Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9708185 · Physical review. B, Condensed matter

Two holes in a locally antiferromagnetic background: The role of retardation and Coulomb repulsion effects

José A. Riera, Elbio R. Dagotto

Published Apr 1, 1998• 31 citations
arXivHTMLPDF

Abstract

The problem of two holes in the presence of strong antiferromagnetic fluctuations is revisited using computational techniques. Two-dimensional clusters and two-leg ladders are studied with the Lanczos and truncated Lanczos algorithms. Lattices with up to $2\ifmmode\times\else\texttimes\fi{}16$ and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$ sites are studied. The motivation of the paper is the recently discussed spatial distribution of holes in ladders where the maximum probability for the hole-hole distance is obtained at $d=\sqrt{2}$ in units of the lattice spacing, a counterintuitive result considering that the overall symmetry of the two-hole bound state is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. Here this effect is shown to appear in small ladder clusters that can be addressed exactly, and also in planes. The probability distribution of hole distances $d$ was found to be broad with several distances contributing appreciably to the wave function. The existence of holes in the same sublattice is argued to be a consequence of non-negligible retardation effects in the $t\ensuremath{-}J$ model. Effective models with instantaneous interactions nevertheless capture the essence of the hole pairing process in the presence of short-range antiferromagnetic fluctuations (especially regarding the symmetry properties of the condensate), similarly as the (nonretarded) BCS model contains the basic features of the more complicated electron-phonon problem in low-temperature superconductors. The existence of strong spin singlets in the region where the two-hole bound state is located is here confirmed, and a simple explanation for its origin in the case of planes is proposed using the N\'eel state as a background, complementing previous explanations based on a spin-liquid undoped state. It is predicted that these strong singlets should appear regardless of the long distance properties of the spin system under consideration, as long as the bound state is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. In particular, it is shown that they are present in an Ising spin background. The time retardation in the family of $t\ensuremath{-}J$ models leads naturally to low-energy hole states with nonzero momentum and spin one, providing a possible explanation for apparent SO(5)-symmetric features observed recently in this context. Finally, the influence of a short-range Coulombic repulsion is analyzed. Rough estimations suggest that at a distance of one lattice spacing this repulsion is larger than the exchange $J$. The hole distribution in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ bound state is reanalyzed in the presence of such repulsion. Very short hole-hole distances lose their relevance in the presence of a realistic hole-hole interaction.

DOI
10.1103/physrevb.57.8609

Related papers

Enhancement of antiferromagnetic correlations induced by nonmagnetic impurities: Origin and predictions for NMR experiments

Markus Laukamp, George Balster Martins, C. J. Gazza +5 more

Spin models that have been proposed to describe dimerized chains, ladders, two-dimensional antiferromagnets, and other compounds are studied here when some spins are replaced by spinless vacancies, such as it occurs by Zn doping. A small percentage of vacancies rapidly destroys the spin gap, and their presence induces enhanced antiferromagnetic correlations near those vacancies. The study is performed with computational techniques which includes Lanczos, world-line Monte Carlo, and the density-matrix renormalization-group methods. Since the phenomenon of enhanced antiferromagnetism is found to occur in several models and cluster geometries, a common simple explanation for its presence may exist. It is argued that the resonating-valence-bond character of the spin correlations at short distances of a large variety of models is responsible for the presence of robust staggered spin correlations near vacancies and lattice edges. The phenomenon takes place regardless of the long distance properties of the ground state, and it is caused by a ``pruning'' of the available spin singlets in the vicinity of the vacancies. The effect produces a broadening of the low-temperature NMR signal for the compounds analyzed here. This broadening should be experimentally observable in the structurally dimerized chain systems $\mathrm{Cu}({\mathrm{NO}}_{3}{)}_{2}\ensuremath{\cdot}2.5{\mathrm{H}}_{2}\mathrm{O}$, ${\mathrm{CuWO}}_{4}$, $(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}$, and ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},$ in ladder materials such as ${\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}$, in the spin-Peierls systems ${\mathrm{CuGeO}}_{3}$ and ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$, and in several others since it is a universal effect common to a wide variety of models and compounds. In addition, it is argued that the N\'eel order observed in ${\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}$ upon Zn doping is induced by the local antiferromagnetic order discussed in this paper, enhanced by a favorable ratio between the actual Heisenberg couplings along chains and rungs, as reported in recent experimental literature. Based on this reasoning it is predicted here that other ladder materials such as Zn-doped ${\mathrm{Cu}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}$ will not present N\'eel order at small Zn concentrations.

arXiv:cond-mat/9707261•Physical review. B, Condensed matter•May 1, 1998•89 citations

