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Granet E, Zhang C, Dreyer H. Volume-Law to Area-Law Entanglement Transition in a Nonunitary Periodic Gaussian Circuit. PHYSICAL REVIEW LETTERS 2023; 130:230401. [PMID: 37354425 DOI: 10.1103/physrevlett.130.230401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 04/26/2023] [Indexed: 06/26/2023]
Abstract
We consider Gaussian quantum circuits that alternate unitary gates and postselected weak measurements, with spatial translation symmetry and time p eriodicity. We show analytically that such models can host different kinds of measurement-induced phase transitions detected by entanglement entropy, by mapping the unitary gates and weak measurements onto Möbius transformations. We demonstrate the existence of a log-law to area-law transition, as well as a volume-law to area-law transition at a finite measurement amplitude. For the latter, we compute the critical exponent ν for the Hartley, von Neumann and Rényi entropies exactly.
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Affiliation(s)
- Etienne Granet
- Department of Physics, Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Carolyn Zhang
- Department of Physics, Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Henrik Dreyer
- Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany
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2
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Dey D, Parvej A, Das S, Saha SK, Kumar M, Ramasesha S, Soos ZG. Density matrix renormalization group (DMRG) for interacting spin chains and ladders. J CHEM SCI 2023. [DOI: 10.1007/s12039-023-02140-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/30/2023]
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3
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Ren J, Wang Z, Chen W, You WL. Long-range order and quantum criticality in antiferromagnetic chains with long-range staggered interactions. Phys Rev E 2022; 105:034128. [PMID: 35428105 DOI: 10.1103/physreve.105.034128] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2021] [Accepted: 03/04/2022] [Indexed: 11/07/2022]
Abstract
We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.
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Affiliation(s)
- Jie Ren
- Department of Physics, Changshu Institute of Technology, Changshu 215500, China
| | - Zhao Wang
- Department of Physics, Changshu Institute of Technology, Changshu 215500, China
| | - Weixia Chen
- Department of Physics, Changshu Institute of Technology, Changshu 215500, China
| | - Wen-Long You
- College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China and MIIT Key Laboratory of Aerospace Information Materials and Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
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4
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Chanda T, Lewenstein M, Zakrzewski J, Tagliacozzo L. Phase Diagram of 1+1D Abelian-Higgs Model and Its Critical Point. PHYSICAL REVIEW LETTERS 2022; 128:090601. [PMID: 35302796 DOI: 10.1103/physrevlett.128.090601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2021] [Accepted: 02/08/2022] [Indexed: 06/14/2023]
Abstract
We determine the phase diagram of the Abelian-Higgs model in one spatial dimension and time (1+1D) on a lattice. We identify a line of first order phase transitions separating the Higgs region from the confined one. This line terminates in a quantum critical point above which the two regions are connected by a smooth crossover. We analyze the critical point and find compelling evidence for its description as the product of two noninteracting systems: a massless free fermion and a massless free boson. However, we find also some surprising results that cannot be explained by our simple picture, suggesting this newly discovered critical point is an unusual one.
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Affiliation(s)
- Titas Chanda
- The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
- Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Maciej Lewenstein
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Avenue Carl Friedrich Gauss 3, 08860 Barcelona, Spain
- ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
| | - Jakub Zakrzewski
- Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
- Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Luca Tagliacozzo
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
- Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia, Spain
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5
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Dupont M, Sherman NE, Moore JE. Spatiotemporal Crossover between Low- and High-Temperature Dynamical Regimes in the Quantum Heisenberg Magnet. PHYSICAL REVIEW LETTERS 2021; 127:107201. [PMID: 34533348 DOI: 10.1103/physrevlett.127.107201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2021] [Accepted: 08/10/2021] [Indexed: 06/13/2023]
Abstract
The stranglehold of low temperatures on fascinating quantum phenomena in one-dimensional quantum magnets has been challenged recently by the discovery of anomalous spin transport at high temperatures. Whereas both regimes have been investigated separately, no study has attempted to reconcile them. For instance, the paradigmatic quantum Heisenberg spin-1/2 chain falls at low temperature within the Tomonaga-Luttinger liquid framework, while its high-temperature dynamics is superdiffusive and relates to the Kardar-Parisi-Zhang universality class in 1+1 dimensions. This Letter aims at reconciling the two regimes. Building on large-scale matrix product state simulations, we find that they are connected by a temperature-dependent spatiotemporal crossover. As the temperature T is reduced, we show that the onset of superdiffusion takes place at longer length and timescales ∝1/T. This prediction has direct consequences for experiments including nuclear magnetic resonance: it is consistent with earlier measurements on the nearly ideal Heisenberg S=1/2 chain compound Sr_{2}CuO_{3}, yet calls for new and dedicated experiments.
