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arXiv:1104.1721 · Journal of Physics A Mathematical and Theoretical

Exact ground state of the sine-square deformed XY spin chain

Hosho Katsura

Published May 19, 2011• 48 citations
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Abstract

We study the sine-square deformed quantum XY chain with open boundary conditions in which the interaction strength at the position x in the chain of length L is proportional to the function . The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan–Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino (2011 Phys. Rev. B 83 060414) based on numerical evidence.

DOI
10.1088/1751-8113/44/25/252001

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We study the sine-square deformed quantum XY chain with open boundary conditions in which the interaction strength at the position x in the chain of length L is proportional to the function . The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan–Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino (2011 Phys. Rev. B 83 060414) based on numerical evidence.
We study solvable spin chains, one-dimensional massless Dirac fermions and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function f ( x ) = sin 2 (π x /ℓ), where x is the position and ℓ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac–Moody) algebras, including c = 1 Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.
We study free fermion systems with the sine-square deformation (SSD) in which the energy scale of local Hamiltonians is modified according to the scaling function $f(x)={\mathrm{sin}}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where xxx is the position of the local Hamiltonian and LLL is the length of the system in the xxx direction. It has been revealed that when applied to one-dimensional (1D) critical systems, the SSD realizes the translationally-invariant ground state which is the same as that of the uniform periodic system. In this paper, we propose a simple theory to explain how the SSD maintains the translational invariance in the ground-state wave function. In particular, for a certain 1D system with SSD, it is shown that the ground state is exactly identical to the Fermi sea of the uniform periodic chain. We also apply the SSD to 2D systems and show that the SSD is able to suppress the boundary modulations from the open edges extremely well, demonstrating that the SSD works in any dimension and in any direction.
We investigate the sine-square deformation (SSD) of free fermions in one-dimensional continuous space. On the basis of supersymmetric quantum mechanics, we prove the correspondence between the many-body ground state of the system with SSD and that of the uniform system with periodic boundary conditions. We also discuss the connection between the SSD in the continuous space and its lattice version, where the geometric correction due to the real-space deformation plays an important role in relating the eigenstates of the lattice SSD with those of the continuous SSD.
We consider the non-Hermitian XY spin chain with open boundary conditions when the anisotropy parameter is extended to complex values. By analyzing the quasi-Hamiltonian matrix, we demonstrate that the free-fermion structure of the quasi-energy spectrum coincides with that of the Hermitian model and construct the corresponding biorthogonal fermionic basis away from exceptional points (EPs). We make use of an explicit Chebyshev-polynomial representation of the open-boundary eigenvectors in which the quasi-energy ε\varepsilonε is the natural spectral variable. This quasi-energy polynomial form is particularly useful at EPs, because EPs correspond to repeated roots of the same boundary polynomial, making the construction of generalized eigenvectors by ε\varepsilonε-differentiation transparent. At EPs, where the quasi-Hamiltonian becomes defective, we derive the Jordan normal form and construct the associated generalized eigenvectors, which yields the correct counting of independent many-body eigenstates. We further show that EPs act as branch points in the complex anisotropy plane, leading to the characteristic permutation of eigenenergies and eigenstates upon encirclement. The branch-cut structure of the biorthogonal eigenstates provides direct evidence for the exchange of eigenstates when an EP is encircled. These results provide an analytically controlled many-body platform for studying EP physics and non-Hermitian topology beyond momentum-space descriptions.
In this paper we analyze the ground-state phase diagram of a class of fermionic Hamiltonians by looking at the fidelity of ground states corresponding to slightly different Hamiltonian parameters. The Hamiltonians under investigation can be considered as the variable range generalization of the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the XYXYXY spin chain in a transverse magnetic field. Under periodic boundary conditions, the matrices of the problem become circulant and the models are exactly solvable. Their free-ends counterparts are instead analyzed numerically. In particular, we focus on the long-range model corresponding to a fully connected directed graph, providing asymptotic results in the thermodynamic limit, as well as the finite-size scaling analysis of the second-order quantum phase transitions of the system. A strict relation between fidelity and single particle spectrum is demonstrated, and a peculiar gapful transition due to the long-range nature of the coupling is found. A comparison between fidelity and another transition marker borrowed from quantum information, i.e., single site entanglement, is also considered.
The Jordan–Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of N spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the N − 1 bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan–Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state, but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model ( σ x σ y − σ x σ y ) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.