Self-Similar Qubistic Plot of the Half-Filled Hubbard Model Basis States

This Demonstration provides a "qubistic" representation of basis states in many-body wavefunctions of quantum-mechanical Hamiltonians, such as the Hubbard model for strongly correlated electron systems.
Specifically, basis states of the half-filled Hubbard model with quantum number are considered. Each -particle state is described by the electron occupation of the lattice sites, which can be occupied by a spin-up electron () or a spin-down electron (), doubly-occupied with antiparallel spins (), or empty (). For the half-filled Hubbard model, each state has on average one electron per site and an equal number of spin-up and spin-down electrons over the sites of the system lattice, hence . Accordingly, the tensor basis is composed of four states , , , and , which can in turn be associated to four level-1 squares according to the mapping: upper-left, upper-right, lower-left, lower-right. As shown in [1], the procedure can be iterated by splitting the squares to obtain level-2 qubit representation, and so on up to the desired level- qubits.
In general, for a given qubit level , the tensor product gives states, but states with electrons and quantum number are considerably fewer, namely . These states are colored in the Demonstration qubistic plot, whereas states that do not have are represented by white squares and are associated with zero. For instance, level-2 colored squares map to four states, with an average of one electron on each of the two sites: , , , and .
This 2D mapping displays a self-similar pattern, which shows an increasing resolution and complexity for higher . You can verify this property by selecting the number of qubits and the index related to the available sublevels mesh grids in the array plot. Moreover, by moving the mouse over the array plot you can inspect the occupation patterns of the related states and verify the presence of symmetric occupation patterns at the corners of the fractal features in the plot. Since the states are selected from the qubits, which is already huge for small , this Demonstration is limited to level-6 qubits.
Similar visualization techniques have been applied to get a quantitative characterization of DNA and protein sequences [1].

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See [1] for a discussion of qubism, which means self-similar visualization of many-body wavefunctions starting from qubits, the basic quantum information units.
References
[1] J. Rodriguez-Laguna, P. Migdał, M. Ibáñez Berganza, M. Lewenstein, and G. Sierra, "Qubism: Self-Similar Visualization of Many-Body Wavefunctions," New Journal of Physics, 14, 2012 p. 053028. doi:10.1088/1367-2630/14/5/053028.
[2] "Qubism: Plotting Quantum States." (Oct 4, 2012) qubism.wikidot.com.
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