^{1,a)}, B. Kyung

^{1}and D. Sénéchal

^{1}

### Abstract

This is a short review of the theoretical work on the two-dimensional Hubbard model performed in Sherbrooke in the last few years. It is written on the occasion of the twentieth anniversary of the discovery of high-temperature superconductivity. We discuss several approaches, how they were benchmarked and how they agree sufficiently with each other that we can trust that the results are accurate solutions of the Hubbard model. Then comparisons are made with experiment. We show that the Hubbard model does exhibit -wave superconductivity and antiferromagnetism essentially where they are observed for both hole- and electron-doped cuprates. We also show that the pseudogap phenomenon comes out of these calculations. In the case of electron-doped high temperature superconductors, comparisons with angle-resolved photoemission experiments are nearly quantitative. The value of the pseudogap temperature observed for these compounds in recent photoemission experiments had been predicted by theory before it was observed experimentally. Additional experimental confirmation would be useful. The theoretical methods that are surveyed include mostly the two-particle self-consistent approach, variational cluster perturbation theory (or variational cluster approximation), and cellular dynamical mean-fieldtheory.

The present work was supported by NSERC (Canada), FQRNT (Québec), CFI (Canada), CIAR, and the Tier I Canada Research Chair Program (A.-M. S. T.). We are grateful to our collaborators, G. Albinet, S. Allen, M. Boissonneault, C. Brillon, M. Capone, L. Chen, M. Civelli, A.-M. Daré, B. Davoudi, J.-Y. P. Delannoy, A. Gagné-Lebrun, M. J. P. Gingras, A. Georges, M. Guillot, V. Hankevych, P. C. W. Holdsworth, F. Jackson, S. Kancharla, G. Kotliar, J.-S. Landry, P.-L. Lavertu, F. Lemay, S. Lessard, M.-A. Marois, S. Pairault, D. Perez, M. Pioro-Ladriére, D. Plouffe, D. Poulin, L. Raymond, S. Roy, P. Sahebsara, H. Touchette, and especially Y. M. Vilk. We also acknowledge useful discussions with P. Fournier, M. Greven, I. Herbut, K. Shen, and L. Taillefer, and we are grateful to V. Hankevych and S. Kancharla, for permission to include some of their unpublished figures in this paper.

I. INTRODUCTION

II. METHODOLOGY

A. Weak coupling approach

1. Two-particle self-consistent approach

2. Benchmarks for TPSC

B. Strong-coupling approaches: quantum clusters

1. Cluster perturbation theory

2. Self-energy functional approach

3. Variational cluster perturbation theory, or variational cluster approximation

4. Cellular dynamical mean-fieldtheory

5. The dynamical cluster approximation

6. Benchmarks for quantum cluster approaches

III. RESULTS AND CONCORDANCE BETWEEN DIFFERENT METHODS

A. Weak- and strong-coupling pseudogap

1. Cluster perturbation theory

2. Cellular dynamical mean field theory and dynamical cluster approximation

3. TPSC (including analytical results)

4. Weak- and strong-coupling pseudogaps

B. -wave superconductivity

1. Zero-temperature phase diagram

2. Instability of the normal phase

IV. COMPARISONS WITH EXPERIMENT

A. ARPES spectrum, an overview

B. The pseudogap in electron-doped cuprates

C. The phase diagram for high-temperature superconductors

1. Competition between antiferromagnetism and superconductivity

2. Anomalous superconductivity near the Mott transition

V. CONCLUSION, OPEN PROBLEMS

### Key Topics

- Band gap
- 85.0
- Antiferromagnetism
- 72.0
- Superconductivity
- 65.0
- Doping
- 50.0
- Superconductivity models
- 44.0

## Figures

The and extended -wave susceptibilities obtained from QMC simulations for and a lattice. The solid lines are the noninteracting results. From Ref. 10. The low temperature downturn, however, seems to come from a mistreatment of the sign problem (D.J. Scalapino, private communication).

The and extended -wave susceptibilities obtained from QMC simulations for and a lattice. The solid lines are the noninteracting results. From Ref. 10. The low temperature downturn, however, seems to come from a mistreatment of the sign problem (D.J. Scalapino, private communication).

