Analytical results for a hole in an antiferromagnet

The Green's function for a hole moving in an antiferromagnet is derived analytically in the long-wavelength limit. We find that the infrared divergence is eliminated in two and higher dimensions so that the quasiparticle weight is finite. Our results also suggest that the hole motion is polaronic in nature with a bandwidth proportional to $\frac{{t}^{2}}{J}\mathrm{exp}[\ensuremath{-}c{(\frac{t}{J})}^{2}]$ ( $c$ is a constant) for $\frac{J}{t}\ensuremath{\gtrsim}0.5$. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.The Green's function for a hole moving in an antiferromagnet is derived analytically in the long-wavelength limit. We find that the infrared divergence is eliminated in two and higher dimensions so that the quasiparticle weight is finite. Our results also suggest that the hole motion is polaronic in nature with a bandwidth proportional to $\frac{{t}^{2}}{J}\mathrm{exp}[\ensuremath{-}c{(\frac{t}{J})}^{2}]$ ($c$ is a constant) for $\frac{J}{t}\ensuremath{\gtrsim}0.5$. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.

arXiv:cond-mat/9601062Physical review. B, Condensed matterJun 1, 19961 citation

Single-hole dynamics in the t−J model

W. Apel, H. U. Everts, U. Körner

The quasiparticle weight of a single hole in an antiferromagnetic background is studied in the semiclassical approximation. We start from the $t\ensuremath{-}J$ model, generalize it to arbitrary spin S by employing an appropriate coherent state representation for the hole, and derive an effective action for the dynamics in the long-wavelength low-energy limit. In the same limit, we find an expression for the single-hole Green's function which we evaluate in an $1/S$ expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.The quasiparticle weight of a single hole in an antiferromagnetic background is studied in the semiclassical approximation. We start from the $t\ensuremath{-}J$ model, generalize it to arbitrary spin S by employing an appropriate coherent state representation for the hole, and derive an effective action for the dynamics in the long-wavelength low-energy limit. In the same limit, we find an expression for the single-hole Green's function which we evaluate in an $1/S$ expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.

arXiv:cond-mat/0208554Physical review. B, Condensed matterNov 15, 20020 citations

A single hole in a quantum antiferromagnet: selfconsistent Green’s function approach

Gerardo Martı́nez, Peter Horsch

We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.

International Journal of Modern Physics BJan 1, 199112 citations

Spin polarons in the t - J model

Gerardo Martı́nez, Peter Horsch

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 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.

Physical review. B, Condensed matterJul 1, 1991498 citations

Spin polarons in thet-Jmodel: Shape and backflow

A. Ramšak, Peter Horsch

The self-consistent Born approximation for the self-energy and the wave function of a single hole moving in a two-dimensional quantum antiferromagnet is used to calculate various spin-correlation functions that characterize the perturbation induced by the hole. The results for these correlation functions agree with the available small-cluster calculations. The hole-spin correlation functions decay asymptotically with the distance from the hole as a power law and exhibit a dipolar distortion of the antiferromagnet. The quasiparticle spectral weight remains finite.The self-consistent Born approximation for the self-energy and the wave function of a single hole moving in a two-dimensional quantum antiferromagnet is used to calculate various spin-correlation functions that characterize the perturbation induced by the hole. The results for these correlation functions agree with the available small-cluster calculations. The hole-spin correlation functions decay asymptotically with the distance from the hole as a power law and exhibit a dipolar distortion of the antiferromagnet. The quasiparticle spectral weight remains finite.

