Missouri University of Science and Technology Scholars' Mine Physics Faculty Research & Creative Works Physics 01 Jan 2005 How Generic Scale Invariance Influences Quantum and Classical Phase Transitions D. Belitz T. Kirkpatrick Thomas Vojta Missouri University of Science and Technology, [email protected] Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork Part of the Physics Commons Recommended Citation D. Belitz et al., "How Generic Scale Invariance Influences Quantum and Classical Phase rT ansitions," Reviews of Modern Physics, American Physical Society (APS), Jan 2005. The definitive version is available at https://doi.org/10.1103/RevModPhys.77.579 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. REVIEWS OF MODERN PHYSICS, VOLUME 77, APRIL 2005 How generic scale invariance influences quantum and classical phase transitions D. Belitz* Department of Physics and Materials Science Institute, University of Oregon, Eugene, Oregon 97403, USA T. R. Kirkpatrick† Institute for Physical Science and Technology, and Department of Physics, University of Maryland, College Park, Maryland 20742, USA Thomas Vojta‡ Department of Physics, University of Missouri-Rolla, Rolla, Missouri 65409, USA ͑Published 5 July 2005͒ This review discusses a paradigm that has become of increasing importance in the theory of quantum phase transitions, namely, the coupling of the order-parameter fluctuations to other soft modes and the resulting impossibility of constructing a simple Landau-Ginzburg-Wilson theory in terms of the order parameter only. The soft modes in question are manifestations of generic scale invariance, i.e., the appearance of long-range order in whole regions in the phase diagram. The concept of generic scale invariance and its influence on critical behavior is explained using various examples, both classical and quantum mechanical. The peculiarities of quantum phase transitions are discussed, with emphasis on the fact that they are more susceptible to the effects of generic scale invariance than their classical counterparts. Explicit examples include the quantum ferromagnetic transition in metals, with or without quenched disorder; the metal-superconductor transition at zero temperature; and the quantum antiferromagnetic transition. Analogies with classical phase transitions in liquid crystals and classical fluids are pointed out, and a unifying conceptual framework is developed for all transitions that are influenced by generic scale invariance. CONTENTS 4. Spatial correlations in nonequilibrium steady states 593 B. Quantum systems 594 Preamble 580 1. Examples of generic scale invariance in I. Phase Transitions 580 itinerant electron systems 594 A. Landau theory 581 a. Disordered systems in equilibrium 594 B. Landau-Ginzburg-Wilson theory, the renormalization b. Clean systems in equilibrium 595 group, and scaling 582 2. Soft modes in itinerant electron systems 595 C. Classical versus quantum phase transitions 583 3. Generic scale invariance via explicit D. The soft-mode paradigm 585 calculations 598 II. Generic Scale Invariance 586 4. Generic scale invariance via A. Classical systems 587 renormalization-group arguments 600 1. Goldstone modes 587 a. Disordered electron systems 600 a. Nonlinear ␴ model 587 b. Clean electron systems 601 b. Generic scale invariance from explicit 5. Another example of generic scale calculations 588 invariance: Disordered systems in a c. Generic scale invariance from RG arguments 588 nonequilibrium steady state 602 2. U͑1͒ gauge symmetry 589 6. Quantum Griffiths phenomena: Power laws a. Symmetric phase 589 from rare regions in disordered systems 602 b. The broken-symmetry phase 590 7. Quantum long-time tails from detailed 3. Long-time tails 590 balance 603 a. Fluctuating hydrodynamics 591 III. Influence of Generic Scale Invariance on Classical b. Generic scale invariance from explicit Critical Behavior 604 calculations 591 A. The nematic–smectic-A transition in liquid crystals 604 1. Action, and analogy with superconductors 604 *Electronic address: [email protected] 2. Gaussian approximation 605 †Electronic address: [email protected] 3. Renormalized mean-field theory 605 ‡Electronic address: [email protected] 4. Effect of order-parameter fluctuations 606 0034-6861/2005/77͑2͒/579͑54͒/$50.00 579 ©2005 The American Physical Society 580 Belitz, Kirkpatrick, and Vojta: How generic scale invariance influences phase transitions B. Critical behavior in classical fluids 607 is very well developed. A characteristic feature of con- 1. Critical dynamics in equilibrium fluids 607 tinuous phase transitions, or critical