一维量子多体系统的精确解.pdf

The exact solution of one-dimensional quantum many-body systems Junpeng Cao Institute of Physics, Chinese Academy of Sciences junpengcao@iphy.ac.cn J. Cao (IOP) Integrable models 1 / 49 Contents 1. Introduction to the exactly solvable models 2. Bethe ansatz 3. Off-diagonal Bethe ansatz • Eigenvalues • Eigenstates • Exact physical quantities in the thermodynamic limit • t − W scheme 4. Summary J. Cao (IOP) Integrable models 2 / 49 Introduction to the exactly solvable models J. Cao (IOP) Integrable models 3 / 49 Research object: one-dimensional systems. Correlation effect: many-body effect induced by the short-range Coulomb repulsion between electrons. The correlation effects in low dimensional systems are more prominent. Many interesting phenomena such as fractional elementary excitations, various phases of quantum liquids, nonlinear effects, collective modes, critical behaviors are induced by the strong correlation. J. Cao (IOP) Integrable models 4 / 49 Methods for many-body systems: Numerical: Exact diagonalization, DMRG, Monte Carlo, Tensor network, Machine learning, DFT, MD Analytical: Mean field, Perturbation, Exact solution • Due to the strong correlation, many traditional methods such as mean field and Perturbation are invalid. • No universal quantum many body theory. Exact solution is a good method. Actual physical problems ⇒ Exactly solvable models ⇒ Quantitative results ⇑ Physical mechanism J. Cao (IOP) ⇓ ⇐ Experiments Integrable models ⇐ Universal class theory 5 / 49 • Exact solution can provide the benchmark for many new phenomena and physical concepts, and check the correction of numerical methods and numerical results. Examples: 2D Ising model (thermodynamic phase transition), 1D Hubbard model (Mott insulator), Heisenberg model (spinon, fractional charge), Hydrogen atom (quantum mechanics). • It is an important branch of condensed matter physics, statistical physics, theoretical and mathematical physics. J. Cao (IOP) Integrable models 6 / 49 Exact solvable model: Quantum spin chain 1921, Stern Grach • Spin singlet and triplet states, exchanging interactions ~σ1 · ~σ2 • Heisenberg model H=J X ~σj · ~σj+1 j=1 quantum magnetism, anisotropy, quantum phase transition, spinon, Bethe ansatz J. Cao (IOP) Integrable models 7 / 49 • Boundary conditions J. Cao (IOP) Integrable models 8 / 49 • Ising model • J1 − J2 model H= X J1 ~σj · ~σj+1 + J2 ~σj · ~σj+2 j=1 • Dzyloshinsky-Moriya interaction X H= D̂ · (~σj × ~σj+1 ) j=1 • Chiral three spins interaction X H= ~σj · (~σj+1 × ~σj+2 ) j=1 J. Cao (IOP) Integrable models 9 / 49 • Gaudin model Hi = X ~σj · ~σj j=1 i −j • Haldane-Shastry model H= X ~σj · ~σj i,j=1 J. Cao (IOP) (i − j)2 Integrable models 10 / 49 • New model: integrable quantum spin chain with competing interactions H = Hbulk + HL + HR Bulk and periodic Hbulk = 2N−1 Xn J1 ~σj · ~σj+1 +J2 ~σj · ~σj+2 + J3 (−1)j ~σj+1 · (~σj × ~σj+2 ) o j=1 Open boundary HL = 1 − 4a2 [pσ1z − a2 σ1z σ2z − iapD1z · (~σ1 × ~σ2 )] p 2 − a2 HR =  4a2 − 1 x z x z x z q(ξσ2N + σ2N ) − a2 (ξσ2N−1 + σ2N−1 )(ξσ2N + σ2N ) a2 ξ 2 + a2 − q 2  x z −iaq(ξD2N + D2N ) · (~σ2N × ~σ2N−1 ) J. Cao (IOP) Integrable models 11 / 49 • New model: integrable cold atomic models Particles with δ-function interaction: boson; fermion; mixture H=− N X ∂2 ∂xj2 j=1 +c X δ(xj − xl ) j