We here introduce some pieces of our paper on numerical methods.

[Hotta-Shibata PRB

It seems quite usual to consider that the nature of the system is determined solely by the Hamiltonian, and modifying the Hamiltonian means changing the state itself. However, it was once pointed out by Nishino (at around 2010) that if we consider a warped space consisting of real-imaginary time axes, and trying to make a imaginary time evolution without changing the profile of the wave function, one needs to modify the Hamiltonian. This idea has developed since than and nowadays accepted as a protocol called "sine square deformation" (SSD). The grand canonical approach made use of this protocol to get rid of the finite size effect in the quantum many body calculation.

The key ingredient of this analysis is to deform the Hamiltonian in real space. For the uniform (original) Hamiltonian with periodic boundary condition (PBC) the expectation values of the physical quantities of the ground state also remains translationally invariant. In isolated quantum systems, there are several conserved quantities such as particle number N

Once we place the SSD to the Hamiltonian, the translational symmetry is broken. This SSD function takes the form of a single wavelength, with maximum at the center of the system and zero at both edges. The energy scale of the Hamiltonian is modified to scale smoothly down from order-1 at the center toward 0 at the edges.

In the figure we show the case of free fermion in one dimension. For the periodic boundary the particle density is uniform and shifts according to the value of N

In usual DMRG calculations, we always examine several different sizes, N=50,100, 150, ... and perform the size scaling against 1/N to obtain the value at N=∞ . However, in this analysis we no longer need a size scaling.

Another advantage is that, one can deal with the incommensurate state with long periodicity, such as spiral state. In usual open or periodic boundaries one needs to adopt N to be multiples of the periodicity m of the state we are focusing. If m varies with system parameters, the calculation becomes a mess. We showed in our first paper that for the J

Q1:Can we adopt another functional form → NO. SSD is a magic function.

Q2:Which is better for 2D, Cartetian or polar?→ polar is better.

Placing SSD to the Hamiltonian means adding a scattering term. (Maruyama-Katsura-Hikihara2011). As a simplest example, let us consider a free fermionic cosine energy band. The energy eigen state at each k-point of the band is a plane wave. The scattering takes place between the two adjacent discrete energy levels in k-space. This is because the scattering wave number reflects the periodicity of the sine square function, charagerized by the single wavelength 2π/N, corresponding to the discreteness of k-space. As is well known, choosing a bunch of plane waves within the certain range of wave number, and taking linear combination, one can form a wave packet. The wave packet formed at the edge of the system has nearly zero energy, namely in the vicinity of μ. Whereas, those at the system center is formed at the bottom of the band. The edge states are densely distributed near μ (have a large density of states);. By using these edge states as baths, one can finely tune the number of particles at the system center.

Going back to Q1, if we use the functions other than SSD, the way the plane waves are mixed up becomes much more complicated, and the resultant distribution of states is not as ideal as the one we have for SSD.

Let us briefly mention about the size effect.

(a) artificial discreteness of the wave number and the physical quantities accordingly,

(b) boundary effect

(c) cluster shapes/ aspect ratio

Here, (a) is the simplest and well known. The size scaling analysis often works well for 1D and 2D. However, there are some cases where the correlation develops in space, and when using DMRG with open boundary, the effect of Friedel oscillations cannot be easily get rid of. In the paper, [Shibata-Hotta, PRB 84 115116 (2011)] we have studied such difficult case and demonstrated how one can cope with such problem and examine very carefully the numerical results.