DOCSLIB.ORG

Lattice Nonlinear Schrödinger

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lattice Nonlinear Schrödinger Lattice nonlinear Schr¨odinger: history and open problems Kun Hao and Vladimir Korepin Dublin Institute for Advanced Studies January 20, 2021 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 1 / 30 Continuous model The model has many names: Lieb-Liniger, Bose gas with delta interaction, Tonks-Girardeau, nonlinear Sch¨odinger.It has application in quantum optics. It was build in optical lattice by N. J. van Druten https://arxiv.org/pdf/0709.1899.pdf The Hamiltonian is: Z y y y H = dx(@x @x + κψ ); (x); y(y) = iδ(x y) f g − It is integrable both in classical and quantum cases. It has infinitely many conservation laws and Lax representation https://en.wikipedia.org/wiki/Lax_pair. @t Ln = Mn+1(λ)Ln(λ) − Ln(λ)Mn(λ) An integer n is a discrete space variable x = n∆ and ∆ is lattice spacing. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 2 / 30 R matrix We shall discuss the quantum case. Lax representation follows from Yang-Baxter equation. https://arxiv.org/pdf/0910.0295.pdf O O R(λ, µ) Ln(λ) Ln(µ) = Ln(µ) Ln(λ) R(λ, µ) The R(λ, µ) solves the Yang-Baxter equation : C.N. Yang Phys. Rev. Lett. 19, (1967) 1312-1314 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1312 Yang found the following solution: R(λ, µ) = i1 + (µ − λ)Π This is the same R matrix, which describes spin 1=2 XXX Heisenberg chain. Here Π is permutation. C.N. Yang Phys. Rev. 168, (1968) 1920 https://journals.aps.org/pr/abstract/10.1103/PhysRev.168.1920 Note Note that for λ − µ = i the R(λ, µ) = i (1 − Π) is degenerate. It is 1D projector. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 3 / 30 QISM formulation The model was embedded into quantum inverse scattering method https://en.wikipedia.org/wiki/Quantum_inverse_scattering_method. We shall use algebraic Bethe Ansatz https://arxiv.org/abs/hep-th/9605187 The orthogonality and completeness of the Bethe Ansatz eigenstates was proved by Teunis C. Dorlas in 1993. An approximate L operator on a dense lattice was discovered by Faddeev , Sklyanin and Takhtajan in 1979 iλ∆ y 1 2 ipκχn 2 Ln(λ) = − − iλ∆ + O(∆ ); ipκχn 1 + 2 An integer n is a discrete space variable x = n∆ and ∆ is lattice spacing. The χn is the quantum field: y [χn; χm] = ∆ δnm χ = ∆ in the limit ∆ 0. The λ is the spectral parameter! and κ is a coupling constant. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 4 / 30 Lattice nonlinear Schr¨odingerequation Exact lattice Lax operator (all orders in ∆) was constructed by A.G. Izergin and V. E. Korepin in Doklady Akademii Nauk, 1981 https://arxiv.org/pdf/0910.0295.pdf see also Nuclear Physics B 205 [FS5], 401, 1982 iλ∆ κ y y ! 1 2 + 2 χj χj ipκχj %j Lj (λ) = − −iλ∆ κ y : ipκ%j χj 1 + 2 + 2 χj χj y κ y 1 [χ ; χ ] = ∆δ and % = (1 + χ χ ) 2 ; j l j;l j 4 j j here κ > 0, and ∆ > 0. The same R matrix. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 5 / 30 Equivalence of NLS and XXX chain with negative spin We can rewrite the L operator as XXX Heisenberg chain Zeitschrift f¨urPhysik 49, (1928) 619-636 https://link.springer.com/article/10.1007/BF01328601: XXX z k k Lj = −σ Lj = iλ + Sj ⊗ σ + y − z κ y Sj = ipκχ %j ; Sj = ipκ%j χj ; Sj = (1 + χ χj ): − j 2 j representation of SU(2) algebra with negative spin s = 2 . − κ∆ The σ are Pauli matrices. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 6 / 30 Antipode and quantum determinant The monodromy matrix and transfer matrix: A(λ) B(λ) T0(λ) L0L(λ) ::: L01(λ) = ; τ(λ) = tr0 T0(λ): ≡ C(λ) D(λ) N N R(λ, µ)(T0(λ) T0(µ)) = (T0(µ) T0(λ)) R(λ, µ) The antipode of quantum monodromy matrix is: −1 −L y t y T (λ) = dq (λ)σ T (λ + i)σ 2 dq(λ) = ∆ (λ ν)(λ ν + i)=4 ν = 2i=∆ − − − This is similar to Cramer's formula http://pi.math.cornell.edu/~andreim/Lec17.pdf The difference is a shift of the spectral parameter by i. The denominator is the quantum determinant: L detqT (λ) = A(λ)D(λ + i) B(λ)C(λ + i) = dq (λ) This was discovered in 1981− in https://arxiv.org/pdf/0910.0295.pdf Remark When λ − µ = i the R matrix turns into one dimensional projector. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 7 / 30 XXX chain with spin s = 1 − PL The Hamiltonian is: = k=1 Hk;k+1 The density of the HamiltonianH is Hk;k+1 = ( Jk;k+1) (Jk;k+1 + 1) + (2): − − − Here (x) = d ln Γ(x)=dx and ~ ~ Jk;k+1(Jk;k+1 + 1) = 2Sk Sk+1 ⊗ Can be solved by Vieta's formulas. QCD describes deep inelastic scattering. The scattering of a lepton on a baryon is a sum of Feynman diagrams. In leading logarithmic approximation ladder diagrams dominate. Quarks exchange gluons. The Hamiltonian describing interactions of the gluons is this spin chain. L. Lipatov https://arxiv.org/pdf/hep-th/9311037.pdf L. Alvarez-Gaume, https://arxiv.org/pdf/0804.1464.pdf Our paper: https://arxiv.org/pdf/1909.00800.pdf This is a part of larger trend: spin chains are related to 4D Yang-Mills N. A. Nekrasov, S. L. Shatashvili https://arxiv.org/pdf/0908.4052.pdf V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 8 / 30 Bethe equations An example is XXX chain with spin s = 1. The Bethe equations are https://en.wikipedia.org/wiki/Bethe_ansatz− : L N L N λk + is Y λk λj + i s=−1 λk i Y λk λj + i = − − = − λ is λ λ i λ + i λ λ i k j=1 k j −−−! k j=1 k j − j6=k − − j6=k − − k = 1; ; N ··· These are periodic boundary conditions. The energy is: N N X 1 d λj + i X 2 E ln = 2− ; ≡ i dλj λj i λ + 1 j=1 − j=1 j V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 9 / 30 Analysis of Bethe equations Theorem 1. All solutions of Bethe equations for s = −1 are real numbers. L N λk − i Y λk − λj + i = ; k = 1; ··· ; N: λk + i λk − λj − i j=1; j6=k Proof: Let us use the following properties: λ − i λ − i LHS: ≤ 1; when Imλ ≥ 0; ≥ 1; when Imλ ≤ 0; λ + i λ + i λ + i λ + i RHS: ≥ 1; when Imλ ≥ 0; ≤ 1; when Imλ ≤ 0; λ − i λ − i If we denote the one with maximal imaginary part as λmax 2 fλj g, then Im λmax ≥ Im λj ; j = 1; ··· ; N: For λk = λmax L N λmax − i Y λmax − λj + i = ≥ 1: λmax + i λmax − λj − i j=1 Due to LHS, this results in: Im λj ≤ Im λmax ≤ 0 Similarly, one has 0 ≤ Im λmin ≤ Im λj ! so Im λj = 0 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 10 / 30 Solution exists and unique The logarithm of Bethe equations for the model s = 1, − N X 2πnk = θ(λk λj ) + L θ (λk ) ; − j=1 Here nk are different integer (or half integer) numbers: Pauli principle http://insti.physics.sunysb.edu/~korepin/PDF_files/Pauli.pdf iκ + λ θ(λ) = θ( λ) = i ln ; π < θ(λ) < π; Im λ = 0 − − iκ λ − − 2κ θ0(λ µ) = K(λ, µ) = ; K(λ) = K(λ, 0): − κ2 + (λ µ)2 − V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 11 / 30 Yang's action is convex Theorem 2. The solutions of the logarithmic form Bethe equations exist. Logarithmic Bethe equations are the extremums of the Yang action: N N N X 1 X X S = L θ (λ ) + θ (λ − λ ) − 2π n λ ; 1 k 2 1 k j k k k=1 k;j k=1 R λ θ1(λ) = 0 θ(µ)dµ. Bethe equations: @S=@λj = 0. N @2S X = δjl [LK(λj ) + K(λj ; λm)] − K(λj ; λl ) @λj @λl m=1 Consider some real vector vj . The quadratic form is positive: 2 N N X @ S X 2 X 2 vj vl = LK(λj )vj + K(λj ; λl )(vj − vl ) ≥ 0 @λj @λl j;l j=1 j>l The K(λj ) are positive. The action is convex: it has unique minimum. Solution of Bethe equation exists and unique. V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 12 / 30 Side remark The same second derivative appears later in the theory. The square of the norm of the Bethe wave function is a determinant: @2S ΦN ΦN = det N h j i @λj @λl M. Gaudin conjectured in 1972. V.E. Korepin proved in 1982 http://insti.physics.sunysb.edu/~korepin/PDF_files/norm.PDF N @2S X = δjl [LK(λj ) + K(λj ; λm)] K(λj ; λl ) @λ @λ − j l m=1 V. E. Korepin (YITP) Lattice nonlinear Schr¨odingerequation January 20, 2021 13 / 30 Open problem: Is to analyze the integral equation in the limit of κ 0 The K(λ, µ) 2πδ(λ µ): the integral cancel the! LHS ...? ! − The thermodynamic limit at zero temperature For positive κ all λj has to be different: Pauli principle in the momentum space is valid.