Hole pairs in a spin liquid: Influence of electrostatic hole-hole repulsion

C. J. Gazza, George Balster Martins, José A. Riera +1 more

The stability of hole bound states in the $t\ensuremath{-}J$ model including short-range Coulomb interactions is analyzed using computational techniques on ladders with up to $2\ifmmode\times\else\texttimes\fi{}30$ sites. For a nearest-neighbor (NN) hole-hole repulsion, the two-holes bound state is surprisingly robust and breaks only when the repulsion is several times the exchange $J.$ At $\ensuremath{\sim}10%$ hole doping the pairs break only for a NN repulsion as large as $V\ensuremath{\sim}4J.$ Pair-pair correlations remain robust in the regime of hole binding. The results support electronic hole-pairing mechanisms on ladders based on holes moving in spin-liquid backgrounds. Implications in two dimensions are also presented. The need for better estimations of the range and strength of the Coulomb interaction in copper oxides is remarked.

arXiv:cond-mat/9803314•Physical review. B, Condensed matter•Jan 1, 1999•18 citations

Hole and pair structures in the t-J model

Steven R. White, Douglas J. Scalapino

Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.

arXiv:cond-mat/9605143•Physical review. B, Condensed matter•Mar 1, 1997•165 citations

Exact dynamics of two holes in two-leg antiferromagnetic ladders

K. Knakkergaard Nielsen

We study the motion of holes in a mixed-dimensional setup of an antiferromagnetic ladder, featuring nearest-neighbor hopping $t$ along the ladders and Ising-type spin interactions along, ${J}_{\ensuremath{\parallel}}$, and across, ${J}_{\ensuremath{\perp}}$, the ladder. We determine exact solutions for the low-energy one- and two-hole eigenstates. The presence of the trans-leg spin coupling, ${J}_{\ensuremath{\perp}}$, leads to a linear confining potential between the holes. As a result, holes on separate legs feature a superlinear binding energy scaling as ${({J}_{\ensuremath{\perp}}/t)}^{2/3}$ in the strongly correlated regime of ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$. This behavior is linked to an emergent length scale $\ensuremath{\lambda}\ensuremath{\propto}{(t/{J}_{\ensuremath{\perp}})}^{1/3}$, stemming from the linear confining potential, and which describes how the size of the two-hole molecular state diverges for ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\ll}t$. On the contrary, holes on the same leg unbind at sufficiently low spin couplings. This is a consequence of the altered short-range boundary condition for holes on the same leg, yielding an effective Pauli repulsion between them, limiting their kinetic energy and making binding unfavorable. Finally, we determine the exact nonequilibrium quench dynamics following the sudden immersion of initially localized nearest-neighbor holes. The dynamics is characterized by a crossover from an initial ballistic quantum walk to an aperiodic oscillatory motion around a finite average distance between the holes due to the confining potential between them. In the strongly correlated regime of low spin couplings, ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$, we find this asymptotic distance to diverge as $t/{J}_{\ensuremath{\perp}}$, showing a much stronger scaling than the eigenstates. The predicted results should be amenable to state-of-the-art quantum simulation experiments using currently implemented experimental techniques.

arXiv:2303.17381•Physical review. B./Physical review. B•Aug 18, 2023•6 citations

Superconductivity in the Two-Dimensional t−J Model

Sandro Sorella, George Balster Martins, Federico Becca +4 more

Using computational techniques, it is shown that pairing is a robust property of hole-doped antiferromagnetic insulators. In one dimension and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting long-range order in this case. However, in two dimensions it is argued-and numerically confirmed using several techniques, especially quantum Monte Carlo-that quantum fluctuations are not strong enough to suppress superconductivity.

arXiv:cond-mat/0110460•Physical Review Letters•Feb 28, 2002•213 citations

Indications of spin-charge separation at short distance and stripe formation in the extendedt-Jmodel on ladders and planes

George Balster Martins, J. C. Xavier, C. J. Gazza +2 more

The recently discussed tendency of holes to generate nontrivial spin environments in the extended two-dimensional $t\ensuremath{-}J$ model [G. Martins, R. Eder, and E. Dagotto, Phys. Rev. B 60, R3716 (1999)] is here investigated using computational techniques applied to ladders with several number of legs. This tendency is studied also with the help of analytic spin-polaron approaches directly in two dimensions. Our main result is that the presence of robust antiferromagnetic correlations between spins located at both sides of a hole either along the x or y axis, observed before numerically on square clusters, is also found using ladders, as well as applying techniques based on a string-basis expansion. This so-called ``across-the-hole'' nontrivial structure exists even in the two-leg spin-gapped ladder system, and leads to an effective reduction in dimensionality and spin-charge separation at short-distances, with a concomitant drastic reduction in the quasiparticle weight Z. In general, it appears that holes tend to induce one-dimensional-like spin arrangements to improve their mobility. Using ladders it is also shown that the very small $J/t\ensuremath{\sim}0.1$ regime of the standard $t\ensuremath{-}J$ model may be more realistic than anticipated in previous investigations, since such regime shares several properties with those found in the extended model at realistic couplings. Another goal of the present paper is to provide additional information on the recently discussed tendencies to stripe formation and spin incommensurability reported for the extended $t\ensuremath{-}J$ model. These tendencies are illustrated with several examples.

