Determinant quantum Monte Carlo study of the enhancement of d-wave pairing by charge inhomogeneity

Mondaini, Rubem; Ying, Tao; Paiva, Thereza; Scalettar, Richard T.

doi:10.1103/PhysRevB.86.184506

Title: Determinant quantum Monte Carlo study of the enhancement of $d$ -wave pairing by charge inhomogeneity

Abstract

Striped phases, in which spin, charge, and pairing correlations vary inhomogeneously in the CuO₂ planes, are a known experimental feature of cuprate superconductors, and are also found in a variety of numerical treatments of the two-dimensional Hubbard Hamiltonian. In this paper, we use determinant quantum Monte Carlo to show that if a stripe density pattern is imposed on the model, the d-wave pairing vertex is significantly enhanced. We attribute this enhancement to an increase in antiferromagnetic order which is caused by the appearance of more nearly half-filled regions when the doped holes are confined to the stripes. Here, we also observe a π-phase shift in the magnetic order.

Authors:

Mondaini, Rubem ^[1]; Ying, Tao ^[2]; Paiva, Thereza ^[1]; Scalettar, Richard T. ^[3]

Univ. Federal do Rio de Janeiro, Rio de Janeiro RJ (Brazil)
Univ. of California, Davis, CA (United States); Harbin Institute of Technology, Harbin (China)
Univ. of California, Davis, CA (United States)

Publication Date:: Wed Nov 07 00:00:00 EST 2012

Research Org.:: Univ. of California, Davis, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1329354

Grant/Contract Number:: FG52-09NA29464

Resource Type:: Accepted Manuscript

Journal Name:: Physical Review. B, Condensed Matter and Materials Physics

Additional Journal Information:: Journal Volume: 86; Journal Issue: 18; Journal ID: ISSN 1098-0121

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats


                    Mondaini, Rubem, Ying, Tao, Paiva, Thereza, and Scalettar, Richard T. Determinant quantum Monte Carlo study of the enhancement of d-wave pairing by charge inhomogeneity.  United States: N. p., 2012. 
Web.  doi:10.1103/PhysRevB.86.184506.

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                    Mondaini, Rubem, Ying, Tao, Paiva, Thereza, & Scalettar, Richard T. Determinant quantum Monte Carlo study of the enhancement of d-wave pairing by charge inhomogeneity.  United States.  https://doi.org/10.1103/PhysRevB.86.184506

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                    Mondaini, Rubem, Ying, Tao, Paiva, Thereza, and Scalettar, Richard T. Wed .  
"Determinant quantum Monte Carlo study of the enhancement of d-wave pairing by charge inhomogeneity".  United States.  https://doi.org/10.1103/PhysRevB.86.184506.  https://www.osti.gov/servlets/purl/1329354.

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@article{osti_1329354,

  title        = {Determinant quantum Monte Carlo study of the enhancement of d-wave pairing by charge inhomogeneity},

  author       = {Mondaini, Rubem and Ying, Tao and Paiva, Thereza and Scalettar, Richard T.},

  abstractNote = {Striped phases, in which spin, charge, and pairing correlations vary inhomogeneously in the CuO2 planes, are a known experimental feature of cuprate superconductors, and are also found in a variety of numerical treatments of the two-dimensional Hubbard Hamiltonian. In this paper, we use determinant quantum Monte Carlo to show that if a stripe density pattern is imposed on the model, the d-wave pairing vertex is significantly enhanced. We attribute this enhancement to an increase in antiferromagnetic order which is caused by the appearance of more nearly half-filled regions when the doped holes are confined to the stripes. Here, we also observe a π-phase shift in the magnetic order.},

  doi          = {10.1103/PhysRevB.86.184506},

  journal      = {Physical Review. B, Condensed Matter and Materials Physics},

  number       = 18,

  volume       = 86,

  place        = {United States},

  year         = {Wed Nov 07 00:00:00 EST 2012},

  month        = {Wed Nov 07 00:00:00 EST 2012}

}

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https://doi.org/10.1103/PhysRevB.86.184506

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Figures / Tables:

FIG. 1: (Color online) 16x16 lattice with period $\mathcal{P}$ = 4, sites with V₀ active are depicted in blue ( lled cricles), whereas the interstripe sites are red (empty circles).

