Bipartite entanglement measures are surprisingly useful tools to investigate quantum phases of correlated electrons. Here, I analyze the entanglement spectrum of gapped two-leg quantum Heisenberg ladders on a periodic ribbon partitioned into two identical periodic chains. The entanglement spectrum closely reflects the low-energy gapless spectrum of each individual edge. This extends the conjecture initially drawn for fractional quantum Hall systems to the field of quantum magnetism, stating a direct correspondence between the low-energy entanglement spectrum of a partitioned system and the true spectrum of the virtual edges. A mapping of the reduced density matrix to a thermodynamic density matrix is also proposed via the introduction of an effective temperature.
Bipartite entanglement measures are surprisingly useful tools to investigate quantum phases of correlated electrons. Here, I analyze the entanglement spectrum of gapped two-leg quantum Heisenberg ladders on a periodic ribbon partitioned into two identical periodic chains. The entanglement spectrum closely reflects the low-energy gapless spectrum of each individual edge. This extends the conjecture initially drawn for fractional quantum Hall systems to the field of quantum magnetism, stating a direct correspondence between the low-energy entanglement spectrum of a partitioned system and the true spectrum of the virtual edges. A mapping of the reduced density matrix to a thermodynamic density matrix is also proposed via the introduction of an effective temperature.
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.
Abstract Entanglement spectroscopy has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix when the system is partitioned into two. For many quantum systems, when computed from the bulk ground-state wavefunction, it gives access to the physics of edge excitations. It is thus a valuable tool to diagnose topological ordering. These lecture notes provide an overview of entanglement spectroscopy, mostly in the context of the fractional quantum Hall effect. The basic concepts are introduced through the example of quantum spin chains. The connection with entanglement entropy and the matrix product state representation is discussed. The entanglement spectrum can be computed for non-interacting topological phases, revealing edge excitation from the ground state. Application to fractional quantum Hall phases shows how much information is encoded within the ground state and how different partitions probe different types of excitations. Application to fractional Chern insulators is discussed.
The entanglement spectroscopy, initially introduced by Li and Haldane in the context of the fractional quantum Hall effects, has stimulated an extensive range of studies. The ent anglement spectrum is the spectrum of the reduced density matrix, when we partition the system into two. For many quantum systems, it unveils a unique feature: Computed from the bulk ground state wave function, the entanglement spectrum give access to the physics of edge excitations. In this manuscript, we give an overview of the entanglement spectroscopy. We introduce the basic concepts through the case of the quantum spin chains. We present an extensive rev iew of the entanglement spectra applied to the fractional quantum Hall phases, showing how much information is encoded within the ground state and how different partitions probe different type of excitations. As a practical application of this technique, we discuss how it can discriminate between the different phases that emerge in the strongly interacti ng Chern insulators.
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.
We show that the entanglement spectrum can be used to define non-local order in gapless spin systems. We find a gap that fully separates a series of generic, high "entanglement energy" levels, from a flat band of levels with specific multiplicities defining the ground state, and remains finite in the thermodynamic limit. We pick the appropriate set of quantum numbers and partition the system in this space, corresponding to a nonlocal real-space cut. Despite the Laughlin state being bulk gapped while the antiferromagnetic spin chain state is bulk gapless, we show that the S=1/2 Heisenberg antiferromagnet in one dimension has an entanglement spectrum almost identical to that of the Laughlin Fractional Quantum Hall state in two dimensions, revealing the similar field theory of their low-energy bulk and edge excitations, respectively. We also discuss the dimerization transition from entanglement gap scaling.
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.