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.
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.
The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 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. 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) is 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 to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.
By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters up to 24 sites, we demonstrate that quantum entanglement between subsystems A and B in a pure ground state of a whole system A + B can induce thermal equilibrium in subsystem A. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem A. An effective temperature T A of subsystem A induced by quantum entanglement is not a parameter but can be determined from the entanglement von Neumann entropy S A and the total energy E A of subsystem A calculated for the ground state of the whole system.We show that temperature T A can be derived by minimizing the relative entropy for the reduced density matrix operator of subsystem A and the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem A with respect to the coupling strength between subsystems A and B. Temperature T A is essentially identical to the thermodynamic temperature, for which the entropy and the internal energy evaluated using the canonical ensemble in statistical mechanics for the isolated subsystem A agree numerically with the entanglement entropy S A and the total energy E A of subsystem A. Fidelity calculations ascertain that the reduced density matrix operator of subsystem A for the pure but entangled ground state of the whole system A + B matches, within a maximally 1.5% error in the finite size clusters studied, the thermodynamic density matrix operator of subsystem A at temperature T A , despite that these density-matrix operators are different in general.We also find that temperature T A evaluated from the ground state of the whole system depends insignificantly on the system sizes, which is consistent with the fact that the thermodynamic temperature is an intensive quantity.We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem.We also provide two simple but nontrivial analytical examples of free bosons and free fermions for which the two density-matrix operators are exactly the same if the effective temperature T A is adopted.We furthermore discuss implications and possible applications of our finding.
In this paper, we study the ground-state entanglement properties of finite XX spin-1/2 chains with random couplings using the Jordan-Wigner transformation. We divide the system into two parts and study the reduced density matrices (RDMs) of its subsystems. Due to the free-fermion nature of the problem, the RDMs take the form of that of a free-fermion thermal ensemble. Finding the spectrum of the corresponding entanglement Hamiltonian and corresponding eigenvectors and comparing them with the real-space renormalization-group (RSRG) treatment, we establish the validity of the RSRG approach for entanglement in the limit of strong disorder but also find its limitations when disorder is weak. In the latter case, our paper provides a way to visualize the ``effective spins'' that form long-distance singlet pairs.
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.
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 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.