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Affiliation(s)
- Maxime Dupont
- Department of Physics, University of California, Berkeley, California 94720, USA and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Nicholas E Sherman
- Department of Physics, University of California, Berkeley, California 94720, USA and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Joel E Moore
- Department of Physics, University of California, Berkeley, California 94720, USA and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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6
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Bhattacharya T, Buser AJ, Chandrasekharan S, Gupta R, Singh H. Qubit Regularization of Asymptotic Freedom. PHYSICAL REVIEW LETTERS 2021; 126:172001. [PMID: 33988408 DOI: 10.1103/physrevlett.126.172001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Accepted: 03/22/2021] [Indexed: 06/12/2023]
Abstract
We provide strong evidence that the asymptotically free (1+1)-dimensional nonlinear O(3) sigma model can be regularized using a quantum lattice Hamiltonian, referred to as the "Heisenberg comb," that acts on a Hilbert space with only two qubits per spatial lattice site. The Heisenberg comb consists of a spin-half antiferromagnetic Heisenberg chain coupled antiferromagnetically to a second local spin-half particle at every lattice site. Using a world-line Monte Carlo method, we show that the model reproduces the universal step-scaling function of the traditional model up to correlation lengths of 200000 in lattice units and argue how the continuum limit could emerge. We provide a quantum circuit description of the time evolution of the model and argue that near-term quantum computers may suffice to demonstrate asymptotic freedom.
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Affiliation(s)
| | - Alexander J Buser
- Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91106, USA
| | | | - Rajan Gupta
- Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Hersh Singh
- Department of Physics, Box 90305, Duke University, Durham, North Carolina 27708, USA
- Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA
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7
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Sulejmanpasic T, Göschl D, Gattringer C. First-Principles Simulations of 1+1D Quantum Field Theories at θ=π and Spin Chains. PHYSICAL REVIEW LETTERS 2020; 125:201602. [PMID: 33258633 DOI: 10.1103/physrevlett.125.201602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2020] [Accepted: 10/13/2020] [Indexed: 06/12/2023]
Abstract
We present a lattice study of a 2-flavor U(1) gauge-Higgs model quantum field theory with a topological term at θ=π. Such studies are prohibitively costly in the standard lattice formulation due to the sign problem. Using a novel discretization of the model, along with an exact lattice dualization, we overcome the sign problem and reliably simulate such systems. Our work provides the first ab initio demonstration that the model is in the spin-chain universality class, and demonstrates the power of the new approach to U(1) gauge theories.
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Affiliation(s)
- Tin Sulejmanpasic
- Department of Mathematical Sciences, Durham University, DH1 3LE Durham, United Kingdom
| | - Daniel Göschl
- University of Graz, Institute for Physics, A-8010 Graz, Austria
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Yang W, Nocera A, Tummuru T, Kee HY, Affleck I. Phase Diagram of the Spin-1/2 Kitaev-Gamma Chain and Emergent SU(2) Symmetry. PHYSICAL REVIEW LETTERS 2020; 124:147205. [PMID: 32338963 DOI: 10.1103/physrevlett.124.147205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2019] [Accepted: 03/18/2020] [Indexed: 06/11/2023]
Abstract
We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.