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We used the determinant Quantum Monte Carlo (DQMC) to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters - a two-dimensional square lattice consisting of elemental 2x2 plaquettes with hopping t and on-site repulsion U coupled by an interplaquette hopping t' ≤ t. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half-filling by the inhomogeneous hopping or not. For U/t = 4, DQMC indicates an optimal t'/t ≈ 0.4 at which themore »« less
Cited by 20
https://doi.org/10.1103/PhysRevB.90.075121

Full Text Available
Evolution of superconducting correlations within magnetic-field-decoupled La $_{2 - x}$ Ba $_{x}$ CuO $_{4}$ ( $x = 0.095$ )

Journal Article Stegen, Z. ; Han, Su Jung ; Wu, Jie ; ... - Physical Review. B, Condensed Matter and Materials Physics

We explore the evolution of superconductivity in La 2 - x Ba x CuO 4 with x = 0.095 in magnetic fields of up to 35 T applied perpendicular to the CuO 2 planes. Previous work on this material has shown that perpendicular fields enhance both charge- and spin-stripe order within the planes. We present measurements of the resistivity parallel and perpendicular to the planes, as well as the Hall effect. Measurements of magnetic susceptibility for fields of up to 15 T applied both parallel and perpendicular to the planes provide complementary measures of the superconductivity. We show that fieldsmore »« less
https://doi.org/10.1103/PhysRevB.87.064509
Vacancy-driven orbital and magnetic order in (K,Tl,Cs) $_{y}$ Fe $_{2 - x}$ Se $_{2}$

Journal Article Lv, Weicheng ; Lee, Wei-Cheng ; Phillips, Philip - Physical Review. B, Condensed Matter and Materials Physics

We investigate the effects of the √ 5 × √ 5 Fe vacancy ordering on the orbital and magnetic order in (K,Tl,Cs) y Fe 2 - x Se 2 using a three-orbital ( t 2 g ) tight-binding Hamiltonian with generalized Hubbard interactions. We find that vacancy order enhances electron correlations, resulting in the onset of a block antiferromagnetic phase with large moments at smaller interaction strengths. In addition, vacancy ordering modulates the kinetic energy differently for the three t 2 g orbitals. This results in a breaking of the degeneracy between the d x z and d y zmore »« less
https://doi.org/10.1103/PhysRevB.84.155107
Light-Induced Melting of Competing Stripe Orders without Introducing Superconductivity in ${La}_{2 - x} {Ba}_{x} {CuO}_{4}$

Journal Article Zhang, S. J. ; Zhou, X. Y. ; Xu, S. X. ; ... - Physical Review. X

The ultrafast manipulation of quantum material has led to many novel and significant discoveries. Among them, the light-induced transient superconductivity in cuprates achieved by melting competing stripe orders represents a highly appealing accomplishment. However, recent investigations have shown that the notion of photoinduced superconductivity remains a topic of controversy, and its elucidation solely through c -axis time-resolved terahertz spectroscopy remains an arduous task. Here, we measure the in-plane and out-of-plane transient terahertz responses simultaneously in the stripe-ordered nonsuperconducting La_2-xBa_xCuO₄ after near-infrared excitations. We find that although a pump-induced reflectivity edge appears in the c -axis reflectance spectrum, the reflectivity alongmore »« less
https://doi.org/10.1103/PhysRevX.14.011036
Magnetic properties and pairing tendencies of the iron-based superconducting ladder ${BaFe}_{2} S_{3}$ : Combined ab initio and density matrix renormalization group study

Journal Article Patel, Niravkumar D. ; Nocera, Alberto ; Alvarez, Gonzalo ; ... - Physical Review B

