Anomalies and the lattice Schwinger model: Paradigm not paradox

Annals of Physics, 1999

We study the strong coupling limit of the 2-flavor massless Schwinger model on a lattice using staggered fermions and the Hamiltonian approach to lattice gauge theories. Using the correspondence between the low-lying states of the 2-flavor strongly coupled lattice Schwinger model and the antiferromagnetic Heisenberg chain established in a previous paper, we explicitly compute the mass spectrum of this lattice gauge model: we identify the low-lying excitations of the Schwinger model with those of the Heisenberg model and compute the mass gaps of other excitations in terms of vacuum expectation values (v.e.v.'s) of powers of the Heisenberg Hamiltonian and spin-spin correlation functions. We find a satisfactory agreement with the results of the continuum theory already at the second order in the strong coupling expansion. We show that the pattern of symmetry breaking of the continuum theory is well reproduced by the lattice theory; we see indeed that in the lattice theory the isoscalar ψ ψ and isovector ψ σ a ψ chiral condensates are zero to every order in the strong coupling expansion. In addition, we find that the chiral condensate < ψ (2) L ψ (1) L ψ (1) R ψ (2) R > is non zero also on the lattice; this is the relic in this lattice model of the axial anomaly in the continuum theory. We compute the v.e.v.'s of the spin-spin correlators of the Heisenberg model which are pertinent to the calculation of the mass spectrum and obtain an explicit construction of the lowest lying states for finite size Heisenberg antiferromagnetic chains.

Physical Review D, 1995

Using the monomer--dimer representation of the lattice Schwinger model, with $N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we evaluate its partition function, $Z$, exactly on finite lattices. By studying the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of small lattices, we find the zeroes closest to the real axis for infinite stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence for the existence of a critical value for the hopping parameter in the thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq 0.39$. By looking at the behaviour of quantities, such as the chiral condensate, the chiral susceptibility and the third derivative of $Z$ with respect to $1/2k$, close to the critical point $k_c$, we find some indications for a continuous phase transition.

Physical Review D, 1996

Cornell University - arXiv, 2022

Journal of Statistical Mechanics: Theory and Experiment, 2011

We consider the general Z 2 -symmetric free-fermion model on the finite periodic lattice, which includes as special cases the Ising model on the square and triangular lattices and Z n -symmetric BBS τ (2) -model with n = 2. Translating Kaufman's fermionic approach to diagonalization of Ising-like transfer matrices into the language of Grassmann integrals, we determine the transfer matrix eigenvectors and observe that they coincide with the eigenvectors of a square lattice Ising transfer matrix. This allows to find exact finite-lattice form factors of spin operators for the statistical model and the associated finite-length quantum chains, of which the most general is equivalent to the XY chain in a transverse field.

Physical Review D, 1998

We revisit the strong coupling limit of the Schwinger model on a lattice using staggered fermions and the hamiltonian approach to lattice gauge theories. Although staggered fermions have no continuous chiral symmetry, they posses a discrete axial invariance which forbids fermion mass and which must be broken in order for the lattice Schwinger model to exhibit the features of the spectrum of the continuum theory. We show that this discrete symmetry is indeed broken spontaneously in the strong coupling limit. Expanding around a gauge invariant ground state and carefully considering the normal ordering of the charge operator, we derive an improved strong coupling expansion and compute the masses of the low-lying bosonic excitations as well as the chiral condensate of the model. We find very good agreement between our lattice calculations and known continuum values for these quantities already in the fourth order of strong coupling perturbation theory. We also find the exact ground state of the antiferromagnetic Ising spin chain with long range Coulomb interaction, which determines the nature of the ground state in the strong coupling limit.

Arxiv preprint hep-th/9611084, 1996

1985

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Nuclear Physics B-Proceedings Supplements, 1998