Entanglement Gap Research Articles

Entanglement gap in 1D long-range quantum spherical models

We investigate the finite-size scaling of the entanglement gap in the one-dimensional long-range quantum spherical model (QSM). We focus on the weak long-range QSM, for which the thermodynamic limit is well-defined. This model exhibits a continuous phase transition, separating a paramagnetic from a ferromagnet phase. The universality class of the transition depends on the long-range exponent α. We show that in the thermodynamic limit the entanglement gap is finite in the paramagnetic phase, and it vanishes in the ferromagnetic phase. In the ferromagnetic phase the entanglement gap is understood in terms of standard magnetic correlation functions. The half-system entanglement gap decays as , where the constant C α depends on the low-energy properties of the model and L is the system size. This reflects that the lower part of the dispersion is affected by the long range physics. Finally, multiplicative logarithmic corrections are absent in the scaling of the entanglement gap, in contrast with the higher-dimensional case.

Entanglement and precession in two-dimensional dynamical quantum phase transitions

Nonanalytic points in the return probability of a quantum state as a function of time, known as dynamical quantum phase transitions (DQPTs), have received great attention in recent years, but the understanding of their mechanism is still incomplete. In our recent work [Phys. Rev. Lett. 126, 040602 (2021)], we demonstrated that one-dimensional DQPTs can be produced by two distinct mechanisms, namely semiclassical precession and entanglement generation, leading to the definition of precession (pDQPTs) and entanglement (eDQPTs) dynamical quantum phase transitions. In this manuscript, we extend and investigate the notion of p- and eDQPTs in two-dimensional systems by considering semi-infinite ladders of varying width. For square lattices, we find that pDQPTs and eDQPTs persist and are characterized by similar phenomenology as in 1D: pDQPTs are associated with a magnetization sign change and a wide entanglement gap, while eDQPTs correspond to suppressed local observables and avoided crossings in the entanglement spectrum. However, DQPTs show higher sensitivity to the ladder width and other details, challenging the extrapolation to the thermodynamic limit especially for eDQPTs. Moving to honeycomb lattices, we also demonstrate that lattices with an odd number of nearest neighbors give rise to phenomenologies beyond the one-dimensional classification.

Quantum correlations, entanglement spectrum, and coherence of the two-particle reduced density matrix in the extended Hubbard model

We study the ground state properties of the one-dimensional extended Hubbard model at half-filling from the perspective of its particle reduced density matrix. We focus on the reduced density matrix of $2$ fermions and perform an analysis of its quantum correlations and coherence along the different phases of the model. Specifically, we study its (i) entanglement entropy, (ii) $\ell_1$ norm of coherence, (iii) irreducible two-body cumulant matrix and (iv) entanglement spectrum. Our results show that these different properties are complementary to each other depending on the phase of the system, exhibiting peculiar behaviors such as discontinuities, maximum or minimum values at the quantum phase transitions, thus providing a qualitative view of the phase diagram of the model. In particular, in the superconducting region, we obtain that the entanglement spectrum signals a transition between a dominant singlet (SS) to triplet (TS) pairing ordering in the system. Moreover, from the analysis of the dominant eigenvector in the reduced state, we can relate the SS-TS transition to the spatial separation between the fermion pairs in the two different pairing orderings. The entanglement gap is also able to highlight a transition - at a few-body level - in the groundstate wavefunction, not discussed previously in the literature. While other quantifiers are less sensitive to few-body defects in the wavefunction, the entanglement gap can work as a magnifying glass for these, capturing such small fluctuations.

Laughlin topology on fractal lattices without area law entanglement

Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin type topology. Here, we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.

Entanglement gap, corners, and symmetry breaking

We investigate the finite-size scaling of the lowest entanglement gap \delta\xiδξ in the ordered phase of the two-dimensional quantum spherical model (QSM). The entanglement gap decays as \delta\xi=\Omega/\sqrt{L\ln(L)}δξ=Ω/Lln(L). This is in contrast with the purely logarithmic behaviour as \delta\xi=\pi^2/\ln(L)δξ=π2/ln(L) at the critical point. The faster decay in the ordered phase reflects the presence of magnetic order. We analytically determine the constant \OmegaΩ, which depends on the low-energy part of the model dispersion and on the geometry of the bipartition. In particular, we are able to compute the corner contribution to \OmegaΩ, at least for the case of a square corner.

