Ferromagnetic quantum criticality
MPI-CPfS co-workers: M. Brando, M. Baenitz, C. Geibel, S. Hamann, R. Kuechler, T. Luehmann, M. Nicklas, J. Sichelschmidt, F. Steglich, A. Steppke
The general concept of quantum criticality has become one of the foundations for the study of strongly correlated electron physics. In antiferromagnets the situation has been clear and consistent; many antiferromagnetic quantum critical systems, both localized and itinerant, have been identified. For ferromagnetic systems the path to understanding has been more challenging. At first, no theoretical barriers to ferromagnetic quantum criticality were foreseen, but first order transitions rather than critical points were seen in real systems. Then, just over ten years ago, seminal work by Belitz and Kirkpatrick showed why this was the case, and led to speculation that ferromagnetic criticality would never be observable. Using thermodynamic techniques for investigating quantum criticality that were invented in this Institute, and materials grown here, we first demonstrated close proximity to ferromagnetic quantum criticality in some example systems, and then the existence of the first known metallic ferromagnetic quantum critical point in YbNi4(P1-xAsx)2. Furthermore, this project has led to the discovery of a new class of ferromagnets in which the magnetic moments surprisingly prefer to align along the magnetically hard direction instead of ordering in the conventional way along the easy direction, establishing the foundations for a new area of future research.
While classical phase transitions are driven by thermal fluctuations and have been extensively studied, much current interest focuses on continuous quantum phase transitions (QPTs), which occur at zero temperature and are driven by quantum fluctuations between competing ground states of matter. The point at which the QPT takes place is known as quantum critical point (QCP). Here, several novel states of matter, including unconventional superconductivity or spin-liquids were discovered. QCPs can be generally revealed in experiments when a material is continuously tuned by means of an external non-thermal parameter (typically pressure, magnetic field or chemical substitution) between the competing ground states at T = 0. In case of a ferromagnet, a ferromagnetic (FM) QCP exists when it is possible to tune the Curie transition temperature, TC, continuously to zero where a second order QPT takes place. The FM QPT is historically the first one that was studied. Indeed, the earliest theory of a QPT was the Stoner theory of ferromagnetism that was revisited by John Hertz in the early 1970s. For a comprehensive review see Ref. 
Although there is clear evidence for the existence of antiferromagnetic (AFM) QCPs, the FM QCP case is controversial. A long-standing question is whether a FM QCP generally exists and, if not, which are the possible ground states of matter that replace it. In recent years, substantial experimental and theoretical efforts have been made to further investigate this problem in metallic systems, with groups from this Institute playing a major role. According to these recent studies it seems that a FM QCP can exist, but only under special circumstances. On theoretical grounds, it was shown that in 2D and 3D itinerant systems the QPT from the paramagnetic to the ferromagnetic phase in the absence of quenched disorder is inherently unstable, either towards a first order phase transition or towards inhomogeneous magnetic phases (modulated or textured structures). This conclusion, proposed in the late 1990s  has been confirmed by different theoretical approaches . The physics underlying this important result is a coupling of the magnetization to electronic soft modes that exist in any 2D and 3D metal, which leads to a fluctuation-induced first-order transition. Naturally, this mechanism would not work for 1D metals. In real materials, several clean (stoichiometric) magnetic transition-metal ferromagnets, like MnSi or ZrZn2 indeed show that the transition changes to first order as the QCP is approached, as predicted by theory [4,5]. The generic phase diagram of these systems is shown in Fig. 1a in the space spanned by temperature T, magnetic field H and the control parameter pressure p or chemical substitution x. The existence of a first order QPT implies the presence of a tricritical point (TCP) and surfaces of first-order transitions (tricritical wings) that vanish at quantum critical end points at finite magnetic field.
On the other hand, disordered 2D or 3D systems are much more complicated, depending on the disorder strength and the distance from the QPT. In many disordered materials the QPT is continuous (Fig. 1b), or second order, but clear evidence of the presence of a FM QCP is still missing. In other systems the transition from the ferromagnetic state at low temperatures is to a different type of long-range order, such as an antiferromagnetic or a spin-density-wave state (Fig. 1c). In still other materials a transition to a state with glass-like spin dynamics is suspected (Fig. 1d).
