Lattice QCD

Techniques

Monte-Carlo simulations

After Wick rotation, the path integral for the partition function of QCD takes the form

Z = \int \mathcal{D} U , e^{-S[U]} = \int \prod_{x, \mu} dU_\mu(x) , e^{-S[U]}

where the gauge links U_\mu(x) \in \mathrm{SU}(3) range over all the sites x and space-time directions \mu in a 4-dimensional space-time lattice, S[U] denotes the (Euclidean) action and dU_\mu(x) denotes the Haar measure on \mathrm{SU}(3). Physical information is obtained by computing observables

\left\langle \mathcal{O} \right\rangle = \frac{1}{Z} \int \mathcal{D} U , \mathcal{O}(U) e^{-S[U]}

For cases where evaluating observables pertubatively is difficult or impossible, a Monte Carlo approach can be used, computing an observable \mathcal{O} as

\left\langle \mathcal{O} \right\rangle \approx \frac{1}{N} \sum_{i=1}^{N} \mathcal{O}(U_i)

where U_1, \dots, U_{N} are i.i.d random variables distributed according to the Boltzman distribution U_i \sim e^{-S[U_i]}/Z . For practical calculations, the samples {U_i} are typically obtained using Markov chain Monte Carlo methods, in particular Hybrid Monte Carlo, which was invented for this purpose.

Fermions on the lattice

Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use approximate physical constants which are different from their true physical values:

• The lattice discretization means approximating continuous and infinite space-time by a finite lattice spacing and size. The smaller the lattice, and the bigger the gap between nodes, the bigger the error. Limited resources commonly force the use of smaller physical lattices and larger lattice spacing than wanted, leading to larger errors than wanted.
• The quark masses are also approximated. Quark masses are larger than experimentally measured. These have been steadily approaching their physical values, and since the 2000s a few collaborations have used nearly physical values to extrapolate down to physical values.

Lattice perturbation theory

In lattice perturbation theory physical quantities (such as the scattering matrix) are expanded in powers of the lattice spacing, a. The results are used primarily to renormalize Lattice QCD Monte-Carlo calculations. In perturbative calculations both the operators of the action and the propagators are calculated on the lattice and expanded in powers of a. When renormalizing a calculation, the coefficients of the expansion need to be matched with a common continuum scheme, such as the MS-bar scheme, otherwise the results cannot be compared. The expansion has to be carried out to the same order in the continuum scheme and the lattice one.

The lattice regularization was initially introduced by Wilson as a framework for studying strongly coupled theories non-perturbatively. However, it was found to be a regularization suitable also for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant is small, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function, are necessary.

Lattice perturbation theory can also provide results for condensed matter theory. One can use the lattice to represent the real atomic crystal. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed (a UV regulator), and a quantum field theory can be formulated and solved on the physical lattice.

Quantum computing

The U(1), SU(2), and SU(3) lattice gauge theories can be reformulated into a form that can be simulated using "spin qubit manipulations" on a universal quantum computer.

Limitations

The method suffers from a few limitations:

• Currently there is no formulation of lattice QCD that allows us to simulate the real-time dynamics of a quark-gluon system such as quark–gluon plasma.
• It is computationally intensive, with the bottleneck not being flops but the bandwidth of memory access.
• Computations of observables at nonzero baryon density suffer from a sign problem, preventing direct computations of thermodynamic quantities.

References

References

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2. (2004). "High-Precision Lattice QCD Confronts Experiment". [[Physical Review Letters]].
3. "Lattice QCD – MITQCD Collaboration".
4. A. Bazavov. (2010). "Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks". Reviews of Modern Physics.
5. (2008). "Ab Initio Determination of Light Hadron Masses". Science.
6. P. Petreczky. (2012). "Lattice QCD at non-zero temperature". J. Phys. G.
7. (September 2015). "Melting hadrons, boiling quarks". The European Physical Journal A.
8. (2016). "2016 International Conference on High Performance Computing & Simulation (HPCS)".
9. (1987). "Hybrid Monte Carlo". Physics Letters B.
10. (17 February 2006). "Simulating lattice gauge theories on a quantum computer". Physical Review A.
11. Philipsen, O.. (2008). "Lattice calculations at non-zero chemical potential: The QCD phase diagram". Proceedings of Science.