Physical review. B, Condensed matterOct 1, 199352 citations

Dispersion of a single hole in an antiferromagnet

Andrey V. Chubukov, Dirk K. Morr

We revisit the problem of the dispersion of a single hole injected into a quantum antiferromagnet. We applied a spin-density-wave formalism extended to a large number of orbitals and obtained an integral equation for the full quasiparticle Green's function in the self-consistent ``noncrossing'' Born approximation. We found that for $t/J\ensuremath{\gg}1$, the bare fermionic dispersion is completely overshadowed by the self-energy corrections. In this case, the quasiparticle Green's function contains a broad incoherent continuum which extends over a frequency range of $\ensuremath{\sim}6t$. In addition, there exists a narrow region of width $O(\mathrm{JS})$ below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z\ensuremath{\sim}J/t$. The top of the valence band is located at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. We found that the form of the fermionic dispersion, and, in particular, the ratio of the effective masses near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ strongly depend on the assumptions one makes for the form of the magnon propagator. We argue in this paper that two-magnon Raman scattering as well as neutron-scattering experiments strongly suggest that the zone-boundary magnons are not free particles since a substantial portion of their spectral weight is transferred into an incoherent background. We modeled this effect by introducing a cutoff ${q}_{c}$ in the integration over magnon momenta. We found analytically that for small ${q}_{c},$ the strong-coupling solution for the Green's function is universal, and both effective masses are equal to ${(4JS)}^{\ensuremath{-}1}$. We further computed the full fermionic dispersion for $J/t=0.4$ relevant for ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$ , and ${t}^{\ensuremath{'}}=\ensuremath{-}0.4J$ and found not only that the masses are both equal to ${(2J)}^{\ensuremath{-}1}$, but also that the energies at $(0,0)$ and $(0,\ensuremath{\pi})$ are equal, the energy at $(0,\ensuremath{\pi}/2)$ is about half of that at $(0,0)$ , and the bandwidth for the coherent excitations is around $3J$ . All of these results are in full agreement with the experimental data. Finally, we found that weakly damped excitations only exist in a narrow range around $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. Away from the vicinity of $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$, the excitations are overdamped, and the spectral function possesses a broad maximum rather than a sharp quasiparticle peak. This last feature was also reported in photoemission experiments.We revisit the problem of the dispersion of a single hole injected into a quantum antiferromagnet. We applied a spin-density-wave formalism extended to a large number of orbitals and obtained an integral equation for the full quasiparticle Green's function in the self-consistent ``noncrossing'' Born approximation. We found that for $t/J\ensuremath{\gg}1$, the bare fermionic dispersion is completely overshadowed by the self-energy corrections. In this case, the quasiparticle Green's function contains a broad incoherent continuum which extends over a frequency range of $\ensuremath{\sim}6t$. In addition, there exists a narrow region of width $O(\mathrm{JS})$ below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z\ensuremath{\sim}J/t$. The top of the valence band is located at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. We found that the form of the fermionic dispersion, and, in particular, the ratio of the effective masses near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ strongly depend on the assumptions one makes for the form of the magnon propagator. We argue in this paper that two-magnon Raman scattering as well as neutron-scattering experiments strongly suggest that the zone-boundary magnons are not free particles since a substantial portion of their spectral weight is transferred into an incoherent background. We modeled this effect by introducing a cutoff ${q}_{c}$ in the integration over magnon momenta. We found analytically that for small ${q}_{c},$ the strong-coupling solution for the Green's function is universal, and both effective masses are equal to ${(4JS)}^{\ensuremath{-}1}$. We further computed the full fermionic dispersion for $J/t=0.4$ relevant for ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$, and ${t}^{\ensuremath{'}}=\ensuremath{-}0.4J$ and found not only that the masses are both equal to ${(2J)}^{\ensuremath{-}1}$, but also that the energies at $(0,0)$ and $(0,\ensuremath{\pi})$ are equal, the energy at $(0,\ensuremath{\pi}/2)$ is about half of that at $(0,0)$, and the bandwidth for the coherent excitations is around $3J$. All of these results are in full agreement with the experimental data. Finally, we found that weakly damped excitations only exist in a narrow range around $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. Away from the vicinity of $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$, the excitations are overdamped, and the spectral function possesses a broad maximum rather than a sharp quasiparticle peak. This last feature was also reported in photoemission experiments.

arXiv:cond-mat/9709235Physical review. B, Condensed matterMar 1, 199837 citations

c

is a constant) for $\frac{J}{t}\ensuremath{\gtrsim}0.5$. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.

1/S

expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.

O(\mathrm{JS})

below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z\ensuremath{\sim}J/t$. The top of the valence band is located at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. We found that the form of the fermionic dispersion, and, in particular, the ratio of the effective masses near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ strongly depend on the assumptions one makes for the form of the magnon propagator. We argue in this paper that two-magnon Raman scattering as well as neutron-scattering experiments strongly suggest that the zone-boundary magnons are not free particles since a substantial portion of their spectral weight is transferred into an incoherent background. We modeled this effect by introducing a cutoff

{q}_{c}

in the integration over magnon momenta. We found analytically that for small

{q}_{c},

the strong-coupling solution for the Green's function is universal, and both effective masses are equal to ${(4JS)}^{\ensuremath{-}1}$. We further computed the full fermionic dispersion for

J/t=0.4

relevant for

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}

, and ${t}^{\ensuremath{'}}=\ensuremath{-}0.4J$ and found not only that the masses are both equal to ${(2J)}^{\ensuremath{-}1}$, but also that the energies at

(0,0)

and $(0,\ensuremath{\pi})$ are equal, the energy at $(0,\ensuremath{\pi}/2)$ is about half of that at

(0,0)

, and the bandwidth for the coherent excitations is around

3J

. All of these results are in full agreement with the experimental data. Finally, we found that weakly damped excitations only exist in a narrow range around $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. Away from the vicinity of $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$, the excitations are overdamped, and the spectral function possesses a broad maximum rather than a sharp quasiparticle peak. This last feature was also reported in photoemission experiments.