Hyperbolic Deformation on Quantum Lattice Hamiltonians

A group of non-uniform quantum lattice Hamiltonians in one dimension is introduced, which is related to the hyperbolic (1 + 1)-dimensional space. The Hamiltonians contain only nearest neighbor interactions whose strength is proportional to cosh j lambda. where j is the lattice index and where lambda >= 0 is a deformation parameter. In the limit lambda -> 0 the Hamiltonians become uniform. Spacial translation of the deformed Hamiltonians is induced by the corner Hamiltonians. As a simple example, we investigate the ground state of the deformed S = 1/2 Heisenberg spin chain by use of the density matrix renormalization group (DMRG) method. It is shown that the ground state is dimerized when lambda is finite. Spin correlation function show exponential decay, and the boundary effect decreases with increasing lambda.