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Critical Kondo Destruction and the violation of the quantum-to-classical mapping of quantum criticality
Quantum Phase Transitions (QPT) occur when a system changes its state of matter at zero temperature. Such states of matter can be characterized by an order parameter, a quantity that changes across the transition, typically from zero in the 'disordered phase' to a finite value in the 'ordered phase'. Of particular interest are continuous quantum phase transitions where the order parameter vanishes continuously upon approaching the QPT from the ordered side. In this case, the transition is often called quantum critical since a number of physical quantities diverge at the critical point in a powerlaw fashion. The associated exponents are characteristic for the transition. An important quantity that grows to infinity at the transition is the correlation length. Is is a measure for how intertwined or correlated spatially separated parts of the system are. The order parameter fluctuates on all length scales shorter than the correlation length. In many if not all aspects, QPT seem to resemble their classical counterparts of finite temperature phase transitions. This physical intuition is condensed in the theoretical approach to the problem that maps the problem onto a classical one: The quantum critical properties are believed to be entirely captured by a classical Ginzburg-Landau-Wilson type functional of only the order parameter, describing its fluctuations in elevated dimensions. The fact that the fluctuations occur in elevated dimensions captures quantum mechanical aspects of the transition. We will refer to this approach as the “quantum-to-classical mapping of quantum criticality”.
A class of systems in which the existence of a QPT has been experimentally established are Heavy Fermion compounds. Some of these rare earth and actinide metal compounds can be tuned from a paramagnetic state through a QPT into an antiferromagnetic state by changing pressure, composition of the compound or an external field. The quantum-to-classical mapping of quantum criticality in this context is known as the Hertz-Millis or Spin-Density-Wave (SDW) scenario[1,2]. It predicts, that the order parameter susceptibility at the ordering wavevector of the antiferromagnetism depends in a specific way on frequency (ω) and temperature (T):
These predictions can be tested in experiments with the help of e.g. inelastic neutron scattering. It turns out, however, that antiferromagnetic heavy fermion metals close to their quantum critical points display a richness in their physical properties unanticipated by the this approach to quantum criticality. Some compounds do follow the predictions made by the SDW-scenario, but many do not. The order parameter susceptibility of CeCu5.9Au0.1 e.g. displays scaling in terms of ω / T[3].
This has led to the question as to how the Kondo effect, the local screening of f-moments by the conduction electrons, gets destroyed as the system undergoes a phase change. In one approach to the problem, Kondo lattice systems are studied through a self-consistent Bose-Fermi Kondo Model (BFKM) within the Extended Dynamical Mean Field Theory (EDMFT) [4]. Figure 1 shows a sketch of the selfconsistent mapping from the Kondo lattice to an quantum impurity problem, the BFKM.
Two groups recently succeeded in showing that the Kondo lattice indeed allows for a new type of quantum criticality, termed Local Quantum Criticality (LQC). Their approaches were based on variants of a zero temperature method, the Numerical Renormalization Group[5,6]. Quantum Monte Carlo at finite temperatures already suggested that the scaling function of LQC resembles the one observed for CeCu5.9Au0.1 [7,8]:
There is also growing evidence, that this new form of quantum criticality is realized in a number of Heavy Fermion compounds [9,10,11]. How does LQC manage to give results different from the ones obtained through the quantum to classical mapping? The QCP of the Kondo lattice is selfconsistently mapped onto the one of the Bose-Fermi Kondo model. The selfconsistency condition has to be implemented on a microscopic level and this prevents at this stage a formulation in terms of only the order parameter fluctuations.
An interesting question has however emerged regarding the nature of the QCP of the impurity problem without selfconsistency (see Figure 2).
If the bosonic bath spectrum is sub-Ohmic bosonic ( 
) the BFKM can become quantum critical as a function of the coupling constants J and g.
In Reference [12] it was shown, that a large-N limit of the BFKM, with the symmetry group extended from SU(2) to SU(N) does have a critical point with scaling functions that display ω / T-scaling, see Figure 3, where the dynamical susceptibility for ε=2/3 is shown.
This implies that the quantum-to-classical mapping of quantum criticality fails for the BFKM. Similar scaling properties have been found for the Ising-anisotropic BFKM at criticality with the NRG[13,14]. The quantum-to-classical mapping maps the BFKM to a one-dimensional spin-chain with long-ranged interaction related to the sub-Ohmic bath where -for certain parameter ranges- no ω/T-scaling is observed, see Figure 4, and Reference [15].
In the spin rotational invariant case, it was argued that the Berry phase spoils the quantum-to-classical mapping [12]. The reasons for the Ising case are at present less clear and is subject of onging research.
References
[1] J. Hertz: Quantum critical phenomena, Phys. Rev. B 14, 1165 (1976).
[2] A. J. Millis: Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Phys. Rev. B 48, 7183 (1993).
[3] A. Schröder et al.: Onset of antiferromagnetism in heavy-fermion metals, Nature 407, 351, (2000).
[4] Q. Si, and S. Rabello, and K. Ingersent, and J. Smith: Locally critical quantum phase transitions in strongly correlated metals, Nature 413, 804, (2001).
[5] J.-X. Zhu and S. Kirchner and R. Bulla and Q. Si: Zero-Temperature Magnetic Transition in an Easy-Axis Kondo Lattice Model, Phys. Rev. Lett. 99, 227204 (2007).
[6] M. Glossop and K. Ingersent: Magnetic Quantum Phase Transition in an Anisotropic Kondo Lattice, Phys. Rev. Lett. 99, 227203 (2007).
[7] D. Grempel and Q. Si: Locally Critical Point in an Anisotropic Kondo Lattice, Phys. Rev. Lett. 91, 026401 (2003).
[8] J. Zhu and D. Grempel and Q. Si: Continuous Quantum Phase Transition in a Kondo Lattice Model, Phys. Rev. Lett. 91, 026401 (2003).
[9] S. Paschen et al.: Hall-effect evolution across a heavy-fermion quantum critical point, Nature, 432, 881 (2004).
[10] P. Gegenwart et al.: Multiple energy scales at a quantum critical point, Science 315, 1049 (2007).
[11] P. Gegenwart and Q. Si and F. Steglich: Quantum criticality in heavy-fermion metals, Nature Physics 4, 186 (2008).
[12] L. Zhu and S. Kirchner and Q. Si and A. Georges: Quantum critical properties of the Bose-Fermi Kondo Model in a large-N limit, Phys. Rev. Lett. 93, 267201 (2004).
[13] M. Vojta and N.-H. Tong and R. Bulla: Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping, Phys. Rev. Lett. 94, 070604 (2005).
[14] M. Glossop and K. Ingersent: Numerical Renormalization-Group Study of the Bose-Fermi Kondo Model, Phys. Rev. Lett. 95, 067202 (2005).
[15] S. Kirchner and Q. Si: Finite Size Scaling of Classical Long-Ranged Ising Chains and the Criticality of Dissipative Quantum Impurity Models, arXiv:0808.0916,(2008).