Load more

Recommended publications
  • (Oxon) on the THEORY of MODERN QUANTUM ALGORITHMS
    SKOLKOVO INSTITUTE OF SCIENCE AND TECHNOLOGY draft version Jacob Daniel Biamonte, BS, DPhil (Oxon) ON THE THEORY OF MODERN QUANTUM ALGORITHMS Specialization 01.01.03 Mathematical Physics Dissertation submitted for the degree of Doctor of Physical and Mathematical Sciences arXiv:2009.10088v1 [quant-ph] 21 Sep 2020 Moscow | 2020 2 Contents Page Chapter 1. Introduction: Programming Ground States .......5 1.1 Computation and the Ising model . .9 1.2 Low-energy subspace embedding . 15 1.3 Two-body reductions . 19 1.4 P- vs. NP problems and physics . 23 1.5 Computational phase transitions . 24 1.5.1 Thermal states at the SAT phase transition . 29 Chapter 2. Quantum vs Probabilistic Computation ......... 32 2.1 Defining Mechanics . 32 2.1.1 Stochastic time evolution . 33 2.1.2 Quantum time evolution . 33 2.1.3 Properties in stochastic mechanics . 34 2.1.4 Hamiltonian properties in quantum mechanics . 35 2.1.5 Observables . 36 2.2 Walks on Graphs: quantum vs stochastic . 44 2.2.1 Normalized Laplacians . 47 2.2.2 Stochastic walk . 50 2.2.3 Quantum walk . 51 2.2.4 Perron's theorem . 54 2.3 Google page rank|a ground eigenvector problem . 59 2.4 Kitaev's quantum phase estimation algorithm . 61 2.5 Finding the ground state . 66 Chapter 3. Tensor Networks and Quantum Circuits ......... 69 3.1 Clifford gates . 69 3.2 Tensor network building blocks . 72 3.2.1 Reversible logic . 75 3.2.2 Heisenberg picture . 77 3.2.3 Stabilizer tensor theory . 79 3 Chapter 4. Variational Search and Optimization ..........
    [Show full text]
  • Algebraic Bethe Ansatz and Tensor Networks
    Algebraic Bethe Ansatz and Tensor Networks A Dissertation presented by You Quan Chong to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University May 2015 Stony Brook University The Graduate School You Quan Chong We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Vladimir E. Korepin - Dissertation Advisor Professor, C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Tzu-Chieh Wei - Chairperson of Defense Assistant Professor, C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Dominik A. Schneble - Committee Member Associate Professor, Department of Physics and Astronomy Scott Sutherland - Outside Member Associate Professor, Department of Mathematics This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation Algebraic Bethe Ansatz and Tensor Networks by You Quan Chong Doctor of Philosophy in Physics Stony Brook University 2015 We consider several exactly solvable models of strongly correlated elec- trons in one dimension, such as the Heisenberg XXX model, the supersym- metric t-J model and the Hubbard model. These models can be solved by using the method of graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a Matrix Product State. This overcomes a major shortcoming of current density matrix renormal- ization group (DMRG) methods which work well on the ground states, but have difficulty working with the excited states of such models. In addition, observables such as correlation functions are important as they are experimentally measurable, but have been analytically described in the double scaling limit only.
