This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid. The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space. Since the time variable lies in the finite interval [0,- iβ ], where β is the inverse temperature, the resulting description of a d -dimensional system shares some features with that of a ( d + 1)-dimensional classical system which has finite extent in one dimension. However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional m the same way. The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply. Of these, the itinerant ferromagnet is studied in greater detail. The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but 4 − z dimensions, where z depends on the way the frequency enters the generalized free-energy functional. When it does so m the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; z = 1. It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds z = 2, 3, and 4, respectively, (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d < 4) very close to the critical value of the coupling U c , when ( U − U c ) / U c ≪ ( T / U c ) 2 / z z / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2970.tif"/> is therefore the quantum-to-classical crossover exponent.
This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid. The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space. Since the time variable lies in the finite interval [0,- iβ ], where β is the inverse temperature, the resulting description of a d -dimensional system shares some features with that of a ( d + 1)-dimensional classical system which has finite extent in one dimension. However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional m the same way. The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply. Of these, the itinerant ferromagnet is studied in greater detail. The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but 4 − z dimensions, where z depends on the way the frequency enters the generalized free-energy functional. When it does so m the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; z = 1. It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds z = 2, 3, and 4, respectively, (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d < 4) very close to the critical value of the coupling U c , when ( U − U c ) / U c ≪ ( T / U c ) 2 / z z / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2970.tif"/> is therefore the quantum-to-classical crossover exponent.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .
The Landau Fermi liquid theory of metals has been so successful that exceptions to it draw a fair amount of attention. The theory rests on the assumption that the low‐energy properties of interacting electrons in a metal in three dimensions may be described in terms of weakly repulsive charged spin 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_1.tif"/> fermionic excitations— the quasiparticles. This is of course not generally expected to hold following a phase transition, for example, in the form of a Pomeranchuk instability or a superconducting transition. In Chapter 2 we showed that Pomeranchuk instabilities occur whenever some Landau F function satisfies the condition F l s ( a ) / ( 2 l + 1 ) = - 1 $ F_{l}^{s(a)} /(2l + 1) = - 1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_2.tif"/> , where l = 0, 1, 2, . . . denotes the relevant angular momentum channel where the instability occurs. The l = 0 instability in the spin‐symmetric channel is associated with a divergence of compressibility that leads to phase separation and consequent breakdown of the Fermi liquid. In the spin‐antisymmetric channel, the l = 0 instability corresponds to ferromagnetism. In the ferromagnetic quantum critical region (“classical” Gaussian regime in Hertz theory), we found in Sec. 9.6.2.3 that spin fluctuations lead to the destruction of the Landau Fermi liquid and this is evident, for example, in the ln ( 1 / T ) $ {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_3.tif"/> divergence of the specific heat coefficient γ ( T ) . Proximity to an electronic instability is however not a necessary condition for the breakdown of the Fermi liquid. In Sec. 10.2 we found that transverse electromagnetic fluctuations are only weakly screened by conduction electrons and that results in a γ ( T ) ∼ ln ( 1 / T ) $ \gamma (T) \sim {\text{ln }}(1/T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math13_4.tif"/> divergence for the specific heat coefficient without needing to be in the vicinity of a ferromagnetic quantum critical point 1 . The lack of screening of such electromagnetic fluctuations is a feature shared with low energy spin fluctuations in the Hertz theory of quantum critical ferromagnets. Another route to non Fermi‐liquid behavior, not reliant on the existence of long‐range interactions, appeared in Sec. 11.4 where we analyzed non‐optimally screened Kondo models. It was found that singularities in properties such as the susceptibility and specific heat coefficient may be expected in the event that residual low‐energy degeneracies associated with spin‐flip scattering with conduction electrons persist down to the lowest temperatures. Finally, weak potential disorder was found to preserve the Fermi liquid (in three dimensions) down to the lowest temperatures (see Chapter 8 ).
We discuss some problems related to quantum critical phenomena in superfluids and superconductors. In Ch. 1, we apply generalizations of hyperuniversality to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. In Ch. 2, we apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial fixed point. The onset of mean-field behavior for dimensions d > d[subscript]c = 4 is unconventional, yielding discontinuous exponents. Including positive temperatures, we give a clear picture of various crossover regimes. In Ch. 3, We give a detailed derivation of the critical thermodynamics of O(n) spin models, correct to O([epsilon] = 4 - d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique introduced by Rudnick and Nelson. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n [...] 2), deriving detailed expressions for the renormalized spin-wave stiffness and longitudinal susceptibility.
