We study the momentum space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain with analytical methods and exact diagonalization. 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 reveals that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entanglement gap observed in [Thomale et al., Phys. Rev. Lett. 105, 116805 (2010)] depends on the scaling dimension of the conformal field theory as varied by the XXZ 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 these shifted transition points in the entanglement spectra may support the numerical study of phase transitions in the momentum space density matrix renormalization group.PACS numbers: 71.10. Pm, 03.67.Mn, 11.25.Hf Introduction -Quantum information ideas applied to condensed matter systems have revealed novel insights into exotic phases of matter [1]. Quantitatively, quantum information between two regions, A and B, can be characterized by the groundstate reduced density matrix of A, ρ A , and analogously B, ρ B . For example, the entanglement entropy (EE) is given by Tr(ρ A lnρ A ) = Tr(ρ B lnρ B ). The entanglement spectrum (ES) [2] (defined as the set of eigenvalues of a fictitious entanglement Hamiltonian, H e , with ρ A written as e −H e ) is a useful tool in understanding topological states of matter and strongly correlated systems, including fractional quantum Hall (FQH) systems [2-10], quantum spin chains [11][12][13][14][15][16][17] and ladders [18][19][20][21][22][23][24][25][26], topological insulators [27][28][29][30][31][32], symmetry broken phases [33,34], and other systems in one [35][36][37][38][39] and two [40][41][42][43][44][45][46][47][48][49][50][51][52] spatial dimensions. These studies predominantly focused on real / orbital space entanglement. For many gapped systems, the energy spectrum of the edge states and ES are equivalent. This was proven by X.L. Qi et al. [53] and elaborated on in Refs. [23,[54][55][56]. There is no universal understanding of systems with a gapless bulk, where long range correlations are present [57].The ES in momentum space has been explored in quantum spin chains [11] and ladders [20]. A momentum partition is natural and physically relevant, as the low-energy formulation of one-dimensional systems involves the splitting of particles into left and right movers [58]. A deeper understanding of the momentum space ES could help identify the most fruitful applications of momentum space density matrix renormalization group (DMRG) algorithms [59][60][61][62]. Gapless spin chains are one promising candidate for momentum space DMRG. For example, for chains with higher symmetry groups characterizing the parameters of the critical theories is a challenge for real space DMRG and motivates different ...

We advocate that in critical spin chains, and possibly in a larger class of 1D critical models, 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. To this end, we provide affirmative evidence from multicritical spin chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model.Introduction.-Quantum entanglement has become a key concept in contemporary condensed matter physics. This is due in part to its ability to probe intrinsic topological order [1][2][3]. Consider a density matrix, ρ, represented by the projector onto a many-body ground state. If the associated Hilbert space is partitioned into two non-overlapping regions A and B, the entanglement between the regions A and B can be quantified either through the entanglement entropy (EE) or the entanglement spectrum (ES) obtained by the reduced density matrix ρ A = Tr B ρ. The EE is given by S A = −Tr(ρ A lnρ A ), and the ES is defined as the spectrum of the entanglement Hamiltonian H E = −lnρ A [4]. By definition, EE and ES depend on the chosen basis to partition (cut) the manybody Hilbert space. To resolve bulk and edge features of topological order, some form of spatial cut [4-8] (along with a particle cut [9,10]) is the predominantly used choice. This works well in systems with a bulk energy gap and hence an associated length scale. Upon partitioning, the ES then mimics the spectral features located along the cut of an edge termination [7]. In particular, a set of universal entanglement modes related to the edge can be identified as distinct from generic entanglement weight through the entanglement gap (EG), which can be employed to define topological adiabaticity in the entanglement spectral flow [5]. The spatial EG evolves in a way similar to the physical bulk gap of the topologically ordered phase, even though bulk gap closures occur at points of parameter space different from EG closures due to the unit fictitious temperature in H E [11].

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