## RESEARCH

### Quantum Critical Intinerant Antiferromagnets and Critical Kondo Destruction

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. It 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 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 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, where lattice degrees of freedom are well separated from those of electronic and magnetic correlations.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):

In many 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 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, where lattice degrees of freedom are well separated from those of electronic and magnetic correlations.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 CeCu

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 quantum impurity problem, the Bose-Fermi Kondo Model (BFKM). This approach has been termed the Extended Dynamical Mean Field Theory (EDMFT) [5].

Mapping of the Kondo lattice model onto an effective quantum impurity model augmented with self-consistency conditions within the

extended dynamical mean field theory.

Two groups recently succeeded in showing that the Kondo lattice indeed allows for a new type of quantum criticality, called Local Quantum Criticality (LQC). Their approaches were based on variants of a zero temperature method, the Numerical Renormalization Group (NRG) [6,7]. Quantum Monte Carlo at finite temperatures already suggested that the scaling function of LQC resembles the one observed for CeCu

.
_{5.9}Au_{0.1}e.g. displays scaling in terms of ω/T [3]. For YbRh2Si2, a similar scalingform for the single-particle Green function has recently been established [4].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 quantum impurity problem, the Bose-Fermi Kondo Model (BFKM). This approach has been termed the Extended Dynamical Mean Field Theory (EDMFT) [5].

Mapping of the Kondo lattice model onto an effective quantum impurity model augmented with self-consistency conditions within the

extended dynamical mean field theory.

Two groups recently succeeded in showing that the Kondo lattice indeed allows for a new type of quantum criticality, called Local Quantum Criticality (LQC). Their approaches were based on variants of a zero temperature method, the Numerical Renormalization Group (NRG) [6,7]. Quantum Monte Carlo at finite temperatures already suggested that the scaling function of LQC resembles the one observed for CeCu

_{5.9}Au_{0.1}[8,9]:There is also growing evidence, that
this new form of quantum criticality is realized in a number of heavy
fermion compounds [10,11,12]. 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. For sub-Ohmic bosonic baths, the model undergoes a Quantum Phase Transition as a function of the coupling constant to the bosonic degrees of freedom. 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 [13] 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. 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 [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 Reference [15]. In the spin rotational invariant case, it was argued that the Berry phase spoils the quantum-to-classical mapping [16]. The reasons for the Ising case are at present less clear and is subject of onging research [15].

An interesting question has however emerged regarding the nature of the QCP of the impurity problem without selfconsistency. For sub-Ohmic bosonic baths, the model undergoes a Quantum Phase Transition as a function of the coupling constant to the bosonic degrees of freedom. 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 [13] 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. 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 [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 Reference [15]. In the spin rotational invariant case, it was argued that the Berry phase spoils the quantum-to-classical mapping [16]. The reasons for the Ising case are at present less clear and is subject of onging research [15].

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] S. Friedemann, N. Oeschler, S. Wirth, C. Krellner, C. Geibel, F. Steglich, S. Paschen, S. Kirchner, and Q. Si: Fermi-surface collapse and dynamical scaling near a heavy-fermion quantum critical point,

unpublished (2009).

[5] Q. Si, and S. Rabello, and K. Ingersent, and J. Smith: Locally critical quantum phase transitions in strongly

correlated metals, Nature 413, 804, (2001).

[6] 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).

[7] M. Glossop and K. Ingersent: Magnetic Quantum Phase Transition in an Anisotropic Kondo Lattice, Phys. Rev. Lett. 99, 227203 (2007).

[8] D. Grempel and Q. Si: Locally Critical Point in an Anisotropic Kondo Lattice, Phys. Rev. Lett. 91, 026401 (2003).

[9] J. Zhu and D. Grempel and Q. Si: Continuous Quantum Phase Transition in a Kondo Lattice Model, Phys. Rev. Lett. 91, 026401 (2003).

[10] S. Paschen et al.: Hall-effect evolution across a heavy-fermion quantum critical point, Nature, 432, 881 (2004).

[11] P. Gegenwart et al.: Multiple energy scales at a quantum critical point, Science 315, 1049 (2007).

[12] P. Gegenwart and Q. Si and F. Steglich: Quantum criticality in heavy-fermion metals, Nature Physics 4, 186 (2008).

[13] 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).

[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, Q. Si, and K. Ingersent: Finite Size Scaling of Classical Long-Ranged Ising Chains and the Criticality of Dissipative Quantum Impurity Models, Phys. Rev. Lett. 102 026403 (2009).

[16] S. Kirchner and Q. Si: Berry Phase and the Breakdown of the Quantum to Classical Mapping for the Quantum Critical Point of the Bose-Fermi Kondo model, arXiv:0808.2647 (2008).

### Quantum Criticality out of Equilibrium

Quantum phase transitions are of
extensive current interest in

a variety of strongly correlated electronic and atomic systems [1]. A quantum critical point occurs when such transitions are second order [2]. In a quantum critical state, there is no intrinsic energy scale in the excitation spectrum. External probes will readily drive the system out of equilibrium, such that the standard notion of linear response

breaks down. The response of such a system to an external drive cannot be directly related to its intrinsic fluctuations in equilibrium.

Compared to their classical counterparts, quantum critical systems are much more difficult to study already at the equilibrium level. A quantum critical point involves the mixing of statics and dynamics. This complicates the determination of the equilibrium fluctuation spectrum, especially in the long-time ``quantum relaxational'' regime (frequency ω << temperature T) at finite temperatures [2]. To make progress, we need approaches that can, on the one hand, capture the scaling properties of a quantum critical point in equilibrium, and, on the other hand, be extended to settings out of equilibrium. There is a considerable need for insights into many open issues regarding quantum critical systems out of equilibrium:

How do we connect the scaling of the
equilibrium fluctuationa variety of strongly correlated electronic and atomic systems [1]. A quantum critical point occurs when such transitions are second order [2]. In a quantum critical state, there is no intrinsic energy scale in the excitation spectrum. External probes will readily drive the system out of equilibrium, such that the standard notion of linear response

breaks down. The response of such a system to an external drive cannot be directly related to its intrinsic fluctuations in equilibrium.