Although one-dimensional problems are only an idealisation of naturally occurring physical situations, and hence may seem somewhat artificial, motion of a single particle in one dimension yields exact solutions of the Schrödinger equation which highlight most of the features of the three-dimensional quantum approach, avoiding, however, its complexity. Because the time-dependent Schrödinger equation in one dimension: 6.1 i ℏ ∂ Ψ ( x , t ) ∂ t = − ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + U ( x ) Ψ ( x , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_1_B.tif"/> has the general solution, Eq.(5.62) , which is expressed in terms of the eigenvalues and eigenfunctions of the Hamiltonian operator only: 6.2 Ψ ( x , t ) = ∑ n a n e − i E n t / ℏ ψ n ( x ) + ∫ a ( E ) e − i E t / ℏ ψ ( x , E ) d E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_2_B.tif"/> the problem is always reduced to solving the Schrödinger wave equation: 6.3 − ℏ 2 2 m ∂ 2 ψ ( x ) ∂ x 2 + U ( x ) ψ ( x ) = E ψ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_3_B.tif"/> This is the energy eigenvalue equation, which can also be written as: 6.4 d 2 ψ ( x ) d x 2 + 2 m ℏ 2 [ E − U ( x ) ] ψ ( x ) = d 2 ψ ( x ) d x 2 + k 2 ( x ) ψ ( x ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_4_B.tif"/> by setting: 6.5 k 2 ( x ) = 2 m ℏ 2 [ E − U ( x ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_5_B.tif"/> 203 It follows that the energy eigenvalue problem is similar to the calculation of monochromatic waves in a medium with non-uniform refractive index. If E < U ( x ), it is clear that k ( x ) becomes imaginary, namely k { x ) = iγ (x), where: 6.6 γ 2 ( x ) = 2 m ℏ 2 [ U ( x ) − E ] , E < U ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_6_B.tif"/> For any given eigenvalue E, the second-order differential equation (6.4) yields two linearly independent solutions ψ 1 ( x ) and ψ 2 ( x ), suchthat: 6.7 ψ ( x ) = a 1 ψ 1 ( x ) + a 2 ψ 2 ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_7_B.tif"/> It follows that any eigenvalue might be at most two-fold degenerate. The solutions of interest are those which satisfy certain regularity conditions which make them admissible eigenfunctions: i ψ ( x ) and its first derivative should be continuous functions of x, for all x. This implies that we allow for finite discontinuities of the potential energy U ( x ) at some points, where d 2 ψ/dx 2 should exhibit an opposite finite discontinuity, so preventing the departure from continuity of the right-hand side of Eq.(6.1) , whose left-hand side ∂ Ψ /∂t is a continuous function. A step-like behaviour of d 2 ψ/dx 2 requires the continuity of dψ/dx at the same points, otherwise the discontinuity becomes infinite. ii ψ ( x ) should be finite or single-valued for all x. This ensures the possibility of normalizing the wave functions, according to Eq.(1.37) , and having a uniquely defined probability density || ψ ( x ) | 2 . The eigenfunctions ψ ( x ) can always be chosen to be real, and this becomes apparent by conjugation of Eq.(6.4) , which gives: d 2 ψ * ( x ) d x 2 + k 2 ( x ) ψ * ( x ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_8_B.tif"/> where k 2 ( x ) is always real. Because ψ ( x ) and ψ* ( x ) are eigenfunctions with the same eigenvalue E, their real and imaginary parts, namely: 6.8 ψ r ( x ) = ψ ( x ) + ψ * ( x ) 2 , ψ i ( x ) = ψ ( x ) − ψ * ( x ) 2 i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn6_9_B.tif"/> are real eigenfunctions corresponding to E. Hence, we will limit ourselves to the real solutions ψ ( x ) to Eq.(6.4) .
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.
The solution of Schrödinger’s equation for systems composed of a large number of strongly interacting particles is an unsolved problem. Indeed, even the classical three-body problem cannot be solved in general form. Fortunately, the problem of obtaining the Ψ-function of such a system is not only insoluble but unnecessary; such a detailed description of the many-body problem is not required by any experiment that is possible in real life. Any real experimental set-up contains a comparatively small number of particle indicators, and hence, in cases where a large number of particles take part in a given process, can give only their averaged characteristics. Any attempt to determine the coordinates of every particle of such a system would lead (because of the uncertainty relation) to a complicated excited state and would indeed change the properties of the system. Thus, the description of a macroscopic system by means of a wave function is an inadequate approach * : we need 262 instead methods of incomplete description, which determine only the relations between averaged quantities. An example of such a description is hydrodynamics, the equations of which determine only the average velocity of the particles at each point (i. e. the velocity field). A second example, closer to our problem, is the kinetic equation, which allows us to find the distribution of particles with respect to velocity and position. A knowledge of the one-particle distribution function f ( r ˜ , p ˜ , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1304.tif"/> , which depends on the coordinate r ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1305.tif"/> and velocity p ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1306.tif"/> of the particles, allows us to calculate the mean values of additive quantities such as the density n ( r ˜ ) = ∑ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1307.tif"/> or the momentum per unit volume j ˜ ( r ˜ ) = ∑ i p ˜ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1308.tif"/> . The two-particle distribution function f ( r ˜ 1 , r ˜ 2 ; p ˜ 1 , p ˜ 2 ; t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1309.tif"/> allows us to find the correlations between the coordinates and the velocities of two particles and to determine the averages of quantities which depend on the coordinates of two particles, for instance the mean value of the pairwise interaction energy V = ∑ ik V ( r ˜ i - r ˜ k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1310.tif"/> . The periodic or weakly damped solutions of the equation for the distribution function give the eigenfrequencies of the system.