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Quantum Phase Transitions Quantum Phase Transitions G. H. Lai May 5, 2006 Abstract A quantum phase transition (QPT) is a zero-temperature, generi- cally continuous transition tuned by a parameter in the Hamiltonian at which quantum fluctuations of diverging size and duration (and van- ishing 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 character- ized 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 param- eter 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 uni- versal 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 fer- romagnet 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 func- tion 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 thermo- dynamics(i.e.
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