Compared to their classical counterparts, quantum critical systems are much more difficult to study already at the equilibrium level. A quantum critical point involves the mixing of statics and dynamics. This complicates the determination of the equilibrium fluctuation spectrum, especially in the long-time ``quantum relaxational'' regime (frequency ω << temperature T) at finite temperatures [2]. To make progress, we need approaches that can, on the one hand, capture the scaling properties of a quantum critical point in equilibrium, and, on the other hand, be extended to settings out of equilibrium. There is a considerable need for insights into many open issues regarding quantum critical systems out of equilibrium:

spectrum with that of the out-of-equilibrium properties?

Is there any problem in which the universal out-of-equilibrium scaling functions of a quantum critical point can be determined?

Is the fluctuation-dissipation theorem violated in a universal

way in the quantum critical regime?

A unique probe to explore quantum
criticality in and out of equilibrium has recently been described in
the Proceedings of the National Academy of Sciences [3], a transistor
with an active channel measuring just a few billionths of meter across.

Schematic representation of the single electron transistor with ferromagnetic leads. The magnetization in the source and drain gates are anti-aligned (red arrows).

It uses a pair of electrodes made of ferromagnetic metal, between which a single electron will be trapped. When the electron repeatedly "tunnels" between the electrodes, it can create a quantum critical state in the

electrodes where the particle is trapped. The usage of the ferromagnetic electrodes in the proposed probe brings in spin waves,

which couple to the local magnetic moment of the molecule as a fluctuating magnetic field. It is this coupling that gives rise to the ability to tune the degree of -- and even destroy -- the magnetic quantum entanglement. The effect is manifested in the unique way that the electrical conductance of the transistor depends on temperature and frequency. The traditional theory of metals breaks down completely in matter that exists in such a 'quantum critical state'. This quantum criticality is characterized by the inherent quantum effect of entanglement, and the nanoscale magnetic probe we've proposed could provide a controlled and tunable setting to study entanglement at a quantum critical point.

Phase diagram of the magnetic single electron transistor (SET). J and g denote the coupling constants to the lead electrons and spin waves respectively. The red line separates the flow to the strong coupling fixed point (Kondo) from that to the critical local moment fixed point (LM). The gate voltage (blue dashed line) takes the SET from one phase to the other through a quantum phase transition.

Current carrying (i.e. out-of-equilibrium) quantum critical states are straightforwardly created in such a probe by applying a bias voltage across the two electrodes. This identifies a sytem in which quantum

criticality out of equilibrium can be systematically studied. In this model system, universal scaling is obeyed by the steady state current, the pertinent spectral densities, and the associated fluctuation-dissipation ratios [4].

Our theoretical approach can be extended to study the transient behavior and other non-equilibrium probes of quantum criticality.

Schematic representation of the single electron transistor with ferromagnetic leads. The magnetization in the source and drain gates are anti-aligned (red arrows).

It uses a pair of electrodes made of ferromagnetic metal, between which a single electron will be trapped. When the electron repeatedly "tunnels" between the electrodes, it can create a quantum critical state in the

electrodes where the particle is trapped. The usage of the ferromagnetic electrodes in the proposed probe brings in spin waves,

which couple to the local magnetic moment of the molecule as a fluctuating magnetic field. It is this coupling that gives rise to the ability to tune the degree of -- and even destroy -- the magnetic quantum entanglement. The effect is manifested in the unique way that the electrical conductance of the transistor depends on temperature and frequency. The traditional theory of metals breaks down completely in matter that exists in such a 'quantum critical state'. This quantum criticality is characterized by the inherent quantum effect of entanglement, and the nanoscale magnetic probe we've proposed could provide a controlled and tunable setting to study entanglement at a quantum critical point.

Phase diagram of the magnetic single electron transistor (SET). J and g denote the coupling constants to the lead electrons and spin waves respectively. The red line separates the flow to the strong coupling fixed point (Kondo) from that to the critical local moment fixed point (LM). The gate voltage (blue dashed line) takes the SET from one phase to the other through a quantum phase transition.

Current carrying (i.e. out-of-equilibrium) quantum critical states are straightforwardly created in such a probe by applying a bias voltage across the two electrodes. This identifies a sytem in which quantum

criticality out of equilibrium can be systematically studied. In this model system, universal scaling is obeyed by the steady state current, the pertinent spectral densities, and the associated fluctuation-dissipation ratios [4].

Our theoretical approach can be extended to study the transient behavior and other non-equilibrium probes of quantum criticality.

References

[1] Focus issue: Quantum phase transitions, Nature Physics 4, pp167-204,(2008).

[2] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999).

[3] S. Kirchner et al., Proc.Natl.Acad.Sci. USA 102, 18824 (2005).

[4] S. Kirchner and Q. Si, Quantum Criticality out of Equilibrium: Steady State in a Magnetic Single-Electron Transistor,

arXiv:0805.3717 (2008).

CONTACT

Stefan Kirchner

Max-Planck-Institute for the

Physics of Complex Systems

Noethnitzer Str. 38

D-01187 Dresden

Germany

kirchner(at)mpipks-dresden.mpg.de

Stefan Kirchner

Max-Planck-Institute for the

Physics of Complex Systems

Noethnitzer Str. 38

D-01187 Dresden

Germany

kirchner(at)mpipks-dresden.mpg.de