Quantum Phase Transitions

Home , Quantum critical point, Quantum phase transition

G. H. Lai May 5, 2006

Abstract A quantum phase transition (QPT) is a zero-temperature, generically continuous transition tuned by a parameter in the Hamiltonian at which quantum ﬂuctuations of diverging size and duration (and vanishing energy) take the system between two distinct ground states [4]. This short review discusses the characteristic aspects of QPT and il- lustrates possible applications in physics and biology.

Contents

1 What are quantum phase transitions? 1

2 Features of a quantum phase transition 3

3 Relevance in experimental Physics and Biology 5

4 Further work 9

1 What are quantum phase transitions?

A phase transition is a fundamental change in the state of a system when one of the parameters of the system (the order parameter) passes through its critical point. The states on opposite sides of the critical point are characterized by diﬀerent types of ordering, typically from a symmetric or disordered state, which incorporates some symmetry of the Hamiltonian, to a broken- symmetry or ordered state, which does not have that symmetry, although the Hamiltonian still possesses it.

1 As we approach the phase transition, the correlations of the order parameter become long-ranged. Fluctuations of diverging size and duration (and vanishing energy) take the system between two distinct ground states across the critical point. When are quantum effects significant? Surprisingly, all non-zero temperature transitions are considered “classical”, even in highly quantum-mechanical systems like superfluid helium or superconductors. It turns out that while quantum mechanics is needed for the existence of an order parameter in such systems, it is classical thermal fluctuations that gov- ern the behaviour at long wave-lengths. The fact that the critical behaviour is independent of the microscopic details of the actual Hamiltonian is due to the diverging correlation length and correlation time: close to the critical point, the system performs an average over all length scales that are smaller than the very large correlation length. As a result, to correctly describe universal critical behaviour, it should suffice to work with an effective theory that keeps explicitly only the asymptotic long-wavelength behaviour of the original Hamiltonian (for instance, the phenomenological Landau-Ginsburg free-energy functional). So far, we have shown that classical theory suffices. However, consider what happens when the temperature around the critical point is below some characteristic energy of the system under consideration [2]. For example, the characteristic energy of an atom would be the Ryberg energy. We see a characteristic frequency ωc and it follows that quantum mechanics should be important when kbT < ~ωc. In the same spirit, if kbT >> ~ωc close to the transition, the critical fluctuations should behave classically. This argument also shows that zero-temperature phase transitions, where Tc = 0, are qualitatively different and their critical fluctuations have to be treated quantum mechanically. In such zero-temperature or quantum phase transitions (QPT), instead of varying the temperature through a critical point, we tune a dimensionless coupling constant J in the controlling Hamiltonian H(J). Generically [1], the ground state energy of H(J) will be a smooth, analytic function of g for finite lattices. However, suppose we have the Hamiltonian H(J) = H0+JH1, where H0 and H1 commute. This means that H0 and H1 can be simultaneously diagonalized and the eigenstates of the system will remain the same even though the eigen-energies will vary with J. A level crossing is possible where an excited level crosses the ground energy level at some coupling constant Jc, creating a point of non-analyticity in the ground state energy as a function of J, which manifests as a quantum phase transition.

2 Quantum Phase Transitions 11

10 D.BELITZ AND T.R.KIRKPATRICK critical region T T PM Tc (J) FM classical

PM FM

QM QM

Jc J Jc J (A) (B) Figure 1. Schematic phase diagram showing a paramagnetic (PFMig)uarend2.a feSrrcohmemagantiectipchase diagram as in Fig. 1 showing the vicinity of the quantum (FM) phase. The path indicated by the dashed line path reprcersietnictasl apocilnatss(iJca=l pJhc,aTse= 0). Indicated are the critical region, and the regions dominated transition, and the one indicated by the solid line a quantum phbaysethteracnlsaistsiiocna.l and quantum mechanical critical behavior, respectively. A measurement Figure 1: (A) Schematic phase diagramalong the path showingshown will obse arve paramagneticthe crossover from quantu (PM)m critical b andehavior away from the transition to classical critical behavior asymptotically close to it. 1.3a. A ferromagneticN EXAMPLE: THE PAR (FM)AMAGNE phase.T-TO-FERR TheOMAGN dottedET path represents a classical phase TRtransitionANSITION while the solid line indicates a quantum phase transition. (B) speaking, related to the classical one in a different spatial dimensionality, Let us illustrate the concepts introduced above by means of a concrete and since we know that changing the dimensionality usually means chang- exaVicinitymple. For defi ofnite theness, w quantume consider a m criticaletallic or itin pointerant ferr (oJmag=net.Jc,T = Tc). Indicated are the 5 ing the universality class, we expect the critical behavior at this quantum critical region, as well as the regionscritical dominatedpoint to be diffe byrent thefrom t clasicalhe one obser andved at quantumany other point on the coexistence curve. 1.3mechanical.1. The phase diag criticalram behaviour(QM) [2]. Figure 1 shows a schematic phase diagram in the T -J plTanheis, bwriintghsJustthoe the question of how continuity is guaranteed when one strength of the exchance coupling that is responsiblemfoorvefesrarloomngagtnheeticsomex. istence curve. The answer, which was found by Suzuki The coexistence curve separates the paramagnetic ph[9a]s,ealasto leaxrpgelaiTnsawndhy the behavior at the T = 0 critical point is relevant sm2all J from Featuresthe ferromagnetic one ofat sma all T quantumand larfgoer oJb.sFeorvraatiognivsenatJ phases,mall but non-ze transitionro temperatures. Consider Fig. 2, which there is a critical temperature, the Curie temperaturesThco,wwshaenreetnhlaerpghedasesection of the phase diagram near the quantum criti- transition occurs. This is the usual situation: A particaullaprominat.teTrihael chraitsicaal region, i.e. the region in parameter space where the givToen va illustratelue of J, and t thehe cla concepts,ssical transition weis tr consideriggecrreidticbayl ploow theweerrinlga examplewtshecan be obs oferve ad, metallicis bounded b ory th itinerante two dashed lines. temperature through Tc. Alternatively, however, we caInt timheangetuchrnasngoiuntg tJhat the critical region is divided into two subregions, at ferromagnetzero temperature (e. [2].g. by a Figurelloying the 1Amagn showset withdseonmo ateed schematicnobny-m‘QagMn’etaicnd ‘ phaseclassical’ i diagramn Fig. 2, in w inhich thethe oTbser−vedJcritical maplane,terial). Th withen we wTill ethencoun temperatureter the paramagnet- andtob-feehrraovJmioargtheinseptretrd strengthaonmsii-nantly qu ofantu them mec exchangehanical and cla couplingssical, respectively. tion at the critical value Jc. This is the quantum phaTsehetrdainvsiistiioonn bweetwaereen these two regions (shown as a dotted line in Fig. 2) intthaterested isin. responsibleSince we have seen forthat ferromagnetism.the quantum tisrannostitsiohnarpis The,(alonodsenl coexistenceeyither is the bound curveary of the separatescritical region), thebut rather

