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arXiv:2606.14374

Bond-operator analytical approach for the $t$-$J$ model

A. V. Syromyatnikov

Published Jun 12, 2026• 0 citations
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Abstract

We present a bond-operator theory (BOT) for analytical consideration of the $t$-$J$ model and its extensions with longer-rage hopping terms. This technique is based on previously suggested representation of electron operators via localized spins 1/2 and spinless Fermi-operators of mobile holons which requires no constraint between them. We introduce a representation of operators of spins, holons, and electrons in (extended) unit cell containing several lattice sites via a zoo of Bose- and Fermi-operators acting in the Hilbert space of all quantum states of the whole unit cell. BOT provides a regular expansion of physical quantities in powers of $1/n$ using conventional diagrammatic technique, where $n\ge1$ is the maximum number of introduced quasiparticles (bosons and fermions) which can occupy a unit cell. The suggested representation reproduces commutation algebra of all operators at any $n>0$ and allows to consider both magnetically ordered and disordered phases. Some elementary excitations described in the BOT by separate bosons or fermions appear in common approaches as bound states of conventional quasiparticles. In particular, there are two-hole bound states (Cooper pairs of two holes) which are described within the BOT by separate bosons. We discuss in detail properties of the $t$-$J$ model on the square lattice with no more than two holes (polarons). Although the expansion parameter $1/n$ is not small in the physically meaningful case of $n=1$, we obtain a good quantitative agreement with previous numerical findings at $n=1$ even in the first order in $1/n$ after taking into account a few simple diagrams. Self-consistent calculations in the first order in $1/n$ bring our results to a very good quantitative agreement with previous numerical findings of the ground state energy, staggered magnetization, and spectra of magnons, polarons, and lowest-energy two-hole bound states.