We have contributed by investigating several 3d- and 4f-electron systems. Here, a few examples: In the 4f-electron systems CeRuPO and Yb(Rh1-xCox)2Si2 [6,7] and in the 3d-electron system Nb1−yFe2+y  we discovered that FM order changes into AFM when the order temperature is suppressed to zero by chemical substitution or hydrostatic pressure. We found no FM QCP in another 4f-electron compound, CeFePO, but rather some short-range order with a particular texture whose nature is the subject of ongoing investigations [9,10]. More importantly, we discovered the first FM stoichiometric system, YbNi4P2, with the lowest Curie temperature TC = 0.15 K ever observed [11,12]. This material possesses such a low TC because of a delicate interplay between the Kondo effect, the magnetocrystalline anisotropy and, more importantly, its crystalline and electronic quasi-1D structures. The fact that the mechanism responsible for the presence of a first order phase transition is weak in quasi-1D systems motivated us to look for the presence of a FM QCP in YbNi4(P1-xAsx)2 in which As substitution acts as negative pressure. We indeed found the first FM QCP in a metal  and demonstrated that the 1D character might be the key towards FM quantum criticality.
Furthermore, in CeRuPO, Yb(Rh1-xCox)2Si2 and YbNi4(P1-xAsx)2 the FM order is of a particular type: Despite the strong crystalline electric field anisotropy, the magnetic moments surprisingly align along the magnetically hard direction. This behavior is quite uncommon in magnetic systems and has been observed in just other two ferromagnetic Kondo-lattice systems to date, CeAgSb2 and YbNiSn. The reason for it is not yet clear. It might be the result of competing interactions  or, more excitingly, due to the action of spin fluctuations  which are strong near QCPs. Similar effects are known to exist in nature. A typical example is the 'Kapitza pendulum', a multiply-jointed pendulum that can be stabilized in an inverted configuration if it is driven with random horizontal fluctuating forces at its support. The fact that these five ferromagnets are all Kondo-lattice systems close to a FM instability at zero temperature may be not just a pure accident. The results to date have laid the foundations for a new area of future research on the stabilization of order by fluctuations, which we will pursue during the coming years.
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 Avoided Ferromagnetic Quantum Critical Point: Unusual Short-Range
Ordered State in CeFePO; S. Lausberg, J. Spehling, A. Steppke, A. Jesche,
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 Ferromagnetic quantum criticality in the quasi-one-dimensional heavy
fermion metal YbNi4P2; C. Krellner, S. Lausberg, A. Steppke, M. Brando,
L. Pedrero, H. Pfau, S. Tence, H. Rosner, F. Steglich, and C. Geibel,
New J. Phys. 13, 103014 (2011). MPG.PuRe
 Ferromagnetic Quantum Critical Point in the Heavy-Fermion Metal
YbNi4(P1-xAsx)2; A. Steppke, R. Küchler, S. Lausberg, E. Lengyel,
L. Steinke, R. Borth, T. Lühmann, C. Krellner, M. Nicklas, C. Geibel,
F. Steglich and M. Brando, Science 339, 933 (2013). MPG.PuRe
 Competing orders, competing anisotropies, and multicriticality: The case
of Co-doped YbRh2Si2; E. C. Andrade, M. Brando, C. Geibel, and M. Vojta,
Phys. Rev. B 90, 075138 (2014). MPG.PuRe
 Fluctuation-Driven Magnetic Hard-Axis Ordering in Metallic Ferromagnets;
F. Krueger, C. J. Pedder, and A. G. Green,
Phys. Rev. Lett. 113, 147001 (2014).
In collaboration with:
- D. Belitz, Department of Physics and Materials Science Institute, University of Oregon, Eugene, Oregon 97403, USA
- F. M. Grosche, University of Cambridge, Cavendish Laboratory, CB3 0HE Cambridge, UK
- A. Jesche, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, D-86159 Augsburg, Germany
- T. R. Kirkpatrick, Institute for Physical Science and Technology, and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
- H. H. Klauss, Tech Univ Dresden, Inst Solid State Phys, D-01069 Dresden, Germany
- C. Krellner, Goethe Univ Frankfurt, Inst Phys, Kristall & Mat Lab, Max von Laue Str 1, D-60438 Frankfurt, Germany
- C. Pfleiderer, Physik Department E21, Technische Universitaet Muenchen, D-85748 Garching, Germany
- S. Suellow, Institute of Condensed Matter Physics, Technische Universitaet Braunschweig, D-38106 Braunschweig, Germany
- M. Vojta, Institut fuer Theoretische Physik, Technische Universitaet Dresden, 01062 Dresden, Germany