## Series Expansion:

Linked-cluster series expansion method is sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. It has several advantages over other methods:

- The results obtained is for infinite lattice, so there is no finite lattice corrections.
- There is no mimus sign problem that quantum Monte Carlo simulations have.
- Unlike DMRG, the calculation can be done in higher dimensions.
- It can directly compute excitation dispersion, multi-particle bound states and dynamical structure factors. Controlled and systematic numerical calculation of the multi-particle bound states and dynamical structure factors of quantum models remains a challenging computational task. Despite much progress in developing computational methods there are few methods that can accurately calculate even the single-particle spectrum. Calculations of the multi-particle continuum, bound states and their dynamical structure factors remain even more daunting, with series expansion methods clearly leading the way.
- There is very strong self-consistency check in linked-cluster series expansions, to make sure the series we get is correct, unless we are using the wrong Hamiltonian or wrong operator which is not common.

But is has its own problem, it relies on series extrapolation, and it need a suitable no degenerate initial start state to perform the expansions. To extend the series by one order, the CPU time and memory required usually increased by a factor of 6-20, depending on the model. Also the techniques are far more complicate than other methods. Basically the calculations involve following three procedures which need three different sophisticated computer codes:

- It requires generation of a list of clusters for the lattice considered. The number of clusters usually increase exponentially fast with size of lattice. For example, to compute the dispersion for square lattice Heisenberg antiferromagnetic model to order 14, one need to generate a list of 4654284 cluster, up to 15 sites, and data takes about 5 GB disk space.
- It need a very effective method to generate series for individual clusters. To compute T=0 excitation dispersions, this is a procedure to block diangonalize the Hamiltonian to get the effective Hamiltonian.
- It requires series analysis.

##### For example, one can obtained the dispersion for square lattice Heisenberg antiferromagnetic model:

where the lines are the results of first-order (dotted line), second-order (dashed line), third-order (solid line) spin-wave theory.

##### One can also obtained the 2-particle excitation spectrum of alternating Heisenberg chain:

where beside the two-particle continuum (gray shaded), there are two singlet

bound states (S1 and S2), two triplet bound states (T1 and T2) and two

quintet antibound states (Q1 and Q2). The inset enlarges the region near ka=π/2 so

we can see S2, T2 and Q2 below/above the continuum.

##### and its dynamical structure factor for 2-particle continuum:

where the bold solid red line is the dispersion relation for the triplet bound-state T1.

##### neutron scattering intensity:

##### The detail on linked-cluster series expansion technique can be found on the Book "Series Expansion Methods for Strongly Interacting Lattice Models" published by Cambridge University Press, source code for series expansions can be found in my source code page.