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arXiv:cond-mat/0009201 · Physical review. B, Condensed matter

Effects of dissipation on quantum phase transitions

A. Kapitulnik, Nadya Mason, Steven Allan Kivelson, Sudip Chakravarty

Published Mar 12, 2001• 101 citations
arXivHTMLPDF

Abstract

We discuss the effect of dissipation on quantum phase transitions. In particular, we concentrate on the superconductor to insulator and quantum Hall to insulator transitions. By invoking a phenomenological parameter $\ensuremath{\alpha}$ to describe the coupling of the system to a continuum of degrees of freedom representing the dissipative bath, we obtain phase diagrams for the quantum Hall and superconductor-insulator problems. Our main result is that, in two dimensions, the metallic phases observed in finite magnetic fields (possibly also strictly zero field) are adiabatically deformable from one to the other. This is plausible, as there is no broken symmetry that differentiates them.

DOI
10.1103/physrevb.63.125322

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•Jan 1, 2012•0 citations
We discuss the effect of dissipation on quantum phase transitions. In particular, we concentrate on the superconductor to insulator and quantum Hall to insulator transitions. By invoking a phenomenological parameter \ensuremathα\ensuremath{\alpha}\ensuremathα to describe the coupling of the system to a continuum of degrees of freedom representing the dissipative bath, we obtain phase diagrams for the quantum Hall and superconductor-insulator problems. Our main result is that, in two dimensions, the metallic phases observed in finite magnetic fields (possibly also strictly zero field) are adiabatically deformable from one to the other. This is plausible, as there is no broken symmetry that differentiates them.
Zero-temperature or quantum phase transitions in itinerant electronic systems both with and without quenched disordered are discussed. Phase transitions considered include, the ferromagnetic transition, the antiferromagnetic transition, the superconductor-metal transition, and various metal-insulator transitions. Emphasis is placed on how to determine the universal properties that characterize these quantum phase transitions. For the first three of the phase transitions listed above, one of the main physical ideas established is that in zero-temperature systems there are soft or slow modes that exist in addition to the soft order parameter fluctuations, and that these modes can couple to the critical modes. These extra soft modes are shown to have a profound effect on the quantum critical properties. For quantum phase transitions involving zero wavenumber order parameters, i.e., the ferromagnetic and superconductor-metal transitions, these extra modes effectively lead to long-ranged effective interactions between order parameter fluctuations, which in turn lead to exactly soluble critical behaviors. For the antiferromagnetic case, we argue that while in low enough dimensions disorder fluctuation effects tend to destroy long-range order, quantum fluctuations counteract this effect and in some parameter regions manage to re-establish antiferromagnetic long-range order. For the metal-insulator transition, some recent new ideas are reviewed. In particular, it is pointed out that for interacting disordered electrons, one expects that in high dimensions the metal-insulator transition is related to the phase transition that occurs in random-field magnets in high dimensions. If the analogy also holds in three dimensions this suggests that the metal-insulator transition might have glassy characteristics.
The influence of gapless dissipative degrees of freedom on the superconductor-insulator transition in two dimensions is investigated. We develop a series expansion for the free energy of a (2+1)-dimensional XY model coupled to a bosonic heat bath that can be approximately summed to all orders. The calculation explicitly conserves topological excitations. We derive the zero temperature phase diagram and the free energy critical exponent, and find a transition from universal to non-universal scaling behavior as the coupling to the dissipative environment is increased, implying the existence of a new universality class.
In the previous chapter, we discussed both the insulator–superconductor and the insulator–metal transitions. As such transitions are disorder or magnetic-field tuned, thermal fluctuations play no role. Phase transitions of the insulator–superconductor or insulator–metal type are called quantum phase transitions. Such phase transitions are not controlled by changing the temperature, as in the melting of ice or the λ-point of liquid helium, but rather by changing some system parameter, such as the number of defects or the concentration of charge carriers. In all such instances, the tuning parameter transforms the system between quantum mechanical states that either look different topologically (as in the transition between localized and extended electronic states) or have distinctly different magnetic