Quantum Criticality

College of Science and Engineering

School of Physics & Astronomy

Quantum Criticality

Keith Clydesdale

Chiara Decaroli

Richard Duffy

Eric Edmund

September 2013 – February 2014

Supervisor: Dr. Chris Stock

Abstract

Quantum criticality arises in a material at temperatures low enough that thermalfluctuations are dominated by quantum mechanical fluctuations. Following years oftheoretical postulations it has only recently started to be rigorously tested experi-mentally and as a result much controversy still surrounds the field.

Many theoretical and experimental studies on previously-explained materials haveattempted to determine whether quantum criticality could be the correct explana-tion for certain exotic behaviours. In the field of condensed matter physics, quantumcriticality is attracting a significant amount of interest, due to the numerous po-tential real-world applications of novel materials in areas such as memory storageand magnetic refrigeration. In this report an account of quantum criticality will begiven and some of the examples discussed.

Contents

1 Introduction 11.1 Classical Phases of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quantum Critical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Non-Classical Phases of Matter . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theoretical Models of Quantum Criticality 52.1 Universal Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The Kondo/Anderson Impurity Model . . . . . . . . . . . . . . . . . . . . 62.5 Luttinger Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Concentration-Induced Quantum Phases 93.1 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Memory Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Magnetically-Induced Quantum Phases 114.1 The Triangular-Lattice Antiferromagnet . . . . . . . . . . . . . . . . . . . 114.2 The Graphene-like BC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 The Spin-half Antiferromagnet K2NaCrO8 . . . . . . . . . . . . . . . . . . 144.4 Magnetic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Pressure-Induced Quantum Phases 165.1 Pressure-Dependent Ising Coupling . . . . . . . . . . . . . . . . . . . . . . 175.2 The Antiferromagnetic Transition in TlCuCl3 . . . . . . . . . . . . . . . . 175.3 The MnSi Ferromagnetic State . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 The UGe2 Superconducting Phase . . . . . . . . . . . . . . . . . . . . . . . 215.5 Superconducting Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Chemically-Induced Quantum Phases 226.1 Spontaneous Magnetic Ordering due to Pair Breaking . . . . . . . . . . . . 226.2 Doping-Induced Quantum Critical Phenomena . . . . . . . . . . . . . . . . 236.3 The Cuprate Pseudogap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 Cuprate Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Applications of Cuprate Superconductors . . . . . . . . . . . . . . . . . . . 27

7 Conclusions 28

Acknowledgements 29

References 29

Quantum Criticality Introduction

1 Introduction

As a result of recent developments in the theoretical modelling of quantum criticality,condensed matter researchers have been inspired to experimentally investigate the effectof quantum phase transitions in materials. Despite the theoretical background not beingcompletely understood, there are already many examples of experiments proving theexistence of quantum criticality and assessing the properties of materials at quantumphase transitions. In many cases it has been observed that instead of adhering to well-known states, novel phases of matter arise near quantum critical points. Such phasesexhibit characteristics which are unusual and unconventional, and are therefore regardedas exotic and novel.

Experimental studies in this field have been developed substantially in recent years andseveral Nobel Prizes have been awarded already for these discoveries. However, practicalapplications of quantum criticality are currently few and far between, as the subject stillremains largely theoretical. Among the fields where the theory may prove very useful inthe coming years include memory storage, magnetic refrigeration and superconductingnanowires for use in quantum computers.

In order to describe novel quantum phases of matter, a closer look at what a phaseof matter is and what distinguishes quantum phases from classical phases is required.The following sections will describe the main features of phases of matter, introducing adistinction between classical phases of matter and quantum phases of matter.

The term ‘phase’ is usually used to denote a thermodynamic system in which the physicalproperties remain uniform. Often a phase of matter is identified as a state of matter. Afurther distinction can be made between classical and non-classical states of matter.

Among the classical phases of matter are the states which are encountered in everydaylife, such as liquids, solids, gases and plasmas. The non-classical states of matter are lesswell-known and have only recently been studied.

1.1 Classical Phases of Matter

Despite the wide variety of phases which fall under the classical regime, it is possibleto identify several common properties. Considering the phase diagram for a material inthe pressure-temperature (P-T) plane in Fig. 1 below, many general features may beobserved.

As it can be seen from Fig. 1, the common features of classical phases of matter are:the presence of a co-existence curve – a line separating two phases differing by theirmacroscopic properties; and the presence of a critical point – the point at which theline of co-existence terminates. A phase transition occurs when there is a change inthe macroscopic properties of the system as a tuning parameter, such as temperature,pressure or chemical potential is varied.

The order parameter, O, is used to distinguish one phase from another before and afterthe transition. It is usually a thermodynamic quantity that characterises the system.For instance, considering a transition going from a liquid to a vapour or vice versa,i.e. crossing the blue co-existence curve in Fig. 1, the order parameter would be the

1


density difference. The tuning parameter in this case is either pressure (if crossing theco-existence curve vertically) or temperature (if crossing horizontally).

Figure 1. A classical phase diagram for a single material. The co-existence curves arecoloured red, blue and green [1].

1.2 Quantum Critical Behaviour

As in classical phase transitions, quantum phase transitions can be furthermore dis-tinguished into two categories. In quantum systems, first-order or discontinuous phasetransitions display a quantum critical region that may be either a co-existence of the twophases or a completely new phase. Second-order or continuous quantum phase transi-tions display quantum critical points. In a first-order transition, the tuning parameter,g, jumps discontinuously crossing the co-existence curve, whereas in second-order phasetransitions, g tends to zero in the passage through the critical point, g = gc [2].

Quantum phase transitions occur in correlated electron materials – materials in whichelectrons occupy d or f -orbitals, in which Coulomb repulsion is increased as these or-bitals have a smaller spatial extent. The electrons in these materials must therefore becorrelated so that they stay apart from each other. These materials can be driven into adifferent phase by varying a tuning parameter such as the concentration, strength of anexternal magnetic field, the pressure, or the density of the electrons in the solid (variedby chemical doping). These will all be discussed in detail in the following sections.

For a continuous phase transition, the wavefunction at g = gc is different from a productstate wavefunction. It is described by a non-trivial superposition of an exponentially largeset of configurations fluctuating at all length scales with long range quantum entanglement[3]. Points not too far from g = gc are characterised by a ground state spin coherence

2


length, ξ. At length scales smaller than ξ, the electrons have not ‘decided’ which side ofthe phase transition they are on and so the wavefunction resembles the entangled state.At length scales larger than ξ the wavefunction is a simple product.

Continuous phase transitions are parametrised by the critical exponents. These describethe way different properties behave in second-order phases. It has been observed thatdifferent systems often show the same set of critical exponents, this property is knownas universality. The coherence length is an example of such a critical exponent whichdescribes the dependence of the order parameter and its correlations with the distancefrom the critical point and with the field conjugate to the order parameter [4].

A simple model of second-order phase transitions is given by Landau theory. This isan expansion of the Helmholtz free energy, f, in terms of the order parameter of thetransition. From minimising the expression for the energy, the critical exponents can bedetermined. For example, considering a magnetic system, the free energy is given by thefollowing equation:

f(m) = am2 + bm4 +O(m6) (1)

where m is the magnetisation and a and b are coefficients [2]. Notice that this is anexpansion containing only the even-exponent magnetisation terms as all second-ordertransitions must exhibit time-reversal symmetry.

In contrast to classical phase transitions, where the system goes from a disordered toan ordered phase due to thermal fluctuations, quantum phase transitions are driven byquantum fluctuations of the order of hω. At finite temperature they compete with thermalfluctuations, which are of the order of kBT. As a result, quantum fluctuations dominatefor hω > kBT. The effect of this competition is that it is possible to identify signals of aquantum phase transition at finite but low temperatures, close to absolute zero.

Figure 2. An illustration of a quantum phase transition (in this case between a superfluidand an insulator). The quantum critical region is identified (in red) along with the quan-tum critical point where the dashed lines meet at gc. The region of thermal fluctuationsis illustrated by the blue TkT zone [5].

