Renormalization is often described as the removal or ‘integrating out’ of high energy degrees of freedom. In the context of quantum matter, one might suspect that quantum entanglement provides a sharp way to characterize such a loss of degrees of freedom. Indeed, for quantum many-body systems with Lorentz invariance, such entanglement monotones have been proven to exist in one, two, and three spatial dimensions. In each dimension d , a certain term in the entanglement entropy of a d -ball decreases along renormalization group (RG) flows. Given that most quantum many-body systems available in the laboratory are not Lorentz invariant, it is important to generalize these results if possible. In this work we demonstrate the impossibility of a wide variety of such generalizations. We do this by exhibiting a series of counterexamples with understood renormalization group flows which violate entanglement RG monotonicity. We discuss bosons at finite density, fermions at finite density, and majorization in Lorentz invariant theories, among other results.
Renormalization is often described as the removal or ‘integrating out’ of high energy degrees of freedom. In the context of quantum matter, one might suspect that quantum entanglement provides a sharp way to characterize such a loss of degrees of freedom. Indeed, for quantum many-body systems with Lorentz invariance, such entanglement monotones have been proven to exist in one, two, and three spatial dimensions. In each dimension d , a certain term in the entanglement entropy of a d -ball decreases along renormalization group (RG) flows. Given that most quantum many-body systems available in the laboratory are not Lorentz invariant, it is important to generalize these results if possible. In this work we demonstrate the impossibility of a wide variety of such generalizations. We do this by exhibiting a series of counterexamples with understood renormalization group flows which violate entanglement RG monotonicity. We discuss bosons at finite density, fermions at finite density, and majorization in Lorentz invariant theories, among other results.
The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincaré-invariant theories is captured by a monotonic c -function. For such theories, holographic c -functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, these constructions generally fail, reflecting both the violation of entanglement RG monotonicity and its limitations in capturing essential features of nonrelativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is important to identify a quantity that captures the decrease of effective degrees of freedom along such flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c -function applicable to all such theories, which moreover obeys a geometric upper bound on its rate of change. We further demonstrate the existence of this c -function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy condition and thermodynamic stability are satisfied, it obeys the expected RG monotonicity.
A bstract Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d ≥ 4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu-Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
We employ the recent results on the generalization of the central charge theorem to three spacetime dimensions to derive nonperturbative results for several strongly interacting quantum field theories, including quantum electrodynamics (QED-3), and the theory corresponding to certain quantum phase transitions in condensed matter systems. In particular, by demanding that the universal constant part of the entanglement entropy decreases along the renormalization group flow (F theorem), we find sufficient conditions for the stability of QED-3 against chiral symmetry breaking and confinement. Using similar ideas, we derive strong constraints on the nature of quantum critical points in condensed matter systems with topological order.
One of the most important predictions of quantum mechanics is the existence of quantum entanglement. Since the inception of quantum theory, the study of entanglement has become ubiquitous in several branches of physics, from high energy to condensed matter physics and quantum information. To this avail, different measures of entanglement in quantum systems have been introduced. The most successful ones are the von Neumann and the Rényi entanglement entropies, which have been heavily studied both in lattice models and in quantum field theories (QFTs). A different, more complete characterization of entanglement is provided by the entanglement (or modular) Hamiltonian, defined as the logarithm of the reduced density matrix. One of the key property of this operator is that, under generic conditions, in 1+1-dimensional conformal field theories (CFTs) presents a local and few-body structure. Beyond just being of theoretical interest, this property has been recently used to efficiently construct the ground state of spin chains in quantum simulators. A recently introduced generalisation of the entanglement Hamiltonian is the negativity Hamiltonian, a non-Hermitian operator which is more useful for mixed states and tripartite entanglement. In this thesis we study these quantifications in many-body quantum systems and in QFTs. In the first part of the thesis, we focus on fermionic systems at equilibrium, and we obtain analytical expressions for the entanglement and the negativity Hamiltonians in the massless Dirac fermion QFT, both in presence of boundaries and at finite temperature. We compare these predictions in the continuum QFT with numerical calculations on the lattice, showing that a proper comparison requires a non-trivial continuum limit of the lattice expression, which we detail. We also investigate a lattice realisation of a non-unitary CFT, the bc-ghost theory, showing that the entanglement Hamiltonian presents an additional term with respect to the unitary case. In the second part, we study the entanglement entropies in an integrable QFT, the massless renormalisation group flow from the tricritical to the critical Ising CFT, demonstrating that the entropy is monotonically decreasing under the renormalisation group flow, as expected from the entropic c-theorem. In the third part we consider out-of-equilibrium problems. We study the melting of a domain wall configuration, computing analytically the entanglement Hamiltonian and showing that the insertion of a defect dramatically modifies the entanglement properties, changing the growth of the entropies from logarithmic to linear in time. Finally, we study generic quantum quenches in free fermionic theories, using the celebrated quasiparticle picture to obtain a prediction for the evolution of the entanglement Hamiltonian during the quench.
Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d≥4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu--Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
One of the most powerful and popular quantum entanglement-based computation methods for the simulation of one-dimensional and quasi-two-dimensional strongly correlated materials is the Density Matrix Renormalization Group (DMRG) technique. A lower quantum entanglement between the system’s degrees of freedom will result in a higher accuracy of the method. However, quantum entanglement between two complementary subsystems of a closed system depends on how the degrees of freedom have been distributed between them, and therefore, it is not in general invariant under arbitrary unitary transformations. Entanglement is invariant only under those unitary transformations which do not mix the degrees of freedom associated with the two subsystems. Consequently, a fundamental question is that if it is possible to construct, algorithmically, unitary transformations which minimize entanglement between degrees of freedom of the system and therefore maximize the DMRG technique? A general solution to this problem is highly nontrivial in general. In this paper, we study interacting fermionic models and by considering single-particle unitary transformations only. We define a novel and efficient method to find the optimal single-particle unitary transformations. In this framework, we show that using our algorithm, the accuracy of the DMRG method at a given bond dimension is increased and the ground-state energy is decreased and approaches to its exact value. Our method paves the way toward finding more general (many-particle) optimized unitary transformations and helps us to better understand the behavior of fermions in strongly correlated materials.