paramagnetic5For the purposes of this su phasebsection we (largemight as weTll c,on smallsider localizJed)spi fromns, but th thee ferromagnetic phase (small theory discussed in Sec. 2 below applies to itinerant magnets only. T , large J). For any given J, there is an associated Curie temperature Tc and classical phase transition occurs if we vary the temperature through Tc. On the other hand, imagine changing J at zero temperature, for instance, by alloying/doping the magnet with some non-magnetic material. Here, we encounter a quantum phase transition between the paramagnet and the ferromagnet at the critical value Jc. The behaviour at the T = 0,J = Jc is very diﬀerent from T 6= 0. In order to understand what happens in the vicinity of a QPT, a special mathematical trick which allows us to connect QPT with classical statistical mechanics is needed [3]. Consider the partition function of a d-dimension classical system governed by a Hamiltonian H,

3 Z(β) = Tr e−βHˆ (1) which can be written, via Feynman’s path integral formulation, as a function integral of the form (generalized for quantum many-body systems) Z Z = D[ψ,¯ ψ] eS[ψ,ψ¯ ]. (2)

Here, we let Hˆ be the Hamiltonian operator and S is the action of the system,

1 1 Z Z kB T ∂ Z kB T S[ψ,¯ ψ] = dx dτ ψ¯(x, τ)[− +µ]ψ(x, τ)− dτ H(ψ¯(x, τ), ψ(x, τ)). 0 ∂τ 0 (3) ψ¯ and ψ are the conjugate fields isomorphic to the creation and annihilation operators in the second quantized formulation of the Hamiltonian. We have sneaked in the trick of Wick rotation: analytically continuing the inverse temperature β into imaginary time τ via τ = −i~β, so that the operator ˆ ˆ density matrix of Z, eβH , looks like the time-evolution operator e−iHτ/~. The end result is seen in the action, which looks like that of a d + 1 Euclidean space-time integral, except that the extra temporal dimension is finite in extent (from 0 to β). As T → 0, we get the same (infinite) limits for a d + 1 effective classical system. This equivalent mapping between a d-dimension quantum system and a d + 1-dimensional classical system allows for great simplifications in our understanding of QPT. Since we know that the quantum transition is related to a classical analog in a different spatial dimensionality, and since changing the dimensionality usually means changing the universality class, the critical behaviour at the quantum critical point Jc should be different from that observed at any other point along the coexistence curve in Figure 1. Also, another interesting aspect of QPT is that dynamics and thermodynamics(i.e. statics, as thermodynamics is very much a misnomer) cannot be independently analyzed, unlike the case for classical statistical mechanics. This loss of freedom is due to the non-commutability of coordinates and momenta in the quantum problem. As a result, both the form of the Hamiltonian as well as the equations of motion are required, meaning that one cannot solve the thermodynamics without also solving the dynamics –

4 a feature that makes quantum statistical mechanics much harder to solve. Hence, even though it was mentioned in the earlier paragraph that a quantum transition can, in some fashion, be mapped into a classical thermodynamic transition, information about the correlations in excited states which drive the system out of the ground state cannot be extracted from this map.

3 Relevance in experimental Physics and Bi- ology

Quantum phase transitions attract intense attention because they are relevant to a host of experimental issues, despite the fact that T = 0 cannot be achieved experimentally. This is because associated with the quantum critical point, there is a sizable region where quantum critical behaviour is observable (labeled QM in Fig. 1B). Some examples are:

• Anderson-Mott models and metal-insulator transitions [4],

• superconductor-insulator (SI) transition in granular superconductors [5],

• transitions between quantum hall states [3],

• the physics of vortices in the presence of columnar disorder [6].

Besides the above, knowledge of dissipative eﬀects [7] on quantum coher- ence and QPT is essential to assess the reliability of mesoscopic quantum devices in performing tasks that strongly depend on their ability to maintain entanglement (for instance, in quantum computation). Among quantum devices widely used, many are based on a collection of regularly arranged or single small Josephson junctions (JJ). A Josephson junction comprises two superconductors linked by a very thin insulating oxide barrier and the current that tunnels through the barrier is the Josephson current. Josephson junction arrays also constitute a particularly attractive testing ground for the superconducting-insulating (SI) transition, because all parameters are well under control and are widely tunable. Cooper pairs of electrons are able to tunnel back and forth between grains and hence communicate about the quantum state on each grain. If the Cooper pairs are able to move freely from grain to grain throughout the array, the system behaves like a superconductor. If the grains are very small, a large charging energy is incurred to

