1322 Brazilian Journal of Physics, vol. 34, no. 4A, December, 2004
Applications of Quantum Field Theory in Condensed Matter
E. C. Marino Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Cx.P. 68528, Rio de Janeiro RJ 21941-972, Brazil
Received on 18 December, 2003
We present a brief review of some applications of quantum field theory in condensed matter systems. These include isotropic and anisotropic antiferromagnetic chains, strongly correlated organic conductors, such as the Bechgaard salts, carbon nanotubes and high-Tc superconductors. The review is by no means exhaustive and points to a vast range of new interesting possible applications.
1 Introduction dimensional quantum magnetic systems, strongly correlated low-dimensional conductors, carbon nanotubes and high-Tc Quantum field theory, since its birth, has been the basic fra- superconductors. mework for describing the physics of elementary particles and their interactions. Firstly, its success in quantum elec- trodynamics has been enormous, leading to the best theoreti- 2 Quantum magnetic chains cal predictions of the whole physics. Later on, use of quan- tum field theory in the description of the weak and strong 2.1 Isotropic Heisenberg antiferromagnet interactions, was also extremely successful, converging for Let us consider the one-dimensional AF Heisenberg hamil- the standard model, which has a vast and impressive body of tonian (XXZ) experimental support. It was in the 80’s, however, that the X £ ¤ efforts to produce a quantum field theory based grand uni- H = J SxSx + SySy + δSzSz . (1) fied model for the weak, strong and electromagnetic interac- i i+1 i i+1 i i+1 i tions, failed. This was the starting point for the attempts to use new methods such as string theory in high-energy phy- The isotropic case corresponds to δ = 1. We can map this sics. into a fermionic system through the Jordan-Wigner transfor- Coincidentally, three paradigms of condensed matter mation [1] physics fell in the 80’s, namely, the independent electron ap- proximation, the Landau theory of the Fermi liquid and the Xi−1 Sz = ψ†ψ − 1 ; S+ = ψ† exp iπ ψ†ψ . (2) BCS theory of superconductivity. The first one is the basis i i i i i j j upon which the so called solid state physics has been built j and allows a thorough understanding of metals, insulators Taking the continuum limit, we can use bosonization to map and semiconductors. The second one is a systematic pro- the original spins into a bosonic system [2], namely cedure for introducing interactions among the electrons and describes quite well a large variety of systems in condensed β (−1)i Sz = ∂ φ + cos βφ, (3) matter. Both of them failed to provide an adequate descrip- i 2π x β tion of new complex materials presenting strong electronic correlations, found in the 80’s and 90’s. BCS theory, on the and 2π £ ¤ + −i β θ i other hand, for 30 years was the fundamental framework for Si = e cos βφ + (−1) , (4) understanding the phenomenon of superconductivity. Ne- where φ and θ are dual bosonic fields satisfying vertheless, it was unable to provide an explanation for the mechanism underlying the new high-temperature supercon- K K ∂ θ = ∂ φ ; ∂ φ = − ∂ θ, (5) ducting cuprates discovered in the mid 80’s. x v t x v t Quantum field theory, being the natural framework for 2 the quantum description of interacting many-particle sys- where K = β /4π, tems, found a fertile field of applications, precisely in the s description of those condensed matter systems, which could 2π β = 1 (6) not be described by traditional methods. A new and ac- 1 − π arccos δ tive area of applications of quantum field theory in conden- πJ sed matter has thus been opened and is the object of this and v = 2 is the characteristic velocity (spin-wave velo- brief review. We are going to concentrate basically in low- city). Observe that the isotropic case (δ = 0) corresponds to E. C. Marino 1323