3


The effects of a quantum critical phase can be observed in the quantum critical region.As in classical phase transitions, the system also undergoes a transition from disorderedto ordered. When the temperature is high enough, the system is entirely disordered andclassical. However, as the temperature gets lower, the system becomes more and moreordered and the fluctuations diverge at the quantum critical point as shown in Fig. 2above.

1.3 Non-Classical Phases of Matter

In order to show the variety of non-classical phases which exist, a brief description ofsome of the best-understood phases encountered around a quantum critical transitionwill be given in this section. Among the best best understood non-classical phases ofmatter there are: magnetically-ordered states, quantum spin liquids, spin-density wavestates and topological Fermi-liquids.

Magnetically-ordered states arise in materials which are magnetic and therefore have amagnetic moment associated with them. The magnetic moments are a result of the spinsand their alignment. According to the alignment of the spins, four distinct magneticphases can be distinguished. Firstly, the ferromagnetic phase in which all the magneticmoments are aligned within a domain. Secondly, the antiferromagnetic phase where thetotal magnetisation is zero. This is due to the presence of two magnetic networks inopposite directions, where half of the magnetic moments point in one direction with theother half directed in the opposite direction, therefore cancelling each other out. Thirdly,a special case of the antiferromagnetic phase is the ferrimagnetic phase where there aretwo different networks of magnetic moment, but as they are unequal there is a resultingmagnetisation. Finally, the paramagnetic phase occurs when the magnetic moments ina material align with the direction of an applied magnetic field, but when this field isremoved they orientate with no preferred direction.

Quantum spin liquids are so-called because they are a phase of matter only present whenthere is an interaction of spins. They are denoted liquids due to the fact that thesequantum spins maintain a disordered state even at very low temperatures. There is anoticeable correlation between the amount of order in liquids and solids compared withthe amount of order in quantum spin liquids and other phases of matter where the spinsare ordered, such as the magnetically-ordered systems described above [6].

Spin-density wave states are low ordered states which are related to spin-density waves(SDWs). They are a novel phase of matter which occur in solids at low temperatureconsisting of a periodic modulation in the density of electronic spins. SDW states arevery difficult to detect, however it is possible to observe them using neutron scatteringand magnetic susceptibility measurements [7].

Topological Fermi-liquids are materials which behave according to Landau’s theory ofFermi-liquids. This theory describes the interaction between fermions in metals and aimsto give a unified account of these effects in the low temperature regime [8]. Materialswhich do not behave according to Fermi-liquid theory are called non-Fermi-liquids, anexample of which are the Luttinger liquids.

4

Quantum Criticality Theoretical Models of Quantum Criticality

2 Theoretical Models of Quantum Criticality

This section will attempt to give a brief discussion of the theoretical models imple-mented to explain quantum criticality. The tight-binding model will be discussed inorder to examine a few central models to quantum criticality. The Hubbard Model andKondo/Anderson impurity models will also be discussed as a basis for certain classesof quantum phase transitions. Finally, the theoretical case of Luttinger liquids will beexamined.

Note that the following sections will be written in second quantisation notation, wherethe total particle number of the system, N, is given by:

N =∑i

ni =∑i

(c†ici) (2)

where c†i represents the particle creation operator, ci represents the particle annihilationoperator and ni is the particle number for a given wavefunction, Ψ.

2.1 Universal Scaling

A highly attractive feature of the theory of quantum criticality is the idea of universalscaling, which means that for materials tuned to a quantum critical point the properties ofthe material depend entirely on a dimensionless ratio of hω/kBT, where ω is the frequencyof the quantum fluctuations of the system and T is the temperature [9]. The fluctuationsoccur when the system is tuned to a quantum critical region, and allows materials withcompletely different microscopic details to exhibit universal properties [4]. Due to thelong range fluctuations inherent to phase transitions this causes large-scale propertieswhich are completely quantum mechanical in nature. The Hubbard, Kondo/Andersonimpurity and Luttinger liquid models all have potential quantum critical regions whichdisplay universal scaling.

2.2 The Tight-Binding Model

Conventional band theory tends towards the usage of the tight-binding model, where theelectrons are treated as non-interacting (with no Coulomb or spin effects), with a transferintegral related to the overlap of electron wavefunctions with their nearest neighbours asshown in Eqn. 3 below.

H = −t∑<i,j>

(c†icj) + h.c. (3)

Here i and j represent the labels of the electrons; h.c. is a constant – the energy of theunstabilised orbital; and t is the transfer integral, the energy reduction due to electronwavefunction overlap (also referred to as the exchange interaction). Only nearest neigh-bours are considered as the wavefunction overlap is assumed to be small. The constanth.c. will generally be ignored in this discussion and in the presentation of other related

5


models as it does not contribute to the physics of the model aside from determination ofthe total energy [10]. While this model can qualitatively predict the properties of a widerange of solids, there are many extensions which add other effects to match the systemsbeing studied.

2.3 The Hubbard Model

A simple extension of the tight-binding model is the Hubbard model. The Hubbard modeladds a term to the Hamiltonian which accounts for the Coulomb repulsion of electronsin the same orbital, e.g. repulsion between paired electrons in a d -orbital [11] (see Eqn.4 below).

H = −t∑

<i,j>,σ

(c†i,σcj,σ) + U∑i

ni↑ni↓ (4)

The first term is the same as in Eqn. 3, with an additional term σ representing spin polar-isation. The second term has energy U representing the energy of Coulombic repulsion,with ni↑ and ni↓ representing electrons occupying the same atomic orbital. The Hub-bard model has been used extensively to study metal-to-insulator transitions, and is thequalitative basis for the phenomenon of Mott insulators (insulators which should conductin conventional band theory) [4]. Unfortunately it involves strong competition betweenthe transfer integral, t, and the Coulombic repulsion, U. This results in the inability tosolve the Hubbard model with perturbative methods, as taking U to be in either thelarge or small limit cannot provide information on the transitions between conductingand insulating states. This leaves renormalisation as the only source of analysis and it iscomputationally expensive, requiring small lattice sizes. Additionally this model assumesthat the electrons occupy their lowest energy configuration and so it breaks down at hightemperatures. Additional extensions have been made such as the t-J Model which is usedto study high-Tc superconductivity in doped antiferromagnets [12].

In a similar manner to that used in particle physics, the techniques of renormalisationhave a wide variety of uses in systems where there are large fluctuations around someequilibrium parameter [4]. In the context of solid state physics, renormalisation is used toaverage the electron interactions in such a way that to a certain cut-off scale, the physicalobservables are independent of system size. As the scale of the system becomes smaller,beyond the cut-off the results start to diverge and become meaningless. Calculationsinvolving a quantum critical region relate this scale parameter to the coherence length ofthe system and form an effective field theory to describe its properties.

2.4 The Kondo/Anderson Impurity Model

The Anderson impurity model was formulated in an attempt to understand the effectsof magnetic impurities on a non-magnetic lattice, taking the tight-binding wavefunctionand incorporating additional terms to account for the mixing of character (hybridisation)of atomic orbital wavefunctions (s, p, d or f ) and the coupling of the impurity spin toits local environment [12]. The Kondo Model is the Anderson impurity model in the lowenergy limit – where orbital hybridisation is small compared to the interaction of the

6


impurity of the spin with the conduction band [13]. The Kondo model is given in Eqn.5 below [12].

H = −t∑

<i,j>,σ

(c†i,σcj,σ)− JS(0) · S (5)

The first term of the Hamiltonian in Eqn. 5 represents a tight-binding wavefunctionand the second term represents the antiferromagnetic coupling of this wavefunction tothe magnetic impurity with coupling constant, J . The spin of the conduction band isrepresented by S(0) and S is the impurity spin. These two models are used extensivelyin the study of strongly correlated materials and disorder(doping)-driven quantum phasetransitions [4].

