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Garros A, Alcoba DR, Capuzzi P, Lain L, Torre A, Oña OB, Dukelsky J. Determination of electronic excitation energies within the doubly occupied configuration interaction space by means of the Hermitian operator method. J Chem Phys 2023; 159:124107. [PMID: 38127377 DOI: 10.1063/5.0168585] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Accepted: 09/06/2023] [Indexed: 12/23/2023] Open
Abstract
In this work, we formulate the equations of motion corresponding to the Hermitian operator method in the framework of the doubly occupied configuration interaction space. The resulting algorithms turn out to be considerably simpler than the equations provided by that method in more conventional spaces, enabling the determination of excitation energies in N-electron systems under an affordable polynomial computational cost. The implementation of this technique only requires to know the elements of low-order reduced density matrices of an N-electron reference state, which can be obtained from any approximate method. We contrast our procedure against the reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev integrable models, pointing out the reliability of our proposal.
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Affiliation(s)
- Adán Garros
- Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- CONICET-Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Diego R Alcoba
- Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- CONICET-Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Pablo Capuzzi
- Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- CONICET-Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Luis Lain
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
| | - Alicia Torre
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
| | - Ofelia B Oña
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, Diag. 113 y 64 (S/N), Sucursal 4, CC 16, 1900 La Plata, Argentina
| | - Jorge Dukelsky
- Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain
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2
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Rigo M, Hall B, Hjorth-Jensen M, Lovato A, Pederiva F. Solving the nuclear pairing model with neural network quantum states. Phys Rev E 2023; 107:025310. [PMID: 36932590 DOI: 10.1103/physreve.107.025310] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.
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Affiliation(s)
- Mauro Rigo
- Physics Department, University of Trento, via Sommarive 14, I-38123 Trento, Italy
| | - Benjamin Hall
- Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
- Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
| | - Morten Hjorth-Jensen
- Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
- Department of Physics and Center for Computing in Science Education, University of Oslo, N-0316 Oslo, Norway
| | - Alessandro Lovato
- Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
- Computational Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
- INFN-TIFPA Trento Institute for Fundamental Physics and Applications, Via Sommarive, 14, 38123 Trento, Italy
| | - Francesco Pederiva
- Physics Department, University of Trento, via Sommarive 14, I-38123 Trento, Italy
- INFN-TIFPA Trento Institute for Fundamental Physics and Applications, Via Sommarive, 14, 38123 Trento, Italy
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3
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Alcoba DR, Oña OB, Lain L, Torre A, Capuzzi P, Massaccesi GE, Ríos E, Rubio-García A, Dukelsky J. Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach. J Chem Phys 2021; 154:224104. [PMID: 34241224 DOI: 10.1063/5.0051793] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.
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Affiliation(s)
- Diego R Alcoba
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Ofelia B Oña
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad de la Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, Diag. 113 y 64 (S/N), Sucursal 4, CC 16, 1900 La Plata, Argentina
| | - Luis Lain
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
| | - Alicia Torre
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
| | - Pablo Capuzzi
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Gustavo E Massaccesi
- Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Elías Ríos
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad de la Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, Diag. 113 y 64 (S/N), Sucursal 4, CC 16, 1900 La Plata, Argentina
| | - Alvaro Rubio-García
- Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
| | - Jorge Dukelsky
- Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
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4
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Kumar A, Ortiz G, Richerme P, Seradjeh B. Floquet Gauge Pumps as Sensors for Spectral Degeneracies Protected by Symmetry or Topology. PHYSICAL REVIEW LETTERS 2021; 126:206602. [PMID: 34110211 DOI: 10.1103/physrevlett.126.206602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2020] [Accepted: 04/16/2021] [Indexed: 06/12/2023]
Abstract
We introduce the concept of a Floquet gauge pump whereby a dynamically engineered Floquet Hamiltonian is employed to reveal the inherent degeneracy of the ground state in interacting systems. We demonstrate this concept in a one-dimensional XY model with periodically driven couplings and transverse field. In the high-frequency limit, we obtain the Floquet Hamiltonian consisting of the static XY and dynamically generated Dzyaloshinsky-Moriya interaction (DMI) terms. The dynamically generated magnetization current depends on the phases of complex coupling terms, with the XY interaction as the real and DMI as the imaginary part. As these phases are cycled, the current reveals the ground-state degeneracies that distinguish the ordered and disordered phases. We discuss experimental requirements needed to realize the Floquet gauge pump in a synthetic quantum spin system of interacting trapped ions.