5 Sondhi 318et al.: Continuous quantum phase transitionsSondhi et al.: Continuous quantum phase317transitions

rameter on the metallic elements connected by the junctions7 and their conjugate variables, the charges (excess Cooper pairs, or equivalently the voltages) on each FIG. 1. Schematic representation of a 1D Josephson-junctiongrain. TheSondhiintermediateet al.: Continuousstate ͉mjquantum͘, at timephase␶jϵtransitionsj␦␶, that 317 array. The crosses represent the tunnel junctionsentersbetweenthe quantumsu- partition function of Eq. (5) can perconducting segments, and ␪i are the phasesthusof thebe super-defined by specifying the set of phase angles conducting order parameter in the latter. ͕␪(␶j)͖ϵ͓␪1(␶j),␪2(␶j), . . . ,␪L(␶j)͔, where ␪i(␶j) is the phase angle of the ith grain at time ␶j . Two typical paths A. Partition functions and path integrals or time histories on the interval ͓0,ប␤͔ are illustrated in FIG. 1. Schematic representationFigs. of2 aand1D Josephson-junction3, where the orientation of the arrows Let us focus forarray.nowTheoncrossesthe expressionrepresent forthe Ztunnel. Noticejunctions between su- Ϫ␤H (‘‘spins’’) indicates the local phase angle of the order that the operatorperconductingdensity matrixsegments,e isandtheparameter.␪samei are theas theThephasesstatisticalof the super-weight of a given path, in the conductingϪorderiH /ប parameter in the latter. time-evolution operator e T , providedsumwe assignin Eq.the(5), is given by the product of the matrix imaginary value ϭϪiប␤ to the time interval over A.TPartition functions and pathelementsintegrals which the system evolves. More precisely, when the FIG. 2. Typical path or time history of a FIG.1D Josephson-3. Typical path or time history of a 1D Josephson- (A) Ϫ͑1/ប͒␦␶H(B) (C) trace is written in terms of a complete set of states,͕͗␪͑␶jϩ1͖͉͒junctione array.͉͕␪͑␶Notej͖͒͘, that this is equivalent(6) to junctionone of thearraycon-in the insulating phase, where correlations fall Let us focus for now on the͟j expression for Z. Notice Ϫ␤H figurations of a 1+1D classical XY model. Theofflong-rangeexponentiallycor- in both space and time. This corresponds to thatϪ␤theH operator density wherematrix e is the same as the Z͑␤͒ϭ ͗n͉e ͉n͘, ϪiH /ប (3) relations shown here are typical of the superconductingthe disorderedphasephase in the classical model. ͚ time-evolution operator e T , providedC we assign the n 2 of the 1Dˆ arrayˆ or, equivalently, of the ordered phase of the imaginaryFigurevalue 2:ϭϪ (A)iប␤ toH Representationϭthe ͚timeVj ϪintervalEJcos͑␪overjϪ␪jϩ1 of͒ a 1D Josephson(7) Junction array. Crosses Z takes the form of a sum of imaginary-timeT transition2 j classical model. easily identified, finding the classical Hamiltonian for which the system evolves. More precisely, when the FIG. 2. Typical path or time history of a 1D Josephson- amplitudes for therepresentsystem to start in junctionssomeisstatethe quantum͉n͘ and betweenHamiltonian superconductingof the Josephson-junction segments,this X-Y model is andsomewhatθi aremore work theand phasesrequires an trace is written in terms of a completeˆ set of states, junction array. Note that this is equivalent to one of the con- return to the same state after an imaginaryarray.time Hereinterval␪j is thewilloperatorobserverepresentingthat the expressionthe phaseforofthe quantumexplicit evaluationparti- of the matrix elements, which inter- figurations of a 1+1D classical8 XY model. The long-range cor- Ϫiប␤. Thus we seeofthat thecalculating superconductingtheϪ␤Hthermodynamicsthe superconductingtion orderorderfunctionparameter parameter.in Eq. on(5) thehas jtheth grain,form (B)of a estedclassical Typicalreadersparti-can pathfind in the orAppendix. time history Z͑␤͒ϭ ͗n͉e ͉n͘, (3) relations shown9 here are typical of the superconducting phase tion␪ ) ifunction,s conjugatei.e.,toathesumphaseover configurationsand is It expressedis shown in the Appendix that, in an approximationץ/ץ)(of a quantum system is the same͚n as calculatingV ϵϪtransitioni(2e/C j j of the 1D array or, equivalently, of the ordered phase of the 10 amplitudes for its ofevolution thein 1Dimaginary JJthetime, array.voltagewith theon Noticetheinjthtermsjunction,of itaand istransferE equivalentismatrix,the Josephsonif we think tothatof theimaginarypreserves configurationthe universality class ofof the aproblem, 1+1Dthe Z takes the form of a sum of imaginary-time transition classicalJ model. total time interval fixed by the temperaturecouplingof interest.energy. Thetimetwoas termsan additionalin the Hamiltonianspatial dimension.then Inproductparticular,of matrixif elements in Eq. (6) can be rewritten in amplitudes for the system to start in some state ͉n and ϪHXY The fact that the timeclassicalinterval happens XYtorepresent model.be imaginarythe charging Theour quantum long-rangedenergy͘ ofsystemeach grainlivesandin correlationsdthedimensions,Jo- thetheformexpres- aree typical, where the Hamiltonian of the orderedof the equiva- return to the same state after an imaginary time interval is not central. The key idea we hope to transmitsephsontocouplingthe sionof theforphaseits partitionacrosswill functionobservethe junctionthatlooksthebe-likeexpressiona lentclassicalclassicalforparti-theX-quantumY model parti-is Ϫiប␤. Thus we see that calculating the thermodynamics reader is that Eq. (3)(superconductingshould evoke an imagetweenof quantumgrains. for 1Dtion JJ)function phase.fortiona systemfunction (C)withind TypicalEq.