Introduction to Nuclear Lattice Methods
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1 Introduction to Nuclear Lattice Methods Lecture 2: Lattice Theory and Formalism, Scattering on Lattice Dean Lee (NC State) Nuclear Lattice EFT Collaboration Present Status of the Nuclear Interaction Theory Kavli Institute for Theoretical Physics China Chinese Academy of Sciences September 3,
2 Lectures Lecture 1: Overview, Computational Strategies, Algorithms Lecture 2: Lattice Theory and Formalism, Scattering on Lattice 2
3 Outline of Lecture 2 Exact equivalence of lattice formulations 1. Grassmann path integral without auxiliary field 2. Transfer matrix operator without auxiliary field 3. Grassmann path integral with auxiliary field 4. Transfer matrix operator with auxiliary field Scattering on the lattice Lüscher s finite volume method Spherical wall method Solution to homework 3
4 Exact equivalence of lattice formulations We show the exact equivalence of several different lattice formulations. We prove the equivalence following the steps in the diagram below. For simplicity we discuss the example of two-component fermions on the lattice with contact interactions D.L., Prog. Part. Nucl. Phys. 63 (2009) 117 4
5 Grassmann path integral without auxiliary fields This formulation is perhaps the most fundamental. Convenient for the simple derivation of exact conservation laws, Noether currents, and lattice Feynman rules. Let us consider anticommuting Grassmann fields for two-component fermions on a spacetime lattice The Grassmann fields are periodic in the spatial directions 5
6 and antiperiodic in the temporal direction We use the standard definition for the Grassmann integration (no sum on i) We note the equivalence of integration and differentiation with respect to a Grassmann variable 6
7 We use the following shorthand notation for the full integration measure over all Grassmann variables Define the local Grassmann spin densities and the total Grassmann density Define the lattice kinetic energy hopping coefficients 7
8 These are defined to give a quadratic kinetic energy as function of momentum 2ν is the order of lattice improvement for the kinetic energy In the actual simulations of nucleons, we typically use fourth-order improvement for the kinetic energy, but for illustrative simplicity we continue the discussion with the simplest case, 8
9 We use lattice units where everything is divided or multiplied by powers of the spatial lattice spacing to make it dimensionless. We also define the ratio of temporal to spatial lattice spacings The free nonrelativistic particle lattice action is 9
10 With a contact interaction between the two components, the lattice action is We are interested in the path integral of the exponential of the action 10
11 Transfer matrix operator without auxiliary field Consider now fermion annihilation and creation operators. For the moment we consider just one operator each For any function of the annihilation and creation operators we note that the quantum-mechanical trace of the normal-ordered product satisfies the following identity relating it to a Grassmann integral Creutz, Found. Phys. 30 (2000)
12 The straightforward proof consists of testing all four linearly independent functions of the annihilation and creation operators Let us rewrite the identity in a form that starts to resemble the lattice Grassmann path integral This identity can generalized in straightforward manner to normal-ordered products of several annihilation and creation operators. 12
13 with antiperiodic time boundary conditions We now define the free nonrelativistic lattice Hamiltonian 13
14 We also define the following density operators So now the same Grassmann path integral we had defined before can be rewritten in terms of the quantum-mechanical trace of the product of normal-ordered transfer matrices 14
15 This demonstrates the exact equivalence of the two lattice formulations for arbitrary lattice spacings. 15
16 Grassmann path integral with auxiliary fields We can rewrite the same lattice Grassmann path integral using an auxiliary field There are many ways to introduce the auxiliary-field integral measure and coupling. The simplest is a Gaussian measure and linear coupling corresponding with the original Hubbard-Stratonovich transformation 16
17 But there are many choices. The only requirements for exact equivalence are This can be used to derive several varieties of the discrete Hubbard-Stratonovich transformation as well as new transformations such as a compact continuous auxiliary field transformation. D.L., PRC 78 (2008) ; Drut, Lähde, Ten, PRL 106 (2011)
18 This demonstrates the exact equivalence of the three lattice formulations for arbitrary lattice spacings. 18
19 Transfer matrix operator with auxiliary field We use the equivalence of the Grassmann path integral and normal-ordered transfer matrix and apply it to the case of the auxiliary-field Grassmann path integral. We find 19
20 This gives the exact equivalence of the four lattice formulations for arbitrary lattice spacings. 20
21 Two-body scattering in center-of-mass frame Incoming and outgoing waves 21
22 Partial wave decomposition of the time-independent Schrödinger equation All possible integers such that Define 22
23 interacting free 23
24 outgoing wave incoming wave Partial wave decomposition of the scattering amplitude 24
25 Physical scattering data Unknown operator coefficients Lüscher s finite-volume formula Lüscher, Comm. Math. Phys. 105 (1986) 153; NPB 354 (1991) 531 Two-particle energy levels near threshold in a periodic cube related to phase shifts L L L 25
26 Physical scattering data Unknown operator coefficients Spherical wall method Borasoy, Epelbaum, Krebs, D.L., Meißner, EPJA 34 (2007) 185 Spherical wall imposed in the center of mass frame R wall 26
27 Cubic symmetry group 27
28 28
29 blue = +1 red = -1 29
30 30
31 31
32 32
33 blue = +1 red = -1 33
34 34
35 35
36 blue = +1 red = -2 36
37 37
38 38
39 39
40 blue = +1 red = -1 40
41 41
42 blue = +1 red = -1 42
43 Energy levels with hard spherical wall Energy shift from free-particle values gives the phase shift 43
44 Comparison of spherical wall and periodic cube Toy model: Spherical wall, R wall = 10a Periodic cube, L = 12a Borasoy, Epelbaum, Krebs, D.L., Meißner, EPJA 34 (2007)
45 Nucleon-nucleon phase shifts LO 3 : S waves 45
46 LO 3 : P waves 46
47 Homework problem Let Construct a Markov process with the Metropolis algorithm to compute by sampling over all terms in the matrix product If you can, write a computer code to do this Monte Carlo calculation and compare with the exact result. 47
48 Solution Step 20 Step 19 Step 18 Step 2 Step 1 Step
49 Step k+1 Step k Step k-1 Step k+1 Step k Step k-1 49
50 Step 20 Step
51 51
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