2.5 Luttinger Liquids

As the Fermi-liquid model used to describe metallic solids does not work in one dimension,an alternate model has been developed called the Luttinger-Tomonaga model. The Fermi-liquid model is essentially a gas of non-interacting fermions with interacting quasiparticlesto approximate interactions between fermions [14]. A quasiparticle is either a particlethat has its behaviour affected by the solid it is travelling through (such as an electronwith an effective mass caused by the crystal potential) or a vibration or periodic motionthat behaves like a particle in a solid (such as a phonon). The Luttinger-Tomonagamodel involves chains of interacting fermions in one dimension and the realisations ofthis system are referred to as Luttinger liquids. Luttinger liquids display a wide range ofunique properties, in part due to spin-charge separation and the fact that there are noquasiparticles – all low energy excitations are collective oscillations of the spin or charge-density of the lattice [14]. Spin-charge separation results in charge-density waves andspin-density waves which propagate in phase with each other, but at completely differentvelocities through the lattice.

This model has had a wide array of experimental evidence ranging from carbon nanotubesto the cuprate superconductors (see ‘Chemically-Induced Quantum Phases’ later) [9]. Outof the several different applications of Luttinger liquids, one with very few experimentalrealisations is the Heisenberg antiferromagnetic chain (HAFC). This is an antiferromag-netically coupled chain of spin-half electrons. The Heisenberg model is a magnetic systemwhere the spins are represented by vectors which can orient in any direction aligned withthe spin quantum number. This system exists at a quantum critical point between de-generate ground states where transitions between the two are forbidden by the Pauliexclusion principle, which results in a magnetic system with no long range order.

As an example, the compound KCuF3 is a weakly-coupled magnetic system which exhibitsmagnetic coupling only in a 1D chain along the unpaired spin-half electrons in the copperatoms of the lattice, creating an experimental setup to test the HAFC model [9]. Theend result is a confirmation of hω/kBT scaling for a wide range of temperatures, howeverthis scaling is broken for a large range of parameters as a function of both ω and T, as isseen in Fig. 3 below.

7


Figure 3. Quantum critical region in terms of spin wave energy and temperature. Theblack dashed lines indicate the breakdown of scaling [9].

A common experimental technique used to determine many physical parameters of a ma-terial is inelastic neutron scattering (INS). The fact that neutrons are neutral but have amagnetic moment and non-negligible mass allows them to interact with the magnetic mo-ment of a studied material. By collecting data over a range of different angles, the Braggpeaks that are observed can be related back to the material studied as they correspondto the magnetic ordering of the sample studied.

Using this technique the excitation energy of the spin-waves was detected and plotted inFig. 3. At intermediate energies over a range of temperatures the excitation spectrumcan be approximated as a Luttinger liquid. While it is claimed to exhibit universalscaling [9], this is only valid while the system is antiferromagnetic. For large excitationsadditional lattice effects alter the spectrum beyond the Luttinger liquid model. Thecrossover region represents the area at which inter-chain coupling becomes significantenough that the ferromagnetic inter-chain effects start to become non-negligible, and atlow temperatures and spin-wave energies, the excitations are described by the non-linearsigma model (NLσM) (see the red region in the bottom left-hand corner of Fig. 3) [14].

As well as for a range of energies of the spectra, the scaling behaviour breaks down aboveT = 216K where the system transitions into a paramagnet. While certain regions of thescattering spectra can be attributed to the properties of a Luttinger liquid, a system atthe quantum critical point of this type will likely never be seen experimentally.

8

Quantum Criticality Concentration-Induced Quantum Phases

3 Concentration-Induced Quantum Phases

The following section is concerned with quantum phases induced by concentration asthe tuning parameter. In particular the case of skyrmions will be considered and theirpotential applications in memory storage discussed.

3.1 Skyrmions

Skyrmions are hypothetical particles which can be seen as a distortion of the spins of manyground state electrons. This distortion is shown in Fig. 4 below. Despite skyrmionshaving unitary electric charge, their spin is much larger than unity. By varying theskyrmion concentration, a quantum phase can be demonstrated [15]. When two skyrmionswith opposite phases are considered, it is observed that the attractive force between them,inversely proportional to their separation, always dominates over the Coulomb repulsionat long distances. For this reason the skyrmions form pairs at low concentration. Suchpairs behave like point charges and form a triangular Wigner lattice in order to minimisetheir Coulomb potential energy.

The Wigner lattice is a solid phase of electrons which occurs when the electrons reacha critical value. This occurs at low densities where the potential energy dominates overkinetic energy, so the detailed spatial arrangement of the electrons becomes important.To minimise their potential energy, the electrons form a triangular lattice in the 2Dplane [16].

As their concentration is increased, the separation between the skyrmion pairs decreasesand the interaction becomes stronger, triggering a quantum phase transition in theskyrmion lattice.

Figure 4. A skyrmion is a deformation in the spins of many ground state electrons [17].

Below the transition point the z -component of the spin (Sz) of each pair is well-defined,whereas above the transition point the phase, a variable canonically conjugated with Sz,

9

Quantum Criticality Concentration-Induced Quantum Phases

is well-defined. The arrangement of the skyrmions in the lattice is similar to the so-calledCoulomb blockade phenomenon in Josephson arrays; these latter ones are formed by aseries of superconducting islands. These islands consist of a number of discrete charges,which in a skyrmion lattice corresponds to discrete Sz. The Josephson coupling and thecharging energy of the array compete in order to determine the state of the array.

When the charging energy dominates, an insulating phase occurs; when the Josephsoncoupling dominates, a superconducting phase is observed. This behaviour can be usedto model the skyrmion lattice. At low concentration of skyrmions, the pairs are effec-tively isolated and do not interact: this is identified as the insulating phase. When theconcentration is increased, i.e. more skyrmions are added, the distance between pairsdecreases. The pairs start interacting as the coupling increases and the transition tothe superconducting phase occurs. After conducting numerical analysis, it appears thatfuture experiments may be able to observe the transition using magnetic resonance mea-surements of either the spin gap or the local magnetisation parallel to the planes [17].

3.2 Memory Storage

Skyrmions possess a topological charge which can be viewed as a measure of magnetisationcurvature and is defined by:

S =1

4π

∫A

n ·(dn

dx× dn

dy

)dxdy (6)

where n is the normalised magnetisation vector, A is the surface area and x and y arespatial co-ordinates. If S = 1 a skyrmion phase is present, if S = −1 an anti-skyrmionphase is present and if S = 0 a topologically trivial state is present such as ferromagnet.These states are separated by an energy barrier which can be overcome by altering amagnetic field.

A single atomic layer of iron on iridium exhibits a skyrmion lattice even in the absence ofa magnetic field. Adding an additional layer of palladium (Pd), which is easily polarisedby iron, creates a PdFe bilayer and allows the top surface to be modified. A techniqueknown as spin-polarised Scanning Tunnelling Microscopy (STM) has been used to createand annihilate individual skyrmions. The technique uses a metallic scanning probe totunnel electrons onto the surface of a material. By controlling the voltage and thusenergy of the electrons, individual skyrmions have been created reversibly very close tothe critical value of magnetic field between the ferromagnetic and skyrmion phases. Sincethe STM current is spin-polarised (it contains more electrons of one spin orientation thananother), a spin transfer torque is applied to the magnetic material which allows theswitching to the skyrmion state [18]. Fig. 5 below shows the process for creating anddeleting individual skyrmions using the STM probe.

Manipulation of individual skyrmions means that information can be stored as skyrmions.Current electronics use magnetic domains with all magnetic spins aligned in the samedirection meaning there are fundamental limits to domain size. Skyrmions are muchsmaller leading to a higher information density. Skyrmions also require fewer spins to

10

Quantum Criticality Magnetically-Induced Quantum Phases

switch in order for the device memory to switch from ‘0’ to ‘1’ and this final spin state isnot easily disrupted [19].

Figure 5. Writing process of creating and deleting individual skyrmions. Skyrmions arecreated one by one going from F to A and annihilated one by one going from A to F. Kshows the schematic spin configuration for each skyrmion [18].

4 Magnetically-Induced Quantum Phases

The magnetic field strength is a tuning parameter often used to drive a system to aquantum phase transition. In this section a number of both theoretical and experimentalexamples will be discussed such as the cases of a quantum critical point in triangular-lattice antiferromagnets, the quantum phase transition in the graphene-like BC3 and thequantum phase of the spin-half antiferromagnet K2NaCrO8. Finally, the use of magneticrefrigeration will be discussed.

