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arXiv:0909.2928

Matrix Product States and Quantum Phase Transitions

Khabat Heshami, Sadegh Raeisi

Published Sep 16, 2009• 0 citations
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Abstract

We have developed a new approach based on matrix product representations of ground states to study Quantum Phase Transitions (QPT). As confirmation of the power of our approach we have analytically analyzed the XXZ spin-one chain with uniaxial single-ion-type anisotropy and our results compare favourably with previous numerical studies. In addition, our description lets to know which part of parameters space of the Hamiltonian is most likely to be exactly solvable.

DOI
10.48550/arxiv.0909.2928

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We have developed a new approach based on matrix product representations of ground states to study Quantum Phase Transitions (QPT). As confirmation of the power of our approach we have analytically analyzed the XXZ spin-one chain with uniaxial single-ion-type anisotropy and our results compare favourably with previous numerical studies. In addition, our description lets to know which part of parameters space of the Hamiltonian is most likely to be exactly solvable.
One-dimensional S=1S=1S=1 XXZ\mathrm{XXZ}XXZ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two XY\mathrm{XY}XY phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.
We present a new general and much simpler scheme to construct various quantum phase transitions (QPTs) in spin chain systems with matrix product ground states. By use of the scheme we take into account one kind of matrix product state (MPS) QPT and provide a concrete model. We also study the properties of the concrete example and show that a kind of QPT appears, accompanied by the appearance of the discontinuity of the parity absent block physical observable, diverging correlation length only for the parity absent block operator, and other properties which are that the fixed point of the transition point is an isolated intermediate-coupling fixed point of renormalization flow and the entanglement entropy of a half-infinite chain is discontinuous.
The matrix product state (MPS) is utilized to study the ground-state properties and quantum phase transitions (QPTs) of the one-dimensional extended quantum compass model (EQCM). The MPS wave functions are argued to be very efficient descriptions of the ground states, and are numerically determined by imaginary-time projections. The ground-state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in detail. It is revealed that the von Neumann entanglement entropy, as well as the nearest-neighbor correlation functions, can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wave function and found to be doubly degenerate in disordered phases, where nonzero string order parameters exist. Moreover, with the linearized tensor renormalization group method, the specific-heat curves are evaluated and their low-temperature behaviors are investigated. Compared with the exact solutions, our results verify that these MPS-based numerical methods are very accurate and powerful, and can be employed to investigate other EQCMs which do not permit exact solutions at present.
We investigate quantum phase transitions (QPTs) in spin chain systems characterized by local Hamiltonians with matrix product ground states. We show how to theoretically engineer such QPT points between states with predetermined properties. While some of the characteristics of these transitions are familiar, like the appearance of singularities in the thermodynamic limit, diverging correlation length, and vanishing energy gap, others differ from the standard paradigm: In particular, the ground state energy remains analytic, and the entanglement entropy of a half-chain stays finite. Examples demonstrate that these kinds of transitions can occur at the triple point of "conventional" QPTs.
We explore numerically the inverse participation ratios in the ground state of one-dimensional spin-1 XXZXXZXXZ chains with the rhombic single-ion anisotropy. By employing the techniques of density-matrix renormalization group, effects of the rhombic single-ion anisotropy on various information theoretical measures are investigated, such as the fidelity susceptibility, the quantum coherence, and the entanglement entropy. Their relations with the quantum phase transitions are also analyzed. The phase transitions from the YYY-N\'eel phase to the large-Ex{E}_{x}Ex​ or the Haldane phase can be well characterized by the fidelity susceptibility. The second-order derivative of the ground-state energy indicates all the transitions are of second order. We also find that the quantum coherence, the entanglement entropy, the Schmidt gap, and the inverse participation ratios can be used to detect the critical points of quantum phase transitions. Results drawn from these quantum information observables agree well with each other. Finally we provide a ground-state phase diagram as functions of the exchange anisotropy $\mathrm{\ensuremath{\Delta}}$ and the rhombic single-ion anisotropy EEE.
We study the ground-state phase diagram of the quantum spin-2 XXZ chain in the presence of on-site anisotropy using a matrix-product state based infinite-system density matrix renormalization group (iDMRG) algorithm. One of the interests in this system is in connecting the highly quantum-mechanical spin-1 phase diagram with the classical $S=\ensuremath{\infty}$ phase diagram. Several of the recent advances within DMRG make it possible to perform a detailed analysis of the whole phase diagram. We consider different types of on-site anisotropies, which allows us to establish the validity of the following statements: (1) the spin-2 model can be tuned into a phase, which is equivalent to the ``topologically nontrivial'' spin-1 Haldane phase, and (2) the spin-2 Haldane phase at the isotropic Heisenberg point is adiabatically connected to the ``trivial'' large-DDD phase, with a continuous change of the Hamiltonian parameters. Furthermore, we study the spin-3 XXZ chain to help explain the development of the classical phase diagram. We present details on how to use the iDMRG method to map out the phase diagram and include an extensive discussion of the numerical methods.