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The end of c=1 CFT

K. Nomura

Published May 11, 1996• 2 citations
OpenAlex

Abstract

Conformal field theory (CFT) with the central charge c=1 is important both in the field theory and in the condensed matter physics, since it has the continuous internal symmetry (U(1) or SU(2)) and a marginal operator, and it is an effective theory of many 1D quantum spin and 1D electron systems. So it is valuable to understand how the c=1 CFT models become unstable. In this paper we discuss an instability of the c=1 CFT, that is, the transition to the ferromagnetic state. For the U(1) CFT case, we find that the spin wave velocity v and the critical dimension K behave v,1/K ∝ √ �c − � under reasonable assumptions.

Related papers

The end of c=1 CFT

K. Nomura

Conformal field theory (CFT) with the central charge c=1 is important both in the field theory and in the condensed matter physics, since it has the continuous internal symmetry (U(1) or SU(2)) and a marginal operator, and it is an effective theory of many 1D quantum spin and 1D electron systems. So it is valuable to understand how the c=1 CFT models become unstable. In this paper we discuss an instability of the c=1 CFT, that is, the transition to the ferromagnetic state. For the U(1) CFT case, we find that the spin wave velocity $v$ and the critical dimension $K$ behave $v, 1/K \propto \sqrt{Δ_c -Δ}$ under reasonable assumptions.

arXiv:cond-mat/9605070•May 11, 1996•0 citations

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Spin-1 bosons on a one-dimensional chain, at incommensurate filling with an antiferromagnetic spin interaction between neighboring bosons, may form a spin-1 boson condensed state that contains both a gapless charge and spin excitations. We argue that the spin-1 boson condensed state is unstable, and will become one of two superfluids by opening a spin gap. One superfluid must have a spin-1 ground state on a ring if it contains an odd number of bosons and has no degenerate states at the chain end. The other superfluid has a spin-0 ground state on a ring for any number of bosons and has a spin-1/2 degeneracy at the chain end. The two superfluids have the same symmetry and only differ by a spin-SO(3) symmetry protected topological order. Although Landau theory forbids a continuous phase transition between two phases with the same symmetry, the phase transition between the two superfluids can be generically continuous, which is described by conformal field theory (CFT) su(2)_{2}⊕u(1)_{4}⊕su(2)[over ¯]_{2}⊕u(1)[over ¯]_{4}. Such a CFT has a spin fractionalization: spin-1 excitation can decay into a spin-1/2 right mover and a spin-1/2 left mover. We determine the critical theory by solving the partition function based on emergent symmetries and modular invariance condition of CFTs.

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Conformal field theory (CFT) with the central charge c=1 is important both in the field theory and in the condensed matter physics, since it has the continuous internal symmetry (U(1) or SU(2)) and a marginal operator, and it is an effective theory of many 1D quantum spin and 1D electron systems. So it is valuable to understand how the c=1 CFT models become unstable. In this paper we discuss an instability of the c=1 CFT, that is, the transition to the ferromagnetic state. For the U(1) CFT case, we find that the spin wave velocity v and the critical dimension K behave v,1/K ∝ √ �c − � under reasonable assumptions.
Conformal field theory (CFT) with the central charge c=1 is important both in the field theory and in the condensed matter physics, since it has the continuous internal symmetry (U(1) or SU(2)) and a marginal operator, and it is an effective theory of many 1D quantum spin and 1D electron systems. So it is valuable to understand how the c=1 CFT models become unstable. In this paper we discuss an instability of the c=1 CFT, that is, the transition to the ferromagnetic state. For the U(1) CFT case, we find that the spin wave velocity vvv and the critical dimension KKK behave v,1/K∝Δc−Δv, 1/K \propto \sqrt{Δ_c -Δ}v,1/K∝Δc​−Δ​ under reasonable assumptions.
Spin-1 bosons on a one-dimensional chain, at incommensurate filling with an antiferromagnetic spin interaction between neighboring bosons, may form a spin-1 boson condensed state that contains both a gapless charge and spin excitations. We argue that the spin-1 boson condensed state is unstable, and will become one of two superfluids by opening a spin gap. One superfluid must have a spin-1 ground state on a ring if it contains an odd number of bosons and has no degenerate states at the chain end. The other superfluid has a spin-0 ground state on a ring for any number of bosons and has a spin-1/2 degeneracy at the chain end. The two superfluids have the same symmetry and only differ by a spin-SO(3) symmetry protected topological order. Although Landau theory forbids a continuous phase transition between two phases with the same symmetry, the phase transition between the two superfluids can be generically continuous, which is described by conformal field theory (CFT) su(2)_{2}⊕u(1)_{4}⊕su(2)[over ¯]_{2}⊕u(1)[over ¯]_{4}. Such a CFT has a spin fractionalization: spin-1 excitation can decay into a spin-1/2 right mover and a spin-1/2 left mover. We determine the critical theory by solving the partition function based on emergent symmetries and modular invariance condition of CFTs.
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 consider relativistic U(1) gauge theories in 2+12+12+1 dimensions, with Nb{N}_{b}Nb​ species of complex bosons and Nf{N}_{f}Nf​ species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a conformal field theory (CFT). At large Nb{N}_{b}Nb​ and Nf{N}_{f}Nf​, but for arbitrary values of the ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​, we present computations of various critical exponents and universal amplitudes for these CFTs. We make contact with the different spin liquids, charge liquids, and deconfined critical points of quantum magnets that these field theories describe. We compute physical observables that may be measured in experiments or numerical simulations of insulating and doped quantum magnets.
We compute three-point correlators between the stress-energy tensor and the conserved currents of conformal field theories (CFTs) in 2+1 dimensions. We first compute the correlators in the large-flavor-number expansion of conformal gauge theories and then perform the computation using holography. In the holographic approach, the correlators are computed from an effective action on (3+1)-dimensional anti-de Sitter space (AdS4{\mathrm{AdS}}_{4}AdS4​) and depend upon the coefficient $\ensuremath{\gamma}$ of a four-derivative term in the action. We find a precise match between the CFT and the holographic results, thus, fixing the values of $\ensuremath{\gamma}$. The CFTs of free fermions and bosons take the values $\ensuremath{\gamma}=1/12,\ensuremath{-}1/12$, respectively, and so saturate the bound $|\ensuremath{\gamma}|\ensuremath{\le}1/12$ obtained earlier from the holographic theory; the correlator of the conserved gauge flux of U(1)U(1)U(1) gauge theories takes intermediate values of $\ensuremath{\gamma}$. The value of $\ensuremath{\gamma}$ also controls the frequency dependence of the conductivity and other properties of quantum critical transport at nonzero temperatures. Our results for the values of $\ensuremath{\gamma}$ lead to an appealing physical interpretation of particlelike or vortexlike transport near quantum phase transitions of interest in condensed-matter physics. This paper includes Appendices reviewing key features of the AdS-CFT correspondence for condensed-matter physicists.