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Affiliation(s)
- Abhishek Kumar
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
| | - Gerardo Ortiz
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
- Quantum Science and Engineering Center, Indiana University, Bloomington, Indiana 47405, USA
| | - Philip Richerme
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
- Quantum Science and Engineering Center, Indiana University, Bloomington, Indiana 47405, USA
| | - Babak Seradjeh
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
- Quantum Science and Engineering Center, Indiana University, Bloomington, Indiana 47405, USA
- IU Center for Spacetime Symmetries, Indiana University, Bloomington, Indiana 47405, USA
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5
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Gotta L, Mazza L, Simon P, Roux G. Two-Fluid Coexistence in a Spinless Fermions Chain with Pair Hopping. PHYSICAL REVIEW LETTERS 2021; 126:206805. [PMID: 34110210 DOI: 10.1103/physrevlett.126.206805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Accepted: 04/26/2021] [Indexed: 06/12/2023]
Abstract
We show that a simple one-dimensional model of spinless fermions with pair hopping displays a phase in which a Luttinger liquid of paired fermions coexists with a Luttinger liquid of unpaired fermions. Our results are based on extensive numerical density-matrix renormalization-group calculations and are supported by a two-fluid model that captures the essence of the coexistence region.
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Affiliation(s)
- Lorenzo Gotta
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - Leonardo Mazza
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - Pascal Simon
- Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
| | - Guillaume Roux
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
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6
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Harsha G, Henderson TM, Scuseria GE. Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems. J Chem Phys 2020; 153:124115. [DOI: 10.1063/5.0022702] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Gaurav Harsha
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - Thomas M. Henderson
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
| | - Gustavo E. Scuseria
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
- Department of Chemistry, Rice University, Houston, Texas 77005, USA
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7
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Lapa MF, Levin M. Rigorous Results on Topological Superconductivity with Particle Number Conservation. PHYSICAL REVIEW LETTERS 2020; 124:257002. [PMID: 32639770 DOI: 10.1103/physrevlett.124.257002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2020] [Accepted: 05/27/2020] [Indexed: 06/11/2023]
Abstract
Most theoretical studies of topological superconductors and Majorana-based quantum computation rely on a mean-field approach to describe superconductivity. A potential problem with this approach is that real superconductors are described by number-conserving Hamiltonians with long-range interactions, so their topological properties may not be correctly captured by mean-field models that violate number conservation and have short-range interactions. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the mean-field models, including a finite energy gap in a sector of fixed total particle number, the existence of long-range Majorana-like correlations between the ends of an open wire, and a change in the ground state fermion parity for periodic vs antiperiodic boundary conditions. These results show that many of the remarkable properties of mean-field models of topological superconductivity persist in more realistic models with number-conserving dynamics.
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Affiliation(s)
- Matthew F Lapa
- Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Michael Levin
- Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
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8
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Irfan AAM, Mayer K, Ortiz G, Knill E. Certified Quantum Measurement of Majorana Fermions. PHYSICAL REVIEW. A 2020; A101:10.1103/PhysRevA.101.032106. [PMID: 33313459 PMCID: PMC7727742 DOI: 10.1103/physreva.101.032106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We present a quantum self-testing protocol to certify measurements of fermion parity involving Majorana fermion modes. We show that observing a set of ideal measurement statistics implies anti-commutativity of the implemented Majorana fermion parity operators, a necessary prerequisite for Majorana detection. Our protocol is robust to experimental errors. We obtain lower bounds on the fidelities of the state and measurement operators that are linear in the errors. We propose to analyze experimental outcomes in terms of a contextuality witness W, which satisfies 〈W〉 ≤ 3 for any classical probabilistic model of the data. A violation of the inequality witnesses quantum contextuality, and the closeness to the maximum ideal value 〈W〉 = 5 indicates the degree of confidence in the detection of Majorana fermions.
