Degenerate Entanglement Spectrum Research Articles

Topological Phases of an Interacting Majorana Benalcazar–Bernevig–Hughes Model

We study the effects of Coulomb repulsive interactions on a Majorana Benalcazar–Bernevig–Huges (MBBH) model. The MBBH model belongs to the class of second-order topological superconductors (HOTSC2), featuring robust Majorana corner modes. We consider an interacting strip of four chains of length L and perform a density matrix renormalization group (DMRG) numerical simulation based on a tensor-network approach. Study of the non-local fermionic correlations and the degenerate entanglement spectrum indicates that the topological phases are robust in the presence of interactions, even in the strongly interacting regime.

Resonating dimer\u2013monomer liquid state in a magnetization plateau of a spin-frac{1}{2} kagome-strip Heisenberg chain

Highly frustrated spin systems such as the kagome lattice are a treasure trove of new quantum states with large entanglements. Herein, we study the spin-frac{1}{2} Heisenberg model on a kagome-strip chain, which is a one-dimensional kagome lattice, using the density matrix renormalization group method. Calculating the central charge and entanglement spectrum for the kagome-strip chain, we find a gapless spin liquid state with doubly degenerate entanglement spectra in a 1/5 magnetization plateau. We also obtain a gapless low-lying continuum in the dynamic spin structure calculated using the dynamical density matrix renormalization group method. We then propose a resonating dimer–monomer liquid state that meets these features.

Topological phases arising from attractive interaction and pair hopping in the extended Hubbard model

The extended Hubbard model with an attractive density–density interaction, positive pair hopping, or both, is shown to host topological phases, with a doubly degenerate entanglement spectrum and interacting edge spins. This constitutes a novel instance of topological order which emerges from interactions. When the interaction terms combine in a charge-SU(2) symmetric fashion, a novel partially polarized pseudospin phase appears, in which the topological features of the spin degrees of freedom coexist with long-range η-wave superconductivity. Thus, our system provides an example of an interplay between spontaneous symmetry breaking and symmetry-protected topological order that leads to novel and unexpected properties.

Tensor networks demonstrate the robustness of localization and symmetry-protected topological phases

We prove that all eigenstates of many-body localized symmetry protected topological systems with time reversal symmetry have four-fold degenerate entanglement spectra in the thermodynamic limit. To that end, we employ unitary quantum circuits where the number of sites the gates act on grows linearly with the system size. We find that the corresponding matrix product operator representation has similar local symmetries as matrix product ground states of symmetry protected topological phases. Those local symmetries give rise to a $\mathbb{Z}_2$ topological index, which is robust against arbitrary perturbations so long as they do not break time reversal symmetry or drive the system out of the fully many-body localized phase.

Topological phases in frustrated synthetic ladders with an odd number of legs

The realization of the Hofstadter model in a strongly anisotropic ladder geometry has now become possible in one-dimensional optical lattices with a synthetic dimension. In this work, we show how the Hofstadter Hamiltonian in such ladder configurations hosts a topological phase of matter which is radically different from its two-dimensional counterpart. This topological phase stems directly from the hybrid nature of the ladder geometry, and is protected by a properly defined inversion symmetry. We start our analysis considering the paradigmatic case of a three-leg ladder which supports a topological phase exhibiting the typical features of topological states in one dimension: robust fermionic edge modes, a degenerate entanglement spectrum and a non-zero Zak phase; then, we generalize our findings - addressable in the state-of-the-art cold atom experiments - to ladders with an higher number of legs.

