Level Spectroscopy: Physical Meaning and Application to the Magnetization Plateau Problems

The Berezinskii-Kosterlitz-Thouless 1) 4) (BKT, or simply KT) quantum phase transition in one-dimensional systems is the transition between the gapless state and the gapful state. As is well known, the critical behavior of the BKT transition is highly singular and sometimes called “pathological”. In finite systems, this high singularity appears as severe slowly-converging logarithmic size corrections in various physical quantities. Thus it is very difficult to determine the BKT quantum phase transition point from the numerical data if we use the conventional methods. We note that there are two types in the BKT quantum phase transition. One is the doubly degenerate type, of which examples are (a) the transition between the spin-fluid (SF) and Néel states in the S = 1/2 XXZ spin chain, and (b) the SFdimer transition in the S = 1/2 Heisenberg spin chain with next-nearest-neighbor interactions. 5) 8) In these cases, the gapped states (the Néel state of (a) and the dimer state of (b) are doubly degenerate. The mechanism of the transition of this type is the spontaneous symmetry breaking. Another type is the non-degenerate type. The examples are (c) the SF-Haldane transition in S = 1 XXZ spin chain, and (d) the SF-dimer transition in S = 1/2 bond-alternating XXZ spin chain. 9), 10) In the non-degenerate case, the gapped states (the Haldane state in (c) and the dimer state in (d)) are unique and non-degenerate. The nature of this type of transition is that the gap-formation mechanism (for instance, the bond alternation in (d)) in the Hamiltonian is renormalized to zero in the sense of the renormalization group treatment due to the strong quantum fluctuations. In this paper, we review the level spectroscopy 11), 12) (LS) by use of which we can determine the BKT critical point very accurately (typical accuracy is 10−3 or better) from the numerical diagonalization data, overcoming the above-mentioned difficulties. We mainly focus on the physical meaning 13) of the LS for the doubly degenerate type in the one-dimensional quantum spin models. Also we explain how to apply the level spectroscopy to the magnetization plateau problems.