ϩ1 (5)dimensions,has the pathformexceptof a orclassical timeparti- history of the of a quantum system is the same as calculating transition 1 dynamics and temporal propagation. As indicated previously,that the weextracandimensionmaption thefunction,quantumis finitei.e.,insta-aextent—sum overបH␤configurationsinϭunits cos͑␪expressedϪ␪ ͒ (8) amplitudes for its evolution in imaginary time, with the XY K͚ i j This way of looking1Dat JJthings arraycan be given intistical thea particu-mechanics insulatingofof time.the arrayAs phaseTontoin0 thetermsclassicalsystem (orof statisticala disorderedtransfersize in thismatrix,extraif‘‘time’’we phasethink͗ij͘ of forimaginary 1+1D XY) [3]. total time interval fixed by the temperature of interest. larly beautiful and practical implementationmechanicsin the lan-by identifyingdirection thediverges,statistical→time as anweightandadditionalweof a spatialgetanddimension.athe sumtrulyrunsIn particular,over nearest-neighborif points in the The fact that the time interval happens to be imaginary guage of Feynman’s path-integral formulationspace-timeof quan-path indEq.ϩ1(6)-dimensional,with theourBoltzmanneffectivequantumclassicalweightsystemofsystem.livestwo-dimensionalin d dimensions,(space-time)the expres- lattice.11 The nearest- is not central. The key idea we hope to transmit to the tum mechanics (Feynman, 1972).a two-dimensionalFeynman’s spatialSinceconfigurationthis equivalencesion offorabetweenitsclassicalpartitionasys-d-dimensionalfunctionneighborlooksquan-characterlike a classicalof theparti-couplings identifies the classi- reader is that Eq. (3) should evoke an image of quantum ϩ prescription is that the net transition amplitudetem. Inbetweenthis case thetumclassicalsystemsystemandtiona isdϩfunctiontherefore1-dimensionalfora two-a systemclassicalcalwithmodelsystemd 1asdimensions,isthe 2D X-Yexceptmodel, extensively studied in dynamicsmoveand antemporal excesspropagation. Cooper pair onto a grain. When this energy is large enough, two states of the system can be calculateddimensionalby summingX-Y model,crucial i.e.,to everythingits degreesthat theelseof extrafreedomwe havedimensionareto say theisandfinitecontextsincein Eq.extent—of superfluidប␤ in unitsand superconducting films This way of looking at things can be given a particu- amplitudes for all possible paths betweenplanarthem.spins,Thespecified(5) isbyprobablyangles ␪notof, thatverytime.liveilluminatingAsonT a two-0 thefor readerssystem(Goldenfeld,sizenot usedin this1992;extraChaikin‘‘time’’and Lubensky, 1995). We larlyCooperbeautiful and pairspractical cannotimplementation propagatein the lan- i and becomes→ confined on individual grains, path taken by the system is defined by dimensionalspecifying thesquaretolattice.a daily(Recallregimendirectionthatofattransfertemperaturesdiverges,matrices, itemphasizeandwill beweverythat,getwhilea thetrulystraightforward identification guage of Feynman’s path-integral formulation of quan- state of the system at a sequence of finelyabovespacedzerointer-the latticeusefulhasto considera finitedwidthϩa 1specific-dimensional,ប␤/␦example␶ in theeffectivein orderof theclassicalto degreesbe ablesystem.of freedom of the classical model in this tumleadingmechanics to the(Feynman, quenching1972). Feynman’s of the collective superconducting phase. mediate time steps. Formally we write temporal direction.)to visualizeWhile thewhatdegreesEq.Since(5)of thisfreedommeans.equivalenceare betweenexampleaisdrobust,-dimensionalthis simplicityquan- of the resulting classi- prescription is that the net transition amplitude between tum system and a dϩ1-dimensional classical system is Ϫ Ϫ cal Hamiltonian is something of a minor miracle. e ␤Hϭ͓e ͑1/twoប͒␦␶Hstates͔N,Asof the ansystem example,can be calculated(4) we considerby summing a one-dimensional array of identical JJs (Fig. crucial to everything else weIt haveis essentialto saytoandnotesincethatEq.the dimensionless coupling amplitudes 6for all possible paths betweenB.them.Example:TheOne-dimensional Josephson-junction arrays where ␦␶ is a time interval that is small7Itonis thebelievedtimethat neglecting fluctuations(5) isinprobablythe magnitudenot veryof illuminatingconstant Kforin Hreaders, whichnot usedplays the role of the tempera- path2).taken Theby the essentialsystem is defined degreesby specifying of freedomthe are the phases of theXY complex supercon- scales of interest (␦␶ϭប/⌫, where ⌫ is somethe orderultravioletparameter is a good approximationto a daily(see Bradleyregimenandof transferture inmatrices,the classicalit willmodel,be verydepends on the ratio of the state of the system at a sequence of finely spacedConsiderinter-a one-dimensional array comprising a large cutoff) and N isductinga large integer orderchosenDoniach, parameterso 1984;thatWallin et onal., 1994). the metallicuseful to consider segmentsa specificcapacitiveexample connectedchargingin orderenergyto be byableE ϭ the(2e)2 junctions/C to the Joseph- mediate time steps. Formally we write number L of identical Josephson junctions as illustrated C N␦␶ϭប␤. We then insert a sequence of sums8 over com- to visualize what Eq. (5) means. Our notation hereinis thatFig. ͕1.