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Critical quantum spin chains as conformal field theories

Diptiman Sen

Published Jul 1, 1992• 0 citations
OpenAlex

Abstract

The author presents a series of one-dimensional quantum Hamiltonians which, at their critical points, realise the minimal unitary series of conformal field theories with central charge less than 1. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. He shows how the infinite spin (free boson) limit can be obtained by using the Holstein-Primakoff transformation. The analysis can be generalised to SU(3) quantum chains which realise a different series of conformal field theories at criticality .

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Physical Review Letters•Oct 3, 1988•33 citations
The author presents a series of one-dimensional quantum Hamiltonians which, at their critical points, realise the minimal unitary series of conformal field theories with central charge less than 1. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. He shows how the infinite spin (free boson) limit can be obtained by using the Holstein-Primakoff transformation. The analysis can be generalised to SU(3) quantum chains which realise a different series of conformal field theories at criticality .
We present a series of one-dimensional quantum Hamiltonians that, at certain critical points, realize the minimal series of conformal field theories with central charge less than one. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. The infinite-spin (free-boson) limit is especially studied after performing the Holstein-Primakoff transformation. The analysis can be generalized to SU(3) quantum chains. These realize a different series of conformal field theories at criticality.
We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Quantum criticality in one-dimensional quantum systems is characterized by emergent conformal field theories (CFTs), whose central charge counts independent gapless degrees of freedom. Establishing a microscopic connection between this universal conformal structure and the momentum-space topology of the underlying quasiparticle spectrum remains challenging. Here, we uncover a direct correspondence between Fermi-point topology, conformal criticality, and quantum entanglement in an extended quantum spin chain with competing cluster interactions, exchange anisotropy, and a transverse magnetic field. We show that interaction- and field-driven Lifshitz transitions generate conformal critical phases with effective central charges ceff=1/2c_{\rm eff}=1/2ceff​=1/2, 111, 3/23/23/2, 222, and 333, including a multicritical point where Ising and Luttinger-liquid sectors coexist. Importantly, the central charge is not determined simply by the number of lattice gap closings or Fermi points, but by the number and conformal content of independent low-energy continuum sectors after accounting for lattice symmetries, reciprocal-lattice identifications, and mode equivalences. Thus, Lifshitz transitions may leave the conformal anomaly unchanged or modify the central charge depending on whether spectral reconstruction generates new independent continuum sectors. Real- and momentum-space entanglement spectra provide complementary microscopic signatures, revealing both the conformal content and momentum-space organization of critical modes. Our results establish a microscopic framework linking Fermi-point topology to emergent CFTs and show how interaction-driven spectral reconstruction can generate higher-central-charge criticality and unconventional multicritical behavior.
The eigenenergies for the spin SSS generalization of the critical XXZ\mathrm{XXZ}XXZ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge c=3S(S+1)c=\frac{3S}{(S+1)}c=(S+1)3S​ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.