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arXiv:1105.3012 · Journal of Physics A Mathematical and Theoretical

Ground-state fidelity and Kosterlitz–Thouless phase transition for the spin-1/2 Heisenberg chain with next-to-the-nearest-neighbor interaction

Hong-Lei Wang, Ai-Min Chen, Bo Li, Huan-Qiang Zhou

Published Dec 2, 2011• 9 citations
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Abstract

The Kosterlitz–Thouless transition for the spin-1/2 Heisenberg chain with the next-to-the-nearest-neighbor interaction is investigated in the context of an infinite matrix product state algorithm, which is a generalization of the infinite time-evolving block decimation algorithm (Vidal 2007 Phys. Rev. Lett. 98 070201 ) to accommodate both the next-to-the-nearest-neighbor interaction and spontaneous dimerization. It is found that, in the critical regime, the algorithm automatically leads to infinite degenerate ground-state wavefunctions, due to the finiteness of the truncation dimension. This results in pseudo -symmetry spontaneous breakdown, which is reflected as a catastrophe point in the ground-state fidelity per lattice site. In addition, this allows the introduction of a pseudo-order parameter to characterize the Kosterlitz–Thouless transition. Our work demonstrates that the ground-state fidelity per lattice site is able to capture the Kosterlitz–Thouless transition, which is in sharp contrast to the fidelity susceptibility that fails to detect it.

DOI
10.1088/1751-8113/45/1/015306

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The Kosterlitz–Thouless transition for the spin-1/2 Heisenberg chain with the next-to-the-nearest-neighbor interaction is investigated in the context of an infinite matrix product state algorithm, which is a generalization of the infinite time-evolving block decimation algorithm (Vidal 2007 Phys. Rev. Lett. 98 070201 ) to accommodate both the next-to-the-nearest-neighbor interaction and spontaneous dimerization. It is found that, in the critical regime, the algorithm automatically leads to infinite degenerate ground-state wavefunctions, due to the finiteness of the truncation dimension. This results in pseudo -symmetry spontaneous breakdown, which is reflected as a catastrophe point in the ground-state fidelity per lattice site. In addition, this allows the introduction of a pseudo-order parameter to characterize the Kosterlitz–Thouless transition. Our work demonstrates that the ground-state fidelity per lattice site is able to capture the Kosterlitz–Thouless transition, which is in sharp contrast to the fidelity susceptibility that fails to detect it.
The Kosterlitz–Thouless transition is studied from the representation of the systems’s ground state wavefunctions in terms of matrix product states for a quantum system on an infinite-size lattice in one spatial dimension. It is found that, in the critical regime for a one-dimensional quantum lattice system with continuous symmetry, the newly developed infinite matrix product state algorithm automatically leads to infinite degenerate ground states, due to the finiteness of the truncation dimension. This results in pseudo continuous symmetry spontaneous breakdown, which allows the introduction of a pseudo-order parameter that must be scaled down to zero, in order to be consistent with the Mermin–Wegner theorem. We also show that the ground state fidelity per lattice site exhibits a catastrophe point , thus resolving a controversy regarding whether or not the ground state fidelity is able to detect the Kosterlitz–Thouless transition.
In this article, we study the high order term of the fidelity of the Heisenberg chain with next-nearest-neighbor interaction and analyze its connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless type happened in the system. We calculate the fidelity susceptibility of the system and find that although the phase transition point can't be well characterized by the fidelity susceptibility, it can be effectively picked out by the higher order of the ground-state fidelity for finite-size systems.
A Kosterlitz-Thouless phase transition in the ground state of an antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain with nearest and next-nearest-neighbor interactions is re-investigated from a different perspective: An unequivocal correspondence is found between components of the scalar product ⟨Ψ0∣Ψ0⟩\langle Ψ_0| Ψ_0 \rangle⟨Ψ0​∣Ψ0​⟩ and geometrical objects. One can classify these objects according to whether any two of them can be transformed into each other in a continuous way (belong to the same homotopy class). A finite size scaling of the "connection term`` ⟨Ψ0∣∂Ψ0⟩\langle Ψ_0|\partial Ψ_0 \rangle⟨Ψ0​∣∂Ψ0​⟩ with respect to chain length (16, 18, 20, 22, 24 spins) for each homotopy class of above mentioned objects leads to the critical value of λλλ with rather high accuracy.
We use reduced fidelity to characterize quantum phase transitions in the one-dimensional spin-1/2 antiferromagnetic Heisenberg chain with frustration. For the ground state and the low-lying excited states, explicit results of the reduced fidelities between the nearest-neighbor and next-nearest-neighbor spins are given in both the even-size and odd-size cases. We find the reduced fidelity is an effective tool in detecting the quantum phase transitions associated with level crossings for the finite-size systems. Interestingly, the reduced fidelity between next-nearest-neighbor spins is evidently more sensitive at the quantum phase transition points than the nearest-neighbor case. Moreover, the reduced fidelity of the low-lying excited states is a good indicator both for the first-order and infinite-order quantum phase transitions of the system.
We investigate the ground-state phase diagram for a spin-one quantum Heisenberg antiferromagnetic chain with exchange and single-ion anisotropies in an external magnetic field by using the infinite time-evolving block decimation algorithm to compute the ground-state fidelity per lattice site. We detect all phase boundaries solely by computing the ground-state fidelity per lattice site, with the prescription that a phase transition point is attributed to a pinch point on the ground-state fidelity surface. Furthermore, the results indicate that a magnetization plateau corresponds to a fidelity plateau on the ground-state fidelity surface, thus offering an alternative route for investigating the magnetization processes of quantum many-body spin systems. We characterize all phases by using the local-order parameter, the spin correlation, the momentum distribution of the spin correlation structure factor, and mutual information as a function of the lattice distance. The commensurate and incommensurate phases are distinguished by the mutual information. In addition, the central charges at criticalities are identified by performing a finite-entanglement scaling analysis. The results show that all phase transitions between spin liquids and magnetization plateaus belong to the Pokrovsky-Talapov universality class.
A systematic analysis is performed for quantum phase transitions in a bond-alternative one-dimensional Ising model with a Dzyaloshinskii–Moriya (DM) interaction by using the fidelity of ground state wavefunctions based on the infinite matrix product states algorithm. For an antiferromagnetic phase, the fidelity per lattice site exhibits a bifurcation, which shows spontaneous symmetry breaking in the system. A critical DM interaction is inversely proportional to an alternating exchange coupling strength for a quantum phase transition. Further, a finite-entanglement scaling of von Neumann entropy with respect to truncation dimensions gives a central charge c ≃ 0.5 at the critical point.