␪(␶Such)͖ refersarraysto thehaveconfigurationrecently beensonstudiedcouplingbyEJ in the array, plete sets of intermediateϪ␤H statesϪ͑1/intoប͒␦␶HtheN expression for ˆ ande ϭ their͓e conjugate͔ , of the entire set variables,ofHavilandangle variables(4)and Delsing theat time charges(1996).slice ␶. ATheJosephson␪’s (excessjunction Cooperis a pairs, or equiva- Z(␤): KϳͱEC /EJ, (9) wherelently␦␶ is a thetime interval voltages)appearing6 that inis thesmall onHamiltoniantunnelon eachthejunctiontimein grain.Eq. (7)connectingB.areExample:angular EventwocoordinateOne-dimensionalsuperconducting withoutJosephson-junctionmetallic doing furtherarrays math, from operators, and j is agrains.site label.CooperThe statepairsatoftimeelectronsslice ␶ areis anable toandtunnelhas nothingback to do with the physical temperature (see scales of interestϪ͑(1/␦ប␶͒␦ϭ␶Hប/⌫, where ⌫ is some ultraviolet ˆ ˆ Z͑␤͒ϭ ͗n͉e ͉meigenfunction1͘ of these operators: Considercos(␪jϪ␪ajϩ1one-dimensional)͉͕␪(␶)͖͘ the appendix).array comprisingThe physicsa largehere is that a large Joseph- ͚n mcutoff),m ͚, . . . ,mand N is a large integer chosenand soforththatbetween the grains and hence communicate 1 Fig.2 N 2B andϭ 2C,cos͓␪j(␶)Ϫ one␪jϩ1(␶)͔ can͉͕␪(␶)͖͘. drawnumber an analogyL of identical Josephson betweenson couplingjunctionsproduces ouras illustrated systema small value andof K, which a 2-Dfavors N␦␶ϭប␤. We then insert a sequence of sumsinformationover com- about the quantum state on each grain. If Ϫ͑1/ប͒␦␶H 9It is useful to think of this as a quantumin Fig.rotor1. model.Such arraysThe have recently been studied by ϫ͗m1plete͉e sets of͉mintermediate2͗͘m2͉ ͉mNstates͘ into the expressionthe Cooperforpairs are able to move freely from grain to classical XY••• model. It turnsimj␪j out that, in an approximation that preserves the state with wave function e has mj unitsHavilandof angularand momen-Delsing (1996).10 A Josephson junction is a Z(␤Ϫ)͑1/: ប͒␦␶H grain throughout the array, the system is a superconduc-That is, the approximation is such that the universal aspects ϫ͗m ͉e ͉n͘. tum representing(5) mj excess Cooper tunnelpairs onjunctiongrain jconnecting. The two superconducting metallic N tor. If the grains are very small, however, ofit coststhe criticala largebehavior, such as the exponents and scaling func- universality classCooper-pair of thenumber system,operator in wethisgrains. canrepresentationCooper indeedpairsisof electrons mapare ourable 1Dto tunnel JJback array into a 2D This rather messy expressionϭ actually has aϪ͑rather1/ប͒␦␶H charging energy to move an excess Coopertions,pairwillontobe agiven without error. However, nonuniversal ␪j) (see͉mWallin1͘ et al., 1994).andThe forthcosinebetweenterm in Eq.the grains and hence communicatepץ/ץ)Z͑␤͒ ͚ ͚ njϭ͗nϪ͉ei simple physical interpretation.X-Y systemn Formallym1 ,m2 , . . . [3],inclined,m(7)N is witha readers‘‘torque’’ ourtermgrain. dimensionlesswhichIf transfersthis energyunitsisoflargeconserved couplingenough,an- thequantities,Cooper constantpairssuch as theKcritical∼coupling,Ecwill/EdifferJ play-from an ex- fail to propagateinformationand becomeaboutstuck theon individualquantumact evaluation.grains,state onTechnically,each grain.the neglectedIf terms are irrelevant ϫ͗m ͉eϪ͑1/ប͒␦␶gularH͉m momentum͗͘m ͉ ͉m(Cooper͘ pairs) fromthesiteCooperto site. pairsNote thatare able to move freely from grain to 2 ing the role1 ofthe thepotential2 temperature2 ••energy• Nwhichof thecausesbosonstheis inrepresented,system theto classicalbesomewhatan insulator.at analog,the fixed point whereunderlyingEthe transition.= (2e) /C is grain throughout the array,11the system is a superconduc-c 6For convenience we have chosen ␦␶ toϪ͑1/beប͒␦real,paradoxically,␶H so that theby the kineticThe essentialenergy of thedegreesquantumof freedomrotors andin thisNoticesystemthisarecrucial change in notation from Eq. (7), where ϫ͗mN͉e ͉n͘. (5) tor. If the grains are very small, however, it costs a large small interval of imaginarythetime capacitivethat it representsvice chargingisversa.Ϫi␦␶. the energyphases of andthe complexE superconductingthe Josephsonj refersorderto a pa-point couplingin 1D space, not in1+1D thespace-time. array. This rather messy expression actually has a rather charging energyJ to move an excess Cooper pair onto a simpleThisphysical equivalenceinterpretation. Formally generalizesinclined readers to dgrain.-dimensionalIf this energy is large arraysenough, the andCooperd +pairs 1-dimensional Rev. Mod. Phys., Vol. 69, No. 1, January 1997 Rev. Mod. Phys., Vol. 69, No. 1, January 1997fail to propagate and become stuck on individual grains, classical X-Y models. Among thewhich systemscauses the system studied,to be an insulator. two-dimensional ones are 6For convenience we have chosen ␦␶ to be real, so that the The essential degrees of freedom in this system are smallmostinterval interestingof imaginary time that asit represents no trueis Ϫi␦ long-range␶. the phases orderof the complex is possiblesuperconducting at finiteorder pa- temperatures

