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Title: Ferromagnetism beyond Lieb's theorem

Abstract

The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling (ρ = 1) the ground state has a nonzero spin. In this paper we consider a “CuO2 lattice” (also known as “Lieb lattice,” or as a decorated square lattice), in which “d orbitals” occupy the vertices of the squares, while “ p orbitals” lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. Here, we study both the homogeneous (H) case, Ud = Up, originally considered by Lieb, and the inhomogeneous (IH) case, Ud ≠ Up. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling,more » and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U. For the IH system at half-filling, we argue that the case Up ≠ Ud falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up = 0, Ud = U and Up = U, Ud = 0. We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud = 0; by contrast, Up = 0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density ρ = 1/3, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each d site.« less

Authors:
 [1];  [1];  [1];  [1];  [2]
  1. Univ. Federal do Rio de Janeiro, Rio de Janeiro (Brazil). Inst. de Fisica
  2. Univ. of California, Davis, CA (United States). Dept. of Physics
Publication Date:
Research Org.:
Univ. of California, Davis, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1477007
Alternate Identifier(s):
OSTI ID: 1328225
Grant/Contract Number:  
NA0002908; SC0014671
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physical Review B
Additional Journal Information:
Journal Volume: 94; Journal Issue: 15; Journal ID: ISSN 2469-9950
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 36 MATERIALS SCIENCE

Citation Formats

Costa, Natanael C., Mendes-Santos, Tiago, Paiva, Thereza, Santos, Raimundo R. dos, and Scalettar, Richard T. Ferromagnetism beyond Lieb's theorem. United States: N. p., 2016. Web. doi:10.1103/PhysRevB.94.155107.
Costa, Natanael C., Mendes-Santos, Tiago, Paiva, Thereza, Santos, Raimundo R. dos, & Scalettar, Richard T. Ferromagnetism beyond Lieb's theorem. United States. https://doi.org/10.1103/PhysRevB.94.155107
Costa, Natanael C., Mendes-Santos, Tiago, Paiva, Thereza, Santos, Raimundo R. dos, and Scalettar, Richard T. 2016. "Ferromagnetism beyond Lieb's theorem". United States. https://doi.org/10.1103/PhysRevB.94.155107. https://www.osti.gov/servlets/purl/1477007.
@article{osti_1477007,
title = {Ferromagnetism beyond Lieb's theorem},
author = {Costa, Natanael C. and Mendes-Santos, Tiago and Paiva, Thereza and Santos, Raimundo R. dos and Scalettar, Richard T.},
abstractNote = {The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling (ρ = 1) the ground state has a nonzero spin. In this paper we consider a “CuO2 lattice” (also known as “Lieb lattice,” or as a decorated square lattice), in which “d orbitals” occupy the vertices of the squares, while “ p orbitals” lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. Here, we study both the homogeneous (H) case, Ud = Up, originally considered by Lieb, and the inhomogeneous (IH) case, Ud ≠ Up. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U. For the IH system at half-filling, we argue that the case Up ≠ Ud falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up = 0, Ud = U and Up = U, Ud = 0. We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud = 0; by contrast, Up = 0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density ρ = 1/3, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each d site.},
doi = {10.1103/PhysRevB.94.155107},
url = {https://www.osti.gov/biblio/1477007}, journal = {Physical Review B},
issn = {2469-9950},
number = 15,
volume = 94,
place = {United States},
year = {Thu Oct 06 00:00:00 EDT 2016},
month = {Thu Oct 06 00:00:00 EDT 2016}
}

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Works referenced in this record:

Quantum Monte Carlo study of the two-dimensional fermion Hubbard model
journal, August 2009


Sign problem in the numerical simulation of many-electron systems
journal, May 1990


Generalized Hartree-Fock theory and the Hubbard model
journal, July 1994


Rigorous criteria for ferromagnetism in itinerant electron systems
journal, May 1994


Destruction of superconductivity by impurities in the attractive Hubbard model
journal, October 2005