β2 = 2π. After bosonization, the original hamiltonian (1) The sine-Gordon description of copper benzoate is also becomes very successful in describing the specific heat curves of this Z · ¸ material at low temperature. By means of thermal Bethe an- v 1 H = dx KΠ2 + (∂ φ)2 . (7) saatz, one can obtain the free energy, namely [7] 2K φ K x µ ¶ µ ¶ −2T ∆ X∞ (−1)n+1 n∆ 2TM M A similar expression can be obtained in terms of θ by using f(T ) = K − 1 K 1 , π n 1 T π 1 T (5). n=1 Usually, the most relevant quantity in a magnetic system (13) is the magnetic susceptibilty, which is given by where Z M1 = 2∆ sin (πξ/2) , (14) χ = dxdτ < Sz(x, τ)Sz(0, 0) > (8) with β2 ξ = , (15) in the static homogeneous case. This can be evaluated by 8π − β2 using (3) and (7). In order to obtain finite temperature cor- is the first breather mass. The specific heat, given by c = rections conformal perturbation theory must be used to cor- T ∂2f/∂T 2, thereby obtained is in excellent agreement with rect the conformal invariant hamiltonian (7). The result is the experiment [7]. [3] · ¸ Another successful application of sine-Gordon theory in 1 1 copper benzoate concerns its spectrum of excitations. From χ(T ) = 2 1 + , (9) Jπ 2 ln(T0/T ) (14) we can obtain the ratio of the breather M1 and soliton ∆ masses. This may be compared with neutron scattering where T0 ' 7.7J. This expression for the susceptibility is experiments made with an applied magnetic field of 7 T . in excellent agreement with the experiment for Sr2CuO3, For this value of magnetic field, Bethe ansatz calculations which is the best experimental realization of the antiferro- 2 magnetic quantum Heisenberg chain [4], provided we cho- indicate a value of β /2π = 0.82 [7]. Introducing this value ose the exchange coupling as J = 2200K. in (14), we get M 1 ' 0.78 (16) ∆ 2.2 Heisenberg antiferromagnet in an aniso- The neutron scattering cross sections at 7 T [5] present a tropic field - copper benzoate breather peak at 0.17meV and a soliton peak at 0.22meV whose ratio is precisely ' 0.78! Copper benzoate Cu(C6H5COO)2 · 3 H2O is a quite pe- culiar quantum magnetic chain, because if we apply to it an external magnetic field H in the z-direction, a transverse 3 Strongly correlated organic con- staggered magnetic field proportional to H is generated. In- deed, the hamiltonian is ductors X h i ~ ~ z i x H = JSi · Si+1 − HSi − h(−1) Si , (10) There is a class of quasi one-dimensional systems with very i rich physical propetries. These are the organic conductors known as Bechgaarad salts, namely (TMTSF )2X, where where h << H. X = PF6, ClO4, AsF4. These highly anisotropic materials Specific heat measurements clearly show [5] that a fi- consist in stacks of the planar TMTSF -molecules. At high eld induced gap is generated the spectrum, namely ∆(H) ∝ 0.65±0.03 temperatures, the conductivity along the stack axis is two H . The quantum field theory treatment of this sys- orders of magnitude higher than along the transverse ones. tem provides a beautiful explanation for this effect. Indeed, The electron dynamics along this direction is described by using the bosonization technique described in the previous the Hubbard hamiltonian subsection, we can write the hamiltonian above as the one X ³ ´ X of the sine-Gordon theory [6] † H = −t ciσcjσ + h.c + U ni,↑ni,↓, (17) Z ½ · ¸ µ ¶¾ v 1 2π
where · ¸ In this expression, the right and left moving fields (R and 4U 1/2 L) are obtained by expanding the tight-binding dispersion vc = vF 1 + (19) πvF relation about the two Fermi points at +kF and −kF , res- pectively. The sum in a = 1, 2 runs over the two bands ob- and · ¸ tained by folding the graphite sheet and expanding around 4U −1/2 K = 1 + . (20) the Fermi points [9]. σ =↑, ↓ are the two spin orientations. πvF The effective Coulomb potential along the cylinder axis is