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Affiliation(s)
- Wang Yang
- Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - Alberto Nocera
- Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - Tarun Tummuru
- Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - Hae-Young Kee
- Department of Physics, University of Toronto, Ontario M5S 1A7, Canada
| | - Ian Affleck
- Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
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9
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Rutkevich SB. Scaling in the massive antiferromagnetic XXZ spin-1/2 chain near the isotropic point. Phys Rev E 2020; 101:032115. [PMID: 32289968 DOI: 10.1103/physreve.101.032115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2019] [Accepted: 02/19/2020] [Indexed: 06/11/2023]
Abstract
The scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is studied in the gapped antiferromagnetic phase. For a spin-chain ring having N_{x} sites, the universal Casimir scaling function, which characterizes the leading finite-size correction term in the large-N_{x} expansion of the ground-state energy, is calculated by numerical solution of the nonlinear integral equation of the convolution type. It is shown that the same scaling function describes the temperature dependence of the free energy of the infinite XXZ chain at low enough temperatures in the gapped scaling regime.
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10
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Gozel S, Mila F, Affleck I. Asymptotic Freedom and Large Spin Antiferromagnetic Chains. PHYSICAL REVIEW LETTERS 2019; 123:037202. [PMID: 31386482 DOI: 10.1103/physrevlett.123.037202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2018] [Revised: 03/04/2019] [Indexed: 06/10/2023]
Abstract
Building on the mapping of large-S spin chains onto the O(3) nonlinear σ model with coupling constant 2/S, and on general properties of that model (asymptotic freedom, implying that perturbation theory is valid at high energy, and Elitzur's conjecture that rotationally invariant quantities are infrared finite in perturbation theory), we use the Holstein-Primakoff representation to derive analytic expressions for the equal-time and dynamical spin-spin correlations valid at distances smaller than S^{-1}exp(πS) or at energies larger than JS^{2}exp(-πS), where J is the Heisenberg exchange coupling. This is supported by comparing the static correlations with quantum Monte Carlo simulations for S=5/2.
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Affiliation(s)
- Samuel Gozel
- Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Frédéric Mila
- Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Ian Affleck
- Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada, V6T1Z1
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11
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Abstract
The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.
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12
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James AJA, Konik RM, Lecheminant P, Robinson NJ, Tsvelik AM. Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:046002. [PMID: 29480168 DOI: 10.1088/1361-6633/aa91ea] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We review two important non-perturbative approaches for extracting the physics of low-dimensional strongly correlated quantum systems. Firstly, we start by providing a comprehensive review of non-Abelian bosonization. This includes an introduction to the basic elements of conformal field theory as applied to systems with a current algebra, and we orient the reader by presenting a number of applications of non-Abelian bosonization to models with large symmetries. We then tie this technique into recent advances in the ability of cold atomic systems to realize complex symmetries. Secondly, we discuss truncated spectrum methods for the numerical study of systems in one and two dimensions. For one-dimensional systems we provide the reader with considerable insight into the methodology by reviewing canonical applications of the technique to the Ising model (and its variants) and the sine-Gordon model. Following this we review recent work on the development of renormalization groups, both numerical and analytical, that alleviate the effects of truncating the spectrum. Using these technologies, we consider a number of applications to one-dimensional systems: properties of carbon nanotubes, quenches in the Lieb-Liniger model, 1 + 1D quantum chromodynamics, as well as Landau-Ginzburg theories. In the final part we move our attention to consider truncated spectrum methods applied to two-dimensional systems. This involves combining truncated spectrum methods with matrix product state algorithms. We describe applications of this method to two-dimensional systems of free fermions and the quantum Ising model, including their non-equilibrium dynamics.
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Affiliation(s)
- Andrew J A James
- London Centre for Nanotechnology, University College London, Gordon Street, London WC1H 0AH, United Kingdom
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13
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Dey D, Maiti D, Kumar M. An efficient density matrix renormalization group algorithm for chains with periodic boundary condition. PAPERS IN PHYSICS 2016. [DOI: 10.4279/pip.080006] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022] Open
Abstract
The Density Matrix Renormalization Group (DMRG) is a state-of-the-art numerical technique for a one dimensional quantum many-body system; but calculating accurate results for a system with Periodic Boundary Condition (PBC) from the conventional DMRG has been a challenging job from the inception of DMRG. The recent development of the Matrix Product State (MPS) algorithm gives a new approach to find accurate results for the one dimensional PBC system. The most efficient implementation of the MPS algorithm can scale as O($p \times m^3$), where $p$ can vary from 4 to $m^2$. In this paper, we propose a new DMRG algorithm, which is very similar to the conventional DMRG and gives comparable accuracy to that of MPS. The computation effort of the new algorithm goes as O($m^3$) and the conventional DMRG code can be easily modified for the new algorithm.Received: 2 August 2016, Accepted: 12 October 2016; Edited by: K. Hallberg; DOI: http://dx.doi.org/10.4279/PIP.080006Cite as: D Dey, D Maiti, M Kumar, Papers in Physics 8, 080006 (2016)This paper, by D Dey, D Maiti, M Kumar, is licensed under the Creative Commons Attribution License 3.0.