Effect of Dzyaloshinskii–Moriya interaction on phase diagrams of spin-1 Heisenberg–Ising alternating chains

The effect of the Dzyaloshinskii–Moriya interaction (DMI) on ground-state phase diagrams of spin-1 Heisenberg–Ising alternating chains is investigated by the infinite time-evolving block decimation method. Three rich phase diagrams for three cases with different DMIs are obtained and discussed systematically. The DMI on even bonds plays a key role in the ground-state phase diagram, especially the appearance of the Haldane phase. However, the DMI on odd bonds seems to have very weak effect on the phase diagram. Both the odd- and even-string orders become nonzero in the Haldane phase, and have their maximum values at θ=π. For the odd-dimer phase, the even-string correlator vanishes absolutely despite varying θ, but a double-peak structure of the odd-string correlator is observed. Odd-string correlator becomes maximum at θ=π/2 and 3π/2, but vanishes at θ=π. It indicates that the generalized string correlator can be used to distinguish the odd-dimer from the Haldane phase. Doubly degenerate entanglement spectrum is observed in the Haldane phase, which can be regarded as a clear signature of the existence of topological orders. Strong enough transverse nearest-neighbor correlations are found to be very important for the appearance of the Haldane and the odd-dimer phases.

Localized Majorana-Like Modes in a Number-Conserving Setting: An Exactly Solvable Model

In this Letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting nonlocal zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically nontrivial ground state wave function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizable portion of the phase diagram, the solvable line being one of its boundaries.

Doubly degenerate entanglement spectrum and entanglement plateau in the S=1 bond-alternating chain

Quantum entanglement, entanglement spectrum, magnetization, and ground-state energy of the S=1 bond-alternating antiferromagnetic Heisenberg chain under magnetic field are investigated by the infinite time-evolving block decimation (iTEBD) method. Bipartite entanglement and entanglement spectrum are found to be capable of describing all the quantum phase transitions (QPTs). A rich ground-state phase diagram, which comprises of five different phases, i.e., a singlet-dimer phase, a Haldane phase, a Tomonaga–Luttinger liquid (TLL) phase, a 1/2 plateau phase, and a saturated ferromagnetic phase, is determined. It is interesting that, with the appearance of magnetization plateaus, entanglement plateaus are observed simultaneously. In the Haldane phase, doubly degenerate entanglement spectra on both even and odd bonds are observed. However, in the 1/2 plateau phase, only the entanglement spectra on the even bonds are found to be doubly degenerated.

Quantum phase transitions and string orders in the spin-1/2 Heisenberg–Ising alternating chain with Dzyaloshinskii–Moriya interaction

Quantum phase transitions (QPTs) and the ground-state phase diagram of the spin-1/2 Heisenberg–Ising alternating chain (HIAC) with uniform Dzyaloshinskii–Moriya (DM) interaction are investigated by a matrix-product-state (MPS) method. By calculating the odd- and even-string order parameters, we recognize two kinds of Haldane phases, i.e. the odd- and even-Haldane phases. Furthermore, doubly degenerate entanglement spectra on odd and even bonds are observed in odd- and even-Haldane phases, respectively. A rich phase diagram including four different phases, i.e. an antiferromagnetic (AF), AF stripe, odd- and even-Haldane phases, is obtained. These phases are found to be separated by continuous QPTs: the topological QPT between the odd- and even-Haldane phases is verified to be continuous and corresponds to conformal field theory with central charge c = 1; while the rest of the phase transitions in the phase diagram are found to be c = 1/2. We also revisit, with our MPS method, the exactly solvable case of HIAC model with DM interactions only on odd bonds and find that the even-Haldane phase disappears, but the other three phases, i.e. the AF, AF stripe and odd-Haldane phases, still remain in the phase diagram. We exhibit the evolution of the even-Haldane phase by tuning the DM interactions on the even bonds gradually.

Quantum phase transitions and the 1/3 magnetization plateau in the spin-1/2 Ferromagnetic–Ferromagnetic–Antiferromagnetic chain

Abstract The ground-state properties of the spin-1/2 Ferromagnetic–Ferromagnetic–Antiferromagnetic (F–F–AF) trimerized chain are investigated by the infinite time-evolving block decimation (iTEBD) method. A ground-state phase diagram including three different phases, i.e., a fully polarized phase, a 1/3 plateau phase, and a non-plateau phase, is obtained. All the quantum phase transitions (QPTs) can be described well by the model independent bipartite entanglement. QPTs between the non-plateau phase and the other two phases belong to the second-order category. Doubly degenerate entanglement spectrum and nontrivial string order are observed in the 1/3 plateau phase, which can be used to distinguish it from the other phases. By the scaling behavior of the bipartite entanglement, the central charge of the critical non-plateau phase is determined to be c ≃ 1.