Rev.whileMod. Phys., aVol. genuine69, No. 1, January QPT1997 occurs at T = 0. Moreover, 2D samples can be easily fabricated and experimentally characterized.

6 arXiv:cond-mat/0601462 v1 20 Jan 2006 r i c a g a w p t c m t t o d i e a p t t s P e b a p m t t w t t t H n s h i e r h h i h h h t a h h l s l m f d n e i r o o o o o i e i e a e e u a m o t m n e e e e e e o a a t r t N T j d l i a p c n n i r a t t p p p l y n a h r h e s p r r e b e t c o t h p i r s i r p ” e t e p , d s ( c g g l p v t o u c n r i , w e o o o c e e e i t g i e o m r v e a r t a e e o s l s t d e s r t t [ o e d c u - d c o r a c y h e G n n h a c . 6 , p u l r t e s d c D r i e a n a o d h n e s u p u t o m y a l a p t t - , i l r a a d , + m a e e u t i n n c e d B e e t a a , c u o v n e t l r v = l c 7 F e r u δ s n n N h t p e i e κ n s t o e l e r t e r a n o n o i i z o o e ] t t e w ρ a r r e r t c i g f d t n n v d r . s , Z l a u c t i h n c l e t c l o d d - s e r i ( i e 2 i n e y a t o g s h e e u a i o s o r e [ s , i m t t i s z r , s o o p P t t t m z o s π 0 e I i t b l l g s c m r t , . y s l i “ t u f m o o o r o e i ) a – n s m f f A c l n c h e - , o e c i y i e r / n a p m o o f l n - n i n a i s i e s W i o h T o d r t h 2 c t x o a = d u , a r a s s C s d o T n n n e i d a r n t t h m h t s f n t i l h fi u π r f t h a s “ t i a i n h a n h i s t t e S ) c o a d e h n e , t i i n s e i e h m t e p d = x i i r p a o o n p i ] n o ! t n r t = r r l a i e e w n t t i e s n i s b . a a g n n e a a o h e e v , - e e a u c != g c o p r x i t f l t f a i d o r t i s o u i r e l o r f d i d s n a e e a c s i h e t e r S t l t m a o a c t i 0 . 0 e t l n t c c r b t Z n - m d y r c h i [ d r n l s h c - e t i s e i h P a p b r n g h i o v 4 o e " φ t r n t y d v o a i u s e e c e t e q n o o d g e / a b b t n a n e o o o a s e o i s h e ] n b m y h ( e l f “ e D d a w | t s w n u s ρ e c l e c r a e f r c - r t d n ρ y r z d t l c c y i i c s r a f n e r m p a t a a e P n a e o l 0 g c a o s i m c h s N u o n t t a e ) s o e p f o a fi e r f d h - a s r . - n i t o c h n e q r c t n : h c o t o a n r c c u h l | n g fi o c u o d d i e A e r c o t h r e d i h d e a a e d t a u m e m r t n i t 6 t s a s t n t e o w p e t d s r i i e n a e p u o i s e n s a o m s r l 1 o e r e a o e T t x n l s l c d o l z o e t r , r p a y r e t e y a m o n . t t d y f p o s m i a n l t g i a o o c # a e 2 o i l e f b h a l o n h y n l ” h p g n t h t t w o n ” u t n h e y e r , n 0 l r f G a t n g m d , m i i h t t c e o g d e e n e e e o o a e t – r p - r . a g s e o a n d a e l o s h h s w z a t c Q - b φ l e i h n a e i n s l l w e i , o h n l u e r e d t e e o h a b + n A s a fi o t e e l u e o m a n y m e y d b e h g ( o e g n i x n b c o o w a s r n r n c o e t r u c h a n c n φ z p n s , r s t t e l - - i e u e o p s G p t t r l r i s p o t s e r a v i a d l t o c m t h ( d n a i o e o n ) o u p 8 i l e g t r a W e e r d c r l t z ” y r n r t e d h a r t d n n G l r “ a t r 7 d i i d c e r a f b o t l i h l o l , ) i a e o a t o m . e t t r h r i n t . e u a t e t W i e s s $ i o n c l o l u e l 1 i s c − i l h n s f p p f h o i l b o r c e i e g [ e t y c g l r T t a e a o t s 5 f w e t a w m n n 1 a l h + t a e w n d s a i o q 1 o i l n r u e i o h fl [ t e i n a . r t m b n y o n e h t a o , . g n o a h 5 d - g t e i u s l s s e s g a i r d t i e u i v e m c i t n y l w s t e f q n e h b r c v i , . t n s t e y l t o y a p s e u t p s , h r h t i o g a c e b d r o v a i s c h u 2 e , i u a b h n o n e e h i p i t t r a r 7 e t s g e u 6 t r a . t T r u b q v c c , r e r s h t t n i i M o e , o s e % t i lqet ftebnl atc ssoni B,weetedr lc lines black dark condensed the the in where peaks (B), depict [9]. triangular spheres density in a the charge shown while around backbone is ordering polymer lattice helical the depict bundle The the lattice. antiferromagnetic of triangular the plaquette a of states on ground model chiral XY degenerate, and two bundle The (A) polyelectrolyte array. between Correspondence 3: Figure u s U 4 t h b o h h e ] a n o o l t e p n u r e i u n t o e i w h e e h ( d . f h a B e c a a o t n − d o e e y r s o l 4 n n p t l g m t ( a o d u i 3 e . d D n e r s y o i l i . s y e a t n t t n e 0 r w D s f t i q c , c o o r r ] t t y l s s e i l t i w u r t r n i v b , . “ r t h J . G o u n e e a h t t o u c u s + h a s u l a p e a ρ i t s o c c e - a s e l e o y t i b b c e e y a c l | l n a g n t e d i i q e n t m h a h s r i o 0 f u r c k a t ρ l o l p s o o o t r e m t a n s a y t e i n l s : e s a w , t u i d a a a n e i r d a o e s i r n a t fi e n fi l n n f d i e l t l e n , s c t o p s T t a p s e r i n r e t d s e l e u o l p n n | s o n i : s s t ) h f n l d o y t t s i o i g g m r h c o e t t p c r o v m ( u z l h r o c i g w e f e l i i o e a n b g i h h n J e e e t u e u s 1 a o u t n t a i s e o a o o t a o s r r r e i i e s o n n d d d d a y o o ” e e e e i e h s s ) s s r t r s ------f f f l , s a n s n r a e t y a C a - u p e n m j p s n m a l u a d n t a o s o r e l n y p i i l y l i f F d b s b ( ξ i a a t t p m w d d m t y f c e i s n R e h a o o e z i n t e u o o e k l I t h e r h d o 2 r ) o a o n n G l r n i o n r d e l fi a = . e n T t 3 e o e w c c e d n o s i b f e b d h t q . n s , r n a o c e o i T m L i u n u y i t l c i e e . t a r 1 i m 2 n w r 2 s e j y = l a h t r o r a r a n : 0 β u e , i . i e e o h r e h a ( C s r a e r l e n s g l 0 a o s s 1 T l r C c a i r e t n n t c y 6 0 l p h a t F t G e c e d ) x s ( e o e h l t e t ) h t . L y o e o a t − b l a . m . f e t i e 2 a t i l s t e m o J n s c ) g o B i c i n o t . h t s c T c s b e . B d p t o f h l o 2 i p i i e e a s w e u W s o s n c L e X h t T r A c e (a) t a l l t s n a n o h n a r o s u r F o / e h n h t o n t e e l l p a a Y g d l β w r e s i a h o r i e d r e g i n r h s t p t , e p l C fi p o d n t e , e e e e y u h t d s e m h d n d t o o a e g s l h h w s m " r r a i i e [ p l a d a r f a o e o t n b c e o δ φ e y e s d i e s r e c n s o n n s f e a n y ρ d s , r e t ] t k o m o e t o n e – , m i d h e l g t h e ( d h p L = n e n a r d h r - 0 t l b n . v e a e c a e t t o a d a c o h i t ) e a t s l o p s t i t n W s r l - a l i h u n δ c p c C e r e n r u 2 i o m t r g a J c l o ρ o c u e g , i o e A h a u o e c d k a t l a u ( o n x h u u e c t & s y o e n d n e b z d 7 fi u h o s t n n p p n l l t 0 n t d g n s i z i T y a l ) ˆ i n s o i e r t s n i c e L e e . e t s r o # n v i e d i n a a a c c r p e a t l a n b o l = e e e d D r o k l i r s t i l l n t r r t e u o n b t q s - y ∼ z i a b s s o s r g i h g u , b i d u 0 n u y a u y ˆ o ' t a i o e o p t r u i a e a e u s d i i l C t s n l n o f y h i d o , t l ∂ p o n ∂ n t n i x t l e ˆ a b u t r s A e o e t - n e r i i g φ t z t s − o e T r l t c r n h o s m n t h a h u u a s l o e t | ( i d e n A l o v 9 p p t e a e l z l a n - o a . f s a a = a w t 0 t s 2 | n h o r k t e n s / c h t o r r i h E 0 l r o (b e t n d t i c a e ξ o o u o a , l e n i 2 g a a ff p m r x z s i e s 0 s c r l n l e d a g r 4 t e e s t l - - e e e a p ) c d t o a e , w c t c " o o o o y s u e q fl o a h t δ o U l r n s i u n r n u y i o | s t ρ e d m e v ρ s G S e d f h m l c ( e e y e t 1 n A t s p z t d z r t s e | h u h r t ) e 2 H r t , r l h t e o a c # e e a e i e o s q w a h b t s t − t w r a v p f i e m u s a e a n m o κ n a i s t ( e r r i c i o d n t t h 2 i d g n m c k i s i l i h n n – e e s ) t s - - - - - , T DJJ 2D 0 = 3

