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Lecture III: the Thirring Model S Lecture III: The Thirring Model S. Lukyanov Jordan{Wigner transformation The goal of this lecture is to better understand the scaling limit. To do this, start with the Heisenberg XXZ model. The Heisenberg XXZ model describes a system of N one dimensional spins on the line with nearest neighbor interactions. The Hamiltonian is: N X x x y y z z HXXZ = −J σm σm+1 + σm σm+1 + ∆ σm σm+1 ; (1) m=1 a th where the Pauli Matrices σm act on the m site. The quasiperiodic boundary conditions are: ± ±2iθ ± σN+1 = e σ1 : (2) The Hamiltonian commutes with the total spin operator Sz: N 1 X Sz = σz ; 2 m m=1 so that the total number of up spins n is conserved by the Hamiltonian. The Heisenberg spin chain can be converted to a system of fermions by the Jordan{Wigner transformation. The Pauli Matrices at a particular site satisfy the fermionic anti{commutation rela- tions: + + − − + − fσm; σmg = 0 ; fσm; σmg = 0 ; fσm; σmg = 1 : (3) ± In other words, the σm behave as: + y − σm ∼ m ; σm ∼ m ; where the vector space at site i, C2 is spanned by the basis vectors: 1 0 j+i = ; |−i = : 0 1 However, the Pauli matrices at different sites do not anticommute. In order to really consider the system of spins as fermions, it is necessary to multiply the Pauli matrices by some operator, Cm, so that the anti{commutation relations (3) are preserved and the Pauli matrices at different sites anti{commute. A simple way to satisfy this would be if a a the Cm commute with σj for j ≥ m and anti{commute with σj for j < m. For two spins: y + − Ψ1 = σ1 ; Ψ1 = σ1 ; y iπ z iπ z σ1 + − σ1 + Ψ2 = e 2 σ2 ; Ψ2 = e 2 σ2 ; 1 iπ z σ1 z where e 2 = iσ1. The generalization to N spins is trivial: iπ X Ψy = exp − σz σ+ m 2 j m j<m iπ X Ψ = exp + σz σ− m 2 j m j<m which can be equivalently written as: + X y 1 y σm = exp + iπ Ψj Ψj − 2 Ψm j<m − X y 1 σm = exp − iπ Ψj Ψj − 2 Ψm j<m due to the identity: σz = 2 y − 1. Exercise III.1: Show that the Jordan{Wigner transformation maps the Hamiltonian XXZ-spin chain (1),(2) to N X y y y 1 y 1 HXXZ = −2J i ΨmΨm+1 − Ψm+1 Ψm + 2∆ ΨmΨm − 2 Ψm+1Ψm+1 − 2 ; m=1 where the fermions satisfy the Boundary Conditions (BC) y −iπSz+2iθ y +iπSz−2iθ ΨN+1 = −e Ψ1 ; N+1 = −Ψ1 e : (4) Low energy effective theory for ∆ = 0 Clearly it make sense to start with the case ∆ = 0, where the Hamiltonian describes a system of free fermions and can be easily diagonalized. For simplicity, let us assume that that N is an even number, i.e., Sz is an integer. Apply the Fourier transform i−m N−M X imkj Ψm = p cj e N j=−M+1 i+m N−M y X y −imkj Ψm = p cj e : N j=−M+1 N Here M stands for the integer part of 4 . The expansion coefficients cj satisfy the anti{ commutation relations: y y y fcm; clg = δm;l fcm; cl g = fcm; clg = 0 : 2 ϵ 4J k -π kL kR π -4J Figure 1: A plot of the dispersion relation = −4J cos(k) for the first Brillouin zone. Admissible values for the quasimomentum k follow from the twisted BC (4): π 2π 1 θ 1 z kj = 2 − N j − 2 + π + 2 S : (5) Substituting this into the Hamiltonian HXXZ with ∆ = 0 yields: X y H = (kj) cj cj ; where (k) = −4J cos(k) : (6) j The operators cj should be thought of as the creation and annihilation operators of a particle with definite momentum kj and with the dispersion relation = (k). A plot of the dispersion relation is given in figure 1. For quasimomentum kL ≤ kj ≤ kR, π 2π 1 θ 1 z π 2π 1 θ 1 z kL = − 2 + N 2 − π + 2 S ; kR = + 2 − N 2 + π + 2 S ; (7) the energy of the particle is negative. In the ground state, these energy levels are filled and form a Fermi sea, so that the vacuum energy is given by N z 2 −S (vac) X E = −4J cos(kj) : (8) j=1 Exercise III.2. Show that, as N goes to infinity, 4JN π E(vac) = − − 4J 1 − 12 θ2 − 3 (Sz)2 + O(N −3) : (9) π 6N The excited energy states are particles of positive energy or holes in the Fermi sea. As an illustration, let us focus