[Paper Review] Transverse Field Ising Model Under Hyperbolic Deformation

Hiroshi Ueda, Andrej Gendiar+3 more|arXiv (Cornell University)|Aug 20, 2010

Theoretical and Computational Physics3 citations

TL;DR

This paper investigates the one-dimensional transverse field Ising model under hyperbolic deformation, where bond couplings scale as cosh(jλ), introducing a position-dependent Hamiltonian. Despite non-uniform couplings, the ground state remains nearly uniform and finitely correlated. The model exhibits a first-order quantum phase transition at Γ=J, with discontinuities in magnetization and entanglement entropy that confirm Ising universality, contrasting with the mean-field second-order transition in classical hyperbolic Ising models.

ABSTRACT

Ground state of the one-dimensional transverse field Ising model is investigated under the hyperbolic deformation, where the energy scale of j-th bond is proportional to the function \cosh ( j λ) that contains a parameter λ. Although the Hamiltonian is position dependent, the ground state is nearly uniform and finitely correlated. We observe the energy cross over between the ordered and disordered state with respect to the transverse field. The model shows first order phase transition, and the discontinuities in the magnetization and entanglement entropy at the transition point detect the Ising universality.

Motivation & Objective

• To investigate whether non-uniform Hamiltonians, specifically under hyperbolic deformation, can support uniform ground states.
• To determine the nature of the quantum phase transition in the deformed transverse field Ising model.
• To examine the role of hyperbolic geometry in altering phase transition order compared to classical counterparts.
• To analyze entanglement entropy and magnetization as probes of universality class in the deformed system.

Proposed method

• The hyperbolic deformation is implemented via site-dependent couplings: J_j = J·cosh(jλ) for spin interactions and Γ_j = Γ·cosh((j−½)λ) for transverse fields.
• The density matrix renormalization group (DMRG) method is used to compute ground states for finite-size systems with both ferromagnetic and paramagnetic boundary conditions.
• The system is studied on a finite chain from j = −L/2+1 to L/2, with the center between j=0 and j=1 to preserve symmetry.
• Spontaneous magnetization and bipartite entanglement entropy are computed as functions of the transverse field Γ and deformation parameter λ.
• A duality transformation is applied to the Hamiltonian to explore self-duality and phase transition symmetry, especially at J=Γ.
• The entanglement entropy is fitted to (c/6)log(ξ), with ξ ∝ 1/λ, to extract central charge c=1/2 for the critical regime.

Experimental results

Research questions

• RQ1 Does a non-uniform Hamiltonian with hyperbolic coupling deformation still support a uniform ground state?
• RQ2 What is the nature of the quantum phase transition in the hyperbolic-deformed transverse field Ising model?
• RQ3 How does the hyperbolic deformation affect the universality class of the phase transition compared to the uniform case?
• RQ4 What is the behavior of entanglement entropy in the ordered and disordered phases under hyperbolic deformation?
• RQ5 Does the duality symmetry persist under hyperbolic deformation, and how does it influence the phase transition point?

Key findings

• The ground state of the hyperbolic-deformed transverse field Ising model remains nearly uniform and finitely correlated despite position-dependent couplings.
• The model exhibits a first-order quantum phase transition at Γ=J, marked by a discontinuous jump in spontaneous magnetization.
• The entanglement entropy shows a discontinuity at the transition point, consistent with the Ising universality class.
• The entanglement entropy scales as (c/6)log(ξ) with c=1/2 and ξ∝1/λ, confirming critical behavior in the thermodynamic limit.
• The difference in entanglement entropy between ordered and disordered phases approaches approximately 0.3426 in the small λ limit, roughly half of log 2 ≈ 0.6931.
• The duality transformation reveals self-duality at J=Γ even for λ>0, suggesting robust symmetry protection of the critical point.