Exact diagonalization

After determining the eigenstates $|n\backslash rangle$ and energies $\backslash epsilon\_n$ of a given Hamiltonian, exact diagonalization can be used to obtain expectation values of observables. For example, if $\backslash mathcal\{O\}$ is an observable, its thermal expectation value is

:$\backslash langle\; \backslash mathcal\{O\}\backslash rangle\; =\; \backslash frac\{1\}\{Z\}\; \backslash sum\_n\; e^\{-\backslash beta\; \backslash epsilon\_n\}\; \backslash langle\; n\; |\; \backslash mathcal\{O\}\; |\; n\; \backslash rangle,$

where $Z\; =\; \backslash sum\_n\; e^\{-\backslash beta\; \backslash epsilon\_n\}$ is the partition function. If the observable can be written down in the initial basis for the problem, then this sum can be evaluated after transforming to the basis of eigenstates.

Green's functions may be evaluated similarly. For example, the retarded Green's function $G^R(t)\; =\; -i\; \backslash theta(t)\; \backslash langle\; [A(t),\; B(0)]\; \backslash rangle$ can be written

:$G^R(t)\; =\; -\backslash frac\{i\; \backslash theta(t)\}\{Z\}\; \backslash sum\_\{n,m\}\; \backslash left(e^\{-\backslash beta\; \backslash epsilon\_n\}\; -\; e^\{-\backslash beta\; \backslash epsilon\_m\}\; \backslash right)\; \backslash langle\; n\; |\; A(0)\; |\; m\; \backslash rangle\; \backslash langle\; m\; |\; B(0)\; |\; n\; \backslash rangle\; e^\{-i(\backslash epsilon\_m\; -\; \backslash epsilon\_n)t/\backslash hbar\}.$

Exact diagonalization can also be used to determine the time evolution of a system after a quench. Suppose the system has been prepared in an initial state $|\; \backslash psi\; \backslash rangle$, and then for time $t>0$ evolves under a new Hamiltonian, $\backslash mathcal\{H\}$. The state at time $t$ is

:$|\; \backslash psi(t)\; \backslash rangle\; =\; \backslash sum\_n\; e^\{-i\backslash epsilon\_n\; t/\backslash hbar\}\; \backslash langle\; n\; |\; \backslash psi(0)\; \backslash rangle\; |\; n\; \backslash rangle.$

The dimension of the Hilbert space describing a quantum system scales exponentially with system size. For example, consider a system of $N$ spins localized on fixed lattice sites. The dimension of the on-site basis is 2, because the state of each spin can be described as a superposition of spin-up and spin-down, denoted $\backslash left|\backslash uparrow\; \backslash right\backslash rangle$ and $\backslash left|\backslash downarrow\; \backslash right\backslash rangle$. The full system has dimension $2^N$, and the Hamiltonian represented as a matrix has size $2^N\; \backslash times\; 2^N$. This implies that computation time and memory requirements scale very unfavorably in exact diagonalization. In practice, the memory requirements can be reduced by taking advantage of symmetry of the problem, imposing conservation laws, working with sparse matrices, or using other techniques.

Exact diagonalization is useful for extracting exact information about finite systems. However, often small systems are studied to gain insight into infinite lattice systems. If the diagonalized system is too small, its properties will not reflect the properties of the system in the thermodynamic limit, and the simulation is said to suffer from finite size effects.

Unlike some other exact theory techniques, such as Auxiliary-field Monte Carlo, exact diagonalization obtains Green's functions directly in real time, as opposed to imaginary time. Unlike in these other techniques, exact diagonalization results do not need to be numerically analytically continued. This is an advantage, because numerical analytic continuation is an ill-posed and difficult optimization problem.{{cite journal |last1=Bergeron |first1=Dominic |last2=Tremblay |first2=A.-M. S. |title=Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation |journal=Physical Review E |date=5 August 2016 |volume=94 |issue=2 |page=023303 |doi=10.1103/PhysRevE.94.023303|pmid=27627408 |arxiv=1507.01012 |bibcode=2016PhRvE..94b3303B |s2cid=13294476 }}

• Can be used as an impurity solver for Dynamical mean-field theory techniques.{{cite journal |last1=Medvedeva |first1=Darya |last2=Iskakov |first2=Sergei |last3=Krien |first3=Friedrich |last4=Mazurenko |first4=Vladimir V. |last5=Lichtenstein |first5=Alexander I. |title=Exact diagonalization solver for extended dynamical mean-field theory |journal=Physical Review B |date=29 December 2017 |volume=96 |issue=23 |page=235149 |doi=10.1103/PhysRevB.96.235149|arxiv=1709.09176|bibcode=2017PhRvB..96w5149M |s2cid=119347649 }}