arXiv:cond-mat/0007196•Physical review. B, Condensed matter•Dec 12, 2000•28 citations
The problem of two holes in the presence of strong antiferromagnetic fluctuations is revisited using computational techniques. Two-dimensional clusters and two-leg ladders are studied with the Lanczos and truncated Lanczos algorithms. Lattices with up to 2\ifmmode×\else\texttimes\fi162\ifmmode\times\else\texttimes\fi{}162\ifmmode×\else\texttimes\fi16 and 32\ifmmode×\else\texttimes\fi32\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}32​\ifmmode×\else\texttimes\fi32​ sites are studied. The motivation of the paper is the recently discussed spatial distribution of holes in ladders where the maximum probability for the hole-hole distance is obtained at d=2d=\sqrt{2}d=2​ in units of the lattice spacing, a counterintuitive result considering that the overall symmetry of the two-hole bound state is dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​. Here this effect is shown to appear in small ladder clusters that can be addressed exactly, and also in planes. The probability distribution of hole distances ddd was found to be broad with several distances contributing appreciably to the wave function. The existence of holes in the same sublattice is argued to be a consequence of non-negligible retardation effects in the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. Effective models with instantaneous interactions nevertheless capture the essence of the hole pairing process in the presence of short-range antiferromagnetic fluctuations (especially regarding the symmetry properties of the condensate), similarly as the (nonretarded) BCS model contains the basic features of the more complicated electron-phonon problem in low-temperature superconductors. The existence of strong spin singlets in the region where the two-hole bound state is located is here confirmed, and a simple explanation for its origin in the case of planes is proposed using the N'eel state as a background, complementing previous explanations based on a spin-liquid undoped state. It is predicted that these strong singlets should appear regardless of the long distance properties of the spin system under consideration, as long as the bound state is dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​. In particular, it is shown that they are present in an Ising spin background. The time retardation in the family of t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J models leads naturally to low-energy hole states with nonzero momentum and spin one, providing a possible explanation for apparent SO(5)-symmetric features observed recently in this context. Finally, the influence of a short-range Coulombic repulsion is analyzed. Rough estimations suggest that at a distance of one lattice spacing this repulsion is larger than the exchange JJJ. The hole distribution in the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ bound state is reanalyzed in the presence of such repulsion. Very short hole-hole distances lose their relevance in the presence of a realistic hole-hole interaction.
Spin models that have been proposed to describe dimerized chains, ladders, two-dimensional antiferromagnets, and other compounds are studied here when some spins are replaced by spinless vacancies, such as it occurs by Zn doping. A small percentage of vacancies rapidly destroys the spin gap, and their presence induces enhanced antiferromagnetic correlations near those vacancies. The study is performed with computational techniques which includes Lanczos, world-line Monte Carlo, and the density-matrix renormalization-group methods. Since the phenomenon of enhanced antiferromagnetism is found to occur in several models and cluster geometries, a common simple explanation for its presence may exist. It is argued that the resonating-valence-bond character of the spin correlations at short distances of a large variety of models is responsible for the presence of robust staggered spin correlations near vacancies and lattice edges. The phenomenon takes place regardless of the long distance properties of the ground state, and it is caused by a ``pruning'' of the available spin singlets in the vicinity of the vacancies. The effect produces a broadening of the low-temperature NMR signal for the compounds analyzed here. This broadening should be experimentally observable in the structurally dimerized chain systems $\mathrm{Cu}({\mathrm{NO}}_{3}{)}_{2}\ensuremath{\cdot}2.5{\mathrm{H}}_{2}\mathrm{O}$, CuWO4{\mathrm{CuWO}}_{4}CuWO4​, (VO)2P2O7(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}(VO)2​P2​O7​, and Sr14Cu24O41,{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},Sr14​Cu24​O41​, in ladder materials such as SrCu2O3{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}SrCu2​O3​, in the spin-Peierls systems CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​ and NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​, and in several others since it is a universal effect common to a wide variety of models and compounds. In addition, it is argued that the N\'eel order observed in SrCu2O3{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}SrCu2​O3​ upon Zn doping is induced by the local antiferromagnetic order discussed in this paper, enhanced by a favorable ratio between the actual Heisenberg couplings along chains and rungs, as reported in recent experimental literature. Based on this reasoning it is predicted here that other ladder materials such as Zn-doped Cu2(C5H12N2)2Cl4{\mathrm{Cu}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}Cu2​(C5​H12​N2​)2​Cl4​ will not present N\'eel order at small Zn concentrations.
