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New Journal of Physics

Dynamics of string-like states of a hole in a quantum antiferromagnet: a diagrammatic Monte Carlo simulation

Nikolaos G Diamantis, Efstratios Manousakis

Published Nov 16, 2021• 20 citations
Publisher

Abstract

Abstract The dynamics of a hole motion in a quantum antiferromagnet has been studied in the past three decades because of its relationship to models related to superconductivity in cuprates. The same problem has received significant attention because of its connection to very recent experiments of the dynamics of ultra-cold atoms in optical lattices where models of strongly correlated electrons can be simulated. In this paper we apply the diagrammatic Monte Carlo method to calculate the single-hole Green’s function in the t–J model, where the J term is linearized, in a wide range of imaginary-time with the aim to examine the polaron formation and in particular the details of the contribution of the so-called string excitations found in such recent experiments. We calculate the single-hole spectral function by analytic continuation from imaginary to real time and study the various aspects that constitute the string picture, such as, the energy–momentum dependence of the main quasiparticle peak and its residue, the internal excitations of the string which appear as multiple peaks in the spectral function as well as their momentum dependence. We find that the earlier analysis of the spectral function based on a mobile-hole connected with a string of overturn spins and the contribution of the internal string excitations as obtained from the non-crossing approximation is accurate.

DOI
10.1088/1367-2630/ac39b5

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Abstract The dynamics of a hole motion in a quantum antiferromagnet has been studied in the past three decades because of its relationship to models related to superconductivity in cuprates. The same problem has received significant attention because of its connection to very recent experiments of the dynamics of ultra-cold atoms in optical lattices where models of strongly correlated electrons can be simulated. In this paper we apply the diagrammatic Monte Carlo method to calculate the single-hole Green’s function in the t–J model, where the J term is linearized, in a wide range of imaginary-time with the aim to examine the polaron formation and in particular the details of the contribution of the so-called string excitations found in such recent experiments. We calculate the single-hole spectral function by analytic continuation from imaginary to real time and study the various aspects that constitute the string picture, such as, the energy–momentum dependence of the main quasiparticle peak and its residue, the internal excitations of the string which appear as multiple peaks in the spectral function as well as their momentum dependence. We find that the earlier analysis of the spectral function based on a mobile-hole connected with a string of overturn spins and the contribution of the internal string excitations as obtained from the non-crossing approximation is accurate.
The motion of a single hole in an arbitrary magnetic background is investigated for the two-dimensional $t\ensuremath{-}J$ model. The wave function of the hole is described within a generalized string picture which leads to a modified concept of spin polarons. We calculate the one-hole spectral function using a large-string basis for the limits of a N\'eel ordered and a completely disordered background. In addition we use a simple approximation to interpolate between these cases. For the antiferromagnetic background we reproduce the well-known quasiparticle band. In the disordered case the shape of the spectral function is found to be strongly momentum dependent, the quasiparticle weight vanishes for all hole momenta. Finally, we discuss the relevance of results for the lowest energy eigenvalue and its dispersion obtained from calculations using a polaron of minimal size as found in the literature.
We develop a nonperturbative theory for hole dynamics in antiferromagnetic spin lattices, as described by the t-J model. This is achieved by generalizing the self-consistent Born approximation to nonequilibrium systems, making it possible to calculate the full time-dependent many-body wave function. Our approach reveals three distinct dynamical regimes, ultimately leading to the formation of magnetic polarons. Following the initial ballistic stage of the hole dynamics, coherent formation of string excitations gives rise to characteristic oscillations in the hole density. Their damping eventually leaves behind magnetic polarons that undergo ballistic motion with a greatly reduced velocity. The developed theory provides a rigorous framework for understanding nonequilibrium physics of defects in quantum magnets and quantitatively explains recent observations from cold-atom quantum simulations in the strong coupling regime.
The aim of the present paper is twofold. The first goal is to show that high resolution angle-resolved photoelectron spectra from cuprates indicate the presence of string-like excitations of the quasihole excitation in a quantum antiferromagnet. In order to compare with the experimental intensity plots, we calculate the spectral function of the $t\ensuremath{-}J$ and the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ models within the self-consistent Born approximation for widely accepted values of the parameters of these models. The main features of the high resolution photoelectron spectra are in general agreement with the results based on the above models and can be understood by considering not only the lowest-energy quasiparticle peak but, also, the higher-energy string-like excitations of the hole. These features are (a) the energy-momentum dispersion of the string-like excitations, (b) the momentum dependence of the spectral weight of the quasiparticle peak, and (c) the way in which the spectral weight of the lowest-energy quasiparticle peak is transferred to higher-energy string excitations, including the fact that it vanishes near the $\ensuremath{\Gamma}$ point. The second goal of the present paper is to make the case that a proper analysis of both the numerical results obtained from these models and of the experimental results, suggests a theoretical picture for the internal structure of the hole-quasiparticle and a string-exchange pairing mechanism due to strong antiferromagnetic correlations among the background spins. We find that, using a simple model in which the Hilbert space is restricted to states of only unbroken strings attached to the holes and in which the holes are connected with unbroken strings, we can provide a good quantitative description of the most accurate numerical results. We find that the holes experience an effective interaction due to a string-exchange mechanism which is characterized by a rather large string tension and this can provide pairing energy scales much larger than those suggested by spin-fluctuation-mediated pairing models. In addition, it is argued that such string-exchange interaction tends to bind holes in a bound state with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry.
We present quantum Monte Carlo (QMC) simulations for a single hole in a $t\ensuremath{-}J$ model from J=0.4tJ=0.4tJ=0.4t to J=4tJ=4tJ=4t on square lattices with up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The lower edge of the spectrum is directly extracted from the imaginary time Green's function. In agreement with earlier calculations, we find flat bands around $(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}),$ $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0)$ and the minimum of the dispersion at $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2).$ For small J, both self-consistent Born approximation and series expansions give a bandwidth for the lower edge of the spectrum in agreement with the simulations, whereas for J/t>1,J/t>1,J/t>1, only series expansions agree quantitatively with our QMC results. This band corresponds to a coherent quasiparticle. This is shown by a finite-size scaling of the quasiparticle weight $Z(\stackrel{\ensuremath{\rightarrow}}{k})$ that leads to a finite result in the thermodynamic limit for the considered values of J/t.J/t.J/t. The spectral function $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ is obtained from the imaginary time Green's function via the maximum entropy method. Resonances above the lowest edge of the spectrum are identified, whose J dependence is quantitatively described by string excitations up to J/t=2.J/t=2.J/t=2.
The problem of the single-hole motion in a two-dimensional quantum Heisenberg antiferromagnet is examined in the t-J model. By introducing a kind of slave-fermion representation for Hubbard operators a hole is treated as a spinless fermion strongly interacting with spin-1/2 degrees of freedom. We suggest an analytic approach based on a decoupling procedure for two-time fermionic Green functions that allows us to describe a hole as a magnetic polaron of a minimal size (i.e., involving one-step ``string'' spin excitations) and reproduce the main features characteristic of a one-hole band in a quantum antiferromagnet. A generalization of the procedure in the spirit of the Lanczos approach to extend the internal structure of the magnetic polaron is discussed.
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.