Quantum and classical phase transitions in electronic systems

– The quantum ferromagnetic transition of itinerant electrons is considered. It isshown that the Landau-Ginzburg-Wilson theory described by Hertz and others breaks down dueto a singular coupling between fluctuations of the conserved order parameter. This couplinginduces an effective long-range interaction between the spins of the form1/r. It leadsto unusual scaling behavior at the quantum critical point in 1 < d ≤ 3 dimensions, which isdetermined exactly. One of the most obvious examples of a quantum phase transition is the ferromagnetictransition of itinerant electrons at zero temperature T as a function of the exchange couplingbetween the electron spins. Hertz [1] derived a Landau-Ginzburg-Wilson (LGW) functionalfor this case in analogy to Wilson’s treatment of classical phase transitions, and analyzed it bymeans of renormalization group methods. He found that the critical behavior in dimensionsd = 3, 2 is mean-field like, since the dynamical critical exponent z decreases the upper criticaldimensiondc compared to the classical case. In a quest for nontrivial critical behavior, Hertzstudied a model with a magnetization confined to d < 3 dimensions, while the coefficients inthe LGW functional are those of a 3-d Fermi gas. For this model he concluded thatdc = 1,and performed a 1− ǫ expansion to calculate critical exponents in d < 1. Despite the artificialnature of his model, there is a general belief that the qualitative features of Hertz’s analysis,in particular the fact that there is mean-field–like critical behavior for all d > 1, apply to realitinerant quantum ferromagnets as well. In this letter we show that this belief is mistaken, since the LGW approach breaks down dueto the presence of soft modes in addition to the order parameter fluctuations, viz spin-tripletparticle-hole excitations that are integrated out in the derivation of the LGW functional. Thesesoft modes lead to singular vertices in the LGW functional, invalidating the LGW philosophy c© Les Editions de Physique Europhys. Lett. 36 (1996), 191–6, cond-mat/951014659 192EUROPHYSICS LETTERS of deriving an effective local field theory in terms of the order parameter only (). In Hertz’soriginal model this does not change the critical behavior in d > 1, but it invalidates his 1− ǫexpansion. More importantly, in a more realistic model the same effect leads to nontrivialcritical behavior for 1 < d ≤ 3, which we determine exactly.Our results for realistic quantum magnets can be summarized as follows. The magnetization,m, at T = 0 in a magnetic field H is given by the equation of state tm + vm + um = H ,(1) where t is the dimensionless distance from the critical point, and u and v are finite numbers.From (1) one obtains the critical exponents β and δ, defined by m ∼ t and m ∼ H,respectively, at T = 0. For β and δ, and for the correlation length exponent ν, the orderparameter susceptibility exponent η, and the dynamical exponent z, we find β = ν = 1/(d− 1), η = 3 − d, δ = z = d (1 < d < 3),(2) and β = ν = 1/2, η = 0, δ = z = 3 for d > 3. These exponents “lock into” mean-field valuesat d = 3, but have nontrivial values for d < 3. In d = 3, there are logarithmic corrections topower law scaling. Equation (1) applies to T = 0. At finite temperature, we find homogeneitylaws for m, and for the magnetic susceptibility,χm, m(t, T,H) = bm(tb , T b , Hb) ,(3a) χm(t, T,H) = bχm(tb 1/ν , T b , Hb) ,(3b) where b is an arbitrary scale factor. The exponent γ, defined by χm ∼ t −γ at T = H = 0,and the crossover exponent φ that describes the crossover from the quantum to the classicalHeisenberg fixed point (FP) are given by γ = β(δ − 1) = 1 , φ = ν ,(4) for all d > 1. Notice that the temperature dependence of the magnetization is not given bythe dynamical exponent. However, z controls the temperature dependence of the specific-heat coefficient, γV = cV /T , which has a scale dimension of zero for all d, and logarithmiccorrections to scaling for all d < 3 (), γV (t, T,H) = Θ(3 − d) ln b + γV (tb 1/ν , T b, Hb) .(5) Equations (1)-(5) represent the exact critical behavior of itinerant quantum Heisenberg ferro-magnets for all d > 1 with the exception of d = 3, where additional logarithmic corrections toscaling appear. We are able to obtain the critical behavior exactly, yet it is not mean-field like.The exactness is due to the fact that we work above the upper critical dimensiondc = 1. Thenontrivial exponents are due to a singular coupling between the critical modes which leads,e.g., to the unusual term ∼ v in (1). Experimentally, we predict that for 3-d magnets witha very low Tc there is a crossover from essentially mean-field quantum behavior to classicalHeisenberg behavior. In d = 2, where there is no classical transition, we predict that withdecreasing T , long-range order will develop, and the quantum phase transition at T = 0 willdisplay the nontrivial critical behavior shown above. () We use the term “LGW theory” in the narrow sense, in which it is usually used in the literature,of an effective field theory in terms of the order parameter field only.() Wegner [2] has shown how “resonance” conditions between critical exponents lead to logarithmiccorrections to scaling. Their appearance for a whole range of dimensionalities in (5) is a consequenceof the exact relation z = d. 60Reprint 3: Breakdown of Landau-Ginzburg-Wilson theory . . . t. vojta et al.: breakdown of landau-ginzburg-wilson theory for etc. 193 We now sketch the derivation of these results. A more complete account of the technicaldetails will be given elsewhere [3]. We consider a d-dimensional continuum model of interactingelectrons, and pay particular attention to the particle-hole spin-triplet contribution [4] to theinteraction term in the action, Sint, whose (repulsive) coupling constant we denote by J .Writing only the latter explicitly, and denoting the spin density byns, the action reads S = S0 + S tint = S0 + (J/2)∫ dx ns(x) · ns(x) ,(6) where S0 contains all contributions to the action other than S tint. In particular, it contains theparticle-hole spin-singlet and particle-particle interactions, which will be important for what