d = 3, n = 2 stacked triangular XY anti-ferromagnet 0, ω)/ω where [12]. These results have interesting implications in the context of polyelectrolyte bundles. The control param- 2 −i(q⊥·r⊥−ωτ) Cij (q⊥, iω) = d r⊥dτe eter, α, corresponds to (β2CE )−1 and eiφi to the − ! ij ! " expectation value of the modulated charge density of δ2 ln Z , (7) the Wigner crystal, δρi(z) . This mapping evidently × δa (r, τ)δa (0, 0)" ! " i j "a=0 predicts a continuous freezing transition of the counter- " ions as a function of temperature, even though three- is the analytic continuation of the current-cu"rrent cor- dimensional freezing transitions are in general first-order. relation function to imaginary frequencies [13], a(r⊥, τ) The critical exponents are, in fact, close to those of a tri- is an infinitesimal vector gauge field added to an externally applied vector potential, i.e. A A + a. In the critical point [12] , which does indicate the proximity of → a first-order transition. context of polyelectrolyte bundles, the externally applied vector potential would correspond to the A = π con- A further surprise is due to the fact that an f = 1/2 ij dition for the undeformed bundles while, according to Josephson-junction array in the superconducting phase eq. (5), the effect of applying an infinitesimal gauge field also has broken chiral symmetry with an associated (2π/Φ )a (r , τ) on A corresponds to that of intro- Z U(1) order parameter [10]. This is due, as shown 0 k ⊥ ij 2 ducing a shear strain, (2G)+ (r , z). It follows from the in F×ig. 1, to the fact that there are two degenerate sets zk ⊥ definition of the elastic strain energy and eq. (5) that the of chiral ground states for a d = 2 triangular XY anti- finite-temperature strain correlation function equals ferromagnet, with the phase winding around a particular plaquette in opposite senses, with each spin 2π/3 out of δ2 β−1 ln Gijkl(r) = − Z . (8) phase with its neighbors. In particular, the order pa- δ#+ij(r)δ+kl(0)$" "$ij =0 rameter, Ψ, can be constructed from a pair of complex " numbers, ψ+ and ψ−, related to the expectation value of Comparison of eqs. (7) and (8) " shows that the iφ e i by, Fourier transform Gxzxz(q⊥, qz) corresponds to minus the Cxx(q⊥, iω) component of the current-corrent cor- ∗ ∗ − eiφi ψ+e+iq ·xi + ψ−e−iq ·xi , (6) relation function. Thus, we can make use of established ! " ∝ results for the current-current correlation function of the Josephson array to deduce certain elastic properties of a where q = (4π/3a)xˆ, is the wavevector associated with ∗ polyelectrolyte bundle. the chiral winding of the phase. For α < α the dis- c In the insulating phase (α > α ) of the Josephson ar- crete symmetry is broken and the order parameter spon- c ray, the low-frequency conductivity is linearly propor- taneously aligns along the + or direction. This indi- tional to the frequency and C ( iω) ω2ξ(α), where cates that in the charged-ordered− phase the modulated xx ξ(α) is the correlation length wh−ich di∝ve−rges at the crit- density, δφ , breaks chiral symmetry (see Fig. 1), even i ical point as α α −ν . For the polyelectrolyte bun- when the!poly"mers themselves lack chirality at the molec- c dle, this mean| s −that |in the molten phase with no re- ular level. sponse to uniform sliding deformations (or Cxzxz = 0), The electromagnetic properties of a Josephson array, 2 2 Gxzxz(q⊥, qz) (kB T G )qz ξ(α). Hence, we have a such as conductivity and diamagnetic susceptibility, are columnar liquid∝crystal with a bending stiffness, K, of determined by the response of the array to an applied the bundle which diverges at the critical point as the vector potential. Specifically, the conductivity can be correlation length, ξ(α). Turning next to the supercon- q expressed by the Kubo relation as σ(ω) = ImCxx( = ducting phase (α < αc) of the Josephson array, since the zero-frequency conductivity is infinite the zero-frequency current-current correlation function is proportional to the superfluid density, so that C (q = 0, ω = 0) ξ−1(α) TABLE I: Correspondence between the d = 3, classical poly- xx ⊥ ∝ electrolyte system (left) and the d = 2+1, quantum frustrated [14]. Therefore, the shear modulus of the polyelectrolyte 2 −1 Josephson array system (right). bundle, Cxzxz = Gxzxz(q = 0) (kB T G )ξ (α), is finite in the low-temperature, ch∝arge-ordered phase −1 free-energy functional, βF [φ] imaginary-time action, ¯h S[φ] and vanishes at the critical point. This is the mod- height along polymer, z imaginary time, τ = it ulus corresponding to the uniform elastic deformations longitudinal phonon inverse grain charging pictured in Fig. 2. Finally, right at the critical point −1 (or phase) stiffness, βC energy, ¯hEc (α = αc) the Josephson is predicted to be metallic [14], electrostatic, inter-rod Josephson inter-grain i.e. to have a finite zero-frequency conductivity with −1 ∗ coupling, βEij coupling, ¯h EJ Cxx(q⊥, iω) = σ ω . Moreover, the critical conduc- − | | shear strain, 2G$⊥z vector potential, (2π/Φ0)a⊥ tance is expected to be a universal quantity equal to, e2/2h¯ [10]. The corresponding strain correlation func- elastic response, Gxzxz(qz) current response, −Cxx(−iω) tion, G (q , q ) (k T G2) q , would belong to an xzxz ⊥ z ∝ B | z|

