# Computational Methods

## Quantum Monte Carlo

A plot of the value of the displacement field as a function of Monte Carlo step for a simulation of the Hubbard-Holstein model as a function of electron-phonon coupling λ. Here one sees the initial, rapid, warm-up followed by the fluctuations of the displacement around some central value as a function of Monte Carlo step. Using determinant quantum Monte Carlo (QMC or DQMC) and maximum entropy (MEM) analytic continuation one may determine the single-particle spectral function and multi-particle response functions, like the charge and spin susceptibilities and optical conductivity, as a function of real frequency. From the QMC process, one also can determine the single-particle filling, static (thermal) charge and spin susceptibilities, and in principle correlation lengths and even transition temperatures when used as impurity solvers in DMFT and its cluster extensions and through finite-size scaling.

**References and Further Reading**

- R. Blankenbecler, D. J. Scalapino and R. L. Sugar,
*Phys. Rev. D*,**24**, 2278-2286 (1981). - S. R. White
*et al*,*Phys. Rev. B***40**, 506-516 (1989). - E. Gull
*et al*,*Rev. Mod. Phys.***83**, 349-404 (2011).

Richard Scalettar's How To Guide

Richard Scalettar's Homepage

Useful notes from Richard Scalettar's Michigan Quantum Summer School 2010

## Density-Matrix Renormalization Group

Momentum resolved spectrum of a *t−J* chain of *L* = 64 sites, with *J/t* = 0.5 and density *n* = 0.75, calculated with time-dependent DMRG.
Figure from A. E. Feiguin, "The Time-Dependent Density Matrix Renormalization Group", in *Strongly Correlated Systems*, A. Avella and F. Mancini eds., Springer Series in Solid-State Sciences
**176** (Springer, Berlin, Heidelberg, 2013). The density-matrix renormalization group (DMRG) method is one of the most powerful numerical techniques for studying quantum many-body systems in
condensed matter physics. The time-dependent DMRG is a remarkable and reliable approach to simulate real-time dynamics of strongly correlated systems. It can be used to calculate spectral functions, and
to study systems far from equilibrium, and provides valuable information about transport and thermodynamic properties which can be directly compared with real experiments, such as RIXS.

**References and Further Reading**

- S. R. White,
*Phys. Rev. Lett.***69**, 2863 (1992). - U. Schollwöck,
*Rev. Mod. Phys.***77**, 259 (2005).

## Exact Diagonalization

A schematic representation of a Hamiltonian in matrix form. The system in question has been represented by a Hilbert space with 1028 elements. The blue dots represent non-zero entries in the Hamiltonian matrix with the remaining elements (white space) equal to zero. The sparsity of this representation makes matrix methods like Lancozs or implicitly restarted Arnoldi and packages like ARPACK and PARPACK ideal candidates for finding the eigenvalues and eigenvectors of this Hamiltonian. Krylov subspace time-evolution may be employed to efficiently determine time-evolved states for out-of-equilibrium, time-domain problems with either a quantum quench of Hamiltonian parameters, or time-translation symmetry breaking from interaction with time-dependent applied fields.

**References and Further Reading**

- R. B. Lehoucq, D. C. Sorensen, and C. Yang,
*ARPAC Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods*, SIAM: Software, Environments, and Tools**6**(1998). - C. Moler and C. van Loan,
*Ninteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later*, SIAM Review**45**, 3-49 (2003). - S. R. Manmana, A. Muramatsu, R. M. Noack,
*AIP Conf. Proc.***789**, 269-278 (2005). - M. Balzer, N. Gdaniec, and M. Potthoff,
*J. Phys.: Condens. Matter***24**, 035603 (2012).

## Density Functional Theory

Determination of the total density-of-states (DOS) for TbTe_{3} from the *ab initio*
(DFT/LAPW+LO) program WIEN2k. TbTe_{3} consists of a pair of staggered, parallel, flat Te
sheets separated by a pair of staggered, corrugated Tb-Te layers. In principle these *ab initio*
methods can be used to determine the crystal structure, orbitally projected DOS and bandstructure,
and phonon spectrum in a materials specific fashion. The results can be compared to STM, ARPES, optical,
and various x-ray scattering experiments.

**References and Further Reading**

## Nonequilibrium Keldysh

Schematic of the Keldysh contour used in our formalism. Time runs from some t_{min}, out to a
t_{max} and back again, then down the imaginary time spur to account for finite
temperature effects on the system in equilibrium. The red and blue curves represent the application of
a transient pump pulse at some time after t_{min} and a subsequent probe pulse, like that used in
time-resolved optical spectroscopy or photoemission.

**References and Further Reading**

- L. P. Kadanoff and G. Baym,
*Quantum Statistical Mechanics*(Benjamin, New York, 1962). - J. Rammer and H. Smith,
*Rev. Mod. Phys.***58**, 323-359 (1986). - J. K. Freericks, V. M. Turkowski, and V. Zlatic,
*Phys. Rev. Lett.***97**, 266408 (2006). - J. K. Freericks,
*Phys. Rev. B***77**, 075109 (2008). - J. K. Freericks, H. R. Krishnamurthy, and Th. Pruschke,
*Phys. Rev. Lett.***102**, 136401 (2009). - A. P. Jauho and J. W. Wilkins,
*Phys. Rev. B***29**, 1919-1938 (1984).

## Dynamical Mean-Field Theory

Cartoon depicting the self-consistency scheme of dynamical mean-field theory (DMFT). In the DMFT scheme one embeds a single-site impurity with full interaction in a non-interacting bath that contains information from the remaining lattice degrees of freedom. Unlike classical Weiss mean-field theory, the effective mean-field function retains dynamical or temporal information about quantum fluctations of degrees of freedom between the impurity site and the bath. DMFT can be extended by considering the embedding of a finite-size cluster in an effective medium in which case the mean-field function depends upon spatial or momentum-space degrees of freedom in addition to the temporal or frequency dependence.

**References and Further Reading**

- A. Georges
*et al*,*Rev. Mod. Phys.***68**, 13-125 (1996). - Th. A. Maier
*et al*,*Rev. Mod. Phys.***77**, 1027-1080 (2005).

Strongly Correlated Materials: Insights from Dynamical Mean-Field Theory

Lecture notes on the LDA+DMFT approach to strongly correlated materials

Mark Jarrell's Webpage