4.1 The Triangular-Lattice Antiferromagnet

The triangular-lattice antiferromagnet (TLAF) shows interesting quantum effects whenplaced in a magnetic field. A quantum phase transition arises for small spin values dueto the effect of quantum fluctuations. For intrinsic quantum spins, a stable state is notuniquely determined and the ground state is infinitely degenerate. The magnetisationcurve is monotonic, i.e. a straight line. When quantum fluctuations become importantand a specific spin state is selected as the ground state, a magnetisation plateau appearsat one third of the saturation magnetisation, Ms [20]. This saturation magnetisationoccurs when despite an increase in the applied external magnetic field, the magnetisationof the material does not increase any more. Such a magnetisation plateau is interpretedas a new macroscopic quantum phenomenon. Fig. 6 below illustrates the magnetisationcurve for both the quantum and Heisenberg spins.

11


Figure 6. Magnetisation curves of TLAF for quantum (dashed) and Heisenberg spins(solid line). Notice the clear magnetisation plateau at Ms/3 [20].

There is a lack of experimental data on the magnetisation plateau of TLAF, however astudy of the spin-half TLAF, Cs2CuBr4 has been conducted [20]. In this material, at atemperature of T = 0.4K, a plateau in the magnetisation curve was observed, agreeingwith the theoretical expectations. This plateau, at approximately one third of the satu-ration magnetisation, was observed for H ‖ c and H ‖b, where the spin-half Cu ions forma distorted triangular-lattice parallel to the bc plane. Here a, b and c identify the latticedirections. No plateau was observed for H‖a, but the presence of the plateau in the othertwo directions led to the conclusion that the magnetisation plateau arises from quantumeffects.

An investigation for the spin-1 case was also carried out. The TLAF Ba3NiSb2O9, whosemagnetic Ni ions have unitary spin, has been studied under different magnetic fields [20].In this case, a well-defined plateau was observed, again at one third of the magnetisationsaturation value. The range of the plateau was measured to be 10 Tesla, even at atemperature of 1.3K. This study provides experimental confirmation for the existenceof a magnetisation plateau, which is regarded as a novel quantum phase transition forTLAF.

Further studies have suggested that the magnetisation plateau at Ms/3 is not the onlyplateau present and, in fact, a second plateau was also observed for Cs2CuBr4 at 2/3 ofthe saturation magnetisation [20]. Here, an additional cascade of field-induced quantumphase transitions was observed. In addition, the phase transitions to the A phase andthe very narrow B phase can be observed as small peaks in the differential susceptibil-ity (dM/dH) (see Fig. 7 below). The confirmation of the plateau quantum phase isan exciting step forward in magnetically-induced quantum phases, however the overallexplanation for the cascade of quantum phase transitions still remains an unansweredquestion.

12


Figure 7. Magnetisation curve and dM/dH versus the applied magnetic field, H, measuredat a temperature of 0.4K. The vertical arrows indicate critical fields. The blue linecorresponds to the right-hand vertical axis and the red curve to the left-hand verticalaxis [20].

4.2 The Graphene-like BC3

A theoretical example of the appearance of a novel quantum phase transition is providedin the case of the graphene-like BC3. This is still lacking experimental confirmationand in fact the authors, Xi Chen and Jun Ni, open their paper by stating that: “theinvestigations on the quantum phases driven by the Fermi surface nesting are importantwhile inconclusive in condensed matter physics” [21]. The compound studied showed thatmagnetic quantum phase transitions can be triggered under biaxial strain by tuning thefermiology (the shape and size of the Fermi surface) and competitions among magneticorders [21].

This is a theoretical example of a nesting-driven quantum phase transition. Fermi-nestingconsists of large fragments of a Fermi surface that can be connected to another largesegment of another Fermi surface via a reciprocal lattice vector. Graphene is also acandidate material to exhibit a quantum phase transition, showing unconventional phasesdriven by nesting [22]. However, BC3 can be studied more easily since it has similarproperties to graphene and is a semiconductor.

In order to shed more light onto the properties of graphene and on this type of quantumphase, the nesting-induced magnetic ordering of the graphene-like material can be derivedfrom first principles [22]. The competitions between different magnetic orders is tuned

13


by tailoring the π-band (the band corresponding to the dispersion of the π-orbitals) viabiaxial strain, which means applying a stress in two directions (x and y) on the surface.This type of strain is also known as global strain. In this way rich magnetic quantumphase transitions are induced. Fig. 8 below illustrates the phase diagram of the strain-driven quantum phase transitions as a function of the strain in two different directions.

Figure 8. The phase diagram for BC3 as a function of strain (ε). The components of thestrain are plotted along the x and y-axes. The acronyms FM and SDW correspond toferromagnetic and spin-density wave states, respectively [21].

The magnetic orders in BC3 are due to the divergence of the density of states at halffilling. These orders are competing depending on the properties of the Fermi surface andthe π-band width. Under biaxial strain these factors are tuned such that a first-orderquantum phase transition arises between spin-density wave and ferromagnetic states anda second-order quantum phase transition between spin-density wave and non-magneticstates [21].

4.3 The Spin-half Antiferromagnet K2NaCrO8

The final example of a magnetically-induced quantum phase transition is the case of theantiferromagnet K2NaCrO8. An experimental study on spin-half chromium ions was con-ducted using INS [23]. The nature of the transition is identified via the critical exponentα. The results obtained for the specific heat, magnetocaloric effect (see ‘Magnetic Cool-ing’), magnetic susceptibility and torque magnetometry (a measure of the torque on thesample due to its moment in a magnetic field) were used to obtain the magnetic phasediagram (see Fig. 9 below).

14


Figure 9. The phase diagram for K2NaCrO8 as a function the magnetic field [23].

With the phase diagram established, a clear quantum phase was identified. The observedmagnetic field-tuned antiferromagnetic to spin-polarised phase transition suggests a crit-ical field µ0Hc = 7.270 T and a critical exponent α ≈ 0.48 (with an associated error of asmuch as 20%). However since the experiment was conducted at a finite temperature, theresulting error on α makes it insufficient to distinguish unambiguously between the criti-cal exponent for a Bose-Einstein condensate (α = 2/3) and the one for an Ising-like spin(α = 1/2). Bose-Einstein condensates, together with superfluids, occur at temperaturesclose to absolute zero and are sometimes referred to as the fifth state of matter [24]. In aBose-Einstein condensate a non-negligible fraction of the total number of bosons collapseto the ground state, forming what is defined as the ‘condensate.’

In order to make a less ambiguous distinction, lower values of temperature must beinvestigated. In Fig. 9 above, the phase transition observed is shown and the criticalfield can be extrapolated as the value of H where the phase transition will occur asTN tends to zero. TN is the Neel temperature, the analogous temperature to the Curietemperature in ferromagnets, and is the temperature above which an antiferromagneticmaterial enters the paramagnetic phase.

4.4 Magnetic Cooling

It has been discovered that certain magnetic materials can exhibit temperature changesdue to a change in an external magnetic field without the transfer of heat [25]. Thisis known as the magnetocaloric effect (MCE) and may prove to be an environmentallyfriendly alternative to vapour-cycle refrigeration currently in use.

The extent of the MCE is defined by a magnetic Gruneisen parameter given as:

ΓB =1

T

(∂T

∂B

)S

= − 1

C

(∂S

∂B

)T

(7)

15

Quantum Criticality Pressure-Induced Quantum Phases

where T is the temperature, B the magnetic field, C the heat capacity and S the entropy.Therefore a large change in the entropy of the material over a small change in magneticfield will give rise to a large amount of magnetic cooling. Since a quantum critical pointconsists of a large number of simultaneous possible wavefunctions, the entropy at thispoint is very high and since this theoretically occurs at one possible critical value ofmagnetic field, quantum critical materials exhibit a much enhanced MCE.

