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THE ENTANGLEMENT ENTROPY NEAR LIFSHITZ QUANTUM PHASE TRANSITIONS & THE EMERGENT STATISTICS OF FRACTIONALIZED EXCITATIONS

Marlon A. Rodney

Published Oct 1, 2012• 0 citations
Publisher

Abstract

In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-tc|1/2 whereas the first sub-leading term varies like |t-tc|1/3, where tc is the critical value of the hopping parameter at the transition. In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.

DOI
10.71548/10608

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In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-tc|1/2 whereas the first sub-leading term varies like |t-tc|1/3, where tc is the critical value of the hopping parameter at the transition. In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.
We investigate the scaling of the bipartite entanglement entropy across Lifshitz quantum phase transitions, where the topology of the Fermi surface changes without any changes in symmetry. We present both numerical and analytical results which show that Lifshitz transitions are characterized by a well-defined set of critical exponents for the entanglement entropy near the phase transition. In one dimension, we show that the entanglement entropy exhibits a length scale that diverges as the system approaches a Lifshitz critical point. In two dimensions, the leading and subleading coefficients of the scaling of entanglement entropy show distinct power-law singularities at critical points. The effect of weak interactions is investigated using the density matrix renormalization group algorithm.
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.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area law is proven to hold-or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.
We study a one-dimensional lattice model of generalized statistics in which particles have next-nearest-neighbor hopping between sites which depends on the occupation number at the intermediate site and a statistical parameter $\ensuremath{\phi}$. The model breaks parity and time-reversal symmetries and has four-fermion interactions if $\ensuremath{\phi}\ensuremath{\ne}0$. We first analyze the model using mean field theory and find that there are four Fermi points whose locations depend on $\ensuremath{\phi}$ and the filling $\ensuremath{\eta}$. We then study the modes near the Fermi points using the technique of bosonization. Based on the quadratic terms in the bosonized Hamiltonian, we find that the low-energy modes form two decoupled Tomonaga-Luttinger liquids with different values of the Luttinger parameters which depend on $\ensuremath{\phi}$ and $\ensuremath{\eta}$; further, the right- and left-moving modes of each system have different velocities. A study of the scaling dimensions of the cosine terms in the Hamiltonian indicates that the terms appearing in one of the Tomonaga-Luttinger liquids will flow under the renormalization group and the system may reach a nontrivial fixed point in the long distance limit. We examine the scaling dimensions of various charge density and superconducting order parameters to find which of them is the most relevant for different values of $\ensuremath{\phi}$ and $\ensuremath{\eta}$. Finally, we look at two-particle bound states that appear in this system and discuss their possible relevance to the properties of the system in the thermodynamic limit. Our work shows that the low-energy properties of this model of generalized statistics have a rich structure as a function of $\ensuremath{\phi}$ and $\ensuremath{\eta}$.
The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have an entanglement entropy that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. Here I will discuss recent extensions of these ideas to a class of quantum critical points with dynamic critical exponent z = 2 in two space dimensions and to 2D systems in a topological phase. The application of these ideas to quantum dimer models and fractional quantum Hall states will be discussed.