The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

This is a review of ground-state features of the s=1/2 Heisenberg antiferromagnet on two-dimensional lattices. A central issue is the interplay of lattice topology (e.g. coordination number, non-equivalent nearest-neighbor bonds, geometric frustration) and quantum fluctuations and their impact on possible long-range order. This article presents a unified summary of all 11 two-dimensional uniform Archimedean lattices which include e.g. the square, triangular and kagome lattice. We find that the ground state of the spin-1/2 Heisenberg antiferromagnet is likely to be semi-classically ordered in most cases. However, the interplay of geometric frustration and quantum fluctuations gives rise to a quantum paramagnetic ground state without semi-classical long-range order on two lattices which are precisely those among the 11 uniform Archimedean lattices with a highly degenerate ground state in the classical limit. The first one is the famous kagome lattice where many low-lying singlet excitations are known to arise in the spin gap. The second lattice is called star lattice and has a clear gap to all excitations. 
Modification of certain bonds leads to quantum phase transitions which are also discussed briefly. Furthermore, we discuss the magnetization process of the Heisenberg antiferromagnet on the 11 Archimedean lattices, focusing on anomalies like plateaus and a magnetization jump just below the saturation field. As an illustration we discuss the two-dimensional Shastry-Sutherland model which is used to describe SrCu2(BO3)2.

https://arxiv.org/pdf/cond-mat/0412662.pdf

Quantum magnetism in two dimensions: From semi-classical N\'eel order to magnetic disorder

We study the low-temperature thermodynamic properties 
of a number of frustrated quantum antiferromagnets 
which support localized magnon states
in the vicinity of the saturation field.
For this purpose we use 
(1) a mapping of the low-energy degrees of freedom of spin systems 
onto the hard-core object lattice gases
and 
(2) an exact diagonalization of finite spin systems 
of up to N=30 sites.
The considered spin systems exhibit universal behavior
which is determined by a specific hard-core object lattice gas
representing the independent localized magnon states.
We test the lattice gas description 
by comparing its predictions 
with the numerical results 
for low-lying energy states of finite spin systems. 
For all frustrated spin systems considered we find a strong variation 
of the low-temperature specific heat passing the saturation field and a
maximum in the isothermal entropy at saturation field resulting in an
enhanced magnetocaloric effect.

Universal low-temperature behavior of frustrated quantum antiferromagnets in the vicinity of the saturation field

At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number $N$ of spins and hence they have a finite entropy also in the thermodynamic limit $N\to \infty$ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagom\'{e}, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.

https://arxiv.org/pdf/cond-mat/0604649.pdf

Linear independence of localized magnon states

The purpose of the present paper is two-fold. On the one hand, we review some recent studies on the low-temperature strong-field thermodynamic properties of frustrated quantum spin antiferromagnets which admit the so-called localized-magnon eigenstates. One the other hand, we provide some complementary new results. We focus on the linear independence of the localized-magnon states, the estimation of their degeneracy with the help of auxiliary classical lattice-gas models and the analysis of the contribution of these states to thermodynamics.

/pdf/universal-properties-of-highly-frustrated-quantum-magnets-in-kycgqx9ui2.pdf

Universal properties of highly frustrated quantum magnets in strong magnetic fields

Ground states and excitations of a one-dimensional kagomé-like antiferromagnet

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Ground states and excitations of a one-dimensional kagomé-like antiferromagnet