    [Show full text]
  • Entanglement and Quantum Spin Chains
    ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems ENTANGLEMENT AND QUANTUM SPIN CHAINS V. Korepin YITP of Stony Brook University New exactly solvable spin chain came out of quantum information V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Outline 1 ABitofHistoryofQuantumEntanglement 2 Quantum Information and Dynamical Systems V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The dawn Aug 14, 1935 Shroedinger submitted his paper to Proceedings of Cambridge Philosophical Society. 1964 John Bell: entanglement distinguishes quantum mechanics from classical at fixed value of Plank constant. 1981 Richard Feynman: quantum computation. 1996 Bennett, Bernstein, Popescu and Schumacher introduced entropy of subsystem as a measure of entanglement discovery of entanglement entropy V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Application to physics: area Law 1986 L. Bombelli, R. Koul, J. Lee, R. Sorkin; 1993 M. Srednicki 1994 Holzhey, Larsen and Wilczek: ent entr of block of spins of c size x for 1D gapless models scales as S = 3 log x . 2003 Logarithmic scaling of Renyi entropy. B.-Q.Jin, Korepin, Journal Statistical Physics; arXiv.org April 15 of 2003. ln tr(ρα) −1 S (α)= 1+α ln x R 1−α → 6 c v πTx 2004 For finite temperature S = 3 log πT sinh v Korepin, PRL, vol 92, ei 096402, 05 March 2004,! ! "" arXiv:cond-mat/0311056 V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Environment and Error Correcting Codes Q-bits should be entangled with other qubits in quantum computer, but not entangled with the environment.
    [Show full text]
  • Entanglement Entropy & Spectrum • Holographic Spin Chain (VBS/CFT Correspondence) 3
    Entanglement spectra of two- dimensional solvable models Hosho Katsura (Gakushuin University) Collarborators: Anatol Kirillov (RIMS, Kyoto Univ.) Vladimir Korepin (YITP, Stony Brook) Naoki Kawashima (ISSP, Tokyo Univ.) Lou Jie (ISSP Fudan Univ.) Shu Tanaka (Tokyo Univ.) Ryo Tamura (NIMS) H.K., et al., J. Phys. A 43, 255303 (2010). J. Lou, S. Tanaka, H.K. & N. Kawashima, PRB 84, 245128 (2011). S. Tanaka, R. Tamura, & H.K., PRA 86, 032326 (2012). Outline 1. Introduction • AKLT model & valence-bond-solid (VBS) state • Schmidt decomposition & quantum entanglement 2. Entanglement spectra of 2d AKLT models • Gram matrix & Reflection symmetry • Entanglement entropy & spectrum • Holographic spin chain (VBS/CFT correspondence) 3. Entanglement spectra of quantum hard-square models • Tensor network states for interacting Rydberg atom systems • Entanglement spectra • Holographic minimal models (c<1) A brief history of AKLT and valence-bond-solid (VBS) state • Haldane gap problem Haldane’s conjecture (F. D. M. Haldane, Phys. Rev. Lett. 50 (‘83).) S=integer antiferromagnetic (AFM) Heisenberg chains are gapped. Experiment in S=1 spin chain: Ni(C2H8N2)2NO2(ClO4) (NENP) etc. • Affleck-Kennedy-Lieb-Tasaki (AKLT) model (PRL 59 (‘87), CMP 115 (‘87).) 1. Exact unique ground state valence-bond-solid (VBS) state 2. Rigorous proof of the ‘Haldane’ gap in this model 3. AFM correlation decays exponentially with distance. • Valence-bond-solid (VBS) state : spin singlet : symmetrization VBS state Scwinger boson(SB) rep. of spin operator Arovas, Auerbach a b & Haldane, PRL 60 (‘88). Constraint : S=1 state is spanned by . Matrix product state (MPS) Fannes et al.,(‘89), Kluemper et al.,(‘91). VBS on arbitrary graphs Square lattice Hexagonal lattice • Projected pair entangled state (PEPS), Tensor network state (TNS) Verstraete & Cirac (PRA 70 (R) (‘04)), Gross & Eisert (PRL 98 (‘07)).