on the case Sz = θ = 0. Introduce the notations a 1 = cj ; −M + 1 ≤ j ≤ 0 2 −j y a 1 = c ; 1 ≤ j ≤ M j− 2 j 3 and y N aj− 1 = c N ;M − 2 < j ≤ 0 2 2 +j N b 1 = c N ; 1 ≤ j ≤ − M 2 −j 2 +j 2 (recall that M = [N=4]). The low energy physics is described by the low energy hole and particle excitations that lie close to the \Fermi surface" k 2 fkR; kLg, where the dispersion is approximately linear: 2jνjπ 8πJ 1 3 = 4J sin( N ) ≈ N jνj ; ν = ± 2 ;; ± 2 ::: N: The dynamics of the low-energy excitations is captured by the effective Hamiltonian: 1 4JN 8πJ 1 X H = − + − + jνj ay a + by b + O(N −3) : XXZ π N 12 ν ν ν ν 1 3 ν=± 2 ;± 2 ::: Notice that the parameter a = (4J)−1 has the dimension of length and can be interpreted as a dimensionful lattice spacing. The scaling limit is a ! 0, N ! 1 while the macroscopic length of the system, R = aN ; is kept fixed. The scaling limit lead to a low energy effective Hamiltonian 1 1 R 2π 1 X H = − + − + jνj ay a + by b ; (10) eff π a2 R 12 ν ν ν ν 1 3 ν=± 2 ;± 2 ::: which describes the physics of the system at energy scales significantly below that of the microscopic energy scale J. The first term here diverges as a ! 0. It is proportional to the size of the system and represents the contribution of the modes whose energy exceed −1 the ultraviolet (UV) cut-off scale ΛUV = a . This \quadratic" divergency is absorbed by the so-called counterterm of the identity operator and is ignored in the context of QFT. πc The term − 6R with c = 1 is somewhat universal and can be interpreted as the Casimir energy. The spectrum of the effective Hamiltonian is given by R 2π 1 1 E = − + − + L + L¯ ; L; L¯ = 0; 1; 2;:::: eff a2 R 12 2 The nonnegative integers L and L¯ represent the contributions of the modes with half- integers ν > 0 and ν < 0, respectively. 4 In the scaling limit the lattice fermions are approximated as p h −m 1 +m y 1 i Ψm ≈ a i R(x ) + i L(x ) y p h +m y 1 −m 1 i Ψm ≈ a i R(x ) + i L(x ) ; (11) where x1 = ma and the continuous Fermi fields 1 X x1 x1 1 − 2 2πiν R y −2πiν R R(x ) = R aν e + bν e 1 3 ν=+ 2 ;+ 2 ; ::: 1 X x1 x1 1 − 2 2πiν R y −2πiν R L(x ) = R aν e + bν e 1 3 ν=− 2 ;− 2 ; ::: obey the canonical anticommutation relations y 1 1 y 1 1 1 1 f R(x ); R(y )g = f R( x); R(y )g = δ(x − y ) ; 1 1 y 1 y 1 f R;L(x ); R;L(y )g = f R;L(x ); R;L(y )g = 0 (12) and the antiperiodic boundary conditions1 1 1 ±(x + R) = − ±(x ) : Notice that here we use the QFT notation x1 for the space coordinate. The time variable will be denoted below by x0. The low energy effective Hamiltonian can be expressed in terms of R;L. Exercise III.3.: Show that, ignoring the quadratic divergency, the Hamiltonian (10) can be written in the form Z R 1 Heff = dx H0(x) ; (13) 0 i H (x) = − y @ + @ y − y @ − @ y @ = @ : 0 2 R 1 R R 1 R L 1 L L 1 L 1 @x1 1 3 5 Hint: In the analytical regularization the divergent sum 2 + 2 + 2 + ::: is replaced by 1 1 the Hurwitz zeta function ζ − 1; 2 = 24 . In the Lagrangian description both the canonical (anti)commutation relations and Hamiltonian are encoded by means of the Lagrangian. Exercise III.4.: Show that the Lagrangian density corresponding to the canonical anticommutation relations (12) and the Hamiltonian (13) is given by y y y y 1 L0(x) = i R@+ R + R@+ R + L@− L + L@− L ; where @± = 2 (@0 ± @1) : (14) 1Recall that we are focusing here on the scaling limit of the XXZ spin chain with ∆ = Sz = θ = 0. 5 Notice that L0(x) is a local function built from the local fermionic fields R;L(x) taken at the same space-time point x = (x0; x1). The classical equations of motion corresponding to the Lagrangian density (14) read as @+ R = 0 @− L = 0 : (15) In the case of infinite volume, i.e. R = 1, a general solution of these equation represent a 0 1 traveling wave propagating with a constant speed either to the right R = R(x − x ) , 0 1 2 or to the left R = R(x + x ) .
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