points, is that the 2. Critical dynamics in a fluid under shear 609 free energy, as well as time correlation functions, obey IV. Influence of Generic Scale Invariance on Quantum generalized homogeneity laws. This leads to scale invari- Critical Behavior 610 ance, i.e., power-law behavior of various thermodynamic A. Quantum ferromagnetic transition in clean systems 610 derivatives and correlation functions as functions of the 1. Soft-mode action for itinerant quantum temperature, external fields, wave number, time, etc. ferromagnets 610 Quantum phase transitions, which occur at zero tem- 2. Gaussian approximation 612 perature as a function of some nonthermal control pa- 3. Renormalized mean-field theory 612 rameter such as composition or pressure, are not yet as 4. Effects of order-parameter fluctuations 613 well understood. One feature that has slowed down 5. Critical behavior at the continuous progress has turned out to be a connection between transition 615 quantum phase transitions and the phenomenon known a. Critical behavior in d=3 615 as generic scale invariance, which refers to power-law b. Critical behavior in d3 616 decay of correlation functions in entire phases, not just B. Quantum ferromagnetic transition in disordered at an isolated critical point. It turns out that the critical systems 616 behavior at some classical phase transitions is also 1. Soft-mode action 616 heavily influenced by a coupling between generic and 2. Gaussian approximation 616 critical scale invariance, but these phenomena have usu- 3. Renormalized mean-field theory 616 ally not been cast in this language. The goal of this re- 4. Effects of order-parameter fluctuations 618 view article is to give a unifying discussion of the cou- a. Coupled field theory and power counting 618 pling between critical behavior and generic scale b. Hertz’s fixed point 618 invariance, for both classical and quantum phase transi- c. A marginally unstable fixed point 619 tions and for both static and dynamic critical behavior. d. Exact critical behavior 619 Accordingly, we limit our discussion to phase transitions 5. Effects of rare regions on the phase in which this coupling is known or suspected to be im- transition for an Ising system 621 portant. C. Metal-superconductor transition 621 The structure of this paper is as follows. In Sec. I we D. Quantum antiferromagnetic transition 624 give a brief introduction to phase transitions, both clas- V. Discussion and Conclusion 626 sical and quantum. In Sec. II we discuss the concept of A. Summary of review 626 generic scale invariance and illustrate it by means of B. Open problems and suggested experiments 627 various examples. These concepts are less well known 1. Generic scale invariance and soft modes 627 than those pertaining to phase transitions, so our expo- a. Generic non-Fermi-liquid behavior 627 sition is more elaborate. In Secs. III and IV we return to b. Soft modes due to nested Fermi surfaces 627 phase-transition physics and discuss the influence of ge- 2. Aspects of the quantum ferromagnetic neric scale invariance on classical and quantum transi- transitions 627 tions, respectively. We conclude in Sec. V with a sum- a. Disordered ferromagnets 627 mary and a discussion of open problems. b. Itinerant ferromagnets with magnetic impurities 627 I. PHASE TRANSITIONS c. Clean ferromagnets 628 3. Metal-superconductor transition 628 Phase transitions are among the most fascinating phe- a. Experimental study of critical behavior 628 nomena in nature. They also have far-reaching implica- b. Quantum phase transition to a gapless tions. The liquid-gas and liquid-solid transitions in water, superconductor 628 for instance, are common occurrences of obvious impor- 4. Aspects of the quantum antiferromagnetic tance. The transition from a paramagnetic phase to a transitions 628 ferromagnetic one in the elements iron, nickel, and co- a. Phenomenological theory of the quantum balt made possible the invention of the compass. The antiferromagnetic problem 628 structural transition in tin from the ␤ phase ͑white tin͒ to b. Simple quantum antiferromagnets 628 the ␣ phase ͑gray tin͒ is responsible for the degradation 5. Nonequilibrium quantum phase transitions 628 of tin artifacts below a temperature of about 286 K, 6. Other quantum phase transitions in which known as the tin pest. One could continue with a long generic scale invariance might play a role 628 list, but these three examples may suffice to demonstrate Acknowledgments 629 that phase transitions come in a wide variety of phenom- References 629 enologies with no entirely