Comparisons between the QMC simulations (symbols) and TPSC (solid lines) for the filling dependence of the double occupancy. The results are for as a function of filling and for various values of except for , where the dashed line shows the results of our theory at the crossover temperature . From Ref. 22.

Comparisons between the QMC simulations (symbols) and TPSC (solid lines) for the filling dependence of the double occupancy. The results are for as a function of filling and for various values of except for , where the dashed line shows the results of our theory at the crossover temperature . From Ref. 22.

Wave-vector dependence of the spin and charge structure factors for different sets of parameters. Solid lines are from TPSC and symbols are our QMC data. Monte Carlo data for and are for clusters and ; all other data are for clusters and . Error bars are shown only when significant. From Ref. 32.

Wave-vector dependence of the spin and charge structure factors for different sets of parameters. Solid lines are from TPSC and symbols are our QMC data. Monte Carlo data for and are for clusters and ; all other data are for clusters and . Error bars are shown only when significant. From Ref. 32.

Temperature dependence of at half-filling . The solid line is from TPSC and symbols are Monte Carlo data from Ref. 46. Taken from Ref. 32.

Temperature dependence of at half-filling . The solid line is from TPSC and symbols are Monte Carlo data from Ref. 46. Taken from Ref. 32.

Comparisons between Monte Carlo simulations (BW), FLEX calculations and TPSC for the spin susceptibility at as a function of temperature at zero Matsubara frequency. The filled circles (BWS) are from Ref. 48. Taken from Ref. 22.

Comparisons between Monte Carlo simulations (BW), FLEX calculations and TPSC for the spin susceptibility at as a function of temperature at zero Matsubara frequency. The filled circles (BWS) are from Ref. 48. Taken from Ref. 22.

Single-particle spectral weight for , , , and all independent wave vectors of an lattice. Results obtained from maximum entropy inversion of QMC data are in the left panel, from many-body TPSC calculations with Eq. (6) in the middle panel, and from FLEX in the right panel. From Ref. 36.

Single-particle spectral weight for , , , and all independent wave vectors of an lattice. Results obtained from maximum entropy inversion of QMC data are in the left panel, from many-body TPSC calculations with Eq. (6) in the middle panel, and from FLEX in the right panel. From Ref. 36.

Size dependent results for various types of calculations for , , , , , 6, 8, 10. Upper panels show extracted from maximum entropy on shown in the corresponding lower panels. QMC , TPSC using Eq. (6), FLEX . From Ref. 36.

Size dependent results for various types of calculations for , , , , , 6, 8, 10. Upper panels show extracted from maximum entropy on shown in the corresponding lower panels. QMC , TPSC using Eq. (6), FLEX . From Ref. 36.

Comparisons between the susceptibility obtained from QMC simulations (symbols) and from the TPSC approach (lines) in the two-dimensional Hubbard model. Both calculations are for , a lattice. QMC error bars are smaller than the symbols. Analytical results are joined by solid lines. The size dependence of the results is small at these temperatures. The case is also shown at as the upper line. The inset compares QMC and FLEX at , . From Ref. 41.

Comparisons between the susceptibility obtained from QMC simulations (symbols) and from the TPSC approach (lines) in the two-dimensional Hubbard model. Both calculations are for , a lattice. QMC error bars are smaller than the symbols. Analytical results are joined by solid lines. The size dependence of the results is small at these temperatures. The case is also shown at as the upper line. The inset compares QMC and FLEX at , . From Ref. 41.

The crossover diagram as a function of next-nearest-neighbor hopping from TPSC and from a temperature cutoff renormalization group technique from Ref. 27. The corresponding van Hove filling is indicated on the upper horizontal axis. Crossover lines for magnetic instabilities near the antiferromagnetic and ferromagnetic wave vectors are represented by filled symbols, while open symbols indicate instability towards -wave superconducting. The solid and dashed lines below the empty symbols show, respectively, for and , where the antiferromagnetic crossover temperature would have been in the absence of the superconducting instability. The largest system size used for this calculation is . From Ref. 49.