Figure 4: Correspondence between d = 3 classical polyelectrolyte system (left) and d = 2 + 1 quantum frustrated JJ array system (right) [9].

In biology, nature has a small range of temperatures to play with, due to the protein nature of life chemistry, and most phase transitions occur with the variation of a parameter other than temperature. I hypothesize that it may be possible to draw analogies with QPTs, where the temperature is fixed (at T = 0) and some other parameter is varied, and hence tap into the immerse theoretical and experimental work that has been done on JJ arrays. Although i do not have anything solid to back up my statement, the field of polyelectrolyte condensation may show some promise. Polyelectrolyte chains naturally repel each other, but will nevertheless form condensed bundles in the presence of oppositely charged counterions beyond a certain valency (depending on the nature of the polyelectrolyte). Examples include DNA and F-actin. In fact, polyelectrolytes condense above a certain counterion concentration and dissociate above another counterion concentration. It has become clear that this condensation results from some form of organization of the counterions, either dynamical (correlated charge density fluctuations) or essentially static, in the form of a counterion lattice (positional correlations between condensed counterions). Some theoretical work on the counterion ‘melting’ transition suggest that the melting transition (i.e. the bundle-to-individual polyelectrolyte transition) is continuous and can be shown to be in the universality class of the three-dimensional XY model [8]. In another work [9], researchers managed to establish a mapping between a model for hexagonal polyelectrolyte bundles and a two-dimensional, frustrated Josephson-junction array (Fig.3). They found that the T = 0 SI transition of the quantum system corresponds to a continuous liquid-to-solid transition of the condensed charge in the finite

8 temperature classical polyelectrolyte system. Moreover, the role of the vector potential in the JJ system is played by elastic strain in the classical system (Fig. 4). The general conclusion of this work relates the elastic constants of polyelectrolyte bundles to the phase behaviour of the counterions, which is signiﬁcant as elastic constants are easier to measure than the scattering amplitudes of counterion charge modulation.

4 Further work

Due to constraints, much experimental and theoretical details have been left out. There are interesting instances of quantum phase transitions in two dimensional quantum magnets which cannot be predicted with an order parameter using the GLW (Ginzburg-Landau-Wilson) formalism that we have adopted in this study [10]. Also, studies on the dynamics of QPTs are still active and a quantum counterpart for the classical Kibble-Zurek mechanism of second order thermodynamic phase transitions was recently proposed [11, 12, 13]. It appears that both experimentalists and theorists have their work cut out for them!

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