Numerical studies of photon-based spectroscopies on high-Tc superconductors

doi:10.1016/j.cpc.2010.08.017

Computer Physics Communications

Volume 182, Issue 1, January 2011, Pages 106-108

https://doi.org/10.1016/j.cpc.2010.08.017 Get rights and content

Abstract

A computational algorithm with efficient scaling to tens of thousands of CPUs has been applied to large scale exact diagonalization studies of small clusters. The goal is to simulate various photon-based spectra for effective single- and multi-orbital Hubbard Hamiltonians representative of the high- $T_{c}$ cuprate superconductors. As an example of the utility of this method, we address the optical band gap and the spectral functions calculated from these models which strongly depend on cluster geometry and size.

Introduction

With two decades of intense theoretical and experimental studies, the superconducting mechanism of the cuprate high- $T_{c}$ superdonductors remains a mystery. Nonetheless, it has been generally established that various intriguing phenomena in these materials, such as magnetism and superconductivity, are intimately tied to the strong electronic correlation. The essential ingredient in the cuprates is believed to be the quasi-two-dimensional (2D) copper–oxygen (CuO) plane, and therefore it is important to study the physics revealed by various spectroscopic probes by considering 2D effective Hamiltonians that can capture the interplay between electron itinerancy and interactions. In particular, the single and multi-orbital Hubbard models, in many regards, correctly describe the essential physics of these correlated systems. The insulating cuprate parent compounds have a band gap $\sim 2 eV$ revealed by optical conductivity measurements. This optical gap is well simulated by the single-band Hubbard model [1], and also the multi-orbital Hubbard model on quasi-1D clusters [2].

In this work, we simulate various photon-based spectroscopies using an efficient parallel implementation of the exact diagonalization method [3] on finite-size clusters applied to the Hubbard model. This numerical technique enables an exact treatment of many-body interactions, and provides the eigenstate information necessary to construct various linear response functions and spectra via Fermi's golden rule. We revisit the problem of optical conductivity, and discuss the relation of the optical gap to that in the hole addition/removal spectra. In particular, we explain the absence of the 2 eV feature in the Cu₄O₈ cluster [4], and demonstrate with the Cu₈O₁₆ cluster that a 2 eV feature does emerge.

Section snippets

Computational method

The problems of simulating multi-orbital Hubbard Hamiltonians on small clusters involve solving eigenvalue problems of large scale sparse matrices. To reduce storage requirements, the sparse matrices are written in the compressed sparse row format (CSR) [5], where an $n \times n$ matrix can be represented by three 1D arrays. The first array stores the nonzero values of the matrix, and the corresponding column indices are stored in a second array. These two arrays have a size equal to the number of

Example results and discussion

We study the multi-orbital Hubbard Hamiltonian: $H_{M} = \sum_{〈 i, j 〉 σ} t_{p d}^{i j} (d_{i σ}^{†} p_{j σ} + h.c.) + \sum_{〈 j, j^{'} 〉 σ} t_{p p}^{j j^{'}} (p_{j σ}^{†} p_{j^{'} σ} + h.c.) + U_{d} \sum_{i} n_{i ↑}^{d} n_{i ↓}^{d} + U_{p} \sum_{j} n_{j ↑}^{p} n_{j ↓}^{p} + ϵ_{d} \sum_{i σ} n_{i σ}^{d} + ϵ_{p} \sum_{j σ} n_{j σ}^{p} .$ The first two terms represent the kinetic energy using standard hopping integrals $t_{p d}^{i j}$ / $t_{p p}^{j j^{'}}$ [9]. The third and fourth terms are the on-site Coulomb interactions and the orbital site energies. Here $d_{i σ}^{†}$ creates a $Cu - 3 d_{x^{2} - y^{2}}$ hole with spin σ on site i, and $p_{j σ}^{†}$ creates an $O - 2 p_{x, y}$ hole on site j. $n_{i σ}^{d} \equiv d_{i σ}^{†} d_{i σ}$ and $n_{j σ}^{p} \equiv p_{j σ}^{†} p_{j σ}$ are the copper

Concluding remarks

Finally, we note that applying the down-folded, single-band Hubbard model to the interpretation of spectral profiles revealed by different experimental probes should be used with caution. While in optical conductivity the 1.7 eV optical gap is captured correctly by both single- and multi-band models with large enough clusters, for other probes or for spectra at higher energies the validity of the single-band model should be questioned as the high energy degrees of freedom have been projected

Acknowledgements

This work is supported by the Office of Science of the U.S. Department of Energy (DOE) under Contract Nos. DE-AC02-76SF00515 and DE-FG02-08ER4650 (CMSN). The research used resources of the National Energy Research Scientific Computing Center, supported by DOE under Contract No. DE-AC02-05CH11231.

References (14)

E. Dagotto et al.
Phys. Rev. B
(1991)
Y. Ohta et al.
Phys. Rev. Lett.
(1991)
Elbio Dagotto
Rev. Mod. Phys.
(1994)
J. Wagner et al.
Phys. Rev. B
(1991)
U.W. Pooch et al.
ACM Comput. Surveys
(1973)
NetCDF software, at...
ARPACK software, at...

There are more references available in the full text version of this article.

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Numerical studies of photon-based spectroscopies on high-Tc superconductors

Abstract

Introduction

Section snippets

Computational method

Example results and discussion

Concluding remarks

Acknowledgements

Phys. Rev. B

Phys. Rev. Lett.

Rev. Mod. Phys.

Phys. Rev. B

ACM Comput. Surveys

Numerical studies of photon-based spectroscopies on high- $T_{c}$ superconductors