The crossover diagram as a function of next-nearest-neighbor hopping from TPSC and from a temperature cutoff renormalization group technique from Ref. 27. The corresponding van Hove filling is indicated on the upper horizontal axis. Crossover lines for magnetic instabilities near the antiferromagnetic and ferromagnetic wave vectors are represented by filled symbols, while open symbols indicate instability towards -wave superconducting. The solid and dashed lines below the empty symbols show, respectively, for and , where the antiferromagnetic crossover temperature would have been in the absence of the superconducting instability. The largest system size used for this calculation is . From Ref. 49.

TPSC -wave paring structure factor (filled triangles) and QMC (open circles) for and various temperatures at and at on a lattice. The dashed lines are to guide the eye. From Ref. 44.

TPSC -wave paring structure factor (filled triangles) and QMC (open circles) for and various temperatures at and at on a lattice. The dashed lines are to guide the eye. From Ref. 44.

Chemical potential shifts (open diamonds) and (open squares) with the results of QMC calculations (open circles) for . The momentum dependent occupation number . Circles: QMC calculations from Ref. 51. The solid curve: TPSC. The dashed curve is obtained by replacing by in the self-energy with all the rest unchanged. The long-dashed line is the result of a self-consistent -matrix calculation, and the dotted-dashed line the result of second-order perturbation theory . From Ref. 44.

Chemical potential shifts (open diamonds) and (open squares) with the results of QMC calculations (open circles) for . The momentum dependent occupation number . Circles: QMC calculations from Ref. 51. The solid curve: TPSC. The dashed curve is obtained by replacing by in the self-energy with all the rest unchanged. The long-dashed line is the result of a self-consistent -matrix calculation, and the dotted-dashed line the result of second-order perturbation theory . From Ref. 44.

Comparisons of local density of states and single-particle spectral weight from TPSC (solid lines) and QMC (dashed lines) on a lattice. QMC data for the density of states taken from Ref. 52. Figures from Ref. 44.

Comparisons of local density of states and single-particle spectral weight from TPSC (solid lines) and QMC (dashed lines) on a lattice. QMC data for the density of states taken from Ref. 52. Figures from Ref. 44.

Various tilings used in quantum cluster approaches. In these examples the gray and white sites are inequivalent, since an antiferromagnetic order is possible.

Various tilings used in quantum cluster approaches. In these examples the gray and white sites are inequivalent, since an antiferromagnetic order is possible.

The spectral function of the of the one-dimensional Hubbard model, as calculated from an exact diagonalization of the Hubbard model with on a periodic 12-site cluster; the same, but with CPT, on a 12-site cluster with open boundary conditions; the exact solution, taken from Ref. 72; beware: the axes are oriented differently. In and a finite width has been given to peaks that would otherwise be Dirac -functions.

The spectral function of the of the one-dimensional Hubbard model, as calculated from an exact diagonalization of the Hubbard model with on a periodic 12-site cluster; the same, but with CPT, on a 12-site cluster with open boundary conditions; the exact solution, taken from Ref. 72; beware: the axes are oriented differently. In and a finite width has been given to peaks that would otherwise be Dirac -functions.

Chemical potential as a function of density in the one-dimensional Hubbard model, as calculated by CPT (from Ref. 62). The exact, Bethe-ansatz result is shown as a solid line.

Chemical potential as a function of density in the one-dimensional Hubbard model, as calculated by CPT (from Ref. 62). The exact, Bethe-ansatz result is shown as a solid line.

Comparison (expressed in relative difference) between the ground-state energy density of the half-filled, one-dimensional Hubbard model calculated from the exact, Bethe–ansatz result. The results are displayed as a function of the hopping , for and various cluster sizes (connected symbols). For comparison, the exact diagonalization values of finite clusters with periodic boundary conditions are also shown (dashed lines) for and . Bottom: Same for the double occupancy. An extrapolation of the results to infinite cluster size using a quadratic fit in terms of is also shown, and is accurate to within 0.5% . Taken from Ref. 62.

Comparison (expressed in relative difference) between the ground-state energy density of the half-filled, one-dimensional Hubbard model calculated from the exact, Bethe–ansatz result. The results are displayed as a function of the hopping , for and various cluster sizes (connected symbols). For comparison, the exact diagonalization values of finite clusters with periodic boundary conditions are also shown (dashed lines) for and . Bottom: Same for the double occupancy. An extrapolation of the results to infinite cluster size using a quadratic fit in terms of is also shown, and is accurate to within 0.5% . Taken from Ref. 62.

