Coupled Ladders

class models.coupledLadders.COUPLEDLADDERS(alpha=0.0, global_args=<config.GLOBALARGS object>)[source]
Parameters

  • alpha (float) – nearest-neighbour interaction

  • global_args (GLOBALARGS) – global configuration

Build Hamiltonian of spin-1/2 coupled ladders

\[H = \sum_{i=(x,y)} h2_{i,i+\vec{x}} + \sum_{i=(x,2y)} h2_{i,i+\vec{y}} + \alpha \sum_{i=(x,2y+1)} h2_{i,i+\vec{y}}\]

on the square lattice. The spin-1/2 ladders are coupled with strength \(\alpha\):

y\x
   _:__:__:__:_
..._|__|__|__|_...
..._a__a__a__a_...
..._|__|__|__|_...
..._a__a__a__a_...   
..._|__|__|__|_...   
    :  :  :  :      (a = \alpha)

where

  • \(h2_{ij} = \mathbf{S}_i.\mathbf{S}_j\) with indices of h2 corresponding to \(s_i s_j;s'_i s'_j\)

energy_2x1_1x2(state, env)[source]
Parameters

  • state (IPEPS) – wavefunction

  • env (ENV) – CTM environment

Returns

energy per site

Return type

float

We assume iPEPS with 2x2 unit cell containing four tensors A, B, C, and D with simple PBC tiling:

A B A B
C D C D
A B A B
C D C D

Taking the reduced density matrix \(\rho_{2x1}\) (\(\rho_{1x2}\)) of 2x1 (1x2) cluster given by rdm.rdm2x1() (rdm.rdm1x2()) with indexing of sites as follows \(s_0,s_1;s'_0,s'_1\) for both types of density matrices:

rdm2x1   rdm1x2

s0--s1   s0
         |
         s1

and without assuming any symmetry on the indices of individual tensors a following set of terms has to be evaluated in order to compute energy-per-site:

   0       0       0
1--A--3 1--B--3 1--A--3
   2       2       2
   0       0       0
1--C--3 1--D--3 1--C--3
   2       2       2             A--3 1--B,      A  B  C  D
   0       0                     B--3 1--A,      2  2  2  2
1--A--3 1--B--3                  C--3 1--D,      0  0  0  0
   2       2             , terms D--3 1--C, and  C, D, A, B
eval_corrf_DD_V(coord, direction, state, env, dist, verbosity=0)[source]

Evaluates correlation functions of two vertical dimers DD_v(r)= <(S(0).S(y))(S(r*x).S(y+r*x))>

or= <(S(0).S(x))(S(r*y).S(x+r*y))>

eval_obs(state, env)[source]
Parameters

  • state (IPEPS) – wavefunction

  • env (ENV) – CTM environment

Returns

expectation values of observables, labels of observables

Return type

list[float], list[str]

Computes the following observables in order

  1. average magnetization over the unit cell,

  2. magnetization for each site in the unit cell

  3. \(\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle\) for each site in the unit cell

  4. \(\mathbf{S}_i.\mathbf{S}_j\) for all non-equivalent nearest neighbour bonds

where the on-site magnetization is defined as

\[\begin{split}\begin{align*} m &= \sqrt{ \langle S^z \rangle^2+\langle S^x \rangle^2+\langle S^y \rangle^2 } =\sqrt{\langle S^z \rangle^2+1/4(\langle S^+ \rangle+\langle S^- \rangle)^2 -1/4(\langle S^+\rangle-\langle S^-\rangle)^2} \\ &=\sqrt{\langle S^z \rangle^2 + 1/2\langle S^+ \rangle \langle S^- \rangle)} \end{align*}\end{split}\]

Usual spin components can be obtained through the following relations

\[\begin{split}\begin{align*} S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\ S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-) \end{align*}\end{split}\]