The recent discovery of superconductivity under high pressure in the two-leg ladder compound BaFe₂S₃ [H. Takahashi et al., Nat. Mater. 14, 1008 (2015)] opens a broad avenue of research, because it represents the first report of pairing tendencies in a quasi-one-dimensional iron-based high-critical-temperature superconductor. Similarly, as in the case of the cuprates, ladders and chains can be far more accurately studied using many-body techniques and model Hamiltonians than their layered counterparts, particularly if several orbitals are active. In this publication, we derive a two-orbital Hubbard model from first principles that describes individual ladders of BaFe₂S₃. The model is studied withmore »« less
Cited by 31
https://doi.org/10.1103/PhysRevB.94.075119

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Title: Determinant quantum Monte Carlo study of the enhancement of d -wave pairing by charge inhomogeneity

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The BCS–BEC Crossover and the Unitary Fermi Gas book, October 2011

Charged solitons in the Hartree-Fock approximation to the large- U Hubbard model journal, May 1989

Incommensurate antiferromagnetism in the two-dimensional Hubbard model journal, March 1990

Variational Monte Carlo study of incommensurate antiferromagnetic phases in the two-dimensional Hubbard model journal, December 1990

Binding of holes in one-dimensional Hubbard chains journal, October 1990

Phase separation in the Hubbard model journal, May 1991

Phase separation in the two-dimensional Hubbard model journal, September 1996

Phase separation in the one-band Hubbard model journal, April 1996

Magnetism and phase separation in the ground state of the Hubbard model journal, June 2002

Weak phase separation and the pseudogap in the electron-doped cuprates journal, October 2005

Magnetism in La2CuO4 based compounds journal, April 1989

Phase separation in the t - J model journal, January 1990

Phase Separation at all Interaction Strengths in the t-J Model journal, June 1997

ANTIFERROMAGNETISM, STRIPES, AND SUPERCONDUCTIVITY IN THE t–J MODEL WITH COULOMB INTERACTION journal, March 2001

Phase separation based on a U(1) slave-boson functional integral approach to the t − J model journal, December 2000

Limits on phase separation for two-dimensional strongly correlated electrons journal, July 2000

Phase separation of the two-dimensional t − J model journal, January 1998

Indications of spin-charge separation at short distance and stripe formation in the extended t-J model on ladders and planes journal, December 2000

Frustrated electronic phase separation and high-temperature superconductors journal, May 1993

Study of an Ising model with competing long- and short-range interactions journal, March 1994

Charged magnetic domain lines and the magnetism of high- T c oxides journal, October 1989

Hartree-Fock study of the magnetism in the single-band Hubbard model journal, September 1991

Charged domain walls in the two-dimensional Hubbard model journal, October 1991

Evidence for stripe correlations of spins and holes in copper oxide superconductors journal, June 1995

Coexistence of, and Competition between, Superconductivity and Charge-Stripe Order in La 1 . 6 − x Nd 0.4 Sr x CuO 4 journal, January 1997

Two-Dimensional Superconducting Fluctuations in Stripe-Ordered La 1.875 Ba 0.125 CuO 4 journal, August 2007

Dynamical Layer Decoupling in a Stripe-Ordered High- T c Superconductor journal, September 2007

Theory of the striped superconductor journal, February 2009

The Ground State of the Pseudogap in Cuprate Superconductors journal, November 2006

Evidence for unusual superconducting correlations coexisting with stripe order in La 1.875 Ba 0.125 CuO 4 journal, November 2008

Stripe order in superconducting La 2 − x Ba x CuO 4 ( 0 . 095 ⩽ x ⩽ 0 . 155 ) journal, March 2011

Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy journal, September 2011

Stripe order, depinning, and fluctuations in La 1.875 Ba 0.125 CuO 4 and La 1.875 Ba 0.075 Sr 0.050 CuO 4 journal, September 2004

Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa 2 Cu 3 O 6 + δ Superconductors Identified by Resonant Elastic X-ray Scattering journal, October 2012

Long-Range Incommensurate Charge Fluctuations in (Y,Nd)Ba2Cu3O6+x journal, July 2012

Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67 journal, October 2012

Stripes and Pairing in High Temperature Superconductors journal, April 2012

Where do holes go in doped antiferromagnets and what is their relationship to superconductivity? journal, December 2002

Enhancement of superconductivity by local inhomogeneities journal, August 2005

Effect of inhomogeneity on s -wave superconductivity in the attractive Hubbard model journal, March 2006

Superconductivity in inhomogeneous Hubbard models journal, June 2006

Using inhomogeneity to raise the superconducting critical temperature in a two-dimensional X Y model journal, April 2007

Myriad phases of the checkerboard Hubbard model journal, October 2007

s -wave superconductivity phase diagram in the inhomogeneous two-dimensional attractive Hubbard model journal, November 2007

Sublattice model of atomic scale pairing inhomogeneity in a superconductor journal, October 2008

Optimal inhomogeneity for superconductivity: Finite-size studies journal, June 2008

Suppression of d -wave superconductivity in the checkerboard Hubbard model journal, July 2008

Dynamic Cluster Quantum Monte Carlo Simulations of a Two-Dimensional Hubbard Model with Stripelike Charge-Density-Wave Modulations: Interplay between Inhomogeneities and the Superconducting State journal, June 2010

Density Matrix Renormalization Group Study of the Striped Phase in the 2D t − J Model journal, February 1998

Neutron-scattering study of stripe-phase order of holes and spins in La 1.48 Nd 0.4 Sr 0.12 CuO 4 journal, September 1996

Spatially modulated 'Mottness' in La2-xBaxCuO4 journal, December 2005

Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in La 2 − x Sr x CuO 4 journal, March 1998

Magnetic field dependence of charge stripe order in La 2 − x Ba x Cu O 4 ( x ≈ 1 8 ) journal, May 2008

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

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

Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices journal, June 2012

Fermions in 3D Optical Lattices: Cooling Protocol to Obtain Antiferromagnetism journal, August 2011

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

Attractive and repulsive pairing interaction vertices for the two-dimensional Hubbard model journal, January 1989

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

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

Phase diagram of the two-dimensional negative- U Hubbard model journal, March 1989

Quantum melting instability of a striped domain wall in the two-dimensional t-J model journal, August 1994

Charged domain-wall dynamics in doped antiferromagnets and spin fluctuations in cuprate superconductors journal, April 1996

Quantizing charged magnetic domain walls: Strings on a lattice journal, July 1996

A Four Unit Cell Periodic Pattern of Quasi-Particle States Surrounding Vortex Cores in Bi2Sr2CaCu2O8+delta journal, January 2002

Periodic density-of-states modulations in superconducting Bi 2 Sr 2 CaCu 2 O 8 + δ journal, January 2003

Local Ordering in the Pseudogap State of the High-Tc Superconductor Bi2Sr2CaCu2O8+ journal, March 2004

A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2-xNaxCuO2Cl2 journal, August 2004

Coincidence of Checkerboard Charge Order and Antinodal State Decoherence in Strongly Underdoped Superconducting Bi 2 Sr 2 Ca Cu 2 O 8 + δ journal, May 2005

Local modulations of the spin-fluctuation-mediated pairing interaction by impurities in d -wave superconductors journal, August 2012

Disorder-Induced Freezing of Dynamical Spin Fluctuations in Underdoped Cuprate Superconductors journal, September 2010

Pair density wave at high magnetic fields in cuprates with charge and spin orders journal, July 2020

Phase Diagram of the Two-Dimensional Negative- U Hubbard Model journal, July 1989

Myriad phases of the Checkerboard Hubbard Model text, January 2007

The Optimal Inhomogeneity for Superconductivity: Finite Size Studies text, January 2008

Title: Determinant quantum Monte Carlo study of the enhancement of $d$ -wave pairing by charge inhomogeneity