This type of quantum-critically-enhanced magnetic cooling has been shown to exist in theantiferromagnet [Cu(µ-C2O4)(4-aminopyridine)2·H2O]n up to a saturation field Bs [25].Graph (a) in Fig. 10 below shows negative values of ΓB up to Bs meaning that coolingis achieved through magnetisation. ΓB then goes through a minimum before sharplyincreasing implying that here the entropy is a maximum indicating the presence of aquantum critical point.

Quantum critical systems also exhibit extended hold times – the ability of the system toabsorb heat without warming up too rapidly – compared to paramagnetic systems thatare currently in use. This is also beneficial for magnetic cooling.

Figure 10. (a) Graph of Gruneisen parameter against magnetic field showing experimentalresults (symbols) and model calculations (line). (b) Graph of sample temperature againstmagnetic field showing demagnetisation starting at three different temperatures [26].

5 Pressure-Induced Quantum Phases

In this section the use of pressure as a tuning parameter will be discussed. In particular,the antiferromagnet TlCuCl3, the ferromagnet MnSi and the superconductor UGe2 willbe investigated. The idealisation of second-order quantum phase transitions will alsobe considered, along with the potential use of quantum criticality in superconductingnanowires.

16


5.1 Pressure-Dependent Ising Coupling

A quantum phase transition may be induced by varying the pressure at which the sampleis held. Recent research, particularly concerning magnetic compounds, has illustratedthat fermion interactions in pure metals at very low temperatures are still well definedby conventional Landau theory, whereas certain intermetallic compounds may exhibitnon-Fermi-liquid behaviour [27] at the boundary between two distinct phases [28,29].

At very low temperatures the exchange interactions in quantum magnets can be modifiedby varying pressure. Using INS, a quantum phase transition may be observed as thesample evolves from an ordered magnetic state into a disordered phase. The Ising modelconfigurational energy (Hamiltonian) for such a system may be expressed as:

H = −p∑i

〈Si〉 − J(p)∑<i,j>

〈Si · Sj〉 (8)

where p is the pressure, S is the spin and the coupling constant, J, is now pressure-dependent. This allows the favoured state to change as the pressure of the system isaltered, in order for the material to maintain in a preferred state with the lowest possibleconfigurational energy.

5.2 The Antiferromagnetic Transition in TlCuCl3

The antiferromagnet TlCuCl3 is commonly used to demonstrate such an idealised quan-tum critical point [30, 31], as it has an easily measurable spin energy gap (the energyrequired to transition from a |↑↓〉 to a |↑↑〉 state) of ∆ ≈ 0.7 meV between the singletground state and triplet first excited state of a coupled pair of Cu2+ ions.

Figure 11. Quantum phase transition in the antiferromagnet TlCuCl3. The acronyms SLand AFM correspond to the spin liquid and antiferromagnetic phases, respectively [30].

17


Furthermore, it also has an experimentally accessible critical pressure of pc = 1.07 kbar.As the pressure approaches the critical pressure the spin gap drops rapidly to zero. If theexperiment were to be carried out at zero Kelvin, then the spin gap would theoreticallycollapse to zero at p = pc (see Fig. 11 above).

Another interesting observation is that the Neel temperature, TN , also tends to zero atthe quantum critical point when approaching from pressures above the critical value (seethe right-hand vertical axis in Fig. 11 above and Fig. 12(a) below). This means thatthe material will always be paramagnetic, even at 0K, for pressure values below pc andcannot exist in a magnetically-ordered state. Thus to enter the antiferromagnetic phasethe critical pressure needs to be exceeded.

Figure 12. Quantum phase transition in TlCuCl3 [31]: (a) phase diagram for arbitrarytuning parameter, g, between the disordered magnon state (blue) and the ordered spinwave state (red); (b) INS intensities at various pressures for first Bragg peak; (c) Braggpeak intensity around critical pressure, pc; (d) temperature variance of Bragg peak in-tensities at various pressures.

From the contour plot in Fig. 12(b) above it is observed that the intensity of the INSBragg scattering increases substantially as the pressure is increased with the maximumobtained values occurring around the Q-value of 1 (this corresponds to a perfect period-icity of 2π/a, with a the lattice parameter, since the Q-value is related to the periodicityof the lattice). As can be seen in Fig. 12(c), the critical pressure clearly occurs at pc ≈1.07 kbar. Theoretically, the INS Bragg peak intensities will be maximised at zero Kelvinwhere the phase transition occurs, illustrated in Fig. 12(d), where the different coloursrepresent various pressure values. Notice in particular that no peak is observed for the p= 1.00 kbar experiment, illustrated by red crosses, as it has a pressure below the criticalvalue.

18


5.3 The MnSi Ferromagnetic State

Another example of a pressure-induced quantum phase transition may be seen in themagnetic compound MnSi. MnSi is a highly-correlated cubic ferromagnetic material,which exhibits magnetic ordering below a critical temperature of Tc ≈ 29 K to form aperiodical helical formation [32]. By varying pressure on the magnetic phase it has beenobserved that the transition temperature decreases with pressure and tends to zero atthe critical pressure of pc ≈ 14 kbar [33]. By comparing measurements from muon spinresonance (µSR) (whereby muons are implanted in the sample and emitted preferentiallyalong directions of magnetic ordering), along with thermal expansion (α), specific heatcapacity (Cac), magnetic susceptibility (χac) and resistivity (ρ) measurements on MnSi, anapproximation for the value of the critical pressure may be determined by extrapolatingthe limiting value as the critical temperature approaches zero (see Fig. 13 below).

Figure 13. Variance of critical temperature with increasing pressure in MnSi. The right-hand axis shows the magnetic volume fraction of the sample studied [33].

Below the critical pressure, pc, the inverse helix length scales according to an empirically-determined power law, but at p = pc the inverse helix length becomes independent oftemperature and the chain effectively locks into one fixed position (see Fig. 14 below).

These results are consistent with a phase transition occurring at the critical pressure.Notice also that the magnetic volume fraction remains constant up until the transitionpoint showing that the volume of sample studied does not break up into separate regionsas the experiment is carried out.

In violation to earlier measurements made on MnSi [34], however, for pressures belowthe critical pressure, the helical magnetic structure of MnSi does not obey the expectedbehaviour of a Fermi-liquid [35]. This fact, however, does not necessarily directly inferthe existence of a quantum critical point as there is also evidence that a stable non-Fermi-liquid state in MnSi may exist away from the theoretical quantum critical point [36].

19


Figure 14. Non-linear temperature dependence of inverse helix length in MnSi below thecritical pressure, pc = 14.2 kbar [37].

Using the example of MnSi, it appears from recent experiments that some quantum phasetransitions may actually be first-order, rather than second-order as idealised theoretically(illustrated in Fig. 2 earlier) [38]. These first-order phase transitions are still quantumcritical, but do not have a unique quantum critical point – instead they have a region ofphase co-existence (see Fig. 15 below).

Figure 15. Phase diagram of MnSi showing co-existence region (dark green) betweenFermi-liquid (FL) and non-Fermi-liquid (NFL) states. The pressure p∗ is the pressure atthe tri-critical point (the point at which all three co-existence curves meet). The regionbetween p∗ and pc is that of first-order transitions and above pc the transition to a NFLmay occur [38].

20


With the advent of higher quality experimental equipment it is now possible to probelower and lower temperatures, and further results of very low temperature testing maybe able to challenge the theoretical model of perfect second-order continuous quantumcritical phase transitions tending to one quantum critical point at 0K.

5.4 The UGe2 Superconducting Phase

Further ambiguity over the existence of quantum critical points is illustrated by theexample of the ferromagnet UGe2. Close to critical pressure the material undergoes atransition into a superconducting state [39, 40] (see Fig. 16 below). This transitiondoes however remain consistent within the theory that quantum critical transitions mayexplained by applying appropriate corrections to Landau theory, in the form of Landau-Ginzburg theory. This is used in the case of a superconducting phase transition suchas that in UGe2 [41, 42]. However more recent research has questioned whether thistransition is induced by pressure or whether it is due to the underlying magneto-elasticforce (where changes in magnetic moments alter the dimensions of the lattice). This castsdoubt on the quantum criticality of the transition [43].

Figure 16. Phase diagram of UGe2 showing superconducting phase [40].