    [Show full text]
  • SCIENTIFIC REPORT for the YEAR 2002 ESI, Boltzmanngasse 9, A-1090 Wien, Austria
    The Erwin SchrÄodinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scienti¯c Report for the Year 2002 Vienna, ESI-Report 2002 March 1, 2003 Supported by Federal Ministry of Education, Science, and Culture, Austria 2 ESI{Report 2002 ERWIN SCHRODINGERÄ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2002 ESI, Boltzmanngasse 9, A-1090 Wien, Austria, March 1, 2003 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Maria Windhager, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scienti¯c Advisory board: Jean-Pierre Bourguignon (IHES), Luis A. Ca®arelli (Austin, Texas), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), Table of contents General remarks . 2 Report of the Review Panel . 3 Winter School in Geometry and Physics . 12 Arithmetic Groups and Automorphic Forms . 12 Stability Matters: A Symposium on Mathematical Physics . 12 PROGRAMS IN 2002 . 13 Developed Turbulence . 13 Arithmetic, automata, and asymptotics . 18 Quantum ¯eld theory on curved space time . 20 Aspects of foliation theory in geometry, topology and physics . 22 Noncommutative geometry and quantum ¯eld theory Feynman diagrams in mathematics and physics . 24 Mathematical population genetics and statistical physics . 25 CONTINUATION OF PROGRAMS FROM 2001 and earlier . 26 SENIOR FELLOWS and GUESTS via Director's shares .
    [Show full text]
  • Arxiv:Cond-Mat/0405155V3
    Universality of the one dimensional Bose gas with delta interaction Luigi Amico(1) and Vladimir Korepin(2) (1) MATIS-INFM & Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universit´adi Catania, viale A. Doria 6, I-95125 Catania, Italy and (2) C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, NY 11794-3840 We consider several models of interacting bosons in a one dimensional lattice. Some of them are not integrable like the Bose-Hubbard others are integrable. At low density all of these models can be described by the Bose gas with delta interaction. The lattice corrections corresponding to the different models are contrasted. I. INTRODUCTION duces quantum fluctuations that dominate the zero tem- perature phase diagram. At commensurate number of bosons per site the Bose-Hubbard model predicts a Mott The Bose gas with delta interaction is one of the insulator-superfluid quantum phase transition controlled most famous models for strongly interacting bosons [1, 2] by the ratio between the bosonic tunneling rate t and (the corresponding classical dynamics is described by the Coulomb repulsion. In optical applications the sys- the non-linear Schr¨odinger equation). In one dimension tem parameters are tuned by changing the laser intensity this model possesses infinitely many, mutually commut- [20]; the criticality of the superfluid-insulator quantum ing integrals of the motion and is therefore called inte- phase transition is substantially altered by the confining grable. The integrability of the Bose gas can be for- harmonic potential [21]. mulated within the Quantum Inverse Scattering Method (QISM) [3] that exploit a crucial property of the two- In this paper we focus on one dimension.
    [Show full text]
  • Arxiv:1806.04049V3 [Hep-Th] 20 Sep 2018 So Far
    R´enyi entropy of highly entangled spin chains Fumihiko Sugino∗ and Vladimir Korepiny ∗ Fields, Gravity & Strings Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), 55, Expo-ro, Yuseong-gu, Daejeon 34126, Republic of Korea y C.N.Yang Institute for Theoretical Physics, Stony Brook University, NY 11794, USA [email protected], [email protected] Abstract Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an \area law" meaning the entanglement entropy proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm for over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. Although importance of intensive study of the models is undoubted to reveal novel features of quantum entanglement, it is still far from their complete understanding. In this article, we first analytically compute the R´enyi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The R´enyi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the R´enyi entropy is known as a function of its parameter. We find non-analytic behavior of the R´enyi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied arXiv:1806.04049v3 [hep-th] 20 Sep 2018 so far.
    [Show full text]
  • Measures of Entanglement in Dynamical Systems Density Matrix
    Measures of Entanglement in Dynamical Systems Density matrix of block of spins in the ground state of a Hamiltonian Vladimir Korepin Yang Institute for Theoretical Physics State University of New York at Stony Brook Abstract To build quantum computer or any other quan- tum device we need to maintain quantum con- trol. Entanglement is an important recourse for quantum control. Many-body physics is used to study measures of entanglement in ground state of different dynamical systems. 1 Binary system A&B , which is in a pure state: a { } unique wave function denoted by ΨA,B . it will be a | i unique ground state of a dynamical model: interacting spins, Bose gas or Hubbard model. d ΨA,B = ΨA ΨB , d> 1 | i | j i⊗| j i jX=1 Measure of entanglement. Von Neumann entropy of the subsystem S(ρ ) = T r (ρ ln ρ ) A − A A A ρ = T r ΨA,B ΨA,B A B | ih | John Preskill, Matthew Hastings, Jens Eisert, Frank Verstr aete, John Cardy 2 Another measure is R´enyi entropy of a subsystem: 1 α Sα(ρ ) = ln T r (ρ ) A 1 α A A where α is a parameter. − spectrum of density matrix is also in- teresting, even eigenvectors are impor- tant [measurement]. If whole binary system A&B is in a mixed states [thermodynamics]. Mutual entropy: I = S + S S {A,B} A B − A∪B Another measure of entanglement of mixed states was suggested by Asher Peres. We can describe the mixed binary system by a density matrix ρA,B.
