We study momentum-space entanglement in quantum spin-half ladders consisting of two coupled critical $XXZ$ spin-half chains, using field theoretical methods developed by Lundgren et al. [Phys. Rev. B 88, 245137 (2013)] and exact diagonalization. When the system is gapped, we analytically find that the momentum-space entanglement Hamiltonian is described by a chiral conformal field theory with a central charge of 2. This is in contrast to entanglement Hamiltonians of various real-space partitions of gapped-spin ladders that have a central charge of 1. When the system is gapless, the entanglement Hamiltonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. In the gapless region, we extend the work of Lundgren et al. to include the effect of a certain irrelevant term on the entanglement spectrum. In both the gapless and gapped phases, the momentum-space entanglement entropy obeys a volume law. In the gapless region, we find a subleading constant term in the entanglement entropy which contains information about the underlying field theory of the system. The analytical predictions for the entanglement spectrum are consistent with results from exact diagonalization.
We study momentum-space entanglement in quantum spin-half ladders consisting of two coupled critical XXZ spin-half chains, using field theoretical methods developed by Lundgren et al. [Phys. Rev. B 88, 245137 (2013)] and exact diagonalization. When the system is gapped, we analytically find that the momentum-space entanglement Hamiltonian is described by a chiral conformal field theory with a central charge of 2. This is in contrast to entanglement Hamiltonians of various real-space partitions of gapped-spin ladders that have a central charge of 1. When the system is gapless, the entanglement Hamiltonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. In the gapless region, we extend the work of Lundgren et al. to include the effect of a certain irrelevant term on the entanglement spectrum. In both the gapless and gapped phases, the momentum-space entanglement entropy obeys a volume law. In the gapless region, we find a subleading constant term in the entanglement entropy which contains information about the underlying field theory of the system. The analytical predictions for the entanglement spectrum are consistent with results from exact diagonalization.
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.
We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.
Numerical studies of the reduced density matrix of a gapped spin-1/2 Heisenberg antiferromagnet on a two-leg ladder find that it has the same form as the Gibbs density matrix of a gapless spin-1/2 Heisenberg antiferromagnetic chain at a finite temperature determined by the spin gap of the ladder. We investigate this interesting result by considering a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. a cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1 + 1 dimensions) are discussed using the replica trick and scaling arguments. We show that (1) for a system with a bulk gap the reduced density matrix has the form of a thermal density matrix, (2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, (3) the von Neumann entropy equals the thermodynamic entropy of one chain, and (4) the bulk gap plays the role of effective temperature.
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 suggests 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.
We study the quantum entanglement of the spin and orbital degrees of freedom in the one-dimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy and the entanglement spectra using a variety of partitions or ``cuts'' of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show that the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. The counting of the low-lying entanglement eigenvalues shows that the ``virtual edge'' picture, which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the ``low-energy'' entanglement Hamiltonian, breaks down for this model, even though the equivalence has been shown to hold for a similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement, we present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.
In this paper we study the ground-state properties of a ladder Hamiltonian with chiral SU(2)-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of SU(2) symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a c=1 conformal field theory (CFT) in (1+1) dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-21 Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an SU(2)1 Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for SU(2)-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.