• When combined with finite size scaling, estimating the ground state energy and critical exponents of the 1D transverse-field Ising model.{{cite journal |last1=Hamer |first1=C. J. |last2=Barber |first2=M. N. |title=Finite-lattice methods in quantum Hamiltonian field theory. I. The Ising model |journal=Journal of Physics A: Mathematical and General |date=1 January 1981 |volume=14 |issue=1 |pages=241–257 |doi=10.1088/0305-4470/14/1/024|bibcode=1981JPhA...14..241H }}

• Studying various properties of the 2D Heisenberg model in a magnetic field, including antiferromagnetism and spin-wave velocity.{{cite journal |last1=Lüscher |first1=Andreas |last2=Läuchli |first2=Andreas M. |title=Exact diagonalization study of the antiferromagnetic spin-1/2 Heisenberg model on the square lattice in a magnetic field |journal=Physical Review B |date=5 May 2009 |volume=79 |issue=19 |page=195102 |doi=10.1103/PhysRevB.79.195102|arxiv=0812.3420|bibcode=2009PhRvB..79s5102L |s2cid=117436360 }}

• Studying the Drude weight of the 2D Hubbard model.{{cite journal |last1=Nakano |first1=Hiroki |last2=Takahashi |first2=Yoshinori |last3=Imada |first3=Masatoshi |title=Drude Weight of the Two-Dimensional Hubbard Model –Reexamination of Finite-Size Effect in Exact Diagonalization Study– |journal=Journal of the Physical Society of Japan |date=15 March 2007 |volume=76 |issue=3 |pages=034705 |doi=10.1143/JPSJ.76.034705|arxiv=cond-mat/0701735|bibcode=2007JPSJ...76c4705N |s2cid=118346915 }}

• Studying out-of-time-order correlations (OTOCs) and scrambling in the SYK model.{{cite journal |last1=Fu |first1=Wenbo |last2=Sachdev |first2=Subir |title=Numerical study of fermion and boson models with infinite-range random interactions |journal=Physical Review B |date=15 July 2016 |volume=94 |issue=3 |page=035135 |doi=10.1103/PhysRevB.94.035135|arxiv=1603.05246 |bibcode=2016PhRvB..94c5135F |s2cid=7332664 }}

• Simulating resonant x-ray spectra of strongly correlated materials.{{cite encyclopedia | last1 = Wang

| first1 = Y. | last2 = Fabbris| first2 = G.| last3 = Dean| first3 = M.P.M | last4 = Kotliar | first4 = G. | title = EDRIXS: An open source toolkit for simulating spectra of resonant inelastic x-ray scattering | year = 2019 | pages = 151–165 | journal = Computer Physics Communications | volume = 243 | doi = 10.1016/j.cpc.2019.04.018| arxiv = 1812.05735 | bibcode = 2019CoPhC.243..151W | s2cid = 118949898 }}

Numerous software packages implementing exact diagonalization of quantum Hamiltonians exist. These include [http://alps.comp-phys.org/mediawiki/index.php ALPS]{{Dead link|date=March 2024 |bot=InternetArchiveBot |fix-attempted=yes }}, [https://www.swmath.org/software/8906 DoQo], [https://ma.issp.u-tokyo.ac.jp/en/app/1675 EdLib], [https://github.com/NSLS-II/edrixs edrixs], [https://www.quanty.org Quanty] and many others.

Exact diagonalization results from many small clusters can be combined to obtain more accurate information about systems in the thermodynamic limit using the numerical linked cluster expansion.{{cite journal |last1=Tang |first1=Baoming |last2=Khatami |first2=Ehsan |last3=Rigol |first3=Marcos |title=A short introduction to numerical linked-cluster expansions |journal=Computer Physics Communications |date=March 2013 |volume=184 |issue=3 |pages=557–564 |doi=10.1016/j.cpc.2012.10.008|arxiv=1207.3366|bibcode=2013CoPhC.184..557T |s2cid=11638727 }}

• Lanczos algorithm

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• [https://en.wikiversity.org/wiki/Quantum_Simulation/Exact_diagonalization Quantum Simulation/Exact diagonalization]
• [http://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:ED-06_FullDiagonalization ALPS full diagonalization tutorial] {{Webarchive|url=https://web.archive.org/web/20190723021548/http://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:ED-06_FullDiagonalization |date=2019-07-23 }}
• [http://www.cond-mat.de/events/correl19/manuscripts/koch.pdf Exact Diagonalization and Lanczos Method] in E. Pavarini, E. Koch and S. Zhang (eds.): Many-Body Methods for Real Materials, Jülich 2019, ISBN 978-3-95806-400-3

Category:Correlated electrons

Category:Materials science

Category:Condensed matter physics

Category:Quantum mechanics