There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

We discuss the behavior of quantum and classical pairwise correlations in critical systems, with the quantumness of the correlations measured by the quantum discord. We analytically derive these correlations for general real density matrices displaying ${Z}_{2}$ symmetry. As an illustration, we analyze both the $XXZ$ and the transverse field Ising models. Finite size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.

Classical correlation and quantum discord in critical systems

Strong disorder RG approach of random systems

Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing. This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

https://iopscience.iop.org/article/10.1088/0305-4470/39/22/R01/pdf

Rare region effects at classical, quantum and nonequilibrium phase transitions

The phase diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization-group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization-group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behavior of random models.

Real-space renormalization-group study of the two-dimensional Blume-Capel model with a random crystal field

We employ a real-space renormalization-group (RSRG) approach to study a mixed-spin (spin- 1 2 and spin-1) antiferromagnetic Ising model on the square lattice. The model incorporates next-nearest-neighbor interactions which are relevant to describe ferrimagnetism. We present an RSRG calculation of the spontaneous magnetization and compensation temperatures of the mixed-spin model. The influence of crystal-field and next-nearest-neighbor interactions on compensation temperatures is studied; the important physical ingredient that leads to the presence of these temperatures is the next-nearest-neighbor coupling between spins 1 2 .

Renormalization-group magnetization of a ferrimagnetic Ising system

We investigate the application of a perturbative renormalization group (RG) method to random antiferromagnetic Heisenberg chains with arbitrary spin size. At zero temperature we observe that initial arbitrary probability distributions develop a singularity at J = 0, for all values of spin S. When the RG method is extended to finite temperatures, without any additional assumptions, we find anomalous results for S � 1. These results lead us to conclude that the perturbative scheme is not adequate to study random chains with S � 1. Therefore a random singlet phase in its more restrictive definition is only assured for spin-1/2 chains. Low-dimensional random quantum magnetic systems have become an object of increasing interest in recent years. Despite the apparent simplicity, quantum spin chains show a wealth of physical properties which have attracted the attention of both theoretical and experimental physicists. It is generally accepted that quantum antiferromagnetic spin-1/2 chains in the presence of disorder exhibit, at low temperatures, a random singlet (RS) phase characterized by power law behavior of thermodynamic quantities [1,2]. This has, in fact, been observed in several experiments [3–7]. Theoretically, the new phase was revealed by a real space renormalization group (MDH) approach [1]. These results have motivated many authors to study the physical properties of other random antiferromagnetic quantum (RAQ) chains with S � 1. In this case there are additional new problems, as for example, what happens to the Haldane gap of integer spins chains [8] in the presence of disorder. In fact, the possibility of the energy gap be suppressed by disorder, driving the system to a RS phase has become a very attractive matter for investigation. In particular, the S = 1 RAQ chains has been exhaustively studied by several methods [9–16]. Many authors have employed extended versions of the MDH scheme to explore the ground state properties of the model. In the course of these investigations, some authors suggested the existence of random singlet phase in the spin-1 chain in the strong disorder regime [9–13]. On the other hand, studies based on exact diagonalization [14], quantum Monte Carlo (MC) simulation [15] and density matrix renormalization group (DMRG) techniques [16] found that the Haldane phase is quite robust against randomness. In view of these controversial results and the fact that most calculations for spin-1 chains have been done at T = 0, we were motivated to extend the MDH analysis to finite temperature. Our goal was to generalize the MDH recursion relations to study the thermodynamic behavior of chains with arbitrary spin size. We performed careful numerical procedures to iterate our generalized MDH equations at finite temperature. The anomalous results we have obtained for the free energy and are presented in this communication indicate that the MDH approach breaks down for chains with S � 1. Therefore, we argue that it is impossible to conclude on the existence or not of the RS phase in the S � 1 RAQ chains, as proposed in previous zero temperature studies using the MDH method [9–13]. Actually, within the MDH approximation the existence of the RS phase is assured only for the spin-1/2 chains. The random magnetic systems are described by the Heisenberg Hamiltonian:

/pdf/breakdown-of-the-perturbative-renormalization-group-for-s-1-4wxyiz61j3.pdf

Breakdown of the perturbative renormalization group for S>=1 random antiferromagnetic spin chains

We present a new perturbative real space renormalization group (RG) to study random quantum spin chains and other one-dimensional disordered quantum systems. The method overcomes problems of the original approach which fails for quantum random chains with spins larger than S=1/2. Since it works even for weak disorder, we are able to obtain the zero temperature phase diagram of the random antiferromagnetic Heisenberg spin-1 chain as a function of disorder. We find a random singlet phase for strong disorder. As the disorder decreases, the system shows a crossover from a Griffiths to a disordered Haldane phase.

http://arxiv.org/pdf/cond-mat/0205492.pdf

Phase diagram of the random Heisenberg antiferromagnetic spin-1 chain.

We investigate the mixed-spin (spin-1/2 and spin-1) antiferromagnetic Ising model in two dimensions through a real-space renormalization group transformation. The model includes nearest and next-nearest neighbor interactions and is suitable to describe ferrimagnetic ordering. We present the finite-temperature phase diagram of the model. The presence of the second-neighbor bonds enlarge the ferrimagnetic phase. Our results do not indicate the existence of a tricritical point, contrary to previous effective-field calculations.

B. Boechat

Papers

Real-space renormalization-group study of the two-dimensional Blume-Capel model with a random crystal field

Renormalization-group magnetization of a ferrimagnetic Ising system

Breakdown of the perturbative renormalization group for S>=1 random antiferromagnetic spin chains

Phase diagram of the random Heisenberg antiferromagnetic spin-1 chain.

Ferrimagnetism in two-dimensional mixed-spin ising model