5.5 Superconducting Nanowires

A practical application of superconductivity is in superconducting nanowires which arecurrently under investigation for use in ultra-miniaturised electrical components suchas the quantum bit in a quantum computer. Unfortunately their development progresshas been hindered by so-called quantum phase slips [44]. In aluminium wire at 1K, themetal crystal lattice vibrates to allow electrons to overcome negative repulsion and formCooper pairs and thus a superconducting phase. The electrons must overcome a quantumbarrier to stay travelling in a spiral formation (characteristic of superconductivity) and

21

Quantum Criticality Chemically-Induced Quantum Phases

this produces heat which can lead to a quantum phase slip to an insulating phase. Below1K just one electron overcoming or tunnelling through this barrier can cause millions ofelectrons to do the same in a macroscopic tunnelling event.

The phase slip is a topological event where the superconducting order parameter betweentwo adjacent regions of the superconductor changes by 2π over a spatial distance ofthe order of the coherence length. In identifying a critical temperature at which thesequantum phase slips occur, it is thought that they can be prevented from happening alltogether by only designing systems that operate in a temperature range out with thisvalue [44].

6 Chemically-Induced Quantum Phases

A novel method of inducing a quantum phase transition is to change the chemical con-stituents of a studied sample. In this section the case of pair breaking and the that ofinducing a quantum phase transition by adjusting the doping level will be considered.Further, a brief discussion of the pseudogap in the cuprates, the superconductivity of suchcompounds and potential applications of cuprate superconductivity will be analysed.

6.1 Spontaneous Magnetic Ordering due to Pair Breaking

One method of inducing a quantum phase transition, often referred to as pair breaking, isto break dimer bonds in an intermetallic lattice, resulting in the emergence of a quantumcritical ferromagnetic phase from a paramagnetic state. These bonds may be broken byinterchanging one atomic type for another, however the extra electrons removed or addedby the new atom are generally consumed in the breaking of the dimer.

Using the compound structure SrCo2(Ge1−xPx)2, unexpected ferromagnetism is observedwhen transitioning between the paramagnetic compounds SrCo2Ge2 and SrCo2P2, i.e.evolving the system from x = 0 to x = 1 [45]. Thus the concentration, x, may be usedas a tuning parameter to search for a quantum critical phase transition.

Upon substitution of phosphorus (P) atoms for germanium (Ge), dimer breaking occurswithin the SrCo2Ge2 structure, and at a critical value of x = 0.35 spontaneous magneti-sation effects are observed in the intermediate compound. The process is then continued,whereupon the compound loses its ferromagnetic properties and forms a strongly cor-related paramagnetic final state [46]. As can be seen below in Fig. 17, the magneticsusceptibility (χ) of the compound increases dramatically at x = 0.35. The empirically-determined power-law behaviour of the susceptibility at this value is consistent with thatof a ferromagnetic quantum critical point [45].

22


Figure 17. Magnetic susceptibility versus T−4/3 for different x values in SrCo2(Ge1−xPx)2[45].

6.2 Doping-Induced Quantum Critical Phenomena

The high-Tc superconductivity present in many cuprates and iron-based compounds maycommonly be explained in terms of quantum criticality [47]. However, the precise detailsas to why changes in the doping level may cause the transition to a superconducting stateremain largely unexplained. This is a general problem and it is seen particularly in thecuprates where the charge doping-induced high temperature superconducting transitionappears to resemble a first-order transition rather than the expected clean second-orderquantum transition. This may be explained by the doping introducing random electricfields due to the extra charges, which smear out the transition [48,49].

Considering the example of La2−xSrxCuO4, the effect of doping on the electrical resistivityof the material has been shown to vary linearly with temperature over a wide rangeof doping concentrations [50]. By measuring the electrical resistivity a change in theenergy transport at ρc has been observed [50]. This is contradictory to the quantumcriticality expectation of a single parameter change bringing about the quantum criticaltransition, as the doping evolution is a complex procedure which brings about manyparameter changes and does not specifically lead to one obvious critical tuning parameter.These factors make this particular area of research highly interesting to study whilst stillremaining theoretically challenging to understand.

6.3 The Cuprate Pseudogap

One of the significant standing problems in relation to the high-Tc cuprate supercon-ductors is the presence and nature of the pseudogap. This is a partial gap in the bandstructure which results in conduction along only certain directions and has the samesymmetry as cuprate superconductivity [48].

23


The size of this pseudogap can be changed by altering the carrier doping of the supercon-ductor, and it is seen not only near the onset of superconductivity but also up at roomtemperatures as seen in Fig. 18 below.

In Fig. 18, the pseudogap is shown as a decrease in the differential conductance (ameasure of the change in the accessible density of states with respect to a change inapplied voltage) around the Fermi level of the sample. The pseudogap has represented asignificant challenge in analysing the superconducting gap in cuprates as it obscures thegap formation that occurs at the onset of superconductivity [48].

Additionally, cuprate superconductivity and the pseudogap both have d -wave symmetry,meaning that the electrons are only affected by both of these properties when they movethrough the x -y plane. For the superconducting phase this results in the material beingan insulator for electrons moving in the z -direction, and for the pseudogap this results inconduction between planes along the z -direction.

Figure 18. Differential conductance versus sample voltage for slightly underdopedBi2Sr2CaCu2O8 over a range of temperatures [51].

Because of the similarities between these two effects, they have widely been considered tobe two inseparable aspects of the high-Tc superconductivity mechanism. The inherentlydisordered nature of the doped cuprates has drawn research towards the notion that the

24


pseudogap is a quantum phase described by the Kondo model where doping of CuO2

planes is large enough that the magnetic impurities are not completely screened by theenvironment (Screened moment), but not large enough that long-range impurity effects(Free local moment) are dominant [4, 48]. A phase diagram for the pseudogap Kondomodel is shown in Fig. 20 later.

Although there are many similarities with cuprate superconductivity, there has beenrecent research showing that a pseudogap appears in thin films of the disordered con-ventional superconductors TiN and NbN [52, 53]. These thin films are effectively 2Dsuperconductors, and the analogy has been made to the quasi-2D nature of cuprates –CuO2 planes do not interact with the planes above and below them (see Fig. 19 be-low) [48,54].

Figure 19. Illustration of the CuO2 planar structure [55].

This strongly supports the idea of the pseudogap as a disorder-driven effect amplified bythe low-dimensionality of the materials, and occurs in high-Tc and conventional supercon-ductors. This gives further merit to the previously mentioned pseudogap Kondo Model(see ‘Theoretical Models of Quantum Criticality’) as a mechanism for understanding thecuprates. However this model does not provide any direct explanation for the onset ofsuperconductivity and a definitive answer has yet to be seen.

25


Figure 20. Phase diagram of the pseudogap Kondo Model where J is the coupling constantbetween the magnetic impurity and the conduction electrons of the system. The quantumcritical region represents the possible location of the pseudogap in the cuprates [4].

6.4 Cuprate Superconductivity

Overdoped cuprates possess Fermi surface pockets – a reconstruction of the highest oc-cupied energy states due to the existence of extra electrons or holes. In order for theseto exist, the symmetry of the Brillouin zone must be modified by quantum phase tran-sitions between ground states. The behaviour of these transitions in the electron-dopedsuperconductor La2−xCexCuO4 can be studied through changes in doping and magneticfield [56].

Quantum criticality is categorised by the divergence of an experimental tuning parameteras a function of magnetic field. In La2−xCexCuO4 this manifests itself in three ways.Firstly by the divergence of the quasiparticle-quasiparticle scattering cross section as thetemperature range of Fermi-liquid behaviour is driven to zero at the critical magnetic fieldvalue Bc. Secondly by the divergence of a quadratic temperature coefficient, A2, with thechange in magnetic field from the critical value, ∆B = B −Bc. A2 relates the change inresistivity, ∆ρ, to the temperature, T, by ∆ρ = A2T

2. And finally by the divergence ofa reduced field scale, ∆B/Bc, with a universal critical exponent which appears to be thesame for all doping concentrations suggesting that Bc acts as a line of quantum criticalpoints on the phase diagram (see Fig. 21 below).