Numerical study of the two-dimensional Hubbard model
journal, July 1989


The Hubbard model - an introduction and selected rigorous results
journal, May 1998


Quantum anomalous Hall effect in a flat band ferromagnet
journal, February 2012


Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data
journal, May 1996


Quantum Monte Carlo with directed loops
journal, October 2002


Ferrimagnetic long-range order of the Hubbard model
journal, February 1994


Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms
journal, September 2009


Enhancement of long-range antiferromagnetic order by nonmagnetic impurities in the Hubbard model
journal, June 1998


Two-dimensional Hubbard model: Numerical simulation study
journal, April 1985


Microscopic conditions favoring itinerant ferromagnetism
journal, November 1998


Topological insulators on the Lieb and perovskite lattices
journal, August 2010


Monte Carlo calculations of coupled boson-fermion systems. I
journal, October 1981


Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms
journal, February 2015


Two theorems on the Hubbard model
journal, March 1989


Absence of ferromagnetism in the two-dimensional Hubbard model
journal, July 1985


Flat-band ferromagnetism in the multilayer Lieb optical lattice
journal, October 2014


Antiferromagnetic, charge-transfer, and pairing correlations in the three-band Hubbard model
journal, July 1991


Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the square lattice
journal, January 1993


Introduction to quantum Monte Carlo simulations for fermionic systems
journal, January 2003


Metallic ferromagnetism in a single-band model: Effect of band filling and Coulomb interactions
journal, September 1996


Ferromagnetism in Single-Band Hubbard Models with a Partially Flat Band
journal, May 1999


Disordered two-dimensional superconductors: Roles of temperature and interaction strength
journal, November 2008


Observation of Localized States in Lieb Photonic Lattices
journal, June 2015


Accuracy of the Hartree–Fock approximation for the Hubbard model
journal, April 1997


Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model
journal, September 2016


Two Theorems on the Hubbard Model
journal, April 1989


Observation of a Localized Flat-Band State in a Photonic Lieb Lattice
journal, June 2015


Ferromagnetism in Hubbard Models
journal, December 1995


Antiferromagnetism in the Two-Dimensional Hubbard Model
journal, January 1989


Ferromagnetic ground states for the Hubbard model on line graphs
journal, January 1991


Ferromagnetism in the Hubbard model on fcc-type lattices
journal, February 1998


The Directed-Loop Algorithm
conference, January 2003

  • Sandvik, Anders W.
  • THE MONTE CARLO METHOD IN THE PHYSICAL SCIENCES: Celebrating the 50th Anniversary of the Metropolis Algorithm, AIP Conference Proceedings
  • https://doi.org/10.1063/1.1632141

Microscopic conditions favoring itinerant ferromagnetism: Hund's rule coupling and orbital degeneracy
journal, October 1998


Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states
journal, November 1999


Ferromagnetism of cold fermions loaded into a decorated square lattice
journal, December 2009


Electron correlations in narrow energy bands
journal, November 1963


Characterizing the three-orbital Hubbard model with determinant quantum Monte Carlo
journal, April 2016


Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains
journal, September 2016


Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model
journal, September 2016


Ferromagnetism in the Hubbard model and Hund's rule
journal, March 1993


Works referencing / citing this record:

Experimental realization and characterization of an electronic Lieb lattice
journal, April 2017


Magnon-impurity interaction effect on the magnonic heat capacity of the Lieb lattice
journal, December 2018


Magnetic order in a Fermi gas induced by cavity-field fluctuations
journal, October 2018


Impact of strong correlations on a band topological insulator on the Lieb lattice
journal, July 2019


Magnetization, d -wave superconductivity, and non-Fermi-liquid behavior in a crossover from dispersive to flat bands
journal, September 2019


Quantum phase transitions of multispecies Dirac fermions
journal, June 2018


Magnetic Shell Structure of 2D-Trapped Fermi Gases in the Flat-Band Lieb Lattices
journal, January 2019