"A C++ code for exact diagonalization is attached to this solution sheet, and can be compiled by following the instructions provided with the skeleton code. Numeric results are available in the openchain.dat and periodicchain.dat for an open and periodic boundary conditions, respectively. The figures below can be reproduced with the `plot.py` script."
"A C++ code for exact diagonalizationcan be found in the `heised_cpp` directory, and can be compiled by following the instructions provided with the skeleton code. Numeric results are available in the `openchain.dat` and `periodicchain.dat` for an open and periodic boundary conditions, respectively. The figures below can be reproduced with the `plot.py` script."
]
]
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{
{
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### Exact Diagonalization
### Exact Diagonalization
A C++ code for exact diagonalization is attached to this solution sheet, and can be compiled by following the instructions provided with the skeleton code. Numeric results are available in the openchain.dat and periodicchain.dat for an open and periodic boundary conditions, respectively. The figures below can be reproduced with the `plot.py` script.
A C++ code for exact diagonalizationcan be found in the `heised_cpp` directory, and can be compiled by following the instructions provided with the skeleton code. Numeric results are available in the `openchain.dat` and `periodicchain.dat` for an open and periodic boundary conditions, respectively. The figures below can be reproduced with the `plot.py` script.
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<center>Figure 1: Ground state energy converging to value of −0.44325 for the thermodynamic limit.</center>
<center>Figure 1: Ground state energy converging to value of −0.44325 for the thermodynamic limit.</center>
Figure 2: Strength of the local contribution of each bond to the Heisenberg Hamiltonian
Figure 2: Strength of the local contribution of each bond to the Heisenberg Hamiltonian
for even and odd open chains.
for even and odd open chains.
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In Figure 1 we observe the oscillation of the ground state energy between even and odd number of sites in both lattice topologies. In the case of periodic boundary conditions this is easily understood by the appearance of a frustrated spin in an odd chain, i.e. one spin can align in order to favor the bond with the previous spin, or with the next spin, but not both at the same time.
In Figure 1 we observe the oscillation of the ground state energy between even and odd number of sites in both lattice topologies. In the case of periodic boundary conditions this is easily understood by the appearance of a frustrated spin in an odd chain, i.e. one spin can align in order to favor the bond with the previous spin, or with the next spin, but not both at the same time.
For an open chain there is no apparent frustration, but the increase in energy per site can still be understood from the local contributions to the Heisenberg Hamiltonian as shown in Figure 2. The major contribution to the ground state comes from the dimerized state, i.e. a chain of singlets: this is only possible in an even chain (see left panel). In the odd chain (right panel) the system tries to build singlets from both boundaries, which is again giving rise to frustration in the center where the two incompatible dimer patterns meet. In the thermodynamic limit L → ∞ the difference between even and odd length vanishes.
For an open chain there is no apparent frustration, but the increase in energy per site can still be understood from the local contributions to the Heisenberg Hamiltonian as shown in Figure 2. The major contribution to the ground state comes from the dimerized state, i.e. a chain of singlets: this is only possible in an even chain (see left panel). In the odd chain (right panel) the system tries to build singlets from both boundaries, which is again giving rise to frustration in the center where the two incompatible dimer patterns meet. In the thermodynamic limit L → ∞ the difference between even and odd length vanishes.
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<center>Figure 3: Extrapolation of the energy gap. Extrapolation is performed only on the last
<center>Figure 3: Extrapolation of the energy gap. Extrapolation is performed only on the last
four lattice sizes.</center>
four lattice sizes.</center>
By targeting the two lowest eigenenergies in the Lanczos solver (or a similar iterative
By targeting the two lowest eigenenergies in the Lanczos solver (or a similar iterative
eigensolver) for each quantum number sector we identify the smallest excitation from the
eigensolver) for each quantum number sector we identify the smallest excitation from the
ground state. Figure 3 shows how the energy gap nicely extrapolates to zero ∼ 1/L in
ground state. Figure 3 shows how the energy gap nicely extrapolates to zero ∼ 1/L in
