Second-order Quantum Phase Transition Research Articles

On a quantum phase transition in a steady state out of equilibrium

Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we show that the nonequilibrium steady state, constructed in the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, coupled to two thermal reservoirs at different temperatures, is exposed to a local external magnetic field, is breaking translation invariance and exhibits a strictly positive entropy production rate. Moreover, we prove that there exists a second-order nonequilibrium quantum phase transition with respect to the strength of the magnetic field as soon as the system is truly out of equilibrium.

Excited-state quantum phase transition in the Rabi model

The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M. -J. Hwang et al, Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which shows a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states leads to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.

Scaling of the local quantum uncertainty at quantum phase transitions

We investigate the local quantum uncertainty (LQU) between a block of L qubits and one single qubit in a composite system of n qubits driven through a quantum phase transition (QPT). A first-order QPT is analytically considered through a Hamiltonian implementation of the quantum search. In the case of second-order QPTs, we consider the transverse-field Ising chain via a numerical analysis through density matrix renormalization group. For both cases, we compute the LQU for finite-sizes as a function of L and of the coupling parameter, analyzing its pronounced behavior at the QPT.

Spin-orbit coupled repulsive Fermi atoms in a one-dimensional optical lattice

Motivated by recent experimental development, we investigate spin–orbit coupled repulsive Fermi atoms in a one-dimensional optical lattice. Using the density-matrix renormalization group method, we calculate the momentum distribution function, gap, and spin-correlation function to reveal rich ground-state properties. We find that spin–orbit coupling (SOC) can generate unconventional momentum distribution, which depends crucially on the filling. We call the corresponding phase with zero gap the SOC-induced metallic phase. We also show that SOC can drive the system from the antiferromagnetic to ferromagnetic Mott insulators with spin rotating. As a result, a second-order quantum phase transition between the spin-rotating ferromagnetic Mott insulator and the SOC-induced metallic phase is predicted at the strong SOC. Here spin rotating means the spin orientations of the nearest-neighbor sites are not parallel or antiparallel, i.e., they have an intersection angle Finally, we show that the momentum kpeak, at which the peak of the spin-structure factor appears, can also be affected dramatically by SOC. The analytical expression of this momentum with respect to the SOC strength is also derived, which suggests that the predicted spin-rotating ferromagnetic (kpeak < π /2 ) and antiferromagnetic (π /2 < kpeak < π) correlations can be detected experimentally by measuring the SOC-dependent spin-structure factor via time-of-flight imaging.

Quantum correlations in Heisenberg XY spin-1 and spin-1/2 model with Dzyaloshinskii–Moriya interactions

We study the quantum correlations in a spin-1/2 (qubit) and spin-1 (qutrit) Heisenberg XY model separately, based on quantum discord (QD) and measurement-induced disturbance (MID) respectively. We find the evidence of the first- and second-order quantum phase transition (QPT) in both spin-1/2 and spin-1 systems. The effect of the temperature, magnetic field and Dzyaloshinskii–Moriya (DM) interactions on QPT and quantum correlation are also investigated. Finally, we observed that the QD and MID are not only vigorous for higher spin systems but also more robust than entanglement.

Quantum phase transitions and hidden topological order in one-dimensional S=1 Heisenberg–Ising alternating model

By the infinite time-evolving block decimation method, the ground state properties of the =1 Heisenberg–Ising alternating chain have been investigated in the thermodynamic limit. A ground-state phase diagram, which includes three different phases, i.e., an antiferromagnetic strip phase, a dimer phase, and an antiferromagnetic phase, is obtained. A modified definition of the string order is adopted to describe the dimer phase, and it is found to have nonzero antiferromagnetic topological string order. The quantum phase transitions (QPTs) in such a model can be well described by the singularities of bipartite entanglement and fidelity bifurcations, and two second-order QPTs points are determined to be at . The central charges at both QPTs points are determined to be 1/2. Therefore, both QPTs belong to the Ising universality class with central charges 1/2.

Effects of Heisenberg perturbation on the ground-state properties of one-dimensional extended quantum compass model

The ground-state properties and quantum phase transitions (QPTs) of the one-dimensional extended quantum compass model (EQCM) with Heisenberg perturbation are investigated by the infinite time-evolving block decimation (iTEBD) algorithm. The ground-state properties are found to be affected distinctively by the Heisenberg perturbation and rich phase diagrams are obtained. The first-order QPTs line at J1=0 disappears even with an infinitesimal Heisenberg perturbation and two interesting intermediate phases, such as a disordered phase and a transverse Néel phase, are induced. Both bipartite entanglement and fidelity per site are capable of describing all the second-order QPTs.