This quantum criticality appears to both optimise superconductivity as well as limit itsextent. Two quantum critical points exist on the La2−xCexCuO4 phase diagram as shownin Fig. 21. The first is near optimal doping and gives spin fluctuations that stabiliseunconventional superconductivity as shown by the line Bc2 in graph B. The second at Bc

owes its existence to the first as it is born out of the suppression of superconductivity andthe emergence of the normal Fermi-liquid state. Further studies will need to be carriedout in order to see whether this quantum critical behaviour appears elsewhere in thecuprates and other possible superconducting materials [56].

26


Figure 21. Three phase diagrams showing existence of superconducting, spin fluctuation(SF), Fermi-liquid and non-Fermi-liquid states in La2−xCexCuO4. Each phase is charac-terised by its resistivity power law given by ραTn. Graphs B and C show cross sectionsof A through T = 0K and B = 4T, respectively. Each transition is characterised bya critical temperature and field strength given by Bc, Bc2, B1, Tc, T1 and TFL whereB1 is the extrapolated limit of the spin fluctuation phase (a phase of strong low energylocalised fluctuations) [56].

6.5 Applications of Cuprate Superconductors

The existence of quantum critical points in certain high temperature superconductorscould be the key to understanding what allows superconductivity to exist in these mate-

27

Quantum Criticality Conclusions

rials at such high temperatures and thus allow a more systematic approach to creatingsuperconductors at even higher temperatures. If superconductivity can be induced at am-bient temperatures then the possible applications are vast. Magnetic levitation caused bysuperconductors is already being used in the SCMaglev train system in Japan althoughthis system uses a niobium-titanium alloy cooled with liquid helium to 4K [57]. Removingthe need to cool the magnets would save a lot of money and energy. Other applicationsinclude MRI scanners, power generation and transportation and particle colliders suchas the Large Hadron Collider at CERN [58].

7 Conclusions

The distinction between quantum and classical phases of matter lies in the significance ofquantum mechanical fluctuations at very low temperatures, leading to many new novelphases of matter that exist purely as a result of quantum phase transitions.

Theoretical modelling to explain such quantum phase transitions and the existence of aquantum critical point has been developed over the last few decades, with many mod-els becoming well-established. Among these: the Hubbard model, an extension to thetight-binding model which accounts for Coulomb repulsion between electrons of the samesymmetry; the Kondo/Anderson impurity model, which includes coupling of a magneticimpurity to the conduction band of electrons in a non-magnetic lattice; and the Luttingerliquid model, which is a model for a 1D Fermi-liquid were explained.

In condensed matter physics the field of quantum criticality has attracted a lot of interest,now that the theoretical models are finally able to be experimentally tested. Recentexperimental research in the field of quantum criticality has provided highly valuableinformation to test the theoretical models of quantum phase transitions. This has led tothe original idea of an idealised quantum critical point appearing not to hold in manycases, however there are several examples which currently work well under this model.

This report has attempted to highlight the main features of quantum criticality as iden-tified by different tuning parameters. The first example considered was the case ofskyrmions, whose concentration gives rise to a quantum phase transition between su-perconducting and insulating phases. Next, several examples of magnetically-inducedquantum phases were taken into account, including the triangular-lattice antiferromagnet,the graphene-like BC3 and the spin-half antiferromagnet, K2NaCrO8. Pressure-inducedquantum phases were also considered, in particular the TlCuCl3 antiferromagnetic transi-tion, the MnSi ferromagnetic state and the UGe2 superconducting phase were discussed.Finally, the case of chemically-induced quantum phases was analysed, among which werethose triggered by pair breaking in SrCo2(Ge1−xPx)2 and doping in the cuprates.

Many real-life applications are getting close to being able to utilise the power of the novelquantum phases of matter. The applications considered here were in memory storage,magnetic refrigeration and superconducting nanowires. These are all viable possibilities,along with the many uses high-Tc superconductors have in magnetic train systems, MRIscanners and particle colliders.

28

Quantum Criticality References

Acknowledgements

We would like to express our thanks to Dr. Chris Stock for his valuable advice andguidance on the field of quantum criticality and for his continuous assistance throughoutthe duration of this project.

We would also like to thank Prof. Andrew Huxley for providing further insight into thisfascinating field and for particular assistance with the UGe2 superconducting phase.

Finally, we thank Dr. Mark de Vries for helping to clarify the difference between orderand tuning parameters and Mr. Mark Harley for his explanation of renormalisation.

References

[1] McKinnell M, Verhein J, Yu P, Chan L. Phase Diagrams. [Online]. Available from:http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_

of_Matter/Phase_Transitions/Phase_Diagrams. [Accessed on 14th January2014].

[2] Berera A. Statistical Physics Course Notes: Section 10 – Mean-Field Theory of theIsing Model. Edinburgh: University of Edinburgh; 2013.

[3] Sachdev S, Keimer B. Quantum Criticality. Physics Today. 2011; 64(2): 29–35.

[4] Vojta M. Quantum Phase Transitions. Reports on Progress in Physics. 2003; 66(1):2069–2110.

[5] Sachdev S. Quantum Magnetism and Criticality. Nature Physics. 2008; 4(3): 173–185.

[6] Wilkins A. A Strange New Quantum State of Matter. [Online]. Available from: http://io9.com/5831111/a-strange-new-quantum-state-of-matter-spin-liquids.[Accessed on 23th January 2014].

[7] Gruner G. The Dynamics of Charge-Density Waves. Reviews of Modern Physics.1988; 60(4): 1129–1181.

[8] Kittel C. Introduction to Solid State Physics. 8th Ed. New York: Wiley; 2004.

[9] Lake B, Tennant D, Frost C, Nagler S. Quantum Criticality and Universal Scalingof a Quantum Antiferromagnet. Nature Materials. 2005; 4(4): 329–334.

[10] McComb D. Renormalisation Methods: A Guide For Beginners. 1st Ed. Oxford:Oxford University Press; 2007.

[11] Hubbard J. Electron Correlations in Narrow Energy Bands. Proceedings of the RoyalSociety of London, Series A: Mathematical and Physical Sciences. 1963; 276(1365):238–257.

[12] Fabrizio M. Clusters of Anderson Impurities and Beyond: How Kondo Effect Diesand What can we Learn About the Mott Transition. Salerno: XI Training Course inthe Physics of Strongly Correlated Systems; 2006.

29

http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/Phase_Transitions/Phase_Diagrams

http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/Phase_Transitions/Phase_Diagrams

http://io9.com/5831111/a-strange-new-quantum-state-of-matter-spin-liquids

http://io9.com/5831111/a-strange-new-quantum-state-of-matter-spin-liquids


[13] Schrieffer J. Relation Between the Anderson and Kondo Hamiltonians. PhysicalReview. 1966; 149(2): 491–492.

[14] Schulz H, Cuniberti G, Pieri P. Fermi-Liquids and Luttinger Liquids. Chia: ChiaLaguna Summer School Lecture Notes; 1997.

[15] Nazarov Y, Khaetskii A. Quantum Phase Transition in the Skyrmion Lattice. Phys-ical Review Letters. 1997; 80(3): 576–579.

[16] Wigner E. On the Interaction of Electrons in Metals. Physical Review. 1934; 46(11):1002–1011.

[17] Pfleiderer C, Rosch A. Single Skyrmions Spotted. Nature. 2010; 465(7300): 880–881.

[18] Romming N, Hanneken C, Menzel M, Bickel J, et al. Writing and Deleting SingleMagnetic Skyrmions. Science. 2013; 341(6146): 636–639.

[19] Dume B. Skyrmion Spin Control Could Help Speed up Electronics. [Online].Available from: http://physicsworld.com/cws/article/news/2013/aug/09/

skyrmion-spin-control-could-help-speed-up-electronics. [Accessed on 24th

January 2014].

[20] Ono T, Tanaka H, Matsuo A, Kindo K, et al. Magnetic-Field Induced QuantumPhase Transitions in Triangular-Lattice Antiferromagnets. Journal of Physics: Con-ference Series. 2011; 302(1): 1–6.