The BCS–BEC Crossover and the Unitary Fermi Gas
book, October 2011

Charged solitons in the Hartree-Fock approximation to the large- U Hubbard model
journal, May 1989

Incommensurate antiferromagnetism in the two-dimensional Hubbard model
journal, March 1990

Variational Monte Carlo study of incommensurate antiferromagnetic phases in the two-dimensional Hubbard model
journal, December 1990

Binding of holes in one-dimensional Hubbard chains
journal, October 1990

Phase separation in the Hubbard model
journal, May 1991

Phase separation in the two-dimensional Hubbard model
journal, September 1996

Phase separation in the one-band Hubbard model
journal, April 1996

Magnetism and phase separation in the ground state of the Hubbard model
journal, June 2002

Weak phase separation and the pseudogap in the electron-doped cuprates
journal, October 2005

Magnetism in La2CuO4 based compounds
journal, April 1989

Phase separation in the t - J model
journal, January 1990

Phase Separation at all Interaction Strengths in the t-J Model
journal, June 1997

ANTIFERROMAGNETISM, STRIPES, AND SUPERCONDUCTIVITY IN THE t–J MODEL WITH COULOMB INTERACTION
journal, March 2001

Phase separation based on a U(1) slave-boson functional integral approach to the $t - J$ model
journal, December 2000

Limits on phase separation for two-dimensional strongly correlated electrons
journal, July 2000

Phase separation of the two-dimensional $t - J$ model
journal, January 1998

Indications of spin-charge separation at short distance and stripe formation in the extended t-J model on ladders and planes
journal, December 2000

Frustrated electronic phase separation and high-temperature superconductors
journal, May 1993

Study of an Ising model with competing long- and short-range interactions
journal, March 1994

Charged magnetic domain lines and the magnetism of high- $T_{c}$ oxides
journal, October 1989

Hartree-Fock study of the magnetism in the single-band Hubbard model
journal, September 1991

Charged domain walls in the two-dimensional Hubbard model
journal, October 1991

Evidence for stripe correlations of spins and holes in copper oxide superconductors
journal, June 1995

Coexistence of, and Competition between, Superconductivity and Charge-Stripe Order in ${La}_{1.6 - x} {Nd}_{0.4} {Sr}_{x} {CuO}_{4}$
journal, January 1997

Two-Dimensional Superconducting Fluctuations in Stripe-Ordered ${La}_{1.875} {Ba}_{0.125} {CuO}_{4}$
journal, August 2007

Dynamical Layer Decoupling in a Stripe-Ordered High- $T_{c}$ Superconductor
journal, September 2007

Theory of the striped superconductor
journal, February 2009

The Ground State of the Pseudogap in Cuprate Superconductors
journal, November 2006

Evidence for unusual superconducting correlations coexisting with stripe order in ${La}_{1.875} {Ba}_{0.125} {CuO}_{4}$
journal, November 2008

Stripe order in superconducting La $_{2 - x}$ Ba $_{x}$ CuO $_{4}$ ( $0.095 ⩽ x ⩽ 0.155$ )
journal, March 2011

Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy
journal, September 2011

Stripe order, depinning, and fluctuations in ${La}_{1.875} {Ba}_{0.125} {CuO}_{4}$ and ${La}_{1.875} {Ba}_{0.075} {Sr}_{0.050} {CuO}_{4}$
journal, September 2004

Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered ${YBa}_{2} {Cu}_{3} O_{6 + δ}$ Superconductors Identified by Resonant Elastic X-ray Scattering
journal, October 2012

Long-Range Incommensurate Charge Fluctuations in (Y,Nd)Ba2Cu3O6+x
journal, July 2012

Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67
journal, October 2012

Stripes and Pairing in High Temperature Superconductors
journal, April 2012

Where do holes go in doped antiferromagnets and what is their relationship to superconductivity?
journal, December 2002