properties. As quantum mechanics underlies such phase transitions, all quantum phase transitions obtain at the absolute zero of temperature and thus are governed by a T = 0 quantum critical point. While initially surprising, this state of affairs is expected, because quantum mechanics is explicitly a zero-temperature theory of matter. Of course, this is of no surprise to chemists who have known for quite some time that numerous materials can exhibit vastly distinct properties simply by changing the chemical composition and, most importantly, that such transformations persist down to zero temperature. Common examples include turning insulators such as the layered cuprates into superconductors simply by chemical doping or semiconductors into metals once again by doping, or ferromagnets such as Li(Ho, Tb) x Y 1- x F 4 into a spin glass (AR1998) by altering x .
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
The phase diagram of a system of interacting bosons (Cooper pairs) hoping on a one-dimensional (1D) lattice with onsite phase dissipation describing the Josephson tunneling to a nearby diffusive normal-metal electrode is studied. Starting from the system at commensurate lattice filling, it is shown by a combination of analytical techniques that the phase diagram contains two quantum phases: A dissipative Bose-Einstein condensate (D-BEC) or superconductor with long-range phase coherence, and a dissipative Mott insulator (D-Mott) or "metal" with exponentially decaying phase correlations in space and local imaginary-time correlations decaying as the local pairing correlations of the electrode. The D-Mott/metal phase can be described as a 1D array of dissipative boson puddles, weakly coupled by Josephson tunneling. The puddle size roughly corresponds to the length scale beyond which phase slips suppress phase coherence. The dissipative time-dependent Ginsburg-Landau theory phenomenologically used by Sachdev, Werner, and Troyer [Phys. Rev. Lett. {\bf 92} 237003 (2004)] for the superconductor-metal transition in quasi-1D wires is derived from this microscopic puddle picture. Thus, the criticality of the D-Mott/D-BEC transition is shown to belong to the Wilson-Fisher universality class with dynamical exponent z≈2z\approx 2z≈2. At small doping, the D-Mott/metal phase remains stable due to its finite compressibility, which is computed to leading order in a perturbation expansion of the dissipation strength and the inter-puddle Josephson coupling. At larger doping, using a mapping to a pseudospin chain combined with bosonization, the D-BEC/superconductor phase is the ground state for non-vanishing but arbitrarily small dissipation. Similarities and differences with deconfinement transition of an array 1D bosonic Mott insulators in anisotropic optical lattices are also discussed.
This thesis presents seven research papers on topics in condensed-matter theory. Five of the papers report on Monte Carlo studies of quantum phase transitions in various (d+1)-dimensional statistical mechanics models featuring Caldeira-Leggettlike dissipation. The principal motivation for these studies was to investigate a particular bond-dissipative (2+1)-dimensional XY model of circulating currents in cuprate high-temperature superconductors. It has been proposed that quantum critical fluctuations associated with a local quantum critical point described by this model can explain the marginal-Fermi-liquid behaviour of the normal state of these compounds. We present simulation results for this model for both compact and noncompact phase variables and show unambiguously that the quantum critical point in the compact case is not local. If the phases are taken to be noncompact variables, the model is also a model of resistively shunted Josephson junction arrays. The results in this case reveal a more complicated phase diagram, but we have not been able to establish critical behaviour consistent with the scenario of local quantum criticality.\n\nThe study of extended quantum dissipative models is also motivated by the general effect on condensed-matter systems of the coupling to environmental degrees of freedom. Their influence on quantum critical phenomena is characterized by the dynamical critical exponent z, a measure of spatiotemporal anisotropy, the value of which can be estimated by naive scaling arguments. We confirm by numerical means that such scaling estimates give correct results to a good approximations (with a few reservations), irrespective of system dimensionality, order parameter symmetry, or whether the variables are compact or noncompact. Corrections to the naive scaling estimates have to be invoked for strongly super-Ohmic dissipation for d = 1 due to relatively large values of the anomalous scaling dimension η.\n\nThe two last research papers are concerned with the superconducting pairing state of the recently discovered class of iron-based high-temperature superconductors. Here, we calculate possible signatures of the proposed s±-wave pairing state in conductance-spectroscopy and Josephson-effect experiments.