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Quantum Matter: Coherence and Correlations

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Quantum Matter: Coherence and Correlations Quantum Matter: Coherence And Correlations Henrik Johannesson Jari Kinaret March 26, 2007 2 Contents 1 Quantum Coherence in Condensed Matter 7 1.1 Effects of Phase Coherence . 7 1.1.1 Resonant Tunneling . 7 1.1.2 Persistent Currents . 10 1.2 Coherent transport . 12 1.2.1 Landauer-Buttik¨ er formalism . 12 1.2.2 Integer Quantum Hall Effect . 14 1.3 Localization: coherence effects in disordered systems . 17 1.4 Origins of decoherence . 24 2 Correlation Effects 31 2.1 Correlations in classical systems: Coulomb blockade . 31 2.1.1 Double Barrier Structure: Single Electron Transistor . 31 2.2 Correlations in quantum matter . 39 2.2.1 Interacting electrons: perturbative approach . 42 2.2.2 Fermi liquid theory . 50 2.2.3 Non-Fermi liquids: Broken symmetries, quantum criticality, disorder, and all that... 59 2.2.4 The Importance of Dimensionality: Luttinger liquid . 59 3 Examples 73 3.1 Quantum wires . 73 3.1.1 Carbon nanotubes . 73 3.1.2 Other quantum wires . 78 3.2 Fractional Quantum Hall Effect . 80 3.2.1 Bulk properties . 80 3.2.2 Edges . 84 3.3 Quantum magnets . 85 3.4 Kondo effect . 85 3.5 Mott insulators . 85 3.6 Bose-Einstein condensation . 85 4 Toolbox 87 4.1 Renormalization group . 87 4.1.1 Position space renormalization group . 87 4.1.2 Momentum space renormalization group . 99 3 4 CONTENTS 4.2 Approximate analytic techniques . 99 4.2.1 Mean field theory . 99 4.2.2 Dynamic mean field theory . 99 4.3 Exact analytic techniques . 99 4.3.1 Bethe Ansatz . 99 4.3.2 Bosonization . 99 4.4 Numerical techniques . 99 4.4.1 Density functional theory . 99 4.4.2 Quantum Monte Carlo . 104 A Coulomb Blockade at a Single Barrier 113 B The Minnhagen method 119 C Second quantization 121 D Bosonic form for r(x) 129 E Alternate forms for the Luttinger H 131 Preface This is work in preparation. There are undoubtedly many typos, and the notations between the different chapters may not be fully uniform, but hopefully the notes will improve during the course and the coming years. The document will be updated as errors are detected and corrected. We welcome your comments on explanations that you find confusing or particularly clear, trivial or incomprehensible. 5 6 CONTENTS Chapter 1 Quantum Coherence in Condensed Matter 1.1 Effects of Phase Coherence 1.1.1 Resonant Tunneling Consider a double barrier structure comprising two scattering centers in an otherwise clean one-dimensional structure as shown in Figure 1.1. Classically the transmission probability through the structure can be obtained from the probability of transmission (Tj) and reflection (Rj) for each barrier (Tj + Rj = 1) by accounting for multiple reflections between barriers as T T T T T T = T (1 + R R + R R R R + : : :)T = 1 2 = 1 2 1 1 2 1 2 1 2 1 2 1 R R T + T T T !T2=T1 2 T − 1 2 1 2 − 1 2 − 1 which is always less than T1, i.e. it is harder to get through two barriers than one. Hardly surprising. What may be surprising at first is that the total transmission probability is not 2 proportional to T1 but to only T1 for small transmission. The reason is that if the particle gets through the first barrier, it bounces several times between the barriers and hence has many attempts to get through the second barrier; in the limit of small T1 a particle is almost as likely to exit to the right as it is to exit to the left (once it has passed the first barrier), hence T T =2 as our calculation shows. ≈ 1 In quantum mechanics we cannot sum the probabilities for different trajectories but, in- stead, we have to sum the transmission amplitudes. During each passage between the barriers L T1 T2 Figure 1.1: A double barrier structure comprising two barriers with transmission probabilities T1 and T2, separated by distance L. 7 8 CHAPTER 1. QUANTUM COHERENCE IN CONDENSED MATTER the wave function accumulates phase kL, which