String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains

Generalized string orders and entanglement spectrum of S = 1/2 and S = 1 Heisenberg bond-alternating chains have been investigated by the infinite time-evolving block decimation (iTEBD) method. Generalized string order parameters with appropriate θ are capable of distinguishing all the topological phases. Central charges c ≃ 1 and critical exponents β ≃ 1/12 indicate all the topological QPTs belong to the Gaussian universality class. Interestingly, odd- and even-fold degeneracies of the entanglement spectrum are observed. Even-fold (doubly) degenerate entanglement spectra and the typical two-fold degenerate lowest-lying level are found to exist in both the spin-1/2 dimer and the S = 1 Haldane phases. However, odd-fold degenerate entanglement spectra with three-fold degenerate lowest-lying level are observed in both the S = 1 dimer and the S = 2 Haldane phase. The degeneracy of the lowest-lying entanglement spectrum level, which can be understood by entanglement spectra in the dimer limit (J 1 = 0), is adopted to estimate the lowest boundary of the bipartite entanglement. The entanglement spectrum and the generalized string orders are valuable for uncovering the underlying features of these symmetry-protect topological (SPT) states. Similar entanglement spectrum shows that the S = 1 (S = 2) Haldane phase is essentially the same as the S = 1/2 (S = 1) dimer phase.

Quantum Phase Transitions, Entanglement Spectrum, and Schmidt Gap in Bond-Alternative S = 1 Antiferromagnetic Chain

Bipartite entanglement, entanglement spectrum, and Schmidt gap in S=1 bond-alternative antiferromagnetic Heisenberg chain are investigated by the infinite time-evolving block decimation (iTEBD) method. The quantum phase transition (QPT) from the singlet-dimer phase to the Haldane phase can be detected by the singular behavior of bipartite entanglement, the sudden change of the entanglement spectrum, and the completely vanishing of the Schmidt gap. The critical point is determined to be around rc ≃ 0.587, and the second-order character of the QPT is verified. Doubly degenerate entanglement spectra of both even and odd bonds are observed in the Haldane phase, by which one can distinguish the Haldane phase from the singlet-dimer phase easily. Nearest-neighbor antiferromagnetic correlations and next-nearest-neighbor ferromagnetic correlations are found in the whole parameter region. At the critical massless point, although exponentially decaying antiferromagnetic correlation is observed, it approaches to a constant value finally. Therefore, long-range correlations exist and the correlation length becomes divergent at the critical point.

Nonlocal string orders and entanglement spectrum in the spin-1/2 alternating Heisenberg-Ising chain

By the infinite time-evolving block decimation (iTEBD) technique, the quantum phase transitions (QPTs) and the ground-state properties of a spin-1/2 alternating Heisenberg-Ising chain are investigated. As the Ising coupling is generalized into the ferromagnetic case (J I < 0), in addition to two well-known phases (the antiferromagnetic phase and the disordered dimer phase), an antiferromagnetic stripe phase is induced. Two QPTs at J I = ±2 can be described by the singular behavior of the bipartite entanglement entropy and the ground-state energy, and both are verified to be second-order transitions. Nonzero string orders (O s σ, σ = x, y, and z) and doubly degenerate entanglement spectra are observed in the disordered dimer phase, and these characters distinguish the disordered phase from other two phases. Especially, the string order introduced firstly to describe the Haldane phase of spin-1 chain is generalized to the spin-1/2 alternating Heisenberg-Ising chain.

Doubly degenerate entanglement spectrum and nonzero string order in the spin-1/2 Heisenberg–Ising chain

The ground-state properties and quantum phase transitions (QPTs) in spin-1/2 Heisenberg-Ising alternating chain has been investigated by the iTEBD algorithm. Four different ground-state phases, i.e., a ferromagnetic phase (FM), an antiferromagnetic phase (AF), a stripe phase (SP), and a disordered phase were distinguished. The disordered phase, which has nonzero string orders and the doubly degenerate entanglement spectrum, was observed as Heisenberg coupling JH>0.5. The disordered phase in such a model is found to belong to the same topological phase as the Haldane state. In the disordered phase, every two nearest-neighbor spin-1/2 spins connected by the Ising coupling behave like an integer (S=1) spin. Furthermore, the QPTs from the disordered phase to the AF and SP phases belong to the Ising universality class with central charges c=c¯=1/2.

