A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.
We study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that, for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge
c=1/2 and 1.
We present several new results for the
N-partite information,
IN, of spatial regions in the ground state of
d-dimensional conformal field theories. First, we show that
IN can be written in terms of a single
N-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the
N-partite information scales as
IN∼r−2NΔ, where
r is the typical distance between pairs of regions and
Δ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the
I4 in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three-dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that
IN is always positive for general regions and arbitrary
N for that theory. For the
I4, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the
N-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.
Ground states of 1+1d conformal field theories (CFTs) satisfy a local entropic condition called the vector fixed point equation. This condition is surprisingly well satisfied by ground states of quantum critical lattice models even at small system sizes. We perform a search in the space of states of very small systems (4 qubits and 4 qutrits) and examine the states that satisfy this condition. By reconstructing a local Hamiltonian from each state, we are able to identify many of these solutions with known CFTs; others are gapped fixed points or involve large relevant perturbations, and others are CFTs we have not yet identified. These ideas are also useful for identifying continuous quantum phase transitions in a given family of Hamiltonians and for identifying the nature of the critical theory in small systems.
The entanglement entropy (EE) of quantum systems is often used as a test of low-energy descriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state expressed in the valence-bond basis (where the amplitudes decay as a power law of the bond length and the wave function is the product of all bond amplitudes), we show that even though the EE exhibits the expected CFT behavior, there is no CFT description of this state. We provide numerical tests of the behavior predicted by CFT for the correlation functions in the critical transverse-field Ising chain and the $J\text{\ensuremath{-}}Q$ spin chain, where the conformal structure is well understood. That behavior is not reproduced in the amplitude-product state.