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Affiliation(s)
- Abu Ashik Md Irfan
- Department of Physics, Indiana University, Bloomington, IN 47405-7105, USA
| | - Karl Mayer
- National Institute of Standards and Technology, Boulder, Colorado, USA
- Department of Physics, University of Colorado, Boulder, Colorado, USA
| | - Gerardo Ortiz
- Department of Physics, Indiana University, Bloomington, IN 47405-7105, USA
| | - Emanuel Knill
- National Institute of Standards and Technology, Boulder, Colorado, USA
- Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado, USA
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9
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Alcoba DR, Capuzzi P, Rubio-García A, Dukelsky J, Massaccesi GE, Oña OB, Torre A, Lain L. Variational reduced density matrix method in the doubly occupied configuration interaction space using three-particleN-representability conditions. J Chem Phys 2018; 149:194105. [DOI: 10.1063/1.5056247] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Diego R. Alcoba
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Pablo Capuzzi
- Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
- Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Alvaro Rubio-García
- Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
| | - Jorge Dukelsky
- Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
| | - Gustavo E. Massaccesi
- Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Ofelia B. Oña
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, CCT La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, Diag. 113 y 64 (S/N), Sucursal 4, CC 16, 1900 La Plata, Argentina
| | - Alicia Torre
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
| | - Luis Lain
- Departamento de Química Física, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apdo. 644, E-48080 Bilbao, Spain
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10
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Rubio-García A, Alcoba DR, Capuzzi P, Dukelsky J. Benchmarking the Variational Reduced Density Matrix Theory in the Doubly Occupied Configuration Interaction Space with Integrable Pairing Models. J Chem Theory Comput 2018; 14:4183-4192. [PMID: 29906104 DOI: 10.1021/acs.jctc.8b00387] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The variational reduced density matrix theory has been recently applied with great success to models within the truncated doubly occupied configuration interaction space, which corresponds to the seniority zero subspace. Conservation of the seniority quantum number restricts the Hamiltonians to be based on the SU(2) algebra. Among them there is a whole family of exactly solvable Richardson-Gaudin pairing Hamiltonians. We benchmark the variational theory against two different exactly solvable models, the Richardson-Gaudin-Kitaev and the reduced BCS Hamiltonians. We obtain exact numerical results for the so-called [Formula: see text] N-representability conditions in both cases for systems that go from 10 to 100 particles. However, when random single-particle energies as appropriate for small superconducting grains are considered, the exactness is lost but still a high accuracy is obtained.
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Affiliation(s)
- A Rubio-García
- Instituto de Estructura de la Materia, CSIC, Serrano 123 , 28006 Madrid , Spain
| | - D R Alcoba
- Departamento de Física, Facultad de Ciencias Exactas y Naturales , Universidad de Buenos Aires, Ciudad Universitaria , 1428 Buenos Aires , Argentina.,Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas , Ciudad Universitaria , 1428 Buenos Aires , Argentina
| | - P Capuzzi
- Departamento de Física, Facultad de Ciencias Exactas y Naturales , Universidad de Buenos Aires, Ciudad Universitaria , 1428 Buenos Aires , Argentina.,Instituto de Física de Buenos Aires, Consejo Nacional de Investigaciones Científicas y Técnicas , Ciudad Universitaria , 1428 Buenos Aires , Argentina
| | - J Dukelsky
- Instituto de Estructura de la Materia, CSIC, Serrano 123 , 28006 Madrid , Spain
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11
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Zhang RX, Liu CX. Crystalline Symmetry-Protected Majorana Mode in Number-Conserving Dirac Semimetal Nanowires. PHYSICAL REVIEW LETTERS 2018; 120:156802. [PMID: 29756865 DOI: 10.1103/physrevlett.120.156802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2017] [Revised: 01/10/2018] [Indexed: 06/08/2023]
Abstract
One of the cornerstones for topological quantum computations is the Majorana zero mode, which has been intensively searched in fractional quantum Hall systems and topological superconductors. Several recent works suggest that such an exotic mode can also exist in a one-dimensional (1D) interacting double-wire setup even without long-range superconductivity. A notable instability in these proposals comes from interchannel single-particle tunneling that spoils the topological ground state degeneracy. Here we show that a 1D Dirac semimetal (DSM) nanowire is an ideal number-conserving platform to realize such Majorana physics. By inserting magnetic flux, a DSM nanowire is driven into a 1D crystalline-symmetry-protected semimetallic phase. Interaction enables the emergence of boundary Majorana zero modes, which is robust as a result of crystalline symmetry protection. We also explore several experimental consequences of Majorana signals.
