering which were earlier treated in the many-electron representation of Hubbard’s operators (cf. Ref.338 ).

We mention also the rotor representation Florens

[TABLE]

where the original Hubbard interaction is replaced in by a simple kinetic term for the phase field, $(U/2)\hat{L}_{i}^{2}$, with the angular momentum $\hat{L}=-i\partial/\partial\theta$. This representation is suitable to describe the metal-insulator transition in the paramagnetic phase.

In the case of the Anderson lattice model transport and magnetic properties are separated between different systems, $s$ and $d$ correspondingly:

[TABLE]

$\epsilon_{d}$ is the energy of localized (‘$d$-electron’) state, $V$ is on-site $s$-$d$ hybridization providing the coupling between these subsystems.

Provided that the $d$-level is well below the Fermi energy and Coulomb interaction is sufficiently large ($|V|\ll\epsilon_{\rm F}-\epsilon_{d}$, $U$), this model can be reduced by the Schrieffer-Wolff transformation to the $s$-$d$ exchange model

[TABLE]

with spin $S=1/2$ and the exchange parameter

[TABLE]

where $\epsilon_{\rm F}$ is the Fermi level.

Remember also the three-band model of cuprates

[TABLE]

where $\varepsilon$ and $\Delta$ are positions of $p$- and $d$-levels for O- and Cu-ions, $V_{\mathbf{k}}=2V_{pd}\left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)^{1/2}$ are the matrix elements of $p$—$d$ hybridization (cf. Raimondi ). In the large-$U$ limit we can use the slave boson representation $d^{\dagger}_{i\sigma}\rightarrow X{{}_{i}(\sigma 0)}=f^{{\dagger}}_{i\sigma}e_{i}$.

For $|V_{pd}|\ll\varepsilon-\Delta$ (large charge-transfer gap) the Hamiltonian (12) is again reduced by a canonical transformation to the $t-J$ model with $t_{\text{eff}}=V_{pd}^{2}/(\varepsilon-\Delta)$. It is interesting that the $t-J$ model obtained from the one-band Hubbard model is also formally reduced to a similar structure in the representation (7) with auxiliary rather than physical particles $p_{i}$. Thus the Hubbard model and the model (12) can be considered in a parallel way Raimondi .

To describe doped cuprates, also a representation of the Fermi dopons $d^{\dagger}_{i\sigma}$ was proposed Ribeiro ; Scr

[TABLE]

On substituting (13) into the $t-J$ Hamiltonian (2) we obtain the terms which are linear in spin operators. These can provide hybridization between electrons (dopons) and Fermi spinons to describe nodal-antinodal dichotomy and formation of large Fermi surface in cuprates with the increase of doping Scr . Thus the initial one-band model takes the form of an effective two-band model. Note that a similar structure is obtained in the spin-rotation representation (7).

On the other hand, in the antiferromagnetic case it is convenient to use the Schwinger rather than the spinon representation (see (4)). Such an approach was developed in Punk to describe the formation of spin liquid state in terms of frustrations in localized-spin subsystems. The supersymmetry approach Lavagna mixes the fermionic and the bosonic representation of the spin following the standard rules of superalgebra.

II Results of numerical calculations

After the local rotation in the spin space matching the local magnetization vectors at different sites, say along $z$ axis, (which is necessary for the consideration of magnetic spirals) by the angle $\mathbf{QR}_{i}$ (where $\mathbf{Q}$ is the wave vector of the spiral) the Hubbard Hamiltonian in the slave boson representation (6) takes the form

[TABLE]

with $t_{ij}^{\sigma\sigma^{\prime}}=\exp[\mathrm{i}\mathbf{Q}(\mathbf{R}_{i}-\mathbf{R}_{j})\sigma^{x}]_{\sigma\sigma^{\prime}}t_{ij}$. This form enables us to construct the mean-field approximation where the boson averages do not depend on lattice site.

The calculations were performed for the half-filled Hubbard model to describe the metal-insulator transition and for the magnetic phase diagram for arbitrary filling Timirgazin ; Timirgazin1 ; Igoshev:2015 . Here we present the results for the square (Fig. 1) and simple cubic (Fig. 2) lattices in the Hubbard model within the Hartree–Fock approximations (HFA) and slave boson approach (SBA) in the mean-field version and within HFA for the $s$-$d$ exchange model Igoshev:2015 ; Antipin .

One can see that HFA yields the variety of spiral magnetic phases, as well as FM and AFM ones at sufficiently large $U$ generally and at any $U$ in the vicinity of half-filling. The account of correlations (SBA) leads to a noticeable suppression of magnetically ordered states in comparison with HFA: the corresponding density intervals in the phase diagram decrease strongly, and the variety of the spiral states disappears. Besides that, in SBA there occurs a wide region of paramagnetic (PM) state which is a manifestation of correct treatment of the energy of doubles.

It should be stressed that only SBA provides a correct description of large $U$ case at finite density of current carriers, whereas the ground state energy in HFA is divergent at $U\rightarrow\infty$ due to overestimation of Coulomb interaction energy, as well as in random phase approximation (RPA) 353 .

Within the mean-field approximation, the Hubbard model is equivalent to the $s$-$d$ model with the replacement $IS=Um/2$. However, the phase separation condition and description of the PM phase are different in these models. Thus the phases are strongly redistributed.

Due to existence of localized moments, ferromagnetic ordering in the $s$-$d$ model is favorable already at small $|I|$, whereas within the Hubbard model it occurs at sufficiently large $U$ only. The increase of $|I|$ results in a growith of FM region. At small $|I|$ the wave vector of magnetic phase is specified by the position of the maximum of the Lindhardt function

[TABLE]

calculated in the PM phase ($f_{\mathbf{k}}$ is the Fermi function).

An important difference of the square and simple cubic lattice (2D and 3D) cases is the form of FM region at small $n$. In the former case its width vanishes at $I\rightarrow 0$, but in the latter case the width is finite and sufficiently large, the transition to the spiral $(0,\pi,0)$ phase being of the first order. For the square lattice, the spiral phases are fully suppressed by FM and AFM regions at $|IS|\gtrsim 6t$. For the simple cubic lattice, the spiral phases turn out to be more stable. For small number of carriers in the AFM matrix ($1-n\ll 1$), the phase separation between AFM and spiral ($(Q,\pi)$ for square lattice and $(Q,\pi,\pi)$ for simple cubic lattice) phases is present at small interaction parameter.

The research was carried out within the state assignment of FASO of Russia (theme “Quantum” No. AAAA-A18-118020190095-4). This work was supported in part by Ural Division of RAS (project no. 18-2-2-11) and by the Russian Foundation for Basic Research (project no. 16-02-00995).