Quantum phase transitions of atom-molecule Bose mixtures in a double-well potential.

The ground state and spectral properties of Bose gases in double-well potentials are studied in two different scenarios: (i) an interacting atomic Bose gas, and (ii) a mixture of an atomic gas interacting with diatomic molecules. A ground state second-order quantum phase transition is observed in both scenarios. For large attractive values of the atom-atom interaction, the ground state is degenerate. For repulsive and small attractive interaction, the ground state is not degenerate and is well approximated by a boson coherent state. Both systems depict an excited state quantum phase transition. In both cases, a critical energy separates a region in which all the energy levels are degenerate in pairs, from another region in which there are no degeneracies. For the atomic system, the critical point displays a singularity in the density of states, whereas this behavior is largely smoothed for the mixed atom-molecule system.

A Quantum Critical Point Lying Beneath the Superconducting Dome in Iron Pnictides

Whether a quantum critical point (QCP) lies beneath the superconducting dome has been a long-standing issue that remains unresolved in many classes of unconventional superconductors, notably cuprates, heavy fermions, and, most recently, iron pnictides. The existence of a QCP may offer a route to understanding the origin of unconventional superconductors’ anomalous non-Fermi liquid properties, the microscopic coexistence between unconventional superconductivity and magnetic or some other exotic order, and, ultimately, the mechanism of superconductivity itself. The isovalent substituted iron pnictide BaFe2(As1−xPx)2 offers a new platform for the study of quantum criticality, providing a unique opportunity to study the evolution of the electronic properties in a wide range of the phase diagram. Recent experiments in BaFe2(As1−xPx)2 have provided the first clear and unambiguous evidence of a second-order quantum phase transition lying beneath the superconducting dome.

Phase transitions in three-dimensional bosonic systems in optical lattices

We formulate the collective quantum field theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper et al (2011 Phys. Rev. A 83 053622), and in a lowest-order variational perturbation theory. It is shown that the present mean-field approximation predicts some essential features of the experimentally observed dependence of the critical temperature on the coupling strength and a second-order quantum phase transition. In contrast to a recent prediction for atomic gases given by Cooper et al, our finding indicates no superfluid state with zero condensate fraction.

Matrix product states and variational methods applied to critical quantum field theory

We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in ($1+1$) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related density matrix normalization group method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate noncritical quantum field theories under certain conditions.

Quantum criticality of the imperfect Bose gas in d dimensions

We study the low-temperature limit of the d-dimensional imperfect Bose gas. Relying on an exact analysis of the microscopic model, we establish the existence of a second-order quantum phase transition to a phase involving the Bose–Einstein condensate. The transition is triggered by varying the chemical potential and persists at non-zero temperatures T for d > 2. We extract the exact phase diagram and identify the scaling regimes in the vicinity of the quantum-critical point focusing on the behavior of the correlation length ξ. The length ξ develops an essential singularity exclusively for d = 2. We follow the evolution of the phase diagram varying d. For d > 2 our results agree with renormalization-group-based analysis of the effective bosonic order-parameter models with the dynamical exponent z = 2.

Disorder by disorder and flat bands in the kagome transverse field Ising model

We study the transverse field Ising model on a kagome and a triangular lattice using high-order series expansions about the high-field limit. For the triangular lattice our results confirm a second-order quantum phase transition in the 3d XY universality class. Our findings for the kagome lattice indicate a notable instance of a disorder by disorder scenario in two dimensions. The latter follows from a combined analysis of the elementary gap in the high- and low-field limit which is shown to stay finite for all fields h. Furthermore, the lowest one-particle dispersion for the kagome lattice is extremely flat acquiring a dispersion only from order eight in the 1/h limit. This behaviour can be traced back to the existence of local modes and their breakdown which is understood intuitively via the linked cluster expansion.