Ground-state energy of the half-filled, two-dimensional Hubbard model as a function of , as obtained from various methods: exact diagonalization (ED), CPT and VCPT on a 10-site cluster, quantum Monte Carlo (QMC), and variational Monte Carlo (VMC). Taken from Ref. 73.

Ground-state energy of the half-filled, two-dimensional Hubbard model as a function of , as obtained from various methods: exact diagonalization (ED), CPT and VCPT on a 10-site cluster, quantum Monte Carlo (QMC), and variational Monte Carlo (VMC). Taken from Ref. 73.

CDMFT calculation on a cluster with 8 bath sites of the density as a function of the chemical potential in the one-dimensional Hubbard model for , as compared with the exact solution, DMFT and other approximation schemes. Taken from Ref. 74.

CDMFT calculation on a cluster with 8 bath sites of the density as a function of the chemical potential in the one-dimensional Hubbard model for , as compared with the exact solution, DMFT and other approximation schemes. Taken from Ref. 74.

Single-particle spectral weight, as a function of energy in units of , for wave vectors along the high-symmetry directions shown in the inset. CPT calculations on a cluster with ten electrons (17% hole-doped) ; the same as , with 14 electrons (17% electron-doped). In all cases and . A Lorentzian broadening is used to reveal the otherwise delta peaks . From Ref. 77.

Single-particle spectral weight, as a function of energy in units of , for wave vectors along the high-symmetry directions shown in the inset. CPT calculations on a cluster with ten electrons (17% hole-doped) ; the same as , with 14 electrons (17% electron-doped). In all cases and . A Lorentzian broadening is used to reveal the otherwise delta peaks . From Ref. 77.

Onset of the pseudogap as a function of corresponding to Fig. 19, taken from Ref. 77. Hole-doped case on the left, electron-doped case on the right panel.

Onset of the pseudogap as a function of corresponding to Fig. 19, taken from Ref. 77. Hole-doped case on the left, electron-doped case on the right panel.

MDC from CPT in the , Hubbard model, taken from Ref. 77.

MDC from CPT in the , Hubbard model, taken from Ref. 77.

The Fermi energy–momentum distribution curve . The corresponding EDC in the Hubbard model, calculated on a 16-site cluster in CPT, at . Inset: the pseudogap .

The Fermi energy–momentum distribution curve . The corresponding EDC in the Hubbard model, calculated on a 16-site cluster in CPT, at . Inset: the pseudogap .

EDC in the Hubbard model, with and , calculated on an 8-site cluster for in VCPT. -wave superconductivity is present in the hole-doped case (left) and both antiferromagnetism and -wave superconductivity in the electron-doped case. The resolution is not large enough in the latter case to see the superconducting gap. The Lorentzian broadening is . From Ref. 18.

EDC in the Hubbard model, with and , calculated on an 8-site cluster for in VCPT. -wave superconductivity is present in the hole-doped case (left) and both antiferromagnetism and -wave superconductivity in the electron-doped case. The resolution is not large enough in the latter case to see the superconducting gap. The Lorentzian broadening is . From Ref. 18.

MDC in the , Hubbard model, calculated on a 4-site cluster in CDMFT. Energy resolution, , . Hole-doped dSC (, ) . Electron-doped dSC (, ) . Same as with . Note the particle–hole transformation in the electron-doped case. From Ref. 16.

MDC in the , Hubbard model, calculated on a 4-site cluster in CDMFT. Energy resolution, , . Hole-doped dSC (, ) . Electron-doped dSC (, ) . Same as with . Note the particle–hole transformation in the electron-doped case. From Ref. 16.

EDC in the , Hubbard model, calculated on a 4-site cluster in CDMFT. Top: normal (paramagnetic) state for various densities. Bottom: same for the antiferromagnetic state. From Ref. 67.

EDC in the , Hubbard model, calculated on a 4-site cluster in CDMFT. Top: normal (paramagnetic) state for various densities. Bottom: same for the antiferromagnetic state. From Ref. 67.

EDC in the Hubbard model, , calculated in CPT, VCPT, and QMC. From Ref. 73.

EDC in the Hubbard model, , calculated in CPT, VCPT, and QMC. From Ref. 73.

