arXiv subjects

Murray T. Batchelor

Publications and source records attributed to Murray T. Batchelor.

At least 37 records · Page 2Linked to original sources

arXiv2021

Generalized adiabatic approximation to the quantum Rabi model

The quantum Rabi model (QRM) describes the interaction between a two-level system (qubit) and a quantum harmonic oscillator. In the limit where the qubit frequency is smaller than the harmonic frequency, the QRM can be well approximated by the adiabatic approximation (AA). The AA is widely used due to its simplicity and explicit physical interpretation. However, the level crossings in the spectrum of the QRM predicted by the AA are determined by the zeros of Laguerre polynomials, which deviate from the exact points. We propose a new approximation to the QRM that predicts the level crossings correctly. This is done by exploiting a surprising connection between isolated exact solutions to the QRM and the Laguerre polynomials in the AA. We thus refer to this approach as the generalized adiabatic approximation (GAA). By construction, the GAA always predicts the exact exceptional spectrum and approximates the regular spectrum remarkably well in a much larger parameter regime than the AA. This generalized approach offers a framework to deal with the family of Rabi-type light-matter interaction models in a simple but accurate manner.

quant-ph↗

arXiv2021

Hidden symmetry operators for asymmetric generalised quantum Rabi models

The hidden $\mathbb{Z}_2$ symmetry of the asymmetric quantum Rabi model (AQRM) has recently been revealed via a systematic construction of the underlying symmetry operator. Based on the AQRM result, we propose an ansatz for the general form of the symmetry operators for AQRM-related models. Applying this ansatz we obtain the symmetry operator for three models: the anisotropic AQRM, the asymmetric Rabi-Stark model (ARSM) and the anisotropic ARSM.

quant-ph↗

arXiv2021

Generalized adiabatic approximation to the asymmetric quantum Rabi model: conical intersections and geometric phases

The asymmetric quantum Rabi model (AQRM), which describes the interaction between a quantum harmonic oscillator and a biased qubit, arises naturally in circuit quantum electrodynamic circuits and devices. The existence of hidden symmetry in the AQRM leads to a rich energy landscape of conical intersections (CIs) and thus to interesting topological properties. However, current approximations to the AQRM fail to reproduce these CIs correctly. To overcome these limitations we propose a generalized adiabatic approximation (GAA) to describe the energy spectrum of the AQRM. This is achieved by combining the perturbative adiabatic approximation and the exact exceptional solutions to the AQRM. The GAA provides substantial improvement to the existing approaches and pushes the limit of the perturbative treatment into non-perturbative regimes. As a preliminary example of the application of the GAA we calculate the geometric phases around CIs associated with the AQRM.

quant-ph↗

arXiv2021

Hidden symmetry in the biased Dicke model

The symmetry operators generating the hidden $\mathbb{Z}_2$ symmetry of the asymmetric quantum Rabi model (AQRM) at bias $ε\in \frac{1}{2}\mathbb{Z}$ have recently been constructed by V. V. Mangazeev et al. [J. Phys. A: Math. Theor. 54 12LT01 (2021)]. We start with this result to determine symmetry operators for the $N$-qubit generalisation of the AQRM, also known as the biased Dicke model, at special biases. We also prove for general $N$ that the symmetry operators, which commute with the Hamiltonian of the biased Dicke model, generate a $\mathbb{Z}_2$ symmetry.

quant-ph↗

arXiv2021

The hidden symmetry of the asymmetric quantum Rabi model

The asymmetric quantum Rabi model (AQRM) exhibits level crossings in the eigenspectrum for the values $ε\in\frac{1}{2}\mathbb{Z}$ of the bias parameter $ε$. Such level crossings are expected to be associated with some hidden symmetry of the model. The origin of this hidden symmetry is established by finding the operators which commute with the AQRM hamiltonian at these special values. The construction is given explicitly for the first several cases and can be applied to other related light-matter interaction models for which similar level crossings have been observed in the presence of a bias term.

quant-ph↗

arXiv2020

Non-orthogonal qubit states expansion for the asymmetric quantum Rabi model

We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state, and make a comparison with existing approximations and the exact solution. The results show that the variational expansion is a significant improvement over the existing approximations for the AQRM.