Entanglement View of Dynamical Quantum Phase Transitions.

The analogy between an equilibrium partition function and the return probability in many-body unitary dynamics has led to the concept of dynamical quantum phase transition (DQPT). DQPTs are defined by nonanalyticities in the return amplitude and are present in many models. In some cases, DQPTs can be related to equilibrium concepts, such as order parameters, yet their universal description is an open question. In this Letter, we provide first steps toward a classification of DQPTs by using a matrix product state description of unitary dynamics in the thermodynamic limit. This allows us to distinguish the two limiting cases of "precession" and "entanglement" DQPTs, which are illustrated using an analytical description in the quantum Ising model. While precession DQPTs are characterized by a large entanglement gap and are semiclassical in their nature, entanglement DQPTs occur near avoided crossings in the entanglement spectrum and can be distinguished by a complex pattern of nonlocal correlations. We demonstrate the existence of precession and entanglement DQPTs beyond Ising models, discuss observables that can distinguish them, and relate their interplay to complex DQPT phenomenology.

Closure of the entanglement gap at quantum criticality: The case of the quantum spherical model

The study of entanglement spectra is a powerful tool to detect or elucidate universal behaviour in quantum many-body systems. We investigate the scaling of the entanglement (or Schmidt) gap $\delta\xi$, i.e., the lowest laying gap of the entanglement spectrum, at a two-dimensional quantum critical point. We focus on the paradigmatic quantum spherical model, which exhibits a second-order transition, and is mappable to free bosons with an additional external constraint. We analytically show that the Schmidt gap vanishes at the critical point, although only logarithmically. For a system on a torus and the half-system bipartition, the entanglement gap vanishes as $\pi^2/\ln(L)$, with $L$ the linear system size. The entanglement gap is nonzero in the paramagnetic phase and exhibits a faster decay in the ordered phase. The rescaled gap $\delta\xi\ln(L)$ exhibits a crossing for different system sizes at the transition, although logarithmic corrections prevent a precise verification of the finite-size scaling. Interestingly, the change of the entanglement gap across the phase diagram is reflected in the zero-mode eigenvector of the spin-spin correlator. At the transition quantum fluctuations give rise to a non-trivial structure of the eigenvector, whereas in the ordered phase it is flat. We also show that the vanishing of the entanglement gap at criticality can be qualitatively but not quantitatively captured by neglecting the structure of the zero-mode eigenvector.

Detection of topology via entanglement oscillations

We introduce a characterization of topological order based on bulk oscillations of the entanglement entropy and the definition of an `entanglement gap', showing that it is generally applicable to pure and disordered quantum systems. Using exact diagonalization and the strong disorder renormalization group method, we demonstrate that this approach gives results in agreement with the use of traditional topological invariants, especially in cases where topological order is known to persist in the presence of off-diagonal bond disorder. The entanglement gap is then used to analyze classes of quantum systems with alternating bond types, allowing us to construct their topological phase diagrams. The validity of these phase diagrams is verified in limiting cases of dominant bond types, where the solution is known.

Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems

We numerically investigate the momentum-space entanglement entropy and entanglement spectrum of the random-dimer model and its generalizations, which circumvent Anderson localization, after a quench in the Hamiltonian parameters. The type of dynamics that occurs depends on whether or not the Fermi level of the initial state is near the energy of the delocalized states present in these models. If the Fermi level of the initial state is near the energy of the delocalized states, we observe an interesting slow logarithmic-like growth of the momentum-space entanglement entropy followed by an eventual saturation. Otherwise, the momentum-space entanglement entropy is found to rapidly saturate. We also find that the momentum-space entanglement spectrum reveals the presence of delocalized states in these models for long times after the quench and the many-body entanglement gap decays logarithmically in time when the Fermi level is near the energy of the delocalized states.

