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

AIP conference proceedings

Path-Integral Renormalization Group Method

Masatoshi Imada

Published Jan 1, 2003• 1 citation
Publisher

Abstract

Path‐integral renormalization group (PIRG) method has been developed for studying strongly correlated electron systems. In a wide range of systems including Hubbard‐type models, this method overcomes a number of difficulties known in the Monte‐Carlo‐type methods such as the negative sign problem. This method has been combined with a procedure of quantum number projection and grand canonical ensemble method as well, which contribute to a wider applicability. The quantum‐number projected PIRG enables calculations of excited spectra with a specified momentum or spin. We review recent numerical results on strongly correlated electron systems studied by this method. By using the methods, we determine the phase diagram of the two‐dimensional Hubbard model in the parameter space of the onsite interaction U, strength of the geometrical frustration effects defined by the ratio between next‐nearest to nearest neighbor transfer integrals, t′/t, and the chemical potential. It reveals severe competitions of various phases. The phase diagram contains a remarkable nonmagnetic Mott insulator phase with gapless and dispersionless spin excitations, sandwitched by the first‐order Mott transition and the antiferromagnetic transition. The first‐order character becomes more continuous one with increasing the frustration effect.

DOI
10.1063/1.1632131

Related papers

Algorithms and Applications of Path-Integral Renormalization Group Method

Masatoshi Imada

Path‐integral renormalization‐group (PIRG) method is a rapidly developing tool for computing ground state properties of interacting quantum systems on lattices, particularly models for strongly correlated electrons such as the Hubbard model. It has served in clarifying phase diagrams of the Hubbard model containing quantum spin liquid phase. PIRG has also been implemented as a low‐energy solver of the effective Hamiltonian for realistic systems. This makes it possible to construct a scheme of first‐principles calculation by the hybrid approach combined with the density functional theory.

AIP conference proceedings•Jan 1, 2006•0 citations

Magnetic and Metal–Insulator Transitions through Bandwidth Control in Two-Dimensional Hubbard Models with Nearest and Next-Nearest Neighbor Transfers

Tsuyoshi Kashima, Masatoshi Imada

Numerical studies on Mott transitions caused by the control of the ratio between bandwidth and electron-electron interaction ($U$) are reported. By using the recently proposed path-integral renormalization group(PIRG) algorithm, physical properties near the transitions in the ground state of two-dimensional half-filled models with the nearest and the next-nearest neighbor transfers ($-t$ and $t'$, respectively) are studied as a prototype of geometrically frustrated system. The nature of the bandwidth-control transitions shows sharp contrast with that of the filling-control transitions: First, the metal-insulator and magnetic transitions are separated each other and the metal-insulator (MI) transition occurs at smaller $U$, although the both transition interactions $U$ increase with increasing $t'$. Both transitions do not contradict the first-order transitions for smaller $t'/t$ while the MI transitions become continuous type accompanied by emergence of {\it unusual metallic phase} near the transition for large $t'/t$. A nonmagnetic insulator phase is stabilized between MI and AF transitions. The region of the nonmagnetic insulator becomes wider with increasing $t'/t$. The phase diagram naturally connects two qualitatively different limits, namely the Hartree-Fock results at small $t'/t$ and speculations in the strong coupling Heisenberg limit.

arXiv:cond-mat/0104348•Journal of the Physical Society of Japan•Oct 1, 2001•87 citations

Precise Determination of Phase Diagram for Two-Dimensional Hubbard Model with Filling- and Bandwidth-Control Mott Transitions: Grand-Canonical Path-Integral Renormalization Group Approach

Shinji Watanabe, Masatoshi Imada

A new numerical algorithm for interacting fermion systems to treat the grand-canonical ensemble is proposed and examined by extending the path-integral renormalization group method. To treat the grand-canonical ensemble, the particle-hole transformation is applied to the Hamiltonian and basis states. In the interaction-term projection, the Stratonovich-Hubbard transformation which hybridizes up and down spin electrons is introduced. By using this method, the phase diagram of the two-dimensional Hubbard model with next-nearest-neighbor transfer is accurately determined by treating the filling-control (FC) and bandwidth-control (BC) Mott transitions on the same ground. A V-shaped Mott insulating phase is obtained in the plane of the chemical potential and the Coulomb interaction, where the transitions at the corner (BC) and the edges (FC) show contrasted characters with large critical fluctuations near the edges coexisting with the first-order transition at the corner. This contrasted behavior is shown to be consistent with the V-shape structure of the phase boundary because of a general relation, in which the slope of the metal-insulator transition line in the phase diagram is expressed by thermodynamic quantities. The V-shaped opening of the Mott gap is favorably compared with the experimental results of the transition metal oxides.

arXiv:cond-mat/0312671•Journal of the Physical Society of Japan•May 15, 2004•32 citations

Path-Integral Renormalization Group Method for Numerical Study of Strongly Correlated Electron Systems

Masatoshi Imada, Tsuyoshi Kashima

A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with small number of retained states.

arXiv:cond-mat/0007304•Journal of the Physical Society of Japan•Sep 1, 2000•78 citations

Mott Transition in the Hubbard Model on Checkerboard Lattice

Takuya Yoshioka, Akihisa Koga, Norio Kawakami

We investigate the bandwidth-controlled Mott transition in the Hubbard model on the checkerboard lattice at half filling using the path-integral renormalization group (PIRG) method. It is demonstrated that the system undergoes a first-order phase transition to the plaquette-singlet insulating phase at a finite Hubbard interaction. This conclusion is drawn via a detailed analysis of the spin and charge correlations around the phase transition point by means of the PIRG method aided with a new iteration scheme introduced in this paper.