quant-ph↗

arXiv2020

Hidden symmetry and tunnelling dynamics in asymmetric quantum Rabi models

The asymmetric quantum Rabi model (AQRM) has a broken $\mathbb{Z}_2$ symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases where the asymmetric parameter is a multiple of the cavity frequency, stable level crossings typical of the $\mathbb{Z}_2$-symmetric quantum Rabi model are recovered, however, without any obvious parity-like symmetry. This unknown "symmetry" has thus been referred to as hidden symmetry in the literature. Here we show that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models with an asymmetric qubit bias term. Conditions under which the hidden symmetry exists in these models are determined and discussed. By investigating tunnelling dynamics in the displaced oscillator basis, a strong connection is found between the hidden symmetry and selective tunnelling.

quant-ph↗

arXiv2020

A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain

The hamiltonian of the $N$-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by $N-1$ types of Temperley-Lieb generators. This generalises a previous result for $N=3$ obtained by J. F. Fjelstad and T. Månsson [J. Phys. A {\bf 45} (2012) 155208]. A pictorial representation of a related coupled algebra is given for the $N=3$ case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the $N=3$ SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight $\sqrt{3}$ and weight $2$, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter $ρ= e^{ 2π\mathrm{i}/3}$ for the SICP chain and $ρ=1$ for the staggered XX chain. These $ρ$ values are derived assuming the Kauffman bracket skein relation.

cond-mat.stat-mech↗

arXiv2019

Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model

The ground state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected trivial phases belong to the Gaussian universality class with central charge c = 1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge c = 1/2. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry breaking phase, in sharp contrast to other known cases.

cond-mat.str-el↗

arXiv2019

Some ground-state expectation values for the free parafermion Z(N) spin chain

We consider the calculation of ground-state expectation values for the non-Hermitian Z(N) spin chain described by free parafermions. For N=2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann-Feynman theorem to obtain exact results for particular single site and nearest-neighbour ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N=3, along with how they change as a function of distance from the boundary.

cond-mat.stat-mech↗

arXiv2018

Anomalous bulk behaviour in the free parafermion $Z(N)$ spin chain

We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple $Z(N)$ model for $N \ge 3$ for which the model hamiltonian is non-hermitian. For $N=2$ the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the $Z(N)$ model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing $N$. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.

cond-mat.stat-mech↗

arXiv2018

Asymptotic correlation functions and FFLO signature for the one-dimensional attractive Hubbard model

We study the long-distance asymptotic behavior of various correlation functions for the one-dimensional (1D) attractive Hubbard model in a partially polarized phase through the Bethe ansatz and conformal field theory approaches. We particularly find the oscillating behavior of these correlation functions with spatial power-law decay, of which the pair (spin) correlation function oscillates with a frequency $Δk_F$ ($2Δk_F$). Here $Δk_F=π(n_\uparrow-n_\downarrow)$ is the mismatch in the Fermi surfaces of spin-up and spin-down particles. Consequently, the pair correlation function in momentum space has peaks at the mismatch $k=Δk_F$, which has been observed in recent numerical work on this model. These singular peaks in momentum space together with the spatial oscillation suggest an analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in the 1D Hubbard model. The parameter $β$ representing the lattice effect becomes prominent in critical exponents which determine the power-law decay of all correlation functions. We point out that the backscattering of unpaired fermions and bound pairs within their own Fermi points gives a microscopic origin of the FFLO pairing in 1D.

cond-mat.str-el↗

arXiv2017

A biofilm and organomineralisation model for the growth and limiting size of ooids

Ooids are typically spherical sediment grains characterised by concentric layers encapsulating a core. There is no universally accepted explanation for ooid genesis, though factors such as agitation, abiotic and/or microbial mineralisation and size limitation have been variously invoked. We develop a mathematical model for ooid growth, inspired by work on avascular brain tumours, that assumes mineralisation in a biofilm to form a central core and concentric growth of laminations. The model predicts a limiting size with the sequential width variation of growth rings comparing favourably with those observed in experimentally grown ooids generated from biomicrospheres. In reality, this model pattern may be complicated during growth by syngenetic aggrading neomorphism of the unstable mineral phase, followed by diagenetic recrystallisation that further complicates the structure. Our model provides a potential key to understanding the genetic archive preserved in the internal structures of naturally occurring ooids.

physics.geo-ph↗

arXiv2017

Groundstate fidelity phase diagram of the fully anisotropic two-leg spin-1/2 XXZ ladder

