Fisher–Hartwig determinants, conformal field theory and universality in generalised XX models

Hutchinson, Joanna; Jones, Nicholas Blurton

doi:10.1088/1742-5468/2016/07/073103

Cited by 3 publications

(7 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, observe that the charge-two operator e i(θ 1 (x)+θ 2 (x)) with ∆ = 1/2 dominates the charge-two operator e i(2θ 1 (x)+ϕ 2 (x)) with ∆ = 5/4 (and indeed the charge-two operator e i2θ 1 (x) with ∆ = 1). Then, as argued in [53], in isotropic models σ + = O 1 + iO −1 will be dominated by operators τ µ,ν with j µ j + ν j = 1 (charge condition) and |µ i − ν j | ≤ 1 (dominance condition). These conditions give a sum of terms that are products of e ±iθ or e ±i(ϕ+k j x) in each sector, and hence that can be distinguished by the presence or absence of oscillatory factors e ±ik j x .…”

Section: 44mentioning

confidence: 85%

“…This relation implies that ω = −c. The correlators O α (1)O α (N + 1) with |α| ≤ 1 in isotropic models with general c and ω = −c were derived in references [52,53] using the same methods as this paper. Our results go further by studying a wider class of models, including critical phases with general (c, ω), as well as a wider class of observables: O α (1)O α (N + 1) for all α.…”

Section: 5mentioning

confidence: 99%

“…In this section we calculate the large N asymptotics of O α (1)O α (N + 1) for a system described by (9) with non-zero c. This was solved for isotropic models (i.e. models where f (z) = f (1/z)) and for α ∈ {−1, 0, 1} in [53]. We now explain how to use Theorem 9 to find the answer in the general case.…”

Section: 22mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Jones

Verresen

2019

J Stat Phys

View full text Add to dashboard Cite

Topological phases protected by symmetry can occur in gapped and-surprisingly-in critical systems. We consider the class of non-interacting fermions in one dimension with spinless timereversal symmetry. It is known that the phases in this class are classified by a topological invariant ω and a central charge c. Here we investigate the correlations of string operators in order to gain insight into the interplay between topology and criticality. In the gapped phases, these non-local operators are the string order parameters that allow us to extract ω. More remarkable is that the correlation lengths of these operators show universal features, depending only on ω. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding both ω and c. More generally, we derive the exact long-distance asymptotics of these correlation functions using the theory of Toeplitz determinants. We include physical discussion in light of the mathematical results. This includes an expansion of the lattice operators in terms of the operator content of the relevant conformal field theory. Moreover, we discuss the spin chains which are dual to these fermionic systems. Contents

show abstract

Section: 44mentioning

confidence: 85%

Section: 5mentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Jones

Verresen

2019

J Stat Phys

View full text Add to dashboard Cite

show abstract

“…These zeros are 'removable'-for further discussion see Refs. [41,42,49]. 2 In the case N = 1 we will also use the notation kF for the Fermi momentum.…”

Section: 1mentioning

confidence: 99%

“…For N > 1, we have a low-energy CFT description only when each of the v j are equal: an SO(2N ) 1 Wess-Zumino-Witten (WZW) model with c = N [36]. For general velocities, the case N > 1 can still be understood using CFT concepts: the scaling law (1) holds with c = N [37] and one can identify low-energy descriptions of lattice operators by taking sums and products of CFT operators from each of the different sectors [38][39][40][41][42] (each sector being a copy of the c = 1 CFT). Moreover, we give an explicit argument that for a family of N = 2 Hamiltonians that are not described at low energies by a CFT (since they have different Fermi velocities), the ground state has a parent Hamiltonian that is described at low energies by a CFT.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-resolved entanglement entropy in critical free-fermion chains

Jones

2022

Preprint

View full text Add to dashboard Cite

The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix ρ. This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by N massless Dirac fermions. For the density matrix, ρA, of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of ρA; we derive tr(e, where a, x and µ are universal and σ depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term O(L −µ ) is consistent with but weaker than the field theory prediction O(L −2µ ). However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

show abstract

Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Jones

2022

J Stat Phys

View full text Add to dashboard Cite

The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix $$\rho $$ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by N massless Dirac fermions. For the density matrix, $$\rho _A$$ ρ A , of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of $$\rho _A$$ ρ A ; we derive $$\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))$$ tr ( e i α Q A ρ A n ) = a e i α ⟨ Q A ⟩ ( σ L ) - x ( 1 + O ( L - μ ) ) , where a, x and $$\mu $$ μ are universal and $$\sigma $$ σ depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term $$O(L^{-\mu })$$ O ( L - μ ) is consistent with but weaker than the field theory prediction $$O(L^{-2\mu })$$ O ( L - 2 μ ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

show abstract

Fisher–Hartwig determinants, conformal field theory and universality in generalised XX models

Cited by 3 publications

References 42 publications

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Symmetry-resolved entanglement entropy in critical free-fermion chains

Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Contact Info

Product

Resources

About