Physical review. B, Condensed matter

Hole dynamics in the two-dimensional strong-coupling Hubbard Hamiltonian

Efthimios Kaxiras, Efstratios Manousakis

Published Jul 1, 1988• 66 citations

Abstract

The effective Hamiltonian, obtained from the Hubbard model in the strong-coupling limit, is diagonalized exactly for a periodic two-dimensional square lattice of ten sites. We obtain the ground state of the system for any filling of the lattice and the $K$-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio $\frac{t}{U}$, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing $\frac{t}{U}$, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of $\frac{t}{U}$ exists for which pairing of the holes may be possible.

DOI: 10.1103/physrevb.38.866

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Ground state of the strong-coupling Hubbard Hamiltonian: A numerical diagonalization study

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Ferromagnetic domains in the large-$U$ Hubbard model with a few holes: an FCIQMC study

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arXiv:2210.12049Oct 21, 20221 citation

Physical review. B, Condensed matter

Hole dynamics in the two-dimensional strong-coupling Hubbard Hamiltonian

Efthimios Kaxiras, Efstratios Manousakis

Published Jul 1, 1988• 66 citations

Publisher

Abstract

K

-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio

\frac{t}{U}

, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing

\frac{t}{U}

, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of

\frac{t}{U}

exists for which pairing of the holes may be possible.

DOI: 10.1103/physrevb.38.866

Related papers

Ground state of the strong-coupling Hubbard Hamiltonian: A numerical diagonalization study

Efthimios Kaxiras, Efstratios Manousakis

We exactly diagonalize the effective Hamiltonian obtained from the Hubbard model in the strong-coupling limit for a two-dimensional $\sqrt{10}\ifmmode\times\else\texttimes\fi{}\sqrt{10}$ size square lattice. The effective Hamiltonian operates on a restricted Hilbert space containing only states with singly occupied sites, which makes the diagonalization possible within reasonable computational time for the above lattice. The ground-state energy and wave function are obtained for several values of the coupling ratio $\frac{t}{U}$ and the doping fraction $x$ . We find three different phases in the ( $x,\frac{t}{U}$ ) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.We exactly diagonalize the effective Hamiltonian obtained from the Hubbard model in the strong-coupling limit for a two-dimensional $\sqrt{10}\ifmmode\times\else\texttimes\fi{}\sqrt{10}$ size square lattice. The effective Hamiltonian operates on a restricted Hilbert space containing only states with singly occupied sites, which makes the diagonalization possible within reasonable computational time for the above lattice. The ground-state energy and wave function are obtained for several values of the coupling ratio $\frac{t}{U}$ and the doping fraction $x$. We find three different phases in the ($x,\frac{t}{U}$) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.

Physical review. B, Condensed matterJan 1, 198853 citations

Coupled-cluster method applied to the motion of a single hole in a Hubbard antiferromagnet

F. Petit, M. Röger

The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U ${)}^{0.70}$ , in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole ${\mathit{m}}^{\mathrm{*}}$ =8t/W (W represents the bandwidth) is proportional to U/T ( ${\mathit{m}}^{\mathrm{*}}$ \ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value ${\mathit{m}}^{\mathrm{*}}$ \ensuremath{\approxeq}84 for large U/t.The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U${)}^{0.70}$, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole ${\mathit{m}}^{\mathrm{*}}$=8t/W (W represents the bandwidth) is proportional to U/T (${\mathit{m}}^{\mathrm{*}}$\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value ${\mathit{m}}^{\mathrm{*}}$\ensuremath{\approxeq}84 for large U/t.

Physical review. B, Condensed matterFeb 1, 199419 citations

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Elbio R. Dagotto, Adriana Moreo, Ted Barnes

The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J ${J}_{c}$ \ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge} ${J}_{c}$ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J${J}_{c}$\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}${J}_{c}$ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.

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Superconductivity, hole motion, and spin-charge correlations in thet-Jmodel in two dimensions

K.-H. Luk, Daniel Lee Cox

We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t ${)}_{\mathit{c}}$ \ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t ${)}_{\mathit{c}}$ \ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t${)}_{\mathit{c}}$\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t${)}_{\mathit{c}}$\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.

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Ferromagnetic domains in the large- $U$ Hubbard model with a few holes: an FCIQMC studyFerromagnetic domains in the large-$U$ Hubbard model with a few holes: an FCIQMC study

Yun, Sujun, Werner Dobrautz, Luo Hongjun +3 more

Two-dimensional Hubbard lattices with two or three holes are investigated as a function of $U$ in the large- $U$ limit. In the so-called Nagaoka limit (one-hole system at infinite $U$ ), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing $U$ . The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or $\frac{1}{2}$ , depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.Two-dimensional Hubbard lattices with two or three holes are investigated as a function of $U$ in the large-$U$ limit. In the so-called Nagaoka limit (one-hole system at infinite $U$), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing $U$. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or $\frac{1}{2}$, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.

arXiv:2210.12049Oct 21, 20221 citation

K

-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio

\frac{t}{U}

, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing

\frac{t}{U}

, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of

\frac{t}{U}

exists for which pairing of the holes may be possible.

\frac{t}{U}

and the doping fraction

x

. We find three different phases in the (

x,\frac{t}{U}

) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.

{)}^{0.70}

, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole

{\mathit{m}}^{\mathrm{*}}

=8t/W (W represents the bandwidth) is proportional to U/T (

{\mathit{m}}^{\mathrm{*}}

\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value

{\mathit{m}}^{\mathrm{*}}

\ensuremath{\approxeq}84 for large U/t.

{J}_{c}

\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}

{J}_{c}

the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.

{)}_{\mathit{c}}

\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t

{)}_{\mathit{c}}

\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.

Ferromagnetic domains in the large-

U

Hubbard model with a few holes: an FCIQMC study

Two-dimensional Hubbard lattices with two or three holes are investigated as a function of

U

in the large-

U

limit. In the so-called Nagaoka limit (one-hole system at infinite

U

), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing

U

. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or

\frac{1}{2}

, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.