The stability of hole bound states in the $t\ensuremath{-}J$ model including short-range Coulomb interactions is analyzed using computational techniques on ladders with up to $2\ifmmode\times\else\texttimes\fi{}30$ sites. For a nearest-neighbor (NN) hole-hole repulsion, the two-holes bound state is surprisingly robust and breaks only when the repulsion is several times the exchange J.J.J. At $\ensuremath{\sim}10%$ hole doping the pairs break only for a NN repulsion as large as $V\ensuremath{\sim}4J.$ Pair-pair correlations remain robust in the regime of hole binding. The results support electronic hole-pairing mechanisms on ladders based on holes moving in spin-liquid backgrounds. Implications in two dimensions are also presented. The need for better estimations of the range and strength of the Coulomb interaction in copper oxides is remarked.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.
We study the motion of holes in a mixed-dimensional setup of an antiferromagnetic ladder, featuring nearest-neighbor hopping ttt along the ladders and Ising-type spin interactions along, ${J}_{\ensuremath{\parallel}}$, and across, ${J}_{\ensuremath{\perp}}$, the ladder. We determine exact solutions for the low-energy one- and two-hole eigenstates. The presence of the trans-leg spin coupling, ${J}_{\ensuremath{\perp}}$, leads to a linear confining potential between the holes. As a result, holes on separate legs feature a superlinear binding energy scaling as ${({J}_{\ensuremath{\perp}}/t)}^{2/3}$ in the strongly correlated regime of ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$. This behavior is linked to an emergent length scale $\ensuremath{\lambda}\ensuremath{\propto}{(t/{J}_{\ensuremath{\perp}})}^{1/3}$, stemming from the linear confining potential, and which describes how the size of the two-hole molecular state diverges for ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\ll}t$. On the contrary, holes on the same leg unbind at sufficiently low spin couplings. This is a consequence of the altered short-range boundary condition for holes on the same leg, yielding an effective Pauli repulsion between them, limiting their kinetic energy and making binding unfavorable. Finally, we determine the exact nonequilibrium quench dynamics following the sudden immersion of initially localized nearest-neighbor holes. The dynamics is characterized by a crossover from an initial ballistic quantum walk to an aperiodic oscillatory motion around a finite average distance between the holes due to the confining potential between them. In the strongly correlated regime of low spin couplings, ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$, we find this asymptotic distance to diverge as $t/{J}_{\ensuremath{\perp}}$, showing a much stronger scaling than the eigenstates. The predicted results should be amenable to state-of-the-art quantum simulation experiments using currently implemented experimental techniques.
Using computational techniques, it is shown that pairing is a robust property of hole-doped antiferromagnetic insulators. In one dimension and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting long-range order in this case. However, in two dimensions it is argued-and numerically confirmed using several techniques, especially quantum Monte Carlo-that quantum fluctuations are not strong enough to suppress superconductivity.
The recently discussed tendency of holes to generate nontrivial spin environments in the extended two-dimensional $t\ensuremath{-}J$ model [G. Martins, R. Eder, and E. Dagotto, Phys. Rev. B 60, R3716 (1999)] is here investigated using computational techniques applied to ladders with several number of legs. This tendency is studied also with the help of analytic spin-polaron approaches directly in two dimensions. Our main result is that the presence of robust antiferromagnetic correlations between spins located at both sides of a hole either along the x or y axis, observed before numerically on square clusters, is also found using ladders, as well as applying techniques based on a string-basis expansion. This so-called ``across-the-hole'' nontrivial structure exists even in the two-leg spin-gapped ladder system, and leads to an effective reduction in dimensionality and spin-charge separation at short-distances, with a concomitant drastic reduction in the quasiparticle weight Z. In general, it appears that holes tend to induce one-dimensional-like spin arrangements to improve their mobility. Using ladders it is also shown that the very small $J/t\ensuremath{\sim}0.1$ regime of the standard $t\ensuremath{-}J$ model may be more realistic than anticipated in previous investigations, since such regime shares several properties with those found in the extended model at realistic couplings. Another goal of the present paper is to provide additional information on the recently discussed tendencies to stripe formation and spin incommensurability reported for the extended $t\ensuremath{-}J$ model. These tendencies are illustrated with several examples.