with vF being the Fermi velocity (vF = 2ta at half-filling). such that its Fourier transform is given by [9] The last term in (18) corresponds to the so called umklapp µ ¶ µ ¶ 2e2 1 2e2 L term, describing the interaction of electrons in the vicinity Veff (k) = ln ≈ ln , (26) of one Fermi point with the ones around the opposite Fermi κ kR κ R 1 point, for a filling factor of 2n . The bosonic field is related where R is the cylinder radius, L, its length and κ ≈ 1.4 is to the original fermionic field as follows. Calling ψ1 and ψ2, the (experimentally determined) dieletric constant. respectively, the continuum fields associated to the electrons Applying the standard bosonization method [2] to the ψ- around k = +kF and k = −kF (right and left movers), we fields, we get have 2 Z · ¸ vF 1 j = ψ1ψ1 − ψ2ψ2 = √ ∂tφ. (21) H = dx KΠ2 + (∂ φ)2 , (27) π 2K φ K x A relevant quantity in the electronic system is the dyna- mic conductivity, given by where · ¸ 8e2 −1/2 · ¸ K = 1 + ln(L/R) (28) i 2v K πv κ σ(ω) = lim Π(ω, q) + F , (22) F q→0 ω π and 2e where Π(ω, q) is the Fourier transform of the retarded j = ψ† ψ − ψ† ψ = √ ∂ φ. (29) current-curent correlator, namely, R,a,σ R,a,σ L,a,σ L,a,σ π t Z iqx −iωt Using the experimental value for the Fermi velocity, namely, Π(ω, q) = −i dxdte e h[j(x, t), j(0, 0)]iret. 5 vF ' 8 × 10 m/s and the typical dimensions of a carbon (23) nanotube, R ≈ 0.75nm and L ≈ 1µm, we get K ≈ 0.28. The conductivity (22) can be evaluated by using (18) and We see, therefore that carbon nanotubes, like the organic (21). For frequencies much higher than the sine-Gordon gap conductors considered in the previous section, are strongly (ω >> ∆), one obtains [8] correlated electronic systems. A very interesting experimental set up, consists in ap- 2 σ(ω) ≈ ω4n K−5. (24) plying a voltage difference V to the carbon nanotube and measuring the current through it. For a density of states, This can be comparaed with the experimental results for given as a function of the energy as ρ(ε) = εα, at a tempe- 2 (TMTSF )2X and adjusted to the value 4n K − 5 = −1.3 rature T , the differential conductance is given by [10] [8]. For half-filling (n = 1), we would have K ≈ 0.93, µ ¶ ¯ µ ¶¯ ¯ ¯2 indicating that the system would be almost free in this case dI α eV ¯ 1 + α eV ¯ ∝ T cosh ¯Γ + i ¯ . (K = 1 is the free case). In the quarter filled case, which dV 2kBT 2 4πkBT corresponds to the real situation in the Bechgaard salts, (30) conversely, we have K ≈ 0.23. This is very far from 1 This has been obtained by using tunneling theory [10]. From (0 < K ≤ 1) and implies these materials are quite strongly (30), we can infer that correlated. T α k T >> eV dI B ∼ . (31) dV α 4 Carbon nanotubes V kBT << eV
Carbon nanotubes are extremely interesting highly anisotro- The parameter α associated to the electronic density of sta- pic systems consisting in graphite sheets folded in cylindri- tes in the nanotube may be determined from bosonization cal tubes with a diameter of approximately 1.5nm. The theory and (27), giving [11] electrons corresponding to the π-orbitals of graphite can 1 move along the tube and may be modeled by the continuum α = [K + K−1 − 2], (32) 8 hamiltonian [9] for tunneling into the bulk of the nanotube and Z X h i H = dx v ψ† i∂ ψ − ψ† i∂ ψ F R,a,σ x R,a,σ L,a,σ x L,a,σ α = K−1 − 1, (33) a,σ Z 1 for tunneling into the edge. Eq. (30) with α given by (32) + dxdyρ(x)V (x − y)ρ(y). (25) 2 eff and (33) is in excellent agreement with the experiments [12] E. C. Marino 1325
5 High Tc superconductors for YBCO and
High-temperature superconductors are among the most inte- M(δ, T = 0) = M(0,T = 0)[1 − Bδ − Cδ2]1/4 (38) resting and challenging materials ever investigated. Disco- vered in 1986, the mechanism of superconductivity they pre- for YBCO. The constants A, B and C have been computed sent, resists up to now its comprehension, despite the many in detail from first principles in [15]. Expressions (37) and theories that have been proposed for their description. Here (38), which have no adjustable parameters, are in excellent we present a field theory based model, which has been quite agreement with the experimental results [15]. By studying successful in describing part of the phase diagram of the M 2(δ, T = 0), we can obtain an expression for the Neel´ two best studied superconducting cuprates, namely LSCO temperature TN (δ) as a function of doping, which is also (La2−δSrδCuO4) and YBCO (Y Ba2Cu3O6+δ). in excellent agreemet with the experiments for both com- A common feature in all superconducting cuprates is the pounds [15]. ++ presence of CuO2 