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14
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Cöster K, Schmidt KP. Extracting critical exponents for sequences of numerical data via series extrapolation techniques. Phys Rev E 2016; 94:022101. [PMID: 27627240 DOI: 10.1103/physreve.94.022101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2016] [Indexed: 11/07/2022]
Abstract
We describe a generic scheme to extract critical exponents of quantum lattice models from sequences of numerical data, which is, for example, relevant for nonperturbative linked-cluster expansions or nonperturbative variants of continuous unitary transformations. The fundamental idea behind our approach is a reformulation of the numerical data sequences as a series expansion in a pseudoparameter. This allows us to utilize standard series expansion extrapolation techniques to extract critical properties such as critical points and critical exponents. The approach is illustrated for the deconfinement transition of the antiferromagnetic spin-1/2 Heisenberg chain.
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Affiliation(s)
- Kris Cöster
- Lehrstuhl für Theoretische Physik I, Otto-Hahn-Str. 4, TU Dortmund, D-44221 Dortmund, Germany
| | - Kai Phillip Schmidt
- Lehrstuhl für Theoretische Physik I, Staudtstraße 7, FAU Erlangen-Nürnberg, D-91058 Erlangen, Germany
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15
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Soos ZG, Parvej A, Kumar M. Numerical study of incommensurate and decoupled phases of spin-1/2 chains with isotropic exchange J1, J2 between first and second neighbors. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:175603. [PMID: 27028489 DOI: 10.1088/0953-8984/28/17/175603] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ -4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [-4, - 1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2. The decoupled C phase in [-1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, E m and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.
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Affiliation(s)
- Zoltán G Soos
- Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
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16
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Suwa H, Todo S. Generalized Moment Method for Gap Estimation and Quantum Monte Carlo Level Spectroscopy. PHYSICAL REVIEW LETTERS 2015; 115:080601. [PMID: 26340171 DOI: 10.1103/physrevlett.115.080601] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2014] [Indexed: 06/05/2023]
Abstract
We formulate a convergent sequence for the energy gap estimation in the worldline quantum Monte Carlo method. The ambiguity left in the conventional gap calculation for quantum systems is eliminated. Our estimation will be unbiased in the low-temperature limit, and also the error bar is reliably estimated. The level spectroscopy from quantum Monte Carlo data is developed as an application of the unbiased gap estimation. From the spectral analysis, we precisely determine the Kosterlitz-Thouless quantum phase-transition point of the spin-Peierls model. It is established that the quantum phonon with a finite frequency is essential to the critical theory governed by the antiadiabatic limit, i.e., the k=1 SU(2) Wess-Zumino-Witten model.
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Affiliation(s)
- Hidemaro Suwa
- Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
- Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
| | - Synge Todo
- Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
- Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
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17
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Kumar M, Parvej A, Soos ZG. Level crossing, spin structure factor and quantum phases of the frustrated spin-1/2 chain with first and second neighbor exchange. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:316001. [PMID: 26189667 DOI: 10.1088/0953-8984/27/31/316001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The spin-1/2 chain with isotropic Heisenberg exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(q(m)) phases with periodicity 2π/q(m) but remains finite in gapped phases. S(q(m)) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(q(m)) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between -1.2 < J1/|J2| < 0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
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Affiliation(s)
- Manoranjan Kumar
- S.N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Kolkata 700098, India
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18
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Essler FHL, Läuchli AM, Calabrese P. Shell-filling effect in the entanglement entropies of spinful fermions. PHYSICAL REVIEW LETTERS 2013; 110:115701. [PMID: 25166555 DOI: 10.1103/physrevlett.110.115701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2012] [Revised: 02/04/2013] [Indexed: 06/03/2023]
Abstract
We consider the von Neumann and Rényi entropies of the one-dimensional quarter-filled Hubbard model. We observe that for periodic boundary conditions the entropies exhibit an unexpected dependence on system size: for L=4 mod 8 the results are in agreement with expectations based on conformal field theory, while for L=0 mod 8 additional contributions arise. We explain this observation in terms of a shell-filling effect and develop a conformal field theory approach to calculate the extra term in the entropies. Similar shell-filling effects in entanglement entropies are expected to be present in higher dimensions and for other multicomponent systems.