DOI
10.48550/arxiv.2606.14374

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Bond-operator analytical approach for the ttt-JJJ model
We present a bond-operator theory (BOT) for analytical consideration of the ttt-JJJ model and its extensions with longer-rage hopping terms. This technique is based on previously suggested representation of electron operators via localized spins 1/2 and spinless Fermi-operators of mobile holons which requires no constraint between them. We introduce a representation of operators of spins, holons, and electrons in (extended) unit cell containing several lattice sites via a zoo of Bose- and Fermi-operators acting in the Hilbert space of all quantum states of the whole unit cell. BOT provides a regular expansion of physical quantities in powers of 1/n1/n1/n using conventional diagrammatic technique, where n≥1n\ge1n≥1 is the maximum number of introduced quasiparticles (bosons and fermions) which can occupy a unit cell. The suggested representation reproduces commutation algebra of all operators at any n>0n>0n>0 and allows to consider both magnetically ordered and disordered phases. Some elementary excitations described in the BOT by separate bosons or fermions appear in common approaches as bound states of conventional quasiparticles. In particular, there are two-hole bound states (Cooper pairs of two holes) which are described within the BOT by separate bosons. We discuss in detail properties of the ttt-JJJ model on the square lattice with no more than two holes (polarons). Although the expansion parameter 1/n1/n1/n is not small in the physically meaningful case of n=1n=1n=1, we obtain a good quantitative agreement with previous numerical findings at n=1n=1n=1 even in the first order in 1/n1/n1/n after taking into account a few simple diagrams. Self-consistent calculations in the first order in 1/n1/n1/n bring our results to a very good quantitative agreement with previous numerical findings of the ground state energy, staggered magnetization, and spectra of magnons, polarons, and lowest-energy two-hole bound states.
We present a bond-operator theory (BOT) suitable for the description of both magnetically ordered phases and paramagnetic phases with singlet ground states in spin-1/2 magnets. This technique allows one to trace the evolution of quasiparticles across the transition between the phases. Some elementary excitations described in the theory by separate bosons appear in conventional approaches as bound states of well-known quasiparticles (magnons or triplons). The proposed BOT provides a regular expansion of physical quantities in powers of 1/n1/n1/n, where nnn is the maximum number of bosons that can occupy a unit cell (physical results correspond to n=1n=1n=1). Two variants of BOT are suggested: for two and for four spins in the unit cell (two-spin and four-spin BOTs, respectively). We consider spin-1/2 Heisenberg antiferromagnets (HAF) on a simple square lattice bilayer by the two-spin BOT. The ground-state energy E\mathcal{E}E, the staggered magnetization MMM, and quasiparticle spectra found within the first order in 1/n1/n1/n are in good quantitative agreement with previous results both in paramagnetic and in ordered phases not very close to the quantum critical point between the phases. By doubling the unit cell in two directions, we discuss spin-1/2 HAF on a square lattice using the suggested four-spin BOT. We identify the magnon and the amplitude (Higgs) modes among fifteen spin-2, spin-1, and spin-0 quasiparticles arising in the theory. The magnon spectrum, E\mathcal{E}E, and MMM found in the first order in 1/n1/n1/n are in good quantitative agreement with previous numerical and experimental results. We observe a special moderately damped spin-0 quasiparticle (``singlon'' for short) whose energy is smaller than the energy of the Higgs mode in the most part of the Brillouin zone. By considering HAF with Ising-type anisotropy, we find that both Higgs and singlon modes stem from two-magnon bound states, which merge with two-magnon continuum not far from the isotropic limit. We demonstrate that singlons appear explicitly in ``scalar'' correlators one of which describes the Raman intensity in B1g{B}_{1g}B1g​ symmetry. The latter is expressed in the leading order in 1/n1/n1/n via the singlon Green's function at zero momentum, which shows an asymmetric peak. The position of this peak given by the singlon energy coincides with the position of the so-called ``two-magnon'' peak observed experimentally in, e.g., layered cuprates.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
We show an exact representation of the t-J model by making use of a general Holstein-Primakoff transformation. The single occupancy constraint on the electrons is imposed strictly at every lattice site and the t-J Hamiltonian is transformed into an equivalent interacting boson and fermion system. The lowest-order approximation of a systematic perturbative expansion of the representation is shown to correspond to the mean-field theory of the Schwinger-boson approach. The second-order approximation gives the magnon-magnon, magnon-hole, and hole-hole interactions and the high-order antiferromagnetic spin polaron. In particular the second-order approximation takes further account of the renormalization of the magnon frequency and the antiferromagnetic spin polaron. The spectra of holes and magnons are calculated with the Green's function. The relevance of the obtained results to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is discussed. \textcopyright{} 1996 The American Physical Society.
We discuss the spin-12J1−−J2\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}--{J}_{2}21​J1​−−J2​ model on the triangular lattice using recently proposed bond-operator theory (BOT). In agreement with previous discussions of this system, we obtain four phases upon J2{J}_{2}J2​ increasing: the phase with ${120}^{\ensuremath{\circ}}$ ordering of three sublattices, the spin-liquid phase, the state with the collinear stripe order, and the spiral phase. The ${120}^{\ensuremath{\circ}}$ and the stripe phases are discussed in detail. All calculated static characteristics of the model are in good agreement with previous numerical findings. In the ${120}^{\ensuremath{\circ}}$ phase, we observe the evolution of the quasiparticles' spectra and dynamical structure factors (DSFs) upon approaching the spin-liquid phase. Some of the considered elementary excitations were introduced first in our recent study of this system at J2=0{J}_{2}=0J2​=0 using the BOT. In the stripe phase, we observe that the doubly degenerate magnon spectrum known from the spin-wave theory (SWT) is split by quantum fluctuations, which are taken into account more accurately in the BOT. As compared with other known findings of the SWT in the stripe state, we observe additional spin-1 and spin-0 quasiparticles which give visible anomalies in the transverse and longitudinal DSFs. We obtain also a special spin-0 quasiparticle named the singlon that produces a peak only in the four-spin correlator and which is invisible in the longitudinal DSF. We show that the singlon spectrum lies below energies of all spin-0 and spin-1 excitations in some parts of the Brillouin zone. The singlon spectrum at zero momentum can be probed by the Raman scattering.