Double occupancy in the two-dimensional Hubbard model for , as calculated from TPSC (lines with ) and from DCA (symbols) from Ref. 81. Taken from Ref. 84.

Double occupancy in the two-dimensional Hubbard model for , as calculated from TPSC (lines with ) and from DCA (symbols) from Ref. 81. Taken from Ref. 84.

MDC at the Fermi energy for the two-dimensional Hubbard model for , , at various hole dopings, as obtained from TPSC. The far left from Ref. 85 is the Fermi surface plot for 10% hole-doped .

MDC at the Fermi energy for the two-dimensional Hubbard model for , , at various hole dopings, as obtained from TPSC. The far left from Ref. 85 is the Fermi surface plot for 10% hole-doped .

MDC at the Fermi energy in the electron-doped case with , and two different ’s, and obtained from TPSC. The first column shows the corresponding experimental plots at 10% and 15% doping in Ref. 86. From Refs. 41 and 87.

MDC at the Fermi energy in the electron-doped case with , and two different ’s, and obtained from TPSC. The first column shows the corresponding experimental plots at 10% and 15% doping in Ref. 86. From Refs. 41 and 87.

Cartoon explanation of the pseudogap in the weak-coupling limit. Below the dashed crossover line to the renormalized classical regime, when the antiferromagnetic correlation length becomes larger than the thermal de Broglie wave length, there appear precursors of the zero-temperature Bogoliubov quasiparticles for the long-range-ordered antiferromagnet.

Cartoon explanation of the pseudogap in the weak-coupling limit. Below the dashed crossover line to the renormalized classical regime, when the antiferromagnetic correlation length becomes larger than the thermal de Broglie wave length, there appear precursors of the zero-temperature Bogoliubov quasiparticles for the long-range-ordered antiferromagnet.

MDC plots at the Fermi energy (upper) and corresponding scattering rates (lower) obtained from TPSC. The semicircular dark gray (red) lines on the upper panel indicate the region where the scattering rate in the corresponding lower panels is large.

MDC plots at the Fermi energy (upper) and corresponding scattering rates (lower) obtained from TPSC. The semicircular dark gray (red) lines on the upper panel indicate the region where the scattering rate in the corresponding lower panels is large.

Antiferromagnetic (bottom) and -wave (top) order parameters for , , , as a function of the electron density for -, -, and 10-site clusters, calculated in VCPT. Vertical lines indicate the first doping where only -wave order is nonvanishing. From Ref. 18.

Antiferromagnetic (bottom) and -wave (top) order parameters for , , , as a function of the electron density for -, -, and 10-site clusters, calculated in VCPT. Vertical lines indicate the first doping where only -wave order is nonvanishing. From Ref. 18.

-wave order parameter as a function of for various values of , calculated in CDMFT on a cluster for . The positive case corresponds to the electron-doped case when a particle–hole transformation is performed. From Ref. 16.

-wave order parameter as a function of for various values of , calculated in CDMFT on a cluster for . The positive case corresponds to the electron-doped case when a particle–hole transformation is performed. From Ref. 16.

Antiferromagnetic (bottom) and -wave (top) order parameters as functions of the electron density for and various values of on an 8-site cluster, calculated in VCPT. From Ref. 18.

Antiferromagnetic (bottom) and -wave (top) order parameters as functions of the electron density for and various values of on an 8-site cluster, calculated in VCPT. From Ref. 18.

-wave order parameter as a function of for various values of , and calculated in CDMFT on a 4-site cluster. From Ref. 16.

-wave order parameter as a function of for various values of , and calculated in CDMFT on a 4-site cluster. From Ref. 16.

VCPT calculations for , near half-filling on lattice. Contrary to the strong coupling case, the -wave order parameter survives all the way to half-filling at weak coupling, unless we also allow for antiferromagnetism.

VCPT calculations for , near half-filling on lattice. Contrary to the strong coupling case, the -wave order parameter survives all the way to half-filling at weak coupling, unless we also allow for antiferromagnetism.

as a function of doping, , for calculated in TPSC using the Thouless criterion. From Ref. 41.

as a function of doping, , for calculated in TPSC using the Thouless criterion. From Ref. 41.

MDC at the Fermi energy for 10% hole-doped from Ref. 85.

MDC at the Fermi energy for 10% hole-doped from Ref. 85.