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Affiliation(s)
- Fabian H L Essler
- The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom
| | - Andreas M Läuchli
- Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria
| | - Pasquale Calabrese
- Dipartimento di Fisica dell'Università di Pisa and INFN, 56127 Pisa, Italy
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19
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de Forcrand P, Pepe M, Wiese UJ. Walking near a conformal fixed point: The 2-dO(3)model atθ≈πas a test case. Int J Clin Exp Med 2012. [DOI: 10.1103/physrevd.86.075006] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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20
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Lubasch M, Murg V, Schneider U, Cirac JI, Bañuls MC. Adiabatic preparation of a Heisenberg antiferromagnet using an optical superlattice. PHYSICAL REVIEW LETTERS 2011; 107:165301. [PMID: 22107400 DOI: 10.1103/physrevlett.107.165301] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2011] [Revised: 08/02/2011] [Indexed: 05/31/2023]
Abstract
We analyze the possibility to prepare a Heisenberg antiferromagnet with cold fermions in optical lattices, starting from a band insulator and adiabatically changing the lattice potential. The numerical simulation of the dynamics in 1D allows us to identify the conditions for success, and to study the influence that the presence of holes in the initial state may have on the protocol. We also extend our results to two-dimensional systems.
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Becca F, Capriotti L, Parola A, Sorella S. Variational Wave Functions for Frustrated Magnetic Models. INTRODUCTION TO FRUSTRATED MAGNETISM 2011. [DOI: 10.1007/978-3-642-10589-0_15] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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22
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Sandvik AW. Ground states of a frustrated quantum spin chain with long-range interactions. PHYSICAL REVIEW LETTERS 2010; 104:137204. [PMID: 20481910 DOI: 10.1103/physrevlett.104.137204] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2010] [Revised: 02/12/2010] [Indexed: 05/29/2023]
Abstract
The ground state of a spin-1/2 Heisenberg chain with both frustration and long-range interactions is studied using Lanczos exact diagonalization. The evolution of the well-known dimerization transition of the system with short-range frustrated interactions (the J1-J2 chain) is investigated in the presence of additional unfrustrated interactions decaying with distance as 1/r(alpha). It is shown that the continuous (infinite-order) dimerization transition develops into a first-order transition between a long-range ordered antiferromagnetic state and a state with coexisting dimerization and critical spin correlations at wave number k=pi/2. The relevance of the model to real systems is discussed.
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Affiliation(s)
- Anders W Sandvik
- Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
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Simulations of pure and doped low-dimensional spin-1/2 gapped systems. ACTA ACUST UNITED AC 2008. [DOI: 10.1007/bfb0119595] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/09/2023]
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24
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Chen S, Wang L, Gu SJ, Wang Y. Fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbor interaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061108. [PMID: 18233815 DOI: 10.1103/physreve.76.061108] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2007] [Indexed: 05/25/2023]
Abstract
In this paper, we investigate the fidelity for the Heisenberg chain with the next-nearest-neighbor interaction (or the J1-J2 model) and analyze its connections with quantum phase transition. We compute the fidelity between the ground states and find that the phase transition point of the J1-J2 model cannot be well characterized by the ground-state fidelity for finite-size systems. Instead, we introduce and calculate the fidelity between the first excited states. Our results show that the quantum transition can be well characterized by the fidelity of the first excited state even for a small-size system.
