Studying the perturbed Wess–Zumino–Novikov–Witten SU(2) theory using the truncated conformal spectrum approach

Abstract

We study the SU (2) k Wess-Zumino-Novikov-Witten (WZNW) theory perturbed by the trace of the primary field in the adjoint representation, a theory governing the low-energy behaviour of a class of strongly correlated electronic systems. While the model is non-integrable, its dynamics can be investigated using the numerical technique of the truncated conformal spectrum approach combined with numerical and analytical renormalization groups (TCSA+RG). The numerical results so obtained provide support for a semiclassical analysis valid at k ≫ 1. Namely, we find that the low energy behavior is sensitive to the sign of the coupling constant, λ. Moreover for λ > 0 this behavior depends on whether k is even or odd. With k even, we find definitive evidence that the model at low energies is equivalent to the massive O(3) sigma model. For k odd, the numerical evidence is more equivocal, but we find indications that the low energy effective theory is critical. I. INTRODUCTIONConformal field theories (CFT) describe universal critical behavior and by virtue of this play an enormously important role in the physics of strongly correlated systems. This universality is not completely lost in the presence of perturbations since, as a rule, the number of relevant operators is finite and once restricted by symmetry, often number but a few. Physics of perturbed critical models can be rich and complex, especially when the perturbation is non-integrable. For examples one may look at the quantum Ising model perturbed simultaneously by a longitudinal magnetic field and the thermal operator [1,2] or double sine-Gordon models [3,4].A focus on relevant perturbations of a CFT is most appropriate when the perturbations are strongly relevant. Indeed, the more relevant the perturbation the smaller is the energy scale over 2 which the spectrum is significantly altered. This feature lies at the foundation of the truncated conformal spectrum approach (TCSA) introduced in [5]. In the simplest version of this approach (TCSA), one truncates the spectrum of the unperturbed CFT which reduces the problem to numerical diagonalization of finite size matrices. Later this idea was combined with a numerical renormalization group [6] (TCSA+NRG). The TCSA+NRG has been used to tackle a number of problems ranging from the excitonic spectrum in semiconducting carbon nanotubes [7,8], to quenches in the Lieb-Liniger model [9,10], to studying theories whose fields live on a non-compact manifold [11]. In a further development, the precision of TCSA or TCSA+NRG computations can be improved further upon using perturbative renormalization group techniques [7,8,[12][13][14][15].These same renormalization group techniques allow one to use the TCSA to predict gaps in actual material systems which possess a finite bandwidth/cutoff [8].Below we will apply the TCSA+NRG to study the (1+1)-dimensional SU (2) k Wess-Zumino-Novikov-Witten (WZNW) model perturbed by the trace of the adjoint operator. This is a strongly relevant operator with scaling dimension d = 4 k+2 ideal for appl...

Finite-density phase diagram of a(1+1)−dnon-abelian lattice gauge theory with tensor networks

“…Indeed, this procedure provides results compatible with the previous method t A c = 12 ± 1 and t B c = 33 ± 2. The values we obtained for the central charge are compatible with the SU(2) 2 WZNW model, which is an expected critical behaviour for gauge theories [35]. (1) exhibits a gapless phase with central charge c = 1, is already a strong signature that such phase is a band conductor (albeit an interacting one) and its long-range properties at the thermodynamical limit are captured by the Luttinger liquid paradigm [72,73].…”

Hamiltonian truncation approach to quenches in the Ising field theory

“…The particular version of Hamiltonian truncation used in this work is based on a truncated fermionic space approach developed for the Ising field theory [41,42] abbreviated as TFSA. Note, however, that similar Hamiltonian truncation methods apply to a much wider range of field theory models such as perturbed minimal conformal field theories [43,44], sine-Gordon [45], Φ 4 and Landau-Ginzburg [46][47][48][49], and Wess-Zumino models [50][51][52], and can even be extended to more than one spatial dimension [53]. As a result, there are many potential directions to extend the studies in the present work.…”

NLO renormalization in the Hamiltonian truncation

“…Notice that g = 1, 2, while being smaller than the critical coupling g c ≈ 2.8, are well above the window g 0.2 where perturbation theory is accurate. 12 We could push the NLO-HT cutoff up to E T = 20 for L = 10, corresponding to ∼ 10 4 states. The main numerical bottleneck which prevents us from going higher is the evaluation of ∆H 3 .…”