At quantum critical points (QCP) [1,2,3,4,5,6,7] there are quantum fluctuations on all length scales, from microscopic to macroscopic lengths, which, remarkably, can be observed at finite temperatures, the regime to which all experiments are necessarily confined. A fundamental question is how high in temperature can the effects of quantum criticality persist? That is, can physical observables be described in terms of universal scaling functions originating from the QCPs? Here we answer these questions by examining exact solutions of models of correlated systems and find that the temperature can be surprisingly high. As a powerful illustration of quantum criticality, we predict that the zero temperature superfluid density, ρs(0), and the transition temperature, Tc, of the cuprates are related by Tc ∝ ρs(0) y , where the exponent y is different at the two edges of the superconducting dome, signifying the respective QCPs. This relationship can be tested in high quality crystals.Do quantum critical points (QCPs) provide a powerful framework for understanding complex correlated many body problems? Do they shed new light on quantum mechanics of macroscopic systems, providing, for example, a deeper understanding of entanglement? Answers to many such questions require a precise recognition of the experimentally observable regime of quantum criticality. An important class of QCPs are analogs of classical critical points, but in a dimension higher than the actual spatial dimension of the system; that is, a quantum critical point in d spatial dimensions is equivalent to a classical critical point in (d + z), dimensions, where z is the dynamic critical exponent. We shall be primarily concerned with z = 1, although an example involving z = 1 is given below. This extra dimension (imaginary time) is the sine qua non of the effect of quantum mechanics.It seems paradoxical that often the region of classical critical fluctuations is small and tends to zero as the temperature, T , tends to zero, while the region of quantum critical fluctuations fans out as T increases. The reason is that the quantum critical region is determined by the dominant microscopic energy scale in the Hamiltonian, whereas the classical critical region is determined by the ratio of the transition temperature, T c , to the same scale raised to a positive power (the dimensionality). Since this ratio is usually small compared to unity (it is 10 −5 in a conventional superonductor), the classical critical region is small as well. As a classical critical point is driven to zero by tuning a parameter to reach the QCP, T c itself vanishes as ξ −z , provided the T = 0 spatial correlation length, ξ, is large compared to the lattice spacing, further amplifying the contrast between classical and quantum criticalities.How high in temperature does the effect of a QCP persist? Consider perhaps the simplest possible example of a quantum phase transition described by the Hamiltonian of an Ising model in a transverse field [8] in one dimension,where the usual Pauli matrices,...

We study the entanglement between a qubit and its environment from the spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic Kondo model, we derive the entropy of entanglement of the spin E(α, ∆, h), where α is the dissipation strength, ∆ is the tunneling amplitude between qubit states, and h is the level asymmetry. For 1 − α ≫ ∆/ωc and (∆, h) ≪ ωc, we show that the Kondo energy scale TK controls the entanglement between the qubit and the bosonic environment (ωc is a high-energy cutoff). For h ≪ TK, the disentanglement proceeds as (h/TK ) 2 ; for h ≫ TK, E vanishes as (TK/h) 2−2α , up to a logarithmic correction. For a given h, the maximum entanglement occurs at a value of α which lies in the crossover regime h ∼ TK . We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise. The concept of quantum entropy appears in multiple contexts, from black hole physics 1 to quantum information theory, where it measures the entanglement of quantum states.2 Prompted by the link between entanglement and quantum criticality, 3 a number of researchers have begun to study the entanglement entropy of condensed matter systems. In this Letter, we employ the spin-boson model 4,5 to describe the entanglement between a qubit (two-level system) and an infinite collection of bosons. With an Ohmic bosonic bath, the spin-boson model undergoes a quantum phase transition of Kosterlitz-Thouless type when α − 1 = ∆/ω c , where α is the strength of the coupling to the environment, ∆ is the tunneling amplitude between the qubit states, and ω c ≫ ∆ is an ultraviolet cutoff.6,7 When the two levels of the qubit are degenerate, the entanglement between the qubit and the bosons is discontinuous at this transition.8,9 Here we report the first rigorous analytical results for the entanglement (quantum entropy) in the strongly entangled regime 1 − α ≫ ∆/ω c .We exploit a mapping between the spin-boson model and the anisotropic Kondo model; our results follow from the Bethe ansatz solution of the equivalent interacting resonant level model. 10,11 We show that the entropy of entanglement (E) of the qubit with the environment is controlled by the Kondo energy scale T K

In the thermodynamic limit two distinct states of matter cannot be analytic continuations of each other. Classical phase transitions are characterized by non-analyticities of the free energy. For quantum phase transitions (QPTs) the ground state energy often assumes the role of the free energy. But in a number of important cases this criterion fails to predict a QPT, such as the three-dimensional metal-insulator transition of non-interacting electrons in a random potential (Anderson localization). It is therefore essential that we find alternative criteria that can track fundamental changes in the internal correlations of the ground state wavefunction. Here we propose that QPTs are generally accompanied by non-analyticities of the von Neumann (entanglement) entropy. In particular, the entropy is non-analytic at the Anderson transition, where it exhibits unusual fractal scaling. We also examine two dissipative quantum systems of considerable interest to the study of decoherence and find that non-analyticities occur if and only if the system undergoes a QPT.Comment: 8 pages, 6 figures; Annals of Physics, in press (2006