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Affiliation(s)
- Shu Chen
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
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Azcoiti V, Di Carlo G, Galante A. Critical behavior of CP(1) at theta=pi, Haldane's conjecture, and the relevant universality class. PHYSICAL REVIEW LETTERS 2007; 98:257203. [PMID: 17678052 DOI: 10.1103/physrevlett.98.257203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2006] [Indexed: 05/16/2023]
Abstract
Using an approach to analyze the theta dependence of systems with a theta term we recently proposed, the critical behavior of CP(10 at theta=pi is studied. We find a region outside the strong coupling regime where Haldane's conjecture is verified. The critical line, however, does not belong to the universality class of the Wess-Zumino-Novikov-Witten model at topological coupling k=1 since it shows continuously varying critical exponents.
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Affiliation(s)
- Vicente Azcoiti
- Departamento de Física Teórica, Universidad de Zaragoza, Cl. Pedro Cerbuna 12, E-50009 Zaragoza, Spain
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Sirker J, Laflorencie N, Fujimoto S, Eggert S, Affleck I. Chain breaks and the susceptibility of Sr2Cu1-xPdxO3+delta and other doped quasi-one-dimensional antiferromagnets. PHYSICAL REVIEW LETTERS 2007; 98:137205. [PMID: 17501237 DOI: 10.1103/physrevlett.98.137205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2006] [Indexed: 05/15/2023]
Abstract
We study the magnetic susceptibility of one-dimensional S=1/2 antiferromagnets containing nonmagnetic impurities which cut the chain into finite segments. For the susceptibility of long anisotropic Heisenberg chain segments with open boundaries we derive a parameter-free result at low temperatures using field-theory methods and the Bethe ansatz. The analytical result is verified by comparing with quantum Monte Carlo calculations. We then show that the partitioning of the chain into finite segments can explain the Curie-like contribution observed in recent experiments on Sr2Cu(1-x)PdxO(3+delta). Possible additional paramagnetic impurities seem to play only a minor role.
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Affiliation(s)
- J Sirker
- Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada
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Laflorencie N, Sørensen ES, Chang MS, Affleck I. Boundary effects in the critical scaling of entanglement entropy in 1D systems. PHYSICAL REVIEW LETTERS 2006; 96:100603. [PMID: 16605721 DOI: 10.1103/physrevlett.96.100603] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2005] [Indexed: 05/08/2023]
Abstract
We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-1/2 XXZ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.
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Affiliation(s)
- Nicolas Laflorencie
- Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1 Canada
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Affiliation(s)
- T Giamarchi
- University of Geneva, 24 Quai Ernest Ansermet, 1211 Geneva, Switzerland
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Haerter JO, Shastry BS. Kinetic antiferromagnetism in the triangular lattice. PHYSICAL REVIEW LETTERS 2005; 95:087202. [PMID: 16196895 DOI: 10.1103/physrevlett.95.087202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2005] [Indexed: 05/04/2023]
Abstract
We show that the motion of a single hole in the infinite-U Hubbard model with frustrated hopping leads to weak metallic antiferromagnetism of kinetic origin. An intimate relationship is demonstrated between the simplest versions of this problem in one and two dimensions, and two of the most subtle many body problems, namely, the Heisenberg Bethe ring in one dimension and the two-dimensional triangular lattice Heisenberg antiferromagnet.
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Affiliation(s)
- Jan O Haerter
- Physics Department, University of California, Santa Cruz, California 95064, USA
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30
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Korepin VE. Universality of entropy scaling in one dimensional gapless models. PHYSICAL REVIEW LETTERS 2004; 92:096402. [PMID: 15089496 DOI: 10.1103/physrevlett.92.096402] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2003] [Indexed: 05/10/2023]
Abstract
We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.
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Affiliation(s)
- V E Korepin
- C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3840, USA.
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Controzzi D, Mussardo G. Mass spectrum of the two-dimensional O(3) sigma model with a theta term. PHYSICAL REVIEW LETTERS 2004; 92:021601. [PMID: 14753929 DOI: 10.1103/physrevlett.92.021601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2003] [Indexed: 05/24/2023]
Abstract
Form factor perturbation theory is applied to study the spectrum of the O(3) nonlinear sigma model with the topological term in the vicinity of theta=pi. Its effective action near this value is given by the nonintegrable double sine-Gordon model. Using previous results by Affleck and the explicit expressions of the form factors of the exponential operators e(+/-isqrt[8pi]phi(x)), we show that the spectrum consists of a stable triplet of massive particles for all values of theta and a singlet state of higher mass. The singlet is a stable particle only in an interval of values of theta close to pi, whereas it becomes a resonance below a critical value theta(c).