arXiv:0808.1116•Journal of the Physical Society of Japan•Sep 25, 2008•18 citations

Gapless quantum spin liquid, stripe, and antiferromagnetic phases in frustrated Hubbard models in two dimensions

T. Mizusaki, Masatoshi Imada

Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model containing the diagonal transfers ${t}^{\ensuremath{'}}$ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer ${t}^{\ensuremath{'}}$ and a stripe (or collinear) phase for large ${t}^{\ensuremath{'}}$ in the Mott insulating region of the strong on-site interaction $U$. For intermediate ${t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7$ at large $U∕t>7$, we find a longer-period antiferromagnetic-insulator phase with $2\ifmmode\times\else\texttimes\fi{}4$ structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and $^{3}\mathrm{He}$ atoms adsorbed on graphite.

arXiv:cond-mat/0604389•Physical Review B•Jul 18, 2006•107 citations
Path‐integral renormalization group (PIRG) method has been developed for studying strongly correlated electron systems. In a wide range of systems including Hubbard‐type models, this method overcomes a number of difficulties known in the Monte‐Carlo‐type methods such as the negative sign problem. This method has been combined with a procedure of quantum number projection and grand canonical ensemble method as well, which contribute to a wider applicability. The quantum‐number projected PIRG enables calculations of excited spectra with a specified momentum or spin. We review recent numerical results on strongly correlated electron systems studied by this method. By using the methods, we determine the phase diagram of the two‐dimensional Hubbard model in the parameter space of the onsite interaction U, strength of the geometrical frustration effects defined by the ratio between next‐nearest to nearest neighbor transfer integrals, t′/t, and the chemical potential. It reveals severe competitions of various phases. The phase diagram contains a remarkable nonmagnetic Mott insulator phase with gapless and dispersionless spin excitations, sandwitched by the first‐order Mott transition and the antiferromagnetic transition. The first‐order character becomes more continuous one with increasing the frustration effect.
Path‐integral renormalization‐group (PIRG) method is a rapidly developing tool for computing ground state properties of interacting quantum systems on lattices, particularly models for strongly correlated electrons such as the Hubbard model. It has served in clarifying phase diagrams of the Hubbard model containing quantum spin liquid phase. PIRG has also been implemented as a low‐energy solver of the effective Hamiltonian for realistic systems. This makes it possible to construct a scheme of first‐principles calculation by the hybrid approach combined with the density functional theory.
Numerical studies on Mott transitions caused by the control of the ratio between bandwidth and electron-electron interaction (UUU) are reported. By using the recently proposed path-integral renormalization group(PIRG) algorithm, physical properties near the transitions in the ground state of two-dimensional half-filled models with the nearest and the next-nearest neighbor transfers (−t-t−t and t′t't′, respectively) are studied as a prototype of geometrically frustrated system. The nature of the bandwidth-control transitions shows sharp contrast with that of the filling-control transitions: First, the metal-insulator and magnetic transitions are separated each other and the metal-insulator (MI) transition occurs at smaller UUU, although the both transition interactions UUU increase with increasing t′t't′. Both transitions do not contradict the first-order transitions for smaller t′/tt'/tt′/t while the MI transitions become continuous type accompanied by emergence of {\it unusual metallic phase} near the transition for large t′/tt'/tt′/t. A nonmagnetic insulator phase is stabilized between MI and AF transitions. The region of the nonmagnetic insulator becomes wider with increasing t′/tt'/tt′/t. The phase diagram naturally connects two qualitatively different limits, namely the Hartree-Fock results at small t′/tt'/tt′/t and speculations in the strong coupling Heisenberg limit.
A new numerical algorithm for interacting fermion systems to treat the grand-canonical ensemble is proposed and examined by extending the path-integral renormalization group method. To treat the grand-canonical ensemble, the particle-hole transformation is applied to the Hamiltonian and basis states. In the interaction-term projection, the Stratonovich-Hubbard transformation which hybridizes up and down spin electrons is introduced. By using this method, the phase diagram of the two-dimensional Hubbard model with next-nearest-neighbor transfer is accurately determined by treating the filling-control (FC) and bandwidth-control (BC) Mott transitions on the same ground. A V-shaped Mott insulating phase is obtained in the plane of the chemical potential and the Coulomb interaction, where the transitions at the corner (BC) and the edges (FC) show contrasted characters with large critical fluctuations near the edges coexisting with the first-order transition at the corner. This contrasted behavior is shown to be consistent with the V-shape structure of the phase boundary because of a general relation, in which the slope of the metal-insulator transition line in the phase diagram is expressed by thermodynamic quantities. The V-shaped opening of the Mott gap is favorably compared with the experimental results of the transition metal oxides.
A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with small number of retained states.
We investigate the bandwidth-controlled Mott transition in the Hubbard model on the checkerboard lattice at half filling using the path-integral renormalization group (PIRG) method. It is demonstrated that the system undergoes a first-order phase transition to the plaquette-singlet insulating phase at a finite Hubbard interaction. This conclusion is drawn via a detailed analysis of the spin and charge correlations around the phase transition point by means of the PIRG method aided with a new iteration scheme introduced in this paper.
Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model containing the diagonal transfers ${t}^{\ensuremath{'}}$ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer ${t}^{\ensuremath{'}}$ and a stripe (or collinear) phase for large ${t}^{\ensuremath{'}}$ in the Mott insulating region of the strong on-site interaction UUU. For intermediate ${t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7$ at large U∕t>7U∕t>7U∕t>7, we find a longer-period antiferromagnetic-insulator phase with $2\ifmmode\times\else\texttimes\fi{}4$ structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and 3He^{3}\mathrm{He}3He atoms adsorbed on graphite.