    [Show full text]
  • Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases
    Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases Matthew Fahrbach∗ and Dana Randall† School of Computer Science, Georgia Institute of Technology December 23, 2020 Abstract The six-vertex model in statistical physics is a weighted generalization of the ice model on (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains2 will be ecient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertexℤ model in the ordered phases. Specically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the rst rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and signicantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks. arXiv:1904.01495v2 [cs.DS] 22 Dec 2020 ∗Email: [email protected]. Supported in part by an NSF Graduate Research Fellowship under grant DGE-1650044. †Email: [email protected]. Supported in part by NSF grants CCF-1526900, CCF-1637031, and CCF-1733812.
    [Show full text]
  • Arxiv:1804.10133V3 [Cond-Mat.Quant-Gas] 17 Dec 2018
    Rise and fall of a bright soliton in an optical lattice Piero Naldesi1 ∗, Juan Polo Gomez1, Boris Malomed2;3, Maxim Olshanii4, Anna Minguzzi1 and Luigi Amico5;6;7;8;9 1 Universit´eGrenoble-Alpes, LPMMC, F-38000 Grenoble, France and CNRS, LPMMC, F-38000 Grenoble, France 2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel 3 Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel 4 Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA 5 Dipartimento di Fisica e Astronomia, Via S. Sofia 64, 95127 Catania, Italy 6 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 7 MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore 8 CNR-MATIS-IMM & INFN-Sezione di Catania, Via S. Sofia 64, 95127 Catania, Italy and 9 LANEF 'Chaire d'excellence', Universit`eGrenoble-Alpes & CNRS, F-38000 Grenoble, France (Dated: December 19, 2018) We study an ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value of the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system.
    [Show full text]
  • The Hubbard Dimer: a Density Functional Case Study of a Many-Body Problem
    The Hubbard Dimer: A density functional case study of a many-body problem D. J. Carrascal1;2, J. Ferrer1;2, J. C. Smith3 and K. Burke3 1 Department of Physics, Universidad de Oviedo, 33007 Oviedo, Spain 2 Nanomaterials and Nanotechnology Research Center, Oviedo, Spain and 3 Departments of Chemistry and of Physics, University of California, Irvine, CA 92697, USA (Dated: September 3, 2015) This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of Density Functional Theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the `failure' of the Kohn-Sham method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree-Fock and Bethe Ansatz Local Density Approximation) to show their successes and limitations.
    [Show full text]
  • Deforming the Fredkin Spin Chain Away from Its Frustration-Free Point
    Deforming the Fredkin spin chain away from its frustration-free point Khagendra Adhikari∗ and K. S. D. Beachy Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA The Fredkin model describes a spin-half chain segment subject to three-body, correlated-exchange interactions and twisted boundary conditions. The model is frustration-free, and its ground state wave function is known exactly. Its low-energy physics is that of a strong xy ferromagnet with gapless excitations and an unusually large dynamical exponent. We study a generalized spin chain model that includes the Fredkin model as a special tuning point and otherwise interpolates between the conventional ferromagnetic and antiferromagnetic quantum Heisenberg models. We solve for the low-lying states, using exact diagonalization and density-matrix renormalization group calculations, in order to track the properties of the system as it is tuned away from the Fredkin point; we also present exact analytical results that hold right at the Fredkin point. We identify a zero-temperature phase diagram with multiple transitions and unexpected ordered phases. The Fredkin ground state turns out to be particularly brittle, unstable to even infinitesimal antiferromagnetic frustration. We remark on the existence of an “anti-Fredkin” point at which all the contributing spin configurations have a spin structure exactly opposite to those in the Fredkin ground state. I. INTRODUCTION z Stot = 0 The construction of a spin-1 quantum spin chain with an dimerization pattern z AFM exactly solvable ground state [1] and its later generalization to I = spin-S (for all integer S > 1) by Movassagh and Shor [2] have proved to be incredibly fruitful developments.
    [Show full text]