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Affiliation(s)
- D Controzzi
- Institute for Theoretical Physics, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands
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Syljuåsen OF, Sandvik AW. Quantum Monte Carlo with directed loops. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:046701. [PMID: 12443374 DOI: 10.1103/physreve.66.046701] [Citation(s) in RCA: 59] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2002] [Revised: 05/20/2002] [Indexed: 05/24/2023]
Abstract
We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.
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Otsuka H. Ground states of the one-dimensional anisotropic extended hubbard model. PHYSICAL REVIEW LETTERS 2000; 84:5572-5575. [PMID: 10990997 DOI: 10.1103/physrevlett.84.5572] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/1999] [Indexed: 05/23/2023]
Abstract
We investigate a one-dimensional half-filled electron system with on-site and spin-dependent nearest-neighbor-site Coulomb interactions. Non-Tomonaga-Luttinger-type scatterings bring about Gaussian and hidden SU(2) Berezinskii-Kosterlitz-Thouless transitions in the charge and spin parts. We accurately determine these critical points using the level-spectroscopy method. The boundaries deviate from those given by bosonization prediction with the increase of interactions. In particular, in the easy-plain anisotropy region, we find a crossing point of transition lines in the charge and spin parts, which is a multicritical point of four phases. We also check consistencies among excitation levels.
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Affiliation(s)
- H Otsuka
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
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Klumper A, Johnston DC. Thermodynamics of the spin- 1/2 antiferromagnetic uniform heisenberg chain. PHYSICAL REVIEW LETTERS 2000; 84:4701-4704. [PMID: 10990775 DOI: 10.1103/physrevlett.84.4701] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/1999] [Indexed: 05/23/2023]
Abstract
We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.
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Affiliation(s)
- A Klumper
- Institut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strasse 77, 50937 Koln, Germany and Fachbereich Physik, Universitat Dortmund, 44221 Dortmund, Germany
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Itoi C, Mukaida H. Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3688-700. [PMID: 11970202 DOI: 10.1103/physreve.60.3688] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/1998] [Revised: 06/24/1999] [Indexed: 04/18/2023]
Abstract
We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
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Affiliation(s)
- C Itoi
- Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T1Z1.
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Sorensen ES, Eggert S, Affleck I. Integrable versus non-integrable spin chain impurity models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/23/023] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Karbach M, Mutter KH. The antiferromagnetic spin- 1/2 -XXZ model on rings with an odd number of sites. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/16/007] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Klumper A. The spectra of q-state vertex models and related antiferromagnetic quantum spin chains. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/5/023] [Citation(s) in RCA: 72] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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42
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Nomura K, Okamoto K. Critical properties of S= 1/2 antiferromagnetic XXZ chain with next-nearest-neighbour interactions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/17/012] [Citation(s) in RCA: 170] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Frahm H, Rödenbeck C. Properties of the chiral spin liquid state in generalized spin ladders. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/13/005] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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44
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Klumper A, Batchelor MT, Pearce PA. Central charges of the 6- and 19-vertex models with twisted boundary conditions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/24/13/025] [Citation(s) in RCA: 146] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Barzykin V, Affleck I. Finite-size scaling for the spin-Heisenberg antiferromagnetic chain. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/6/001] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Eggert S. Numerical evidence for multiplicative logarithmic corrections from marginal operators. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:R9612-R9615. [PMID: 9984773 DOI: 10.1103/physrevb.54.r9612] [Citation(s) in RCA: 89] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Ng TK, Qin S, Su ZB. Density-matrix renormalization-group study of S=1/2 Heisenberg spin chains: Friedel oscillations and marginal system-size effects. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:9854-9861. [PMID: 9984720 DOI: 10.1103/physrevb.54.9854] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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