Evidence of quantum phase transition in real-space vacuum entanglement of higher derivative scalar quantum field theories

In a bipartite set-up, the vacuum state of a free Bosonic scalar field is entangled in real space and satisfies the area-law— entanglement entropy scales linearly with area of the boundary between the two partitions. In this work, we show that the area law is violated in two spatial dimensional model Hamiltonian having dynamical critical exponent z = 3. The model physically corresponds to next-to-next-to-next nearest neighbour coupling terms on a lattice. The result reported here is the first of its kind of violation of area law in Bosonic systems in higher dimensions and signals the evidence of a quantum phase transition. We provide evidence for quantum phase transition both numerically and analytically using quantum Information tools like entanglement spectra, quantum fidelity, and gap in the energy spectra. We identify the cause for this transition due to the accumulation of large number of angular zero modes around the critical point which catalyses the change in the ground state wave function due to the next-to-next-to-next nearest neighbor coupling. Lastly, using Hubbard-Stratanovich transformation, we show that the effective Bosonic Hamiltonian can be obtained from an interacting fermionic theory and provide possible implications for condensed matter systems.

Magnetic field dependent dynamics and field-driven metal-to-insulator transition of the half-filled Hubbard model: A DMFT+DMRG study

We study the magnetic field driven metal-to-insulator transition in half-filled Hubbard model on the Bethe lattice, using the dynamical mean-field theory by solving the quantum impurity problem with density-matrix renormalization group algorithm. The method enables us to obtain a high-resolution spectral densities in the presence of a magnetic field. It is found that the Kondo resonance at the Fermi level splits at relatively high magnetic field: the spin-up and spin-down components move away from the Fermi level and finally form a spin polarized band insulator. By calculating the magnetization and spin susceptibility, we clarify that an applied magnetic field drives a transition from a paramagnetic metallic phase to a band insulting phase. In the weak interaction regime, the nature of the transition is continuous and captured by the Stoner's description, while in the strong interaction regime the transition is very likely to be metamagnetic, evidenced by the hysteresis curve. Furthermore, we determine the phase boundary by tracking the kink in the magnetic susceptibility, and the step-like change of the entanglement entropy and the entanglement gap closing. Interestingly, the phase boundary determined from these two different ways are largely consistent with each other.

Distilling momentum-space entanglement in Luttinger liquids at finite temperature

While much is known about the entanglement characteristics of ground states, the properties of reduced thermal density matrices have received significantly less attention. Here we investigate the entanglement content of reduced thermal density matrices for momentum-space bipartitioning in Luttinger liquids using analytical and numerical methods. The low lying part of its spectrum contains an "entanglement gap", which persists up to temperatures comparable to the level spacing. With increasing temperature, the low energy modes acquire dispersion and resemble to those in the physical Hamiltonian with an enhanced effective temperature. The momentum-space entanglement is carried by high energy modes (compared to temperature), featuring a completely flat spectrum. The von-Neumann entropy increases with temperature with a universal Sommerfeld coefficient. The momentum-space entanglement Hamiltonian turns out to be as universal as the physical Hamiltonian.

Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly

We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for a layer–layer entanglement Hamiltonian whose spectrum is the entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between the entanglement spectrum and energy spectrum of a monolayer which means that the entanglement spectrum perfectly reflects the edge state properties of the bilayer. We also show that trigonal warping breaks down such a perfect correspondence, however, in -K direction in the hexagonal Brillouin zone, their behaviors are remarkably the same for particular relevances of hopping parameters. In the presence of an on-site energy difference the symmetry of the entanglement spectrum is broken with opening an indirect entanglement gap. We also study the effects of a perpendicular magnetic field on both energy and the entanglement spectrum of the bilayer in the presence of trigonal warping and on-site energy difference. We demonstrate that the entanglement spectrum versus magnetic flux has a self similar fractal structure, known as the Hofstadter butterfly. Our results also show that the on-site energy difference causes a transition from the Hofstadter butterfly to a tree-like picture.

Momentum-Space Entanglement and Loschmidt Echo in Luttinger Liquids after a Quantum Quench.