Fluctuation-Induced Magnetization Dynamics and Criticality at the Interface of a Topological Insulator with a Magnetically Ordered Layer

We consider a theory for a two-dimensional interacting conduction electron system with strong spin-orbit coupling on the interface between a topological insulator and the magnetic (ferromagnetic or antiferromagnetic) layer. For the ferromagnetic case we derive the Landau-Lifshitz equation, which features a contribution proportional to a fluctuation-induced electric field obtained by computing the topological (Chern-Simons) contribution from the vacuum polarization. We also show that fermionic quantum fluctuations reduce the critical temperature T[over ˜](c) at the interface relative to the critical temperature T(c) of the bulk, so that in the interval T[over ˜](c)≤T<T(c) it is possible to have a coexistence of gapless Dirac fermions at the interface with a ferromagnetically ordered layer. For the case of an antiferromagnetic layer on a topological insulator substrate, we show that a second-order quantum phase transition occurs at the interface, and compute the corresponding critical exponents. In particular, we show that the electrons at the interface acquire an anomalous dimension at criticality. The critical behavior of the Néel order parameter is anisotropic and features large anomalous dimensions for both the longitudinal and transversal fluctuations.

Efficient simulation of infinite tree tensor network states on the Bethe lattice

We show that the simple update approach proposed by Jiang et. al. [H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is an efficient and accurate method for determining the infinite tree tensor network states on the Bethe lattice. Ground state properties of the quantum transverse Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The transverse Ising model is found to undergo a second-order quantum phase transition with a diverging magnetic susceptibility but a finite correlation length which is upper-bounded by 1/ln(q-1) even at the transition point (q is the coordinate number of the Bethe lattice). An intuitive explanation on this peculiar "critical" phenomenon is given. The XXZ model on the Bethe lattice undergoes a first-order quantum phase transition at the isotropic point. Furthermore, the simple update scheme is found to be related with the Bethe approximation. Finally, by applying the simple update to various tree tensor clusters, we can obtain rather nice and scalable approximations for two-dimensional lattices.

Study of cooperative breathing-mode in molecular chains

Using a controlled analytic nonperturbative treatment that accounts for the quantum nature of the phonons, we derive a model that generically describes the cooperative breathing-mode at strong electron-phonon interaction in one-band one-dimensional systems. The effective model involves a next-nearest-neighbor hopping (that dominates over the nearest-neighbor hopping at strong coupling) and a nearest-neighbor repulsion that is significantly enhanced due to incompatibility of neighboring dilations/compressions. At non-half-filling, upon tuning the electron-phonon coupling, the system undergoes a period-doubling second-order quantum phase transition from a Luttinger liquid to a conducting commensurate charge-density-wave state: a phenomenon absent in both the Holstein model and the t-V model. Using fidelity to study the nature of the quantum phase transition, we find that the fidelity susceptibility shows a superextensive power law divergence as well as a remarkable scaling behavior; both together establish a second-order transition.

Matrix product state and quantum phase transitions in the one-dimensional extended quantum compass model

The matrix product state (MPS) is utilized to study the ground state properties and quantum phase transitions (QPTs) of the one-dimensional quantum compass model (QCM). The MPS wavefunctions are argued to be very efficient descriptions of QCM 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 details. It is revealed that the bipartite and block entanglement entropies, 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 wavefunction, and found to be doubly degenerate in disordered phases of QCM, where non-local string order parameters exist. Moreover, with linearized tensor renormalization group method, the specific heat curves are evaluated and their low temperature behaviors are investigated.

(De)localization and the mobility edges in a disordered double chain with long-range intrachain correlation and short-range interchain correlation

Correlation effects and phase transitions are central issues in current studies on disordered systems. In this paper, we study the electronic properties of a disordered double chain with long-range intrachain correlation and short-range interchain correlation. Based on detailed numerical calculations, finite size scaling analysis and empirical analytical calculations, we obtain a phase diagram containing rich physics due to the interplay among the disorder, short-range and long-range correlations. Besides the long-range correlation induced localization–delocalization transitions, we find both first-order and second-order quantum phase transitions on changing the short-range correlation. Interestingly, the localization may be suppressed by increasing the disorder strength in some parameter regime and the ‘anti-correlation’ leads to the most delocalized state. Our studies shine some light on the mechanism of the charge transport in DNA molecules, where both types of correlated disorders are present.

Quantum correlations in collective spin systems

Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.

Long-range quantum discord in critical spin systems

We show that quantum correlations as quantified by quantum discord can characterize quantum phase transitions by exhibiting nontrivial long-range decay as a function of distance in spin systems. This is rather different from the behavior of pairwise entanglement, which is typically short-ranged even in critical systems. In particular, we find a clear change in the decay rate of quantum discord as the system crosses a quantum critical point. We illustrate this phenomenon for first-order, second-order, and infinite-order quantum phase transitions, indicating that pairwise quantum discord is an appealing quantum correlation function for condensed matter systems.