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Affiliation(s)
- Rui-Xing Zhang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Chao-Xing Liu
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
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12
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Pérez M, Martínez G. Polynomial description of inhomogeneous topological superconducting wires. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:475503. [PMID: 29035275 DOI: 10.1088/1361-648x/aa93cd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We present the universal features of the topological invariant for p-wave superconducting wires after the inclusion of spatial inhomogeneities. Three classes of distributed potentials are studied, a single-defect, a commensurate and an incommensurate model, using periodic site modulations. An analytic polynomial description is achieved by splitting the topological invariant into two parts; one part depends on the chemical potential and the other does not. For the homogeneous case, an elliptical region is found where the topological invariant oscillates. The zeros of these oscillations occur at points where the fermion parity switches for finite wires. The increase of these oscillations with the inhomogeneity strength leads to new isolated non-topological phases. We characterize these new phases according to each class of spatial distributions. Such phases could also be observed in the XY model, to which our model is dual.
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Affiliation(s)
- Marcos Pérez
- Instituto de Física, UFRGS, 91501-970 Porto Alegre-RS, Brazil
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13
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Shapourian H, Shiozaki K, Ryu S. Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases. PHYSICAL REVIEW LETTERS 2017; 118:216402. [PMID: 28598638 DOI: 10.1103/physrevlett.118.216402] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2016] [Indexed: 06/07/2023]
Abstract
We define and compute many-body topological invariants of interacting fermionic symmetry-protected 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 nonlocal operation, "partial" reflection or transpose. As an application of our scheme, we study the Z_{8} and Z_{16} classification of topological superconductors in one and three dimensions.
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Affiliation(s)
- Hassan Shapourian
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Ken Shiozaki
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Shinsei Ryu
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
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14
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Iemini F, Mazza L, Fallani L, Zoller P, Fazio R, Dalmonte M. Majorana Quasiparticles Protected by Z_{2} Angular Momentum Conservation. PHYSICAL REVIEW LETTERS 2017; 118:200404. [PMID: 28581780 DOI: 10.1103/physrevlett.118.200404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Indexed: 06/07/2023]
Abstract
We show how angular momentum conservation can stabilize a symmetry-protected quasitopological phase of matter supporting Majorana quasiparticles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated by density-matrix-renormalization-group simulations. Our results pave the way toward the observation of Majorana edge modes with alkaline-earth-like fermions in optical lattices, where all basic ingredients for our recipe-spin-orbit coupling and strong interorbital interactions-have been experimentally realized over the last two years.
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Affiliation(s)
- F Iemini
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy
| | - L Mazza
- Departement de Physique, Ecole Normale Superieure/PSL Research University, CNRS, 24 rue Lhomond, F-75005 Paris, France
| | - L Fallani
- Department of Physics and Astronomy, University of Florence, I-50019 Sesto Fiorentino, Italy
- LENS European Laboratory for Nonlinear Spectroscopy, I-50019 Sesto Fiorentino, Italy
| | - P Zoller
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
| | - R Fazio
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy
- NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
| | - M Dalmonte
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy
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15
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Alase A, Cobanera E, Ortiz G, Viola L. Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions. PHYSICAL REVIEW LETTERS 2016; 117:076804. [PMID: 27563986 DOI: 10.1103/physrevlett.117.076804] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2016] [Indexed: 06/06/2023]
Abstract
We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D-1. The key is a Hamiltonian-dependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional 4π-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.
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Affiliation(s)
- Abhijeet Alase
- Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA
| | - Emilio Cobanera
- Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA
| | - Gerardo Ortiz
- Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
| | - Lorenza Viola
- Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA
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Iemini F, Mazza L, Rossini D, Fazio R, Diehl S. Localized Majorana-Like Modes in a Number-Conserving Setting: An Exactly Solvable Model. PHYSICAL REVIEW LETTERS 2015; 115:156402. [PMID: 26550737 DOI: 10.1103/physrevlett.115.156402] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2015] [Indexed: 06/05/2023]
Abstract
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.
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Affiliation(s)
- Fernando Iemini
- Departamento de Física-ICEx-Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil
- NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
| | - Leonardo Mazza
- NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
| | - Davide Rossini
- NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
| | - Rosario Fazio
- NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
- ICTP, Strada Costiera 11, 34151 Trieste, Italy
| | - Sebastian Diehl
- Institute of Theoretical Physics, TU Dresden, D-01062 Dresden, Germany
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