Doping dependence of the MDC from experiments on NCCO with the corresponding EDC. From Ref. 86.

Doping dependence of the MDC from experiments on NCCO with the corresponding EDC. From Ref. 86.

Intensity plot of the spectral function as a function of in units of and wave vector from VCPT for , , , and at the bottom and (electron-doped) at the top. The Lorentzian broadening is in the main figure and in the inset, which displays the -wave gap. Top panel is for the electron-doped case in the right-hand panel of Fig. 23, while the bottom panel is for the hole-doped case on the left of Fig. 23. From Ref. 18.

Intensity plot of the spectral function as a function of in units of and wave vector from VCPT for , , , and at the bottom and (electron-doped) at the top. The Lorentzian broadening is in the main figure and in the inset, which displays the -wave gap. Top panel is for the electron-doped case in the right-hand panel of Fig. 23, while the bottom panel is for the hole-doped case on the left of Fig. 23. From Ref. 18.

Experimental Fermi surface plot (MDC at the Fermi energy) for NCCO (left) and corresponding energy distribution curves (right) for 15% electron-doping. From Ref. 86.

Experimental Fermi surface plot (MDC at the Fermi energy) for NCCO (left) and corresponding energy distribution curves (right) for 15% electron-doping. From Ref. 86.

EDC along the Fermi surface calculated in TPSC (left column) at optimal doping for , , and corresponding ARPES data on NCCO (right column). From Ref. 87.

EDC along the Fermi surface calculated in TPSC (left column) at optimal doping for , , and corresponding ARPES data on NCCO (right column). From Ref. 87.

EDC along two other directions calculated for , , in TPSC (left column) and corresponding ARPES data on NCCO (right column). From Ref. 87.

EDC along two other directions calculated for , , in TPSC (left column) and corresponding ARPES data on NCCO (right column). From Ref. 87.

EDC along the Fermi surface shown for , and corresponding energy distribution curves . Lines are shifted by a constant for clarity. From Ref. 87.

EDC along the Fermi surface shown for , and corresponding energy distribution curves . Lines are shifted by a constant for clarity. From Ref. 87.

Hot spots from quasi-static scatterings off antiferromagnetic fluctuations (renormalized classical regime).

Hot spots from quasi-static scatterings off antiferromagnetic fluctuations (renormalized classical regime).

Semi-log plot of the AFM correlation length (in units of the lattice constant) against inverse temperature (in units of ). Filled symbols denote calculated results and empty ones experimental data of Ref. 126 and Ref. 127. From Ref. 87.

Semi-log plot of the AFM correlation length (in units of the lattice constant) against inverse temperature (in units of ). Filled symbols denote calculated results and empty ones experimental data of Ref. 126 and Ref. 127. From Ref. 87.

Pseudogap temperature (filled circles denote calculated from TPSC, empty ones experimental data extracted from optical conductivity^{128}). Empty triangles are experimental Néel temperatures . The samples are reduced.^{126} From Ref. 87.

Pseudogap temperature (filled circles denote calculated from TPSC, empty ones experimental data extracted from optical conductivity^{128}). Empty triangles are experimental Néel temperatures . The samples are reduced.^{126} From Ref. 87.

The generic phase diagram of high- superconductors, from Ref. 110. There should also be a pseudogap line on the electron-doped side. It was not well studied at the time of publication of that paper.

The generic phase diagram of high- superconductors, from Ref. 110. There should also be a pseudogap line on the electron-doped side. It was not well studied at the time of publication of that paper.

Antiferromagnetic order parameter (dashed) and -wave (solid) order parameter obtained from CDMFT on a cluster. The result obtained by forcing is also shown as a thin dashed line.

Antiferromagnetic order parameter (dashed) and -wave (solid) order parameter obtained from CDMFT on a cluster. The result obtained by forcing is also shown as a thin dashed line.

Phase diagram obtained from DCA for for the two-band model. From Ref. 104.

Phase diagram obtained from DCA for for the two-band model. From Ref. 104.

The gap in the density of states of the dSC as a function of filling for , as calculated in CDMFT on a cluster. From Ref. 16.

The gap in the density of states of the dSC as a function of filling for , as calculated in CDMFT on a cluster. From Ref. 16.

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