The extreme variability of observables across the phase diagram of the cuprate high-temperature superconductors has remained a profound mystery, with no convincing explanation for the superconducting dome. Although much attention has been paid to the underdoped regime of the hole-doped cuprates because of its proximity to a complex Mott insulating phase, little attention has been paid to the overdoped regime. Experiments are beginning to reveal that the phenomenology of the overdoped regime is just as puzzling. For example, the electrons appear to form a Landau Fermi liquid, but this interpretation is problematic; any trace of Mott phenomena, as signified by incommensurate antiferromagnetic fluctuations, is absent, and the uniform spin susceptibility shows a ferromagnetic upturn. Here, we show and justify that many of these puzzles can be resolved if we assume that competing ferromagnetic fluctuations are simultaneously present with superconductivity, and the termination of the superconducting dome in the overdoped regime marks a quantum critical point beyond which there should be a genuine ferromagnetic phase at zero temperature. We propose experiments and make predictions to test our theory and suggest that an effort must be mounted to elucidate the nature of the overdoped regime, if the problem of high-temperature superconductivity is to be solved. Our approach places competing order as the root of the complexity of the cuprate phase diagram.quantum-phase transition ͉ non-Fermi liquid ͉ broken symmetry ͉ quantum order ͉ criticality T he superconducting dome (see Fig. 1), that is, the shape of the superconducting transition temperature T c as a function of doping (added charge carriers), x, is a clue that the high-T c superconductors are unconventional. Conventional superconductors, explained so beautifully by Bardeen, Cooper, and Schrieffer (52), have a unique ground state that is not naturally separated by any nonsuperconducting states. The electronphonon mechanism leads to superconductivity for arbitrarily weak attraction between electrons. To destroy a superconducting state requires a magnetic field or strong material disorder. In the absence of disorder or magnetic field, it is difficult to explain the sharp cutoffs at x 1 and x 2 within the Bardeen, Cooper, and Schrieffer theory. For high-T c superconductors, there is considerable evidence that competing order parameters are the underlying reason. Thus, x 1 and x 2 signify quantum phase transitions, most likely quantum critical points (QCPs) (1). Understanding high-T c superconductors therefore requires an understanding of possible competing orders (2, 3). The QCP at x 1 has been extensively studied (4-8), but little is known about x 2 . Recent work has also emphasized the importance of the maximum of the uniform susceptibility in defining the pseudogap line T* in Fig. 1 that ends at another QCP at x c (8). Here, we attempt, instead, at gaining insight from the possible existence of a QCP at x 2 .Complex materials (cuprates, heavy fermions, and organics)...

The low temperature scanning tunneling microscopy spectra in the underdoped regime is analyzed from the perspective of coexisting d-density wave (DDW) and d-wave superconducting states (DSC). The calculations are carried out in the presence of a low concentration of unitary impurities and within the framework of the fully self-consistent Bogoliubov-de Gennes theory, which allows local modulations of the magnitude of the order parameters in response to the impurities. Our theory captures the essential aspects of the experiments in the underdoped BSCCO at very low temperatures.PACS numbers: PACS numbers: 73.23. Hk, 73.63.Kv, 02.70.Ss A fundamental tension in the field of high temperature superconductors is the notion of a competing order parameter in the underdoped regime, which can provide a natural explanation of why the superconducting dome exists and shed light on the nature of the pseudogap. While a charge ordered state is a candidate [1], one of us has proposed, and extensively studied, a new order parameter, which results in circulating currents arranged in a staggered pattern(DDW) [2]. Many experiments are consistent with this order parameter, as demonstrated in studies of the superfluid density, the polarized neutron scattering, the Hall number in pulsed 60 T magnetic field, the angle resolved photoemission spectroscopy (ARPES), the lack of specific heat anomaly at the pseudogap temperature, the transition temperature in multilayer cuprates, and the infrared Hall angle measurements [3]. So far the clinching direct experiment, the polarized neutron scattering, has remained suggestive [4] because of the difficulty of detecting weak signals from the small magnetic moments generated by the circulating orbital currents. Recently, another novel experimental test has been suggested that takes advantage of the spin-orbit coupling in the DDW state [5].Here we turn to the intriguing scanning tunneling microscopy (STM) measurements [6,7,8,9,10,11]. In spite of numerous theoretical analyses of this problem [1,12,13,14,15,16,17,18,19,20,21,22], no comprehensive theoretical picture has yet emerged, although certain aspects are captured by some of them. For example, earlier measurements in the slightly overdoped samples at very low temperatures have been elegantly explained in terms of a quasiparticle scattering interference model, named the octet model [8,14]. At the same time, an interpretation in terms of dynamic charge fluctuations has also been advanced [1,11]. The focus here, however, is on an extensive set of experiments as a function of doping in BSCCO at very low temperatures [9]. The exciting finding of these experiments is the emergence of a new order, present along with the d-wave superconductivity (DSC). The salient signature is a sudden development of a relatively non-dispersive incommensurate wave vector, q * , at higher energies in the underdoped regime.In this Letter we explain the experiments by adopting a view that is orthogonal to the notion of charge order and consider DDW. At first sight, this w...