Algorithm 923

Wimmer, Michael

doi:10.1145/2331130.2331138

Cited by 149 publications

(89 citation statements)

References 42 publications

(68 reference statements)

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“…Numerical computation of Pfaffians was carried out by the PFAPACK library [61]. This work The supplementary information is organized as follows: In Appendix A, we explain the basics of the cobordism theory, Appendices B -H are devoted to explaining technical details of construction of SPT invariant from a spacetime cross-cap.…”

Section: T1mentioning

confidence: 99%

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

2017

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We define and compute many-body topological invariants of interacting fermionic symmetryprotected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the Z8 and Z16 classification of topological superconductors in one and three dimensions.The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [1, 2] is the prototype for topological characterization of phases of matter. It relates the quantized Hall conductance to the (first) Chern number defined for Bloch wave functions. At the level of many-body physics, the quantized Hall conductance can also be formulated in terms of ground state wave functions in the presence of twisted boundary conditions ("the many-body Chern number") [3]. In contrast to local order parameters, the TKNN integer distinguishes different quantum phases of matter by focusing on their global topological properties.More recently, the discovery of topological insulators and superconductors [4, 5] has led to a new research frontier, generally referred to as symmetry protected topological (SPT) phases. These phases are adiabatically connected to topologically trivial states, i.e., atomic insulators which can be represented as simple product states without any entanglement. Nevertheless, they are topologically distinct once a symmetry condition, e.g., timereversal symmetry, is imposed. A complete classification of the noninteracting fermionic SPT phases protected by non-spatial discrete symmetries [6][7][8], as well as crystalline SPT phases protected by spatial symmetries [9][10][11][12], were achieved.However, it was later discovered that the noninteracting topological classification is not the full story, and can be dramatically altered once interaction effects are taken into account [13]. Since then, there have been several works which discuss the breakdown of the noninteracting classification in the presence of interactions [14][15][16][17][18][19][20][21][22][23][24][25].There are various topological invariants for noninteracting fermionic SPT phases using single-particle states (e.g., Bloch wave functions). For example, the Z 2 -valued topological invariants have been introduced for topological insulators both in two and three spatial dimensions [26][27][28][29]. For topological superconductors protected by time-reversal symmetry, the integer-valued topological invariants ("the winding number") have been introduced [6]. However, the discovered breakdown of non-interacting classification clearly indicates that the situation at the interacting level is more intricate, and a general framework to distinguish interacting fermionic SPT phases is lacking. This should be contrasted from the quantized Hall conductance, which can be formulate...

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Section: T1mentioning

confidence: 99%

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

2017

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“…(15). Therefore, taking the real part in (23) is equivalent to setting H 2m = 0 and calculating a real-valued Pfaffian, which can be performed by efficient numerical algorithms [40].…”

Section: Pfaffian Approachmentioning

confidence: 99%

“…Given the universal result (40) for the magnetization profile in the JW case, the question naturally emerges whether one has a deeper connection to the free-fermion domain-wall problem on the level of the time-evolved state. To answer this question, we shall now consider the entanglement entropy, which carries important information about the state itself.…”

Section: Magnetization Profiles For Domain-wall Excitationmentioning

confidence: 99%

Universal front propagation in the quantum Ising chain with domain-wall initial states

2016

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We study the melting of domain walls in the ferromagnetic phase of the transverse Ising chain, created by flipping the order-parameter spins along one-half of the chain. If the initial state is excited by a local operator in terms of Jordan-Wigner fermions, the resulting longitudinal magnetization profiles have a universal character. Namely, after proper rescalings, the profiles in the noncritical Ising chain become identical to those obtained for a critical free-fermion chain starting from a step-like initial state. The relation holds exactly in the entire ferromagnetic phase of the Ising chain and can even be extended to the zero-field XY model by a duality argument. In contrast, for domain-wall excitations that are highly non-local in the fermionic variables, the universality of the magnetization profiles is lost. Nevertheless, for both cases we observe that the entanglement entropy asymptotically saturates at the ground-state value, suggesting a simple form of the steady state.

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“…The evaluation of the Pfaffians required for calculating dynamic susceptibilities in the one-dimensional models uses the algorithm provided in Ref. [50].…”

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confidence: 99%

Measuring multipartite entanglement through dynamic susceptibilities

et al. 2016

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Entanglement plays a central role in our understanding of quantum many body physics, and is fundamental in characterising quantum phases and quantum phase transitions. Developing protocols to detect and quantify entanglement of many-particle quantum states is thus a key challenge for present experiments. Here, we show that the quantum Fisher information, representing a witness for genuinely multipartite entanglement, becomes measurable for thermal ensembles via the dynamic susceptibility, i.e., with resources readily available in present cold atomic gas and condensed-matter experiments. This moreover establishes a fundamental connection between multipartite entanglement and many-body correlations contained in response functions, with profound implications close to quantum phase transitions. There, the quantum Fisher information becomes universal, allowing us to identify strongly entangled phase transitions with a divergent multipartiteness of entanglement. We illustrate our framework using paradigmatic quantum Ising models, and point out potential signatures in optical-lattice experiments.Entanglement is a central theoretical concept underlying the characterisation of quantum many-body states in condensed-matter and high-energy physics, as well as quantum information. For example, entanglement properties reveal exotic states of matter such as topological spin liquids [1] or many-body localization [2,3], the holographic entanglement entropy identifies confinement/deconfinement transitions in gauge theories [4,5], and entanglement is considered the central resource for quantum-enhanced metrology [6,7] as well as quantum computation [8][9][10][11]. In experiments, entanglement becomes measurable via a tomographic determination of the many-particle quantum state [12][13][14][15], and protocols have been developed [16] and implemented in remarkable experiments [17] to measure entanglement entropies in quench dynamics and quantum phase transitions. However, the resources required by these protocols scale exponentially with the system size, and these experimental efforts are thus limited a priori to few-particle systems.To address the problem of detecting and quantifying multipartite entanglement for large systems, we consider below the quantum Fisher information (QFI) as an entanglement witness [18][19][20]. Our key result is thatfor a many-body system at thermal equilibrium at any temperature-the QFI can be determined directly from a measurement of Kubo linear response functions, in particular the dynamic susceptibility (see Fig. 1). We emphasise that this measurement prescription is independent of microscopic details of the system of interest and that the measurement of linear response is a standard tool in experiments. Importantly, only modest measurement resources are required that do not scale with system size. The presented prescription therefore makes multipartite entanglement observable for a large variety of * philipp.hauke@uibk.ac.at χ (ω, T ) gives the quantum Fisher information (shaded areas). (c) This pro...

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Algorithm 923

Cited by 149 publications

References 42 publications

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Universal front propagation in the quantum Ising chain with domain-wall initial states

Measuring multipartite entanglement through dynamic susceptibilities

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