Entanglement spectrum and quantum phase transitions in one-dimensional [formula omitted] XXZ model with uniaxial single-ion anisotropy

Quantum phase transitions (QPTs) in one-dimensional S=1 XXZ model with uniaxial single-ion anisotropy are investigated. Bipartite entanglement, entanglement spectrum, and Schmidt gap are found to be capable of describing all the QPTs, even the infinite-order Berezinskii–Kosterlitz–Thouless (BKT) transition. According to the singular behavior of the second-order derivative of ground-state energy, the QPT between XY2 and antiferromagnetic phases is a second-order but not a BKT transition. Energy level crossing, accompanied with discontinuous entanglement entropy and entanglement spectrum, is observed at the transition point between the large-D and antiferromagnetic phases, therefore it should be a first-order QPT. In addition, doubly degenerate entanglement spectrum in the Haldane phase is observed.

Magnetization process and quantum entanglement in spin-1 XXZ model with single-ion anisotropy under external field

By the infinite time-evolving block decimation (iTEBD) technique, the magnetization process and the quantum phase transitions (QPTs) in the spin-1 XXZ model with single-ion anisotropy under external field are investigated. It is found that, all the phases will be destroyed by a sufficient strong magnetic field. Before they come into the ferromagnetic (fully polarized) phase, some interesting intermediate phases are induced. A pseudo order Oixy=〈Six〉2+〈Siy〉2 with finite truncation dimension χ can be used to describe the XY1 phase. Especially, the Oxyi with finite χ is found to be nonzero in the XY1 phase, but vanishes in the XY2 phase. It means that two kinds of XY phases can be distinguished by the pseudo order parameter Oxyi. All the QPTs can be described by the behavior of the entanglement entropy and the ground-state energy. QPTs from the XY phase to the large-D, Haldane, and antiferromagnetic phases are found to be infinite-order BKT type transitions, but the QPTs from the XY1 phase to the ferromagnetic phase and XY2 have second-order characters. In addition, doubly degenerate entanglement spectra are observed in the Haldane phase.

Proposal for Verification of the Haldane Phase Using Trapped Ions

A proposal to use trapped ions to implement spin-one XXZ antiferromagnetic chains as an experimental tool to explore the Haldane phase is presented. We explain how to reach the Haldane phase adiabatically, demonstrate the robustness of the ground states to noise in the magnetic field and Rabi frequencies, and propose a way to detect them using their characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order, and a double degenerate entanglement spectrum. Scaling up to higher dimensions and more frustrated lattices, we obtain richer phase diagrams, and we can reach spin liquid phase, which can be detected by its entanglement entropy which obeys the boundary law.

Entanglement spectrum and magnetization plateaus in an S = 1 bond-alternative Heisenberg chain with single-ion anisotropy and magnetic field

Entanglement spectrum and quantum phase transitions (QPTs) in S = 1 bond-alternative antiferromagnetic Heisenberg chain with single-ion anisotropy and magnetic field are investigated by the infinite time-evolving block decimation (iTEBD) method. The combined effects of the single-ion anisotropy and magnetic field have been discussed, and a rich ground-state phase diagram is obtained. We find that the single-ion anisotropy is advantageous to the stability of the 1/2 magnetization plateau. Both entanglement entropy and entanglement spectrum, as two model-independent measures, are capable of describing all the QPTs. Especially, doubly degenerate entanglement spectrum on even bond is observed in the 1/2 plateau phase. Besides constant spontaneous magnetization, three magnetization plateaus (M z = 0, 1/2, and 1) are found to have constant entanglement entropy, entanglement spectrum, and nearest-neighbor correlation. In addition, all the QPTs in such a model have been determined to belong to the second-order category.