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Register a new userEntanglement modes and topological phase transitions in superconductors
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Tharnier Oliveira Puel
Publication date
2015
fields
Physics
and research's language is
English
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Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional $p$-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the $k$-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the $k$-space fidelity operator.
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The importance of models with an exact solution for the study of materials with non-trivial topological properties has been extensively demonstrated. Among these, the Kitaev model of a one-dimensional $p$-wave superconductor plays a guiding role in the search for Majorana modes in condensed matter systems. Also, the $sp$ chain, with an anti-symmetric mixing among the $s$ and $p$ bands provides a paradigmatic example of a topological insulator with well understood properties. There is an intimate relation between these two models and in particular their topological quantum phase transitions share the same universality class. Here we consider a two-band $sp$ model of spinless fermions with an attractive (inter-band) interaction. Both the interaction and hybridization between the $s$ and $p$ fermions are anti-symmetric. The zero temperature phase diagram of the model presents a variety of phases including a Weyl superconductor, topological insulator and trivial phases. The quantum phase transitions between these phases can be either continuous or discontinuous. We show that the transition from the topological superconducting phase to the trivial one has critical exponents different from those of an equivalent transition in Kitaevs model.
Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. In this paper, we develop a unified framework based on variational iPEPS for the quantitative study of both topological and conventional phase transitions through entanglement order parameters. To this end, we employ tensor networks with suitable physical and/or entanglement symmetries encoded, and allow for order parameters detecting the behavior of any of those symmetries, both physical and entanglement ones. First, this gives rise to entanglement-based order parameters for topological phases. These topological order parameters allow to quantitatively probe topological phase transitions and to identify their universal behavior. We apply our framework to the study of the Toric Code model in different magnetic fields, which in some cases maps to the (2+1)D Ising model. We identify 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we moreover find an unknown critical exponent beta=0.021. We then apply our framework of entanglement order parameters to conventional phase transitions. We construct a novel type of disorder operator (or disorder parameter), which is non-zero in the disordered phase and measures the response of the wavefunction to a symmetry twist in the entanglement. We numerically evaluate this disorder operator for the (2+1)D transverse field Ising model, where we again recover a critical exponent hitherto unknown in the model, beta=0.024, consistent with the findings for the Toric Code. This shows that entanglement order parameters can provide additional means of characterizing the universal data both at topological and conventional phase transitions, and altogether demonstrates the power of this framework to identify the universal data underlying the transition.
We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower band eigenvector and the winding number of the Hamiltonians. For exponentially-decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive non-local Dirac fermion localised at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.
Magnetic susceptibility, electrical resistivity and heat capacity data for single crystals of Ce(Rh,Ir)1-x(Co,Ir)xIn5 (0 < x < 1) have allowed us to construct a detailed phase diagram for this new family of heavy-fermion superconductors(HFS). CeRh1-xIrxIn5 displays superconductivity(SC) (Tc < 1 K) over a wide range of composition, which develops out of and coexists (0.30 < x < 0.5) with a magnetically ordered state, with TN ~ 4 K. For CeCo1-xRhxIn5, the superconducting state (Tc ~ 2.3 K for x = 0) becomes a magnetic state (TN ~ 4 K, for x = 1) with two phase transitions observed for 0.40 < x < 0.25. CeCo1-xIrxIn5 also shows two transitions for 0.30 < x < 0.75. For those alloys in which SC is found, a roughly linear relationship between Tc and the lattice parameter ratio c/a, was found, with composition as the implicit parameter. The interplay between magnetism and SC for CeRh1-x(Ir,Co)xIn5 and the possibility of two distinct superconducting states in CeCo1-xIrxIn5 are discussed.
We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate $alpha$. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range ($alpha$ > 1) or long-range ($alpha$ < 1) sector. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with $alpha$ as the quench parameter. In particular, when the lowest upper-half entanglement-spectrum value of the initial Hamiltonian is smaller than the final one, the TPTs can also be diagnosed by the difference between the lowest two upper-half entanglement-spectrum values if the halfway winding number is not equal to that of the initial Hamiltonian. Moreover, we discuss the stability of characterizing the TPTs via entanglement-spectrum crossings against energy dispersion in the long-range model.
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