The fully anisotropic two-leg spin-1/2 $XXZ$ ladder model is studied in terms of an algorithm based on the tensor network representation of quantum many-body states as an adaptation of projected entangled pair states to the geometry of translationally invariant infinite-size quantum spin ladders. The tensor network algorithm provides an effective method to generate the groundstate wave function, which allows computation of the groundstate fidelity per lattice site, a universal marker to detect phase transitions in quantum many-body systems. The groundstate fidelity is used in conjunction with local order and string order parameters to systematically map out the groundstate phase diagram of the ladder model. The phase diagram exhibits a rich diversity of quantum phases. These are the ferromagnetic, stripe ferromagnetic, rung singlet, rung triplet, N$\rm \acute{e}$el, stripe N$\rm \acute{e}$el and Haldane phases, along with the two $XY$ phases $XY1$ and $XY2$.

cond-mat.str-el↗

arXiv2017

The quantum Rabi model: solution and dynamics

This article presents a review of recent developments on various aspects of the quantum Rabi model. Particular emphasis is given on the exact analytic solution obtained in terms of confluent Heun functions. The analytic solutions for various generalisations of the quantum Rabi model are also discussed. Results are also reviewed on the level statistics and the dynamics of the quantum Rabi model. The article concludes with an introductory overview of several experimental realisations of the quantum Rabi model. An outlook towards future developments is also given.

quant-ph↗

arXiv2017

Finite-temperature fidelity and von Neumann entropy in the honeycomb spin lattice with quantum Ising interaction

The finite temperature phase diagram is obtained for an infinite honeycomb lattice with spin-$1/2$ Ising interaction $J$ by using thermal-state fidelity and von Neumann entropy based on the infinite projected entangled pair state algorithm with ancillas. % The tensor network representation of the fidelity, which is defined as an overlap measurement between two thermal states, is presented for thermal states on the honeycomb lattice. % We show that the fidelity per lattice site and the von Neumann entropy can capture the phase transition temperatures for applied magnetic field, consistent with the transition temperatures obtained via the transverse magnetizations, which indicates that a continuous phase transition occurs in the system. In the temperature-magnetic field plane, the phase boundary is found to have the functional form $(k_BT_c)^2 + h_c^2/2 = a J^2$ with a single numerical fitting coefficient $a = 2.298$, where $T_c$ and $h_c$ are the critical temperature and field with the Boltzmann constant $k_B$. For the quantum state at zero temperature, this phase boundary function gives the critical field estimate $h_c = \sqrt{2a} J \simeq 2.1438 J$, consistent with the known value $h_c = 2.13250(4)\, J$ calculated from a Cluster Monte Carlo approach. The critical temperature in the absence of magnetic field is estimated as $k_BT_c = \sqrt{a}J \simeq 1.5159\, J$, consistent with the exact result $k_BT_c = 1.51865...\, J$.

cond-mat.str-el↗

arXiv2016

Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model

The von Neumann entanglement entropy is used to estimate the critical point $h_c/J \simeq 0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H = \sum_i [ J ( X_i X_{i+1}^{\,2} + X_i^{\,2} X_{i+1} ) - h\, R_i ]$, where $X_i$ and $R_i$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($β< 0$) region of the $Δ$--$β$ phase diagram of the three-state quantum chiral clock model, where $Δ$ and $β$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points $β_c \simeq - 0.143(3)$ at $Δ= \frac12$ between incommensurate and commensurate phases and $β_c \simeq - 7.0(1)$ at $Δ= 0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h \le h_c$. The estimate $c \simeq 1$ in this phase is consistent with the known exact value at the particular point $h/J = -1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $η$.

cond-mat.stat-mech↗

arXiv2016

Addendum to `Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model'

In our recent paper (Li and Batchelor J. Phys. A: Math. Theor. 48, 454005 (2015)) we obtained exceptional points in the eigenspectrum of the generalized Rabi model in terms of a set of algebraic equations. We also gave a proof for the number of roots of the constraint polynomials defining these exceptional solutions as a function of the system parameters and discussed the number of crossing points in the eigenspectrum. This approach however, only covered a subset of all exceptional points in the eigenspectrum. In this addendum, we clarify the distinction between the exceptional parts of the eigenspectrum for this model and discuss the subset of exceptional points not determined in our paper.

math-ph↗