Luttinger liquids (LLs) arise by coupling left- and right-moving particles through interactions in one dimension. This most natural partitioning of LLs is investigated by the momentum-space entanglement after a quantum quench using analytical and numerical methods. We show that the momentum-space entanglement spectrum of a LL possesses many universal features both in equilibrium and after a quantum quench. The largest entanglement eigenvalue is identical to the Loschmidt echo, i.e., the overlap of the disentangled and final wave functions of the system. The second largest eigenvalue is the overlap of the first excited state of the disentangled system with zero total momentum and the final wave function. The entanglement gap is universal both in equilibrium and after a quantum quench. The momentum-space entanglement entropy is always extensive and saturates fast to a time independent value after the quench, in sharp contrast to a spatial bipartitioning.

Bulk entanglement spectrum in gapped spin ladders

We study the bulk entanglement of a series of gapped ground states of spin ladders, representative of the Haldane phase. These ground states of spin $S/2$ ladders generalize the valence bond solid ground state. In the case of spin 1/2 ladders, we study a generalization of the Affleck-Kennedy-Lieb-Tasaki and Nersesyan-Tsvelik states and fully characterize the bulk entanglement Hamiltonian. In the case of general spin $S$ we argue that in the Haldane phase the bulk entanglement spectrum of a half integer ladder is either gapless or possess a degenerate ground state. For ladders with integer valued spin particles, the generic bulk entanglement spectrum should have an entanglement gap. Finally, we give an example of a series of trivial states of higher spin $S>1$ in which the bulk entanglement Hamiltonian is critical, signaling that the relation between topological states and a critical bulk entanglement Hamiltonian is not unique to topological systems.

Momentum-space entanglement spectrum of bosons and fermions with interactions.

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 etal. [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.

Identifying symmetry-protected topological order by entanglement entropy

According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations, and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies $S^I$ ($S^H$) of integer (half-integer) representations, we can define an entanglement gap $G = S^I - S^H$, which equals 1 in the SPT phase, and $-1$ in the trivial phase. As application of our proposal, we study the spin-1 models on various 1D and quasi-1D lattices, including the bilinear-biquadratic model on the single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of an SPT phase in the spin-1 tube model, and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge $c=3$ determined by fits to the block entanglement entropy.

Entanglement spectroscopy of SU(2)-broken phases in two dimensions

In magnetically ordered systems the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite size energy spectra, the so called tower of states (TOS). In the present work we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground state entanglement spectrum (ES). Using state-of-the-art DMRG calculations, we examine the ES of the 2D antiferromagnetic J1-J2 Heisenberg model on both the triangular and kagom\'e lattice. At large ferromagnetic J2 the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite size scaling in the thermodynamic limit) sharply reflects tower of states features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.

Understanding the entanglement entropy and spectra of 2D quantum systems through arrays of coupled 1D chains

We describe an algorithm for studying the entanglement entropy and spectrum of two-dimensional (2D) systems, as a coupled array of $N$ one-dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D (both in its disordered phase and near criticality), we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as ${c}_{\mathrm{eff}}\phantom{\rule{0.16em}{0ex}}\mathrm{log}(N)/6$ with ${c}_{\mathrm{eff}}\ensuremath{\approx}1$. Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of the 2D model. When short-range (area law) entanglement dominates we find (numerically and perturbatively) that this spectrum reflects the energy spectrum of a single quantum Ising chain.

Particle entanglement spectra for quantum Hall states on lattices

We use particle entanglement spectra to characterize bosonic quantum Hall states on lattices, motivated by recent studies of bosonic atoms on optical lattices. Unlike for the related problem of fractional Chern insulators, very good trial wave functions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at $\ensuremath{\nu}=1/2$ for the non-Abelian Moore-Read state at $\ensuremath{\nu}=1$. We undertake a comparative study of these trial states to the corresponding ground states of repulsive two-body or three-body contact interactions on the lattice. The magnitude of the entanglement gap is studied as a function of the interaction strength on the lattice, giving insight into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wave functions as possible indicators for the topological order in the system. We discuss how the entanglement spectra allow to detect competing phases such as a Bose-Einstein condensate.