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Zhongguo kexue. Wulixue Lixue Tianwenxue

Quantum phase transition in the U(2) vibron model and the critical point symmetry

Jia XU, Shengxin Li, Yang AN, Yun Peng Zhao, Dongqi Yu, Yu ZHANG

Published Apr 1, 2014• 0 citations
Publisher

Abstract

The quantum phase transition in the U(2) vibron model is investigated in large-N limit with the coherent state method and for finite N in a numerical scheme Through analyzing the potential structure and the corresponding low-lying spectrum, it is found that the dynamical characters at the critical point can be approximately described by the E(1) critical point symmetry, but such an approximation becomes poor with the increasing of the excitation energy. In addition, the scaling behavior of the spectrum at the critical point has been also investigated in a numerical way. The results indicate that the scaling law of the spectrum at the critical point of the 2nd order phase transition may be independent of the dimension of system.

DOI
10.1360/sspma2013-00053

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The quantum phase transition in the U(2) vibron model is investigated in large-N limit with the coherent state method and for finite N in a numerical scheme Through analyzing the potential structure and the corresponding low-lying spectrum, it is found that the dynamical characters at the critical point can be approximately described by the E(1) critical point symmetry, but such an approximation becomes poor with the increasing of the excitation energy. In addition, the scaling behavior of the spectrum at the critical point has been also investigated in a numerical way. The results indicate that the scaling law of the spectrum at the critical point of the 2nd order phase transition may be independent of the dimension of system.
We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios E21/E11, E31/E 11 and E1 transition ratios B(E1;21→1 1)/B(E1;11→11), B(E1;02→ 11)/B(E1;11→01), are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented. © 2010 IOP Publishing Ltd.
We study the details of the U(3)--O(4) quantum phase transition in the U(4) vibron model. Both asymptotic analysis in the classical limit and rigorous calculations for finite boson number systems indicate that a second-order phase transition is still there even for the systems with boson number NNN ranging from tens to hundreds. Two kinds of effective order parameters, including E1E1E1 transition ratios $B(E1:{2}_{1}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$ and $B(E1:{0}_{2}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$, and the energy ratios E21/E02{E}_{{2}_{1}}/{E}_{{0}_{2}}E21​​/E02​​ and E31/E02{E}_{{3}_{1}}/{E}_{{0}_{2}}E31​​/E02​​ are proposed to identify the second-order phase transition in experiments. We also found that the critical point of phase transition can be approximately described by the E(3) symmetry, which persists even for moderate N 10N~10N 10 protected by the scaling behaviors of quantities at the critical point. In addition, a possible empirical example exhibiting roughly the E(3) symmetry is discussed.
In most cases, excited-state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the two-dimensional (2D) limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. This case has been studied in the broken-symmetry phase [P\'erez-Bernal and \'Alvarez-Bajo, Phys. Rev. A 81, 050101 (2010)]. In the present work we delve further into the nature of this excited-state quantum phase transition and we characterize it in the symmetric phase of the model, making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a practical analysis is included with the characterization of the critical energies for the linear isomers HCN and HNC.
This work is concerned with the excited state quantum phase transitions (ESQPTs) defined in Ann. Phys. 323, 1106 (2008). In many‐body models that exhibit such transitions, the ground state quantum phase transition (QPT) occurs in parallel with a singularity in the energy spectrum that propagates to higher energies as the control parameter increases beyond the QPT critical point. The analysis of the spectrum has been a main tool for the detection of these ESQPTs. Studies of the effects of this transition on the system dynamics are more limited. Here, we extend our previous works and show that the evolution of an initial state with energy close to the ESQPT critical point may be extremely slow. This result is surprising, because it may take place in systems with long‐range interactions, where the dynamics is usually expected to be very fast. A timely example is the one‐dimensional spin‐1/2 model with infinite‐range Ising interaction studied in experiments with ion traps. Its Hamiltonian has a U(2) algebraic structure. More generally, the slow dynamics described here occurs in two‐level bosonic or fermionic models with pairing interactions and a Hamiltonian exhibiting a QPT between its limiting and dynamical symmetries. In this work, we compare the results for , and 3.
Singularities in quantum spectra - ground state and excited-state quantum phase transitions - are often connected with singularities in the classical limit of the system and have influence on other properties, such as quantum entanglement, as well. In the first part of the thesis we study quantum phase transitions within the U(2)-based Lipkin model. The relation between quasistationary points of the classical potential and the respective singularities in the spectrum is shown. In the second part, a system of two-level atoms interacting with electromagnetic field in an optical cavity is studied within two simplified models (non-integrable Dicke model and its integrable approximation known as Jaynes-Cummings model). The behaviour of quantum entanglement in these models is shown with a focus on the vicinity of the singular points. Powered by TCPDF (www.tcpdf.org)
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.