Kogut-Susskind Lattice Research Articles

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories.

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Canonical transformations and loop formulation of SU(N) lattice gauge theories

We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space ${\cal H}^p$. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators over the entire lattice are worked out in a self consistent manner. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the $(1/g^2)$ magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the $(g^2)$ electric field terms describe all their interactions. In the weak coupling ($g^2 \rightarrow 0$) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group $SO(4,2)$ in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.

Tensor Networks for Lattice Gauge Theories with Continuous Groups

We discuss how to formulate lattice gauge theories in the Tensor Network language. In this way we obtain both a consistent truncation scheme of the Kogut-Susskind lattice gauge theories and a Tensor Network variational ansatz for gauge invariant states that can be used in actual numerical computation. Our construction is also applied to the simplest realization of the quantum link models/gauge magnets and provides a clear way to understand their microscopic relation with Kogut-Susskind lattice gauge theories. We also introduce a new set of gauge invariant operators that modify continuously Rokshar-Kivelson wave functions and can be used to extend the phase diagram of known models. As an example we characterize the transition between the deconfined phase of the $Z_2$ lattice gauge theory and the Rokshar-Kivelson point of the U(1) gauge magnet in 2D in terms of entanglement entropy. The topological entropy serves as an order parameter for the transition but not the Schmidt gap.

Towards a nonperturbative path integral in gauge theories

We propose a modification of the Faddeev–Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato–Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut–Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang–Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral.

Chern-Simons number diffusion with hard thermal loops

We construct an extension of the standard Kogut-Susskind lattice model for classical 3+1 dimensional Yang-Mills theory, in which ``classical particle'' degrees of freedom are added. We argue that this will correctly reproduce the ``hard thermal loop'' effects of hard degrees of freedom, while giving a local implementation which is numerically tractable. We prove that the extended system is Hamiltonian and has the same thermodynamics as dimensionally reduced hot Yang-Mills theory put on a lattice. We present a numerical update algorithm and study the abelian theory to verify that the classical gauge theory self-energy is correctly modified. Then we use the extended system to study the diffusion constant for Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to hard thermal loop strength. Our numbers correspond to a diffusion constant of Gamma = 29 +- 6 alpha_w^5 T^4 for m_D^2 = 11 g^2 T^2/6.

Negative-energy spinors and the Fock space of lattice fermions at finite chemical potential

Recently it was suggested that the problem of species doubling with Kogut-Susskind lattice fermions entails, at finite chemical potential, a confusion of particles with antiparticles. What happens instead is that the familiar correspondence of positive-energy spinors to particles, and of negative-energy spinors to antiparticles, ceases to hold for the Kogut-Susskind time derivative. To show this we highlight the role of the spinorial ``energy'' in the Osterwalder-Schrader reconstruction of the Fock space of non-interacting lattice fermions at zero temperature and nonzero chemical potential. We consider Kogut-Susskind fermions and, for comparison, fermions with an asymmetric one-step time derivative.

SU(2) Lattice Gauge Theory Formulation of the (2+1)-Dimensional Antiferromagnetic Heisenberg Model11The project supported in part by National Natural Science Foundation of China

The equivalence of 2+1 antiferromagnetic Heisenberg model and the SU(2) Kogut Susskind lattice gauge theory is recapitulated and the naive Euclidean lattice action of the three-dimensional antiferromagnetic Heisenberg model is derived. The three-dimensional lattice gauge fermion theory is formulated to give the consistent lattice gauge theory of antiferromagnetic Heisenberg model. In continuum limit the two copies of two flavor fermions are resulted, which give the negative results of the microscopic derivation of the Chern-Simons terms. The Chern-Simons terms, the gauge invariant problem of effective action and the “chirality” are discussed.

A lattice approach to spinorial quantum gravity

The authors present a new lattice regularisation of quantum general relativity based on Ashtekar's reformulation (1986) of Hamiltonian general relativity. In this form, quantum states of the gravitational field are represented within the physical Hilbert space of a Kogut-Susskind lattice gauge theory. The gauge field of the theory is a complexified SU(2) connection which is the gravitational connection for left-handed spinor fields. The physical states of the gravitational field are those which are annihilated by additional constraints which correspond to the four constraints of general relativity. The authors construct lattice versions of these constraints. Those corresponding to the three-dimensional diffeomorphism generators move states associated with Wilson loops around on the lattice. The lattice Hamiltonian constraint has a simple form, and a correspondingly simple interpretation: it is an operator which cuts and joins Wilson loops at points of intersection.

Calculation of the non-abelian chiral anomaly on the lattice

We check the validity of Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action by calculating perturbatively the non-abelian chiral anomaly on the lattice. We show that the result does not depend on the detailed expression of the coupling term of the lattice fermions to the chiral gauge field, taken as an external field. In two dimensions we recover the consistent expression of the anomaly, up to counterterms which we have computed. In four dimensions, we obtain the correct normalization factors of the anomaly, leaving the problem of the counterterms open.

Anomalies for compact symmetries and the non-abelian anomaly on the lattice

We show that anomalies for compact symmetry groups can always be removed by local counterterms. We use this result in order to complete the proof that Wilson and Kogut-Susskind lattice fermions generate the correct non-abelian chiral anomaly in four dimensions.

Non-abelian anomaly from lattice fermions

The non-abelian anomaly is computed in two and four dimensions, using Wilson and Kogut-Susskind lattice regularizations of the Dirac fermion action.

Finite-size effects due to Goldstone particles in lattice gauge theory

We study the finite-size effects associated with the appearance of a light, i.e. Goldstone, particle in the Kogut-Susskind lattice gauge theory. We analytically study the order parameter ( XX) and the two-meson propagator in the strong coupling limit on a finite lattice. We give their explicit interpolations between the broken phase of the infinite-volume limit and the symmetric one of the finite system. We argue that the results obtained in the strong coupling limit are more general.

Reduction of the Kogut-Susskind fermion determinant

It is shown for Euclidean Kogut-Susskind lattice fermions coupled to non-Abelian gauge fields how the associated determinant of a space-time square matrix can be reexpressed as a determinant of a space square matrix. A numerical scheme is outlined that updates the determinant exactly and thus extends the method of Blankenbecler, Scalapino, and Sugar to Euclidean-invariant gauge theories.

Absence of neutrinos on a lattice: (I). Proof by homotopy theory

It is shown, by a homotopy theory argument, that for a general class of fermion theories on a Kogut-Susskind lattice an equal number of species (types) of left- and right-handed Weyl particles (neutrinos) necessarily appears in the continuum limit. We thus present a no-go theorem for putting theories of the weak interaction on a lattice. One of the most important consequences of our no-go theorem is that is not possible, in strong interaction models, to solve the notorious species doubling problem of Dirac fermions on a lattice in a chirally invariant way.

Gauge theories on a small lattice

We present exact solutions to U(1), SU(2), and SU(3) lattice gauge theories on a Kogut-Susskind lattice consisting of a single plaquette. We demonstrate precise equivalence between the U(1) theory and the harmonic oscillator on an infinite one-dimensional lattice, and between the SU(N) theory and anN-fermion Schrodinger equation.