Enhancement of superconductivity by local inhomogeneities
journal, August 2005

Effect of inhomogeneity on $s$ -wave superconductivity in the attractive Hubbard model
journal, March 2006

Superconductivity in inhomogeneous Hubbard models
journal, June 2006

Using inhomogeneity to raise the superconducting critical temperature in a two-dimensional $X Y$ model
journal, April 2007

Myriad phases of the checkerboard Hubbard model
journal, October 2007

$s$ -wave superconductivity phase diagram in the inhomogeneous two-dimensional attractive Hubbard model
journal, November 2007

Sublattice model of atomic scale pairing inhomogeneity in a superconductor
journal, October 2008

Optimal inhomogeneity for superconductivity: Finite-size studies
journal, June 2008

Suppression of $d$ -wave superconductivity in the checkerboard Hubbard model
journal, July 2008

Dynamic Cluster Quantum Monte Carlo Simulations of a Two-Dimensional Hubbard Model with Stripelike Charge-Density-Wave Modulations: Interplay between Inhomogeneities and the Superconducting State
journal, June 2010

Density Matrix Renormalization Group Study of the Striped Phase in the 2D $t - J$ Model
journal, February 1998

Neutron-scattering study of stripe-phase order of holes and spins in ${La}_{1.48}$ ${Nd}_{0.4}$ ${Sr}_{0.12}$ ${CuO}_{4}$
journal, September 1996

Spatially modulated 'Mottness' in La2-xBaxCuO4
journal, December 2005

Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in ${La}_{2 - x} {Sr}_{x} {CuO}_{4}$
journal, March 1998

Magnetic field dependence of charge stripe order in ${La}_{2 - x} {Ba}_{x} Cu O_{4}$ $(x \approx \frac{1}{8})$
journal, May 2008

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

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

Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices
journal, June 2012

Fermions in 3D Optical Lattices: Cooling Protocol to Obtain Antiferromagnetism
journal, August 2011

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

Attractive and repulsive pairing interaction vertices for the two-dimensional Hubbard model
journal, January 1989

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

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

Phase diagram of the two-dimensional negative- U Hubbard model
journal, March 1989

Quantum melting instability of a striped domain wall in the two-dimensional t-J model
journal, August 1994

Charged domain-wall dynamics in doped antiferromagnets and spin fluctuations in cuprate superconductors
journal, April 1996

Quantizing charged magnetic domain walls: Strings on a lattice
journal, July 1996

A Four Unit Cell Periodic Pattern of Quasi-Particle States Surrounding Vortex Cores in Bi2Sr2CaCu2O8+delta
journal, January 2002

Periodic density-of-states modulations in superconducting ${Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}$
journal, January 2003

Local Ordering in the Pseudogap State of the High-Tc Superconductor Bi2Sr2CaCu2O8+
journal, March 2004

A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2-xNaxCuO2Cl2
journal, August 2004

Coincidence of Checkerboard Charge Order and Antinodal State Decoherence in Strongly Underdoped Superconducting ${Bi}_{2} {Sr}_{2} Ca {Cu}_{2} O_{8 + δ}$
journal, May 2005

Local modulations of the spin-fluctuation-mediated pairing interaction by impurities in $d$ -wave superconductors
journal, August 2012

Disorder-Induced Freezing of Dynamical Spin Fluctuations in Underdoped Cuprate Superconductors
journal, September 2010

Pair density wave at high magnetic fields in cuprates with charge and spin orders
journal, July 2020

Phase Diagram of the Two-Dimensional Negative- $U$ Hubbard Model
journal, July 1989

Myriad phases of the Checkerboard Hubbard Model
text, January 2007

The Optimal Inhomogeneity for Superconductivity: Finite Size Studies
text, January 2008

Theory of the striped superconductor
text, January 2008

Dynamic cluster quantum Monte Carlo simulations of a two-dimensional Hubbard model with stripe-like charge density wave modulations: Interplay between inhomogeneity and superconductivity
text, January 2009

Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices
text, January 2012