yields the total transmission amplitude eikLt t t = t eikL(1 + e2ikLr r + e4ikLr r r r + : : :)t = 1 2 1 2 1 2 1 2 1 2 1 e2ikLr r − 1 2 so that the transmission probability is given by t t 2 t 4 T = j 1 2j j 1j : 1 e2ikLr r 2 !r2=r1;t2=t1 1 e2ikLr2 2 j − 1 2j j − 1j iφ iφ We now write r1 = pR1e R and t1 = pT1e T where T1 + R1 = 1 to get T 2 T 2 T = 1 = 1 : (1 cos(2kL + 2φ )R )2 + sin2(2kL + 2φ )R2 T 2 + 4(1 T ) sin2(kL + φ ) − R 1 R 1 1 − 1 R Now it is not true that T < T , and in particular if kL + φ = nπ, n Z, T = 1 regardless 1 R 2 of the transparency of an individual barrier: the particle is transmitted with unit probability regardless of how reflective the individual barriers are. This is the well-known phenomenon of resonant tunneling. Near a resonance, i.e. for kL nπ φ , we get ≈ − R T 2 1 T 1 = 2 2 1 T1 2 ≈ T1 + 4(1 T1)[kL (nπ φR)] 1 + 4 −2 [kL (nπ φR)] − − − T1 − − showing the Lorentzian line shape of the resonance. It is important to notice that the res- 1 T1 2 T1 onance width is given by 1 = 4 −T 2 [kL (nπ φR)] or [kL (nπ φR)] = p 1 − − − − 2 1 T1 implying that for low-transparency barriers the resonances are very narrow, and in the limit− of vanishing barrier transparency the resonance becomes a δ-peak. Two features are responsible for the difference between the classical and the quantum results: (i) in quantum mechanics we had to add the amplitudes of different ways of arriving at the same final state, and (ii) the phase accumulated during propagation between the barriers is always kL. The first of these is carved in the stone that Bohr & Co. brought down the mountain, but the second is an assumption. Let us now relax the assumption and postulate that, due to some unknown process, the th phase changes by an amount θj during the j round trip between the barriers. We then have ikL+θ0=2 2ikL+θ1 4ikL+θ1+θ2 t = t1e (1 + e r2r1 + e r2r1r2r1 + : : :)t2 so that T ( θ ) f jgj1=1 4 2ikL+θ1 2 4ikL+θ1+θ2 4 2 = t1 1 + e r1 + e r1 + : : : j j 2 2 2ikL+θ1+2φR 4ikL+θ1+θ2+4φR 2 = T1 1 + e R1 + e R1 + : : : or n 2 2 i2n(kL+φR)+i j=1 θj n T = T1 n1=0 e P R1 n m 2 i2(n m)(kL φR)+i j=1 θj i j=1 θj n+m = T1 Pn1=0 m1=0 e − − P − P R1 n 2 2n n 1 i2(n m)(kL φR)+i j=m+1 θj n+m = T1 P n1=0PR1 + 2Re n1=0 m−=0 e − − P R1 where the second termhP describes interferenceP Pbetween different paths. Let us now assumei that all the phase changes θj are independent random variables that follow a Gaussian distribution 1.1. EFFECTS OF PHASE COHERENCE 9 Figure 1.2: Transmission as a function of 2(kL φ ) and σ2 for T = 0:1. − R 1 P (θ) with standard deviation σ. We can then obtain the average transmission probability T for a collection (ensemble) of such systems by h i n 1 1 1 2 1 2n 1 − i2(n m)(kL φ ) 1 (n m)σ2 n+m T = dθ P (θ )T ( θ 1 ) = T R + 2Re e − − R − 2 − R h i j j f jgj=1 1 1 1 "n=0 n=0 m=0 # Z−∞jY=1 X X X 1 2 2 2 where I used 1 dθ P (θ)eiθ = 1 dθ e θ =(2σ )eiθ = e σ =2. This result shows that the p2πσ − − interference terms−∞ are reduced by the −∞phase fluctuations: the exponent has acquired the part R R (n m)σ2=2 the dampens the off-diagonal n = m terms. Proceeding further by evaluating − − 6 the geometric sums we get 1 2 −i2n(kL−φ )+n σ n 2 1 i2n(kL φ ) n 1 σ2 n 1 e R 2 R T = T + 2Re 1 e R 2 R − 1 1 1 R2 n=0 − − 1 −i2(kL−φ )+ 1 σ2 h i 1 1 e R 2 R1 − − σ2 2 2 1 P e R1 1 1 = T 2 1 2 − + 2Re 1 2 1 2 1 1 R σ2 2 σ 2 −i2(kL−φ )+ σ i2(kL−φ )− σ 1 e R +1 2e 2 cos(2(kL φ ))R1 1+R e R 2 R1 e R 2 R1 − 1 − − R 1− − Since there were ample possibilities for errors in this lengthy calculation, let us check the two limits we know: the classical result and deterministic quantum case. The classical limit is obtained by completely smearing out the phase information by setting σ2 , which yields ! 1 2 T1 T1 T σ = 2 = h i !1 1 R 2 T1 − 1 − as above, and the deterministic phase evolution is obtained by setting σ2 = 0, which yields T 2 T = 1 h iσ=0 1 + R2 2 cos(2(kL φ ))R 1 − − R 1 also in agreement with the previous result.
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