Spin Gap of a Spin-1 Heisenberg Chain

In this tutorial we will learn how to use the sparse diagonalization program (Lanczos algorithm) to calculate the energy gaps of a 1D spin-1 Heisenberg chain for various lattice sizes ($L=4, 6, 8$, and 10). The obtained finite-lattice gaps are then used to extrapolate the energy gap in the thermodynamic limit ($L=\infty$).

The Hamiltonian for the spin-1 Heisenberg chain is given by

where $J>0$ for antiferromagnetic interactions between two nearest-neighbor spins $\mathbf{S}^i$ and $\mathbf{S}^j$, and the spin-spin interaction consists of three components, i.e.,

The basis states are usually chosen to be the eigen states of $S_z$ operator. For a spin-1 system, there are three basis states for each lattice site, $|-1\rangle$, $|0\rangle$, and $|+1\rangle$. The application of $S_x$ and $S_y$ operators on these basis states can be expressed in terms of raising $S^{\dagger}$ and lowering $S^{-}$ operators:

who act on the basis states in the following way:

where $S=1$ and $s=-1, 0, +1$.

With the above basis states for each lattice site, the Hamiltonian can be written as a Hermitian matrix. The size of the matrix can be reduced when the total magnetization is fixed, i.e., setting Sz_total = 0 (singlet sector) or Sz_total = 1 (triplet sector) in the simulations.

The Heisenberg exchange Hamiltonian was introduced by W. Heisenberg, Zeitschrift für Physik 49, 619-636 (1928). For an integer-spin chain (such as $S=1$ here), F.D.M. Haldane, Physics Letters A 93, 464-468 (1983) predicted — contrary to the gapless spin-1/2 case — a finite excitation gap in the thermodynamic limit, now known as the Haldane gap.

Parameters

Lattice

The chain lattice is a 1D periodic ring of L spin-1 sites coupled by nearest-neighbour exchange $J$:

 S=1     S=1     S=1           S=1
  o---J---o---J---o--- ... ---o
  |                           |
  +----------- J -------------+
        (periodic ring, L sites)

The bond from the last site back to the first closes the ring. Periodic boundaries matter for this particular calculation: an open spin-1 chain carries effective $S=1/2$ spins localized at its two ends, and the near-degenerate edge states they produce sit inside the bulk gap, so the measured singlet/triplet splitting on an open chain is an edge excitation rather than the Haldane gap. Closing the chain into a ring removes the ends entirely, so the gap extracted below is the bulk gap that the extrapolation is after. See the ALPS lattice library for other built-in lattices, including open chain lattice.

Method Choice

For spin-1, the local Hilbert space has dimension 3, so the full Hilbert space of an $L$-site chain is $3^L$ — e.g. $3^{14}\approx 4.8\times10^6$ before the Sz_total restriction cuts it down substantially. Since only the lowest energy in each Sz_total sector is needed, the Lanczos-based sparsediag is again the appropriate method rather than a full diagonalization: all six lattice sizes used here ($L=4$ to $14$) complete in well under a minute in total.

We first import the required modules.

import pyalps
import numpy as np
import matplotlib.pyplot as plt
import pyalps.plot
import pyalps.fit_wrapper as fw

Then we prepare the input files as a list of Python dictionaries.

parms = []
for l in [4, 6, 8, 10, 12, 14]:
  for sz in [0, 1]:
      parms.append(
        { 
          'LATTICE'                   : "chain lattice", 
          'MODEL'                     : "spin",
          'local_S'                   : 1,
          'J'                         : 1,
          'L'                         : l,
          'CONSERVED_QUANTUMNUMBERS'  : 'Sz',
          'Sz_total'                  : sz
        }
      )

We write the input file and run the simulation.

input_file = pyalps.writeInputFiles('parm2a',parms)
res = pyalps.runApplication('sparsediag',input_file) #, MPI=4)

We next load the spectra for each of the systems sizes and spin sectors:

data = pyalps.loadSpectra(pyalps.getResultFiles(prefix='parm2a'))

To extract the gaps we need to write a few lines of Python, to set up a list of lengths and a Python dictionaries of the minimum energy in each (L,Sz) sector:

lengths = []
min_energies = {}

for sim in data:
  l = int(sim[0].props['L'])
  if l not in lengths: lengths.append(l)
  sz = int(sim[0].props['Sz_total'])
  all_energies = []
  for sec in sim:
    all_energies += list(sec.y)
  min_energies[(l,sz)]= np.min(all_energies)

And finally we make a plot of the gap as a function of 1/L and then show the plot

gapplot = pyalps.DataSet()
gapplot.x = 1./np.sort(lengths)
gapplot.y = [min_energies[(l,1)] -min_energies[(l,0)] for l in np.sort(lengths)]  
gapplot.props['xlabel']='$1/L$'
gapplot.props['ylabel']='Triplet gap (J)'
gapplot.props['label']='S=1'
gapplot.props['line']='.'

plt.figure()
pyalps.plot.plot(gapplot)
plt.legend()
plt.xlim(0,0.25)
plt.ylim(0,1.0)

We then fit the data in the range L=8 to L=14 to obtain the gap in the thermodynamic limit ($L\rightarrow \infty$ or $1/L\rightarrow 0$).

pars = [fw.Parameter(0.411), fw.Parameter(1000), fw.Parameter(1)]
f = lambda self, x, p: p[0]()+p[1]()*np.exp(-x/p[2]())
fw.fit(None, f, pars, np.array(gapplot.y)[2:], np.sort(lengths)[2:])

x = np.linspace(0.0001, 1./min(lengths), 100)
plt.plot(x, f(None, 1/x, pars))

plt.show()

The result of the simulation is shown in the figure:

Results

Running the code above gives the following finite-size triplet gaps and extrapolated value:

Fitting $L=8$ to $14$ to $\Delta(L) = \Delta_\infty + A e^{-L/\xi}$ extrapolates to $\Delta_\infty/J \approx 0.4218$, close to the numerically-known Haldane gap value $\Delta/J \approx 0.4105$ (the ~3% deviation is finite-size fitting error, since only $L\le14$ is used here).

Summary and Outlook

Exact diagonalization of finite spin-1 Heisenberg chains, extrapolated to $L\rightarrow\infty$, confirms the finite Haldane gap predicted for integer-spin antiferromagnetic chains — in sharp contrast to the gapless spin-1/2 chain.

1. How does the extrapolated gap change if you include larger $L$ in the fit, or use only the largest three sizes?
2. What functional form would you expect for the gap in a spin-1/2 chain, where no gap is present?
3. How sensitive is the extrapolated $\Delta_\infty$ to the choice of fit range?