The physics of quantum Ising model (qIm) plays an important role in quantum many body system. We study and present the results of qIm and longer range quantum Ising model (lqIm) in presence of strong correlation. We do the quantum field theoretical renormalization group (RG) calculation to study the behaviour of RG flow lines for different couplings for different region of parameter space. We show how the strong correlation effect enrich the quantum physics of these two systems. We show explicitly that the ordered ferromagnetic (FM) phase to the disorder quantum paramagnet (dqpI) quantum phase transition occurs for only in the strongly correlated regime for qIm and the dqpI phase appears for non-interacting and attractive regime. We show explicitly for lqIm that FM to dqpI transition occurs at the extremely correlated region and also the dqpI phase appears in correlated regime. We show that short range FM coupling and longer range coupling are competiting with each other and also the effect of strong correlation in this competition. We also show the most interesting feature that the transverse field oppose the FM coupling of qIm but it is favour the longer range coupling of lqIm. We find the evidence of another disorder quantum paramagnetic (dqpII) phase due to the relevance of longer range coupling. We also present the existence of another quantum phase transition from dqpII phase to FM phase. We show explicitly that there is no phase transition from dqpI phase to dqpII phase rather they coexists. This work provides a new perspective not only for the statistical physics of quantum Ising model but also for the quantum many body systems.
The physics of quantum Ising model (qIm) plays an important role in quantum many body system. We study and present the results of qIm and longer range quantum Ising model (lqIm) in presence of strong correlation. We do the quantum field theoretical renormalization group (RG) calculation to study the behaviour of RG flow lines for different couplings for different region of parameter space. We show how the strong correlation effect enrich the quantum physics of these two systems. We show explicitly that the ordered ferromagnetic (FM) phase to the disorder quantum paramagnet (dqpI) quantum phase transition occurs for only in the strongly correlated regime for qIm and the dqpI phase appears for non-interacting and attractive regime. We show explicitly for lqIm that FM to dqpI transition occurs at the extremely correlated region and also the dqpI phase appears in correlated regime. We show that short range FM coupling and longer range coupling are competiting with each other and also the effect of strong correlation in this competition. We also show the most interesting feature that the transverse field oppose the FM coupling of qIm but it is favour the longer range coupling of lqIm. We find the evidence of another disorder quantum paramagnetic (dqpII) phase due to the relevance of longer range coupling. We also present the existence of another quantum phase transition from dqpII phase to FM phase. We show explicitly that there is no phase transition from dqpI phase to dqpII phase rather they coexists. This work provides a new perspective not only for the statistical physics of quantum Ising model but also for the quantum many body systems.
The effect of annealed antiferromagnetic bond randomness on the phase transitions of the Quantum Ising Model (QIM) is studied by using mean-field renormalization group method. It is argued that bond randomness drastically alters multicritical phase diagram via transverse field. Multicritical points and coexistence region of ferromagnetic and antiferromagnetic case exist only at weak transverse field, and are entirely eliminated at strong transverse field. The coexistence region diminishes in reducing the fluctuation interaction. This physical picture demonstrates that the competition between transverse field and exchange interaction and fluctuation interaction via bond randomness play an important role in generating multiphase structure. Another consequence of competition is that tricritical points of first-second order phase transitions are not entirely eliminated by bond randomness in two-dimensional QIM.
The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.
Abstract A uniform longitudinal field applied to the transverse Ising model (TIM) distinguishes the antiferromagnetic Ising interaction from its ferromagnetic counterpart. While the ground state of the latter shows no quantum phase transition (QPT), the ground state of the former exhibits rich phases: paramagnetic, antiferromagnetic, and possibly disordered phases. Although the first two are clearly identified, the existence of the disordered phase remains controversial. Here, we use the pattern picture to explore the competition among the antiferromagnetic Ising interaction J , the transverse field h x and the longitudinal field h z , and uncover which patterns are responsible for these three competing energy scales, thereby determining the possible phases and the QPTs among them. The system size ranges from L = 8 to 128 and the transverse field h x is fixed at 1. Under these parameters, our results show the existence of the disordered phase. For a small h z , the system transitions from a disordered phase to an antiferromagnetic phase as J increases. For a large h z , the system undergoes two phase transitions: from paramagnetic to disordered, and then to antiferromagnetic phase. These results not only unveil the rich physics of this paradigmatic model but also stimulate quantum simulation by using currently available experimental platforms.
Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce 2L collective structures, referred to as patterns, for the TIM with L ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength J_{c}=1. We systematically explore this process for small lattice sizes of L=6,8,10,12, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to L=128, the actual QPT point located at J_{c}=1 in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.
Critical properties of an Ising bilayer system, consisting of two triangular planes with antiferromagnetic (AF) JA and ferromagnetic (FM) JB exchange interactions, coupled by the interlayer exchange JAB, are studied by Monte Carlo simulation. The FM plane is found to induce in the AF one some effects similar to those resulting from the joint presence of a next-nearest-neighbor FM coupling and an external magnetic field in the single triangular lattice Ising antiferromagnet. The similarities include the appearance of the quasi-long-range-order BKT phase with algebraically decaying correlations for smaller JAB, as well as the phase transitions belonging to the three-state Potts universality class for larger JAB. On the other hand, no first-order phase transitions are observed. A new phenomenon is a cross-over from the BKT to the Wannier state at higher temperatures, that can be stabilized over a relatively wide temperature range. In the strong JAB limit, the critical behavior of the entire bilayer is governed by the behavior of the plane with a dominant intraplane interaction. If JA is dominant, then the strong JAB coupling makes the system kinetically freeze at lower temperatures.
We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.