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arXiv:1203.2326 · Journal of Statistical Mechanics Theory and Experiment

Exact and scaling form of the bipartite fidelity of the infinite XXZ chain

Robert Weston

Published Apr 12, 2012• 3 citations
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Abstract

We find an exact expression for the bipartite fidelity f = |⟨ vac | vac ⟩'| 2 , where | vac ⟩ is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and | vac ⟩' is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity − ln( f ) which has been put forward as a measure of quantum entanglement, and show that the large correlation length ξ behaviour is consistent with a general conjecture , where c is the central charge of the UV conformal field theory (with c = 1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of for a length L bipartite system with .

DOI
10.1088/1742-5468/2012/04/l04001

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We find an exact expression for the bipartite fidelity f = |⟨ vac | vac ⟩'| 2 , where | vac ⟩ is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and | vac ⟩' is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity − ln( f ) which has been put forward as a measure of quantum entanglement, and show that the large correlation length ξ behaviour is consistent with a general conjecture , where c is the central charge of the UV conformal field theory (with c = 1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of for a length L bipartite system with .
On the logarithmic bipartite fidelity of the open XXZ spin chain at Δ=−1/2Δ=-1/2Δ=−1/2
The open XXZ spin chain with the anisotropy \Delta=-\frac12 Δ = − 1 2 and a one-parameter family of diagonal boundary fields is studied at finite length. A determinant formula for an overlap involving the spin chain’s ground-state vectors for different lengths is found. The overlap allows one to obtain an exact finite-size formula for the ground state’s logarithmic bipartite fidelity. The leading terms of its asymptotic series for large chain lengths are evaluated. Their expressions confirm the predictions of conformal field theory for the fidelity.
We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find . Here ξ is the correlation length and c κ = 3κ/(κ+2) is the central charge associated with the Wess–Zumino–Witten (WZW) model at level κ. ln( g ) is the boundary entropy of the WZW model. Our result extends previous observations and suggests that this is a simple and perhaps rather general way both of extracting the central charge of the ultraviolet conformal field theory (CFT) associated with the scaling limit of a solvable lattice model, and of matching lattice and CFT boundary conditions.
We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find S ≃ cκ 6 ln(ξ) plus a term that depends upon the boundary conditions. Here ξ is the correlation length and cκ = 3κ κ+2 is the central charge associated with the sl2 WZW model at level κ. Our result generalises previous observations and suggests that this is a simple and perhaps rather general way of extracting the central charge of the ultraviolet CFT associated with the scaling limit of a solvable lattice model. In this letter, we consider the spin κ/2 analogue of the XXZ quantum spin chain (which corresponds to the choice κ = 1). We derive an exact formula for the entanglement entropy S (i,κ) associated with splitting the infinite chain into two semi-infinite pieces. We consider the scaling limit and find that S (i,κ) ≃ cκ
We consider correlation functions of the form , where |vac⟩ is the vacuum eigenstate of an infinite antiferromagnetic XXZ chain, |vac⟩′ is the vacuum eigenstate of an infinite XXZ chain which is split in two, and is a local operator. The Hamiltonian of the split chain has no coupling between sites 1 and 0 and has a staggered magnetic field at these two sites; it arises from a tensor product of left and right boundary transfer matrices. We find a simple, exact expression for ⟨vac|vac⟩′ and an exact integral expression for general using the vertex operator approach. We compute the integral when and find a conjectural expression that is analogous to the known formula for the XXZ spontaneous magnetization and reduces to it when the magnetic field is zero. We show that correlation functions obey a boundary qKZ equation of a different level to the infinite XXZ chain with one boundary.
The bipartite entanglement entropy of a segment of length lll in 1+11+11+1-dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\mathrm{ln}l+\ensuremath{\gamma}$, where ccc is the central charge of the CFT and $\ensuremath{\gamma}$ is a cutoff-dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have more bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy, but also have more multipartite entanglement content. In particular, we argue that since a system with a smaller smallest scaling dimension has a larger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in 1+11+11+1 dimensions. Furthermore, we also argue that the QFI decays under renormalization group flow between two unitary CFTs. Finally, we also comment on nonconformal but scale-invariant systems.
We analyze the finite-size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite-size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L\ensuremath{\rightarrow}\ensuremath{\infty}$) exhibit a unique pattern of entanglement, which differ only at leading order (1∕L)2{(1∕L)}^{2}(1∕L)2. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1∕21∕21∕2 XXZXXZXXZ chain with an arbitrary length LLL for both periodic and twisted boundary conditions.