Hyperbolic Deformation on Quantum Lattice Hamiltonians

Abstract

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 $\lambda \ge 0$ is a deformation parameter. In the limit $\lambda \to 0$ 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 $\lambda$.

5 pages, 4 figures

References in corpus (11)

• The density-matrix renormalization group
• Corner Transfer Matrix Renormalization Group Method
• Scaling of entanglement support for Matrix Product States
• Corner Transfer Matrix Algorithm for Classical Renormalization Group
• Numerical Renormalization Group at Criticality
• Smooth Boundary Conditions For Quantum Lattice Systems
• Geometric effects on critical behaviours of the Ising model
• Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane
• Novel scaling behavior of the Ising model on curved surfaces
• Real-space renormalization group approach for the corner Hamiltonian
• Wilson-like real-space renormalization group and low-energy effective spectrum of the XXZ chain in the critical regime

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