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arXiv:0902.1670 · Journal of Statistical Mechanics Theory and Experiment

Kosterlitz–Thouless phase transition and ground state fidelity: a novel perspective from matrix product states

Honglei Wang, Jian-Hui Zhao, Bo Li, Huan-Qiang Zhou

Published Oct 5, 2011• 22 citations
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Abstract

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.

DOI
10.1088/1742-5468/2011/10/l10001

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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.
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.
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.
Spontaneous symmetry breaking occurs in a system when its Hamiltonian possesses a certain symmetry, whereas the ground-state wave functions do not preserve it. This provides such a scenario that a bifurcation, which breaks the symmetry, occurs when some control parameter crosses its critical value. It is unveiled that the ground-state fidelity per lattice site exhibits such a bifurcation for quantum lattice systems undergoing quantum phase transitions. The significance of this result lies in the fact that the ground-state fidelity per lattice site is universal, in the sense that it is model independent, in contrast to (model-dependent) order parameters. This fundamental quantity may be computed by exploiting the developed tensor network algorithms on infinite-size lattices. We illustrate the scheme in terms of the quantum Ising model in a transverse magnetic field and the spin-1/2 XYX model in an external magnetic field on an infinite-size lattice in one spatial dimension.
We establish an intriguing connection between quantum phase transitions and bifurcations in the reduced fidelity between two different reduced density matrices for quantum lattice many-body systems with symmetry-breaking order. Our finding is based on the observation that in the conventional Landau–Ginzburg–Wilson paradigm a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking that is captured by a local-order parameter, which in turn results in an essential change of the reduced density matrix in the symmetry-broken phase. Two quantum systems on an infinite lattice in one spatial dimension, i.e. a quantum Ising model in a transverse magnetic field and a quantum spin-1/2 XYX model in an external magnetic field, are considered in the context of the tensor network algorithm based on the matrix product state representation.
The ground-state fidelity per lattice site is computed for the quantum three-state Potts model in a transverse magnetic field on an infinite-size lattice in one spatial dimension in terms of the infinite matrix product state algorithm. It is found that, on the one hand, a pinch point is identified on the fidelity surface around the critical point, and on the other hand, the ground-state fidelity per lattice site exhibits bifurcations at pseudo critical points for different values of the truncation dimension, which in turn approach the critical point as the truncation dimension becomes large. This implies that the ground-state fidelity per lattice site enables us to capture spontaneous symmetry breaking when the control parameter crosses the critical value. In addition, a finite-entanglement scaling of the von Neumann entropy is performed with respect to the truncation dimension, resulting in a precise determination of the central charge at the critical point. Finally, we compute the transverse magnetization, from which the critical exponent β is extracted from the numerical data.
Spontaneous symmetry breaking in quantum phase transitions leads to a system having degenerate ground states in its broken-symmetry phase. In order to detect all possible degenerate ground states for a broken-symmetry phase, we introduce a quantum fidelity defined as an overlap measurement between a system ground state and an arbitrary reference state. If a system has N-fold degenerate ground states in a broken-symmetry phase, the quantum fidelity is shown to have N different values with respect to an arbitrarily chosen reference state. The quantum fidelity then exhibits an N-multiple bifurcation as an indicator of a quantum phase transition without knowing any detailed broken symmetry between a broken-symmetry phase and a symmetry phase as a system parameter crosses its critical value (i.e., a multiple bifurcation point). Each order parameter, characterizing a broken-symmetry phase from each degenerate ground state reveals an N-multiple bifurcation. Furthermore, it is shown that it is possible to specify how each order parameter calculated from degenerate ground states transforms under a subgroup of a symmetry group of the Hamiltonian. Examples are given through study of the quantum q-state Potts models with a transverse magnetic field by employing tensor network algorithms based on infinite-size lattices. For any q, a general relation between the local order parameters is found to clearly show the subgroup of the Z_{q} symmetry group. In addition, we systematically discuss criticality in the q-state Potts model.