planes, with Cu ions on the sites and We can also describe in our theory the superconducting −− ++ O ions on the links of a square lattice. The Cu ions transition. Since skyrmions are the charge carriers, we must possess a localized electron per site. Upon doping (δ 6= 0) investigate the skyrmion-skyrmion interacting potential pre- holes are introduced in the oxygen ions. These can move dicted by the theory described by (35). For two static char- and interact among themselves and with the Cu++ spin ges of the same sign at positions ~x1 and ~x2, we get [13] background, eventually forming the superconducting Coo- per pairs beyond a certain critical doping and below a criti- Z 2 i~p·(~x1−~x1) cal temperature Tc(x). V (~x1 − ~x2) = d ~p Σ(~p, 0) e , (39) The model we propose is defined by the partition func- tion [13] where Σ(~p, 0) ≡ ΠB(~p) + ΠF (~p) is related to the vacuum Z polarization tensor corresponding to (35) as −S Z = Dz¯DzDψDψDAµ δ[¯zz − 1]e , (34) Σ(p) ¡ ¢ Σµν (p) = p2δµν − pµpν . (40) where p2 Z Z ( β~ X 1 S = dτ d2x |(∂ − iA )z |2 Notice that the bosonic and fermionic contributions to the 2g µ µ i 0 i 0 potential (39) correspond, respectively, to the spin fluctu- ations associated to the localized Cu++ spins (Schwinger X −− + ψ γ (i∂µ − qσ Aµ) ψ (35). bosons) and to the spin part of the O holes (spinons). α,λ µ z α,λ α=↑,↓,λ=1,2 We can evalute the potential (39) as a function of T and δ. In order to find the superconducting temperature, we In this description, the localized spins of the Cu++ ions are study the condition for this potential to have a minimum. † † ~ 1 This leads to the critical temperature Tc(δ). Our expression given by S = zi ~σijzj, where zi (zi zi = 1) are CP fi- elds (Schwinger bosons) describing the fluctuations of the for Tc(δ) is in good agreement with the experimental data spin background, Aµ is a Hubbard-Stratonovich field, g0 is for LSCO and YBCO below optimal doping [13]. Our re- a coupling constant related to the original Heisenberg ex- sults are actually not valid beyond optimal doping because change for these spins. The creation operators for holes in in this region we are outside the pseudogap phase where the −− † † † continuum limit leading to (35) was taken. the O ions are given by ci,α = µi fi,α,(α =↑, ↓) with † † fi,α creating a chargeless spin (spinon) and µi creating a spinless charge (chargon). The Dirac field ψα,λ that appears in (35) is built out of the f fields (λ = Fermi surface branch 6 Concluding remarks ) [13] and q is the spinon-Schwinger boson coupling. The † The few cases described in this review allow us to con- chargon operator µi is identified with the quantum skyrmion (topological excitations) creation operator whose quantum clude that Quantum Field Theory is an extremely power- properties have been described in [14]. ful and efficient method for describing Condensed Mat- The quantum skyrmion correlation function is a power- ter systems. It is specially useful for the case of low- ful tool for evaluating the antiferromagnetic order parameter dimensional and strongly correlated systems where the tra- for the Cu++ spin lattice. Indeed, this is given by [14, 13] ditional methods of Condensed Matter fail. Our review has not been, by no means, exhaustive and many interesting ap- 2 e−2πM (δ,T )|x−y| plications of Quantum Field Theory in the description of < µ(x)µ†(y) >= , (36) q2 Condensed Matter systems have not been covered. These in- |x − y| 2 clude the Chern-Simons-Landau-Ginzburg theory for Quan- where M(δ, T ) is the staggered (sublattice) magnetization tum Hall systems, conjugated polymers like polyacethilene, of the Cu++ spin lattice, as a function of temperature and ferromagnetic chains like the XY-model, Josephson junction doping. Evaluation of (36) gives [15] arrays, superfluid helium, heavy fermions and manganites among many others. We conclude by pointing out that a vast M(δ, T = 0) = M(0,T = 0)[1 − Aδ2]1/2 (37) field of applications of Quantum Field Theory in Condensed 1326 Brazilian Journal of Physics, vol. 34, no. 4A, December, 2004
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