[21] Chen X, Ni J. Fermi Surface Nesting and Magnetic Quantum Phase Transition inGraphene-like BC3: A First-Principles Study. Physical Review B. 2013; 88(11): 1–6.

[22] Castro-Neto A, Guinea F, Peres N, Novoselov K, et al. The Electronic Properties ofGraphene. Reviews of Modern Physics. 2009; 81(1): 1–6.

[23] Nellutla S, Pati M, Jo Y, Moon B, et al. Magnetic Field-Induced Quantum PhaseTransition of the S=1/2 Antiferromagnet K2NaCrO8. Physical Review B. 2010;81(6): 1–6.

[24] Ketterle W, Durfee D, Stamper-Kurn D. Making, Probing and Understanding Bose-Einstein Condensates. Proceedings of the 1998 Enrico Fermi International School ofPhysics. 1998; 1: 1–87.

[25] Lang M, Wolf B, Honecker A, Balents L, et al. Field-Induced Quantum Criticality:Application to Magnetic Cooling. Physica Status Solidi B. 2013; 250(3): 457–463.

[26] Lang M, Wolf B, Honecker A, Tsui Y, et al. Magnetic Cooling Through QuantumCriticality. Journal of Physics: Conference Series. 2012; 400(3): 1–6.

[27] Kittel C. Quantum Theory of Solids. 1st Ed. New York: Wiley; 1963.

[28] Matsumoto M, Normand B, Rice T, Sigrist M. Field and Pressure-Induced MagneticQuantum Phase Transitions in TlCuCl3. Physical Review B. 2004; 69(5): 1–20.

[29] Lohneysen H, Rosch A, Vojta M, Wolfle P. Fermi-Liquid Instabilities at MagneticQuantum Phase Transitions. Reviews of Modern Physics. 2007; 79(3): 1015–1075.

30

http://physicsworld.com/cws/article/news/2013/aug/09/skyrmion-spin-control-could-help-speed-up-electronics

http://physicsworld.com/cws/article/news/2013/aug/09/skyrmion-spin-control-could-help-speed-up-electronics


[30] Ruegg C, Furrer A, Sheptyakov D, Strassle T, et al. Pressure-Induced QuantumPhase Transition in the Spin Liquid TlCuCl3. Physical Review Letters. 2004; 93(25):1–4.

[31] Ruegg C, Normand B, Matsumoto M, Furrer A, et al. Quantum Magnets underPressure: Controlling Elementary Excitations in TlCuCl3. Physical Review Letters.2008; 100(20): 1–4.

[32] Ishikawa Y, Tajoma K, Bloch D, Roth M. Helical Spin Structure in ManganeseSilicide, MnSi. Solid State Communications. 1976; 19(6): 1525–528.

[33] Andreica D, Dalmas de Roetier P, Yaouanc A, Amato A, et al. Absence of MagneticPhase Separation in MnSi under Pressure. Physical Review B. 2010; 81(6): 1–4.

[34] Pfleiderer C, McMullan G, Julian S, Lonzarich G. Magnetic Quantum Phase Tran-sition in MnSi under Hydrostatic Pressure. Physical Review B. 1997; 55(13): 8330–8338.

[35] Shirane G, Cowley R, Majkrzak C. Spiral Magnetic Correlation in Cubic MnSi.Physical Review B. 1983; 28(11): 6251–6255.

[36] Pfleiderer C, Boni P, Keller T, Roßler U, et al. Non-Fermi-Liquid Metal WithoutQuantum Criticality. Science. 2007; 316(5833): 1971–1974.

[37] Fak B, Sadykov R, Flouquet J, Lapertot G. Pressure Dependence of the MagneticStructure of the Itinerant Electron Magnet MnSi. Journal of Physics: CondensedMatter. 2005; 17(10): 1635–1644.

[38] Pfleiderer C. Why First-Order Quantum Phase Transitions are Interesting. Journalof Physics: Condensed Matter. 2005; 17(11): 987–996.

[39] Huxley A, Sheikin I, Braithwaite D. Metamagnetic Behavior Near the QuantumCritical Point in UGe2. Physica B. 2000; 284(2): 1277–1278.

[40] Huxley A, Raymond S, Ressouche E. Magnetic Excitations in the FerromagenticSuperconductor UGe2. Physical Review Letters. 2003; 91(20): 1–4.

[41] Gehring G. Pressure-induced Quantum Phase Transitions. Europhysics Letters.2008; 82(6): 1–6.

[42] Gehring G, Ahmed M. Landau Theory of Compressible Magnets Near a QuantumCritical Point. Journal of Applied Physics. 2010; 107(9): 1–3.

[43] Mineev V. Phase Diagram of UGe2: Whether There are Quantum Phase Transitions?Journal of Physics: Conference Series. 2012; 400(1): 1–6.

[44] Li P, Wu P, Bomze Y, Borzenets I, et al. Switching Currents Limited by Single PhaseSlips in One-Dimensional Superconducting Al Nanowires. Physical Review Letters.2011; 107(13): 1–5.

[45] Jia S, Jiramongkolchai P, Suchomel M, Toby B, et al. Ferromagnetic QuantumCritical Point Induced by Dimer-Breaking in SrCo2(Ge1−xPx)2. Nature Physics.2011; 7(3): 207–210.

[46] Ehrenreich H. Solid State Physics. 33rd Ed. New York: Academic Press Inc; 1978.

31


[47] Ashkenazi J, Johnson N. Quantum Criticality Stabilises High Tc SuperconductivityAgainst Competing Symmetry-Breaking Instabilities. Journal of Superconductivityand Novel Magnetism. 2013; 26(1): 2611–2616.

[48] Timusk T, Statt B. The Pseudogap in High Temperature Superconductors: AnExperimental Survey. Reports on Progress in Physics. 1999; 62(1): 61–122.

[49] Kastner M, Birgeneau R. Magnetic, Transport and Optical Properties of MonolayerCopper Oxides. Review of Modern Physics. 1998; 70(3): 897–928.

[50] Cooper R, Wang Y, Vignolle B, Lipscombe O, et al. Anomalous Criticality in theElectrical Resistivity of La2−xSrxCuO4. Science. 2009; 323(5914): 603–607.

[51] Renner C, Revaz B, Genoud J, Kadowaki K, et al. Pseudogap Precursor of theSuperconducting Gap in Under and Overdoped Bi2Sr2CaCu2O8+δ. Physical ReviewLetters. 1998; 80(1): 149–152.

[52] Sacepe B, Chapelier C, Baturina T, Vinokur V, et al. Pseudogap in a Thin Film ofa Conventional Superconductor. Nature Communications. 2010; 1(140): 1–6.

[53] Kamlapure A, Saraswat G, Chand M, Kumar S, et al. Pseudogap State in theStrongly Disordered Conventional Superconductor NbN. Journal of Physics: Con-ference Series. 2012; 400(2): 1–4.

[54] Kheli J, Goddard A. Origin of the Pseudogap in High Temperature Cuprate Super-conductors. Journal of Physical Chemistry Letters. 2011; 2(18): 2326–2330.

[55] Badoux S. Universal Quantum Oscillations in the Underdoped Cuprate Supercon-ductors. Nature Physics. 2013; 9(12): 761–764.

[56] Butch N, Jin K, Kirshenbaum K, Greene R, et al. Quantum Critical Scaling atthe Edge of Fermi-Liquid Stability in a Cuprate Superconductor. Proceedings of theNational Academy of Sciences of the USA. 2012; 109(22): 8440–8444.

[57] The North East MagLev. The Train. [Online]. Available from: http://www.

northeastmaglev.com/the-train. [Accessed on 19th January 2014].

[58] NSTA SciLinks. Uses for Superconductors. [Online]. Available from: http://www.

superconductors.org/uses.htm. [Accessed on 19th January 2014].

32

http://www.northeastmaglev.com/the-train

http://www.northeastmaglev.com/the-train

http://www.superconductors.org/uses.htm

http://www.superconductors.org/uses.htm

Quantum Criticality

Documents

Transcript of Quantum Criticality