Abstract Although the energy spectrum of the Heisenberg spin chain on a circle defined by $$ H=\frac{1}{4}\sum \limits_{k=1}^M\left({\sigma}_k^x{\sigma}_{k+1}^x+{\sigma}_k^y{\sigma}_{k+1}^y+\Delta {\sigma}_k^z{\sigma}_{k+1}^z\right) $$ H = 1 4 ∑ k = 1 M σ k x σ k + 1 x + σ k y σ k + 1 y + Δ σ k z σ k + 1 z is well known for any fixed M, the boundary conditions vary according to whether M ∈ 4ℕ + r, where r = −1, 0, 1, 2, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with M spins breaking into strings with M1< M and M − M1 spins (or vice versa). We organize the energy spectrum and degeneracies of H in the case ∆ = 0 where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit M → ∞ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate x(σ) ≡ x(σ) + R0. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for ∆ ≠ 0 is a change in the compactification radius R0→ R∆. As ∆ → −1 this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.
Abstract Although the energy spectrum of the Heisenberg spin chain on a circle defined by H=41k=1∑M(σkxσk+1x+σkyσk+1y+Δσkzσk+1z) H = 1 4 ∑ k = 1 M σ k x σ k + 1 x + σ k y σ k + 1 y + Δ σ k z σ k + 1 z is well known for any fixed M, the boundary conditions vary according to whether M ∈ 4ℕ + r, where r = −1, 0, 1, 2, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with M spins breaking into strings with M1< M and M − M1 spins (or vice versa). We organize the energy spectrum and degeneracies of H in the case ∆ = 0 where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit M → ∞ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate x(σ) ≡ x(σ) + R0. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for ∆ ≠ 0 is a change in the compactification radius R0→ R∆. As ∆ → −1 this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.
Although the energy spectrum of the Heisenberg spin chain on a circle defined by H=14Σk=1Mσkxσk+1x+σkyσk+1y+Δσkzσk+1z is well known for any fixed M, the boundary conditions vary according to whether Mϵ 4N + r, where r = -1,0,1,2, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with M spins breaking into strings with M1 < M and M - M1spins (or vice versa). We organize the energy spectrum and degeneracies of H in the case Δ = 0 where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit M → ∞ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate x(σ) ≡ x(σ) + R0. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for Δ ≠ 0 is a change in the compactification radius R0→ RΔ. As Δ→ -1 this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.
Abstract We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h c is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range . To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.
Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions A and B, they can be obtained in terms of the Schmidtvalues, or eigenvalues λα of the reduced density matrix ρA for region A. In this paper we draw attention instead to the Schmidtvectors, or eigenvectors ∣vα⟩ of ρA. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors ∣vα⟩ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums Hn of the lattice Hamiltonian density hj,j+1 over region A and show that the matrix elements ⟨vαHn∣vα′⟩ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for HnCFT=21(Ln+L−n) in the boundary CFT, where Ln's are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.
In this dissertation, the momentum-space entanglement properties of several quantum spin chains are investigated. In the second chapter we re- view important results on many-body entanglement. In the third chapter, we numerically study the momentum-space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain [Phys. Rev. Lett. 113, 256404 (2014)]. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entan- glement gap depends on the scaling dimension of the conformal field theory as varied by the X X Z anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how this may support the numerical study of phase transitions. In the third chapter, we advocate that in certain critical spin chains a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model [arXiv:1512.09030]. We provide affirmative evidence from multicritical spin-1 chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model. In chapter four, we study momentum- space en- tanglement in quantum spin-half ladders [Phys. Rev. B 93, 125107 (2016)]. When the system is gapped, we analytically find the momentum-space entan- glement Hamiltonian is described by a chiral conformal field theory with a central charge of two. When the system is gapless, the entanglement Hamil- tonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. We also analytically include the effect of a certain irrelevant terms (in the renormalization group sense) on the entanglement spectrum. In chapter six, we investigate the momentum- space entanglement spectrum after a quantum quench [arXiv:1603.01997]. We show that the momentum-space entanglement spectrum of the XXZ spin- half chain possesses many universal features both in equilibrium and after a quantum quench.
Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the
In the present thesis we perform a step towards understanding the effec-
tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small
excitations: small number of reversed spins or, equivalently, small number of
fermions per site. The calculation of the first order normal form Hamiltonian
amounts to erasing all the nonresonant interaction terms, and allows to draw
interesting conclusions on the dynamics of the system, e.g. to determine ap-
proximately conserved quantities. Moreover, we get a good correspondence
between our perturbative energy spectrum and the exact one computed via
the Bethe ansatz, the approximation being of course the more accurate the
smaller is the number of fermions per site. The present method, here tested
in an exactly solvable case, can be applied to any nonintegrable system of
weakly interacting fermions (and/or bosons).