During the six days of this valuable Peking University’s summer school, we obtain varieties of knowledge from many fields. Thanks to the generous and brilliant reports by Professor Wang Yinan, Song Huichao, Zhu Huaxing, Liu Jia, Feng Xu, Ma Boqiang, Yang Zhicheng, Huang Huaqing, Chen Bin, Shu Jing, we have the opportunity to get a glimpse of nowadays physics frontier. Next we will make a summary of their report ranging from Dark Matter, Cosmology, Condensed Matter Physics, Quantum Computation and Quantum Many-body Dynamics, QCD, perturbated QCD and Lattice QCD to String Theory.

Dark Matter and Dark energy

What is dark matter? What evidences can prove the existence of dark matter? What properties do dark matter have? What is the possible explanations for dark matter and the experimental observables? How can we detect them? What problems do nowadays research have?

The dark matter is the particle that is stable, cold(weakly-interacting), non-baryons(uncoloured), uncharged. The large scale structure, cosmic microwave background, the rotation velocity curve of galaxy prove its existence at different scale.[2]

One of the most important observables connecting dark matter, cosmology and particle physics is the cosmological density of dark matter, which is called in this context the relic density. Dark matter particles originate from the very early Universe, they have been in interaction with the thermal bath before decoupling from it, they have then decayed or annihilated and we observe today the particles which have survived until now, the relics.[2]

Nowadays the Standard Model for cosmology is the Friedmann-Lemaitre-Robertson-Walker model, also called $ {\Lambda}$CDM. First we take look at Einstein-Hilbert Lagrangian:

\mathscr{L}=\frac{1}{16 \pi G}\left(R+L_{m}\right) \sqrt{-g}

we can obtain the Einstein’s field equations

R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=8 \pi G T_{\mu \nu}

we can get

d \tau^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}=d t^{2}-a^{2}(t)\left\{\frac{d r^{2}}{1-k r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right\}

in a spherical coordinate system $x_{\mu} = (t, r, \theta, \phi)$ , where $a(t)$ is the scale factor of the Universe which characterizes its expansion and is a function of the cosmological time $t$ , and $k$ is a global curvature which is the same at any point of the spacetime.

Figure 1: The observed rotation curve of the dwarf spiral galaxy M33 extends considerably beyond its optical image (shown superimposed)

With them it is possible to understand the expansion of Universe as was realized in the 1920’s through Hubble’s law of expansion. With this FRW metric, the GR cosmological field equations are

H^{2}=\frac{8 \pi G}{3} \rho-\frac{k}{a^{2}}

and

\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}(\rho+3 P)

Considering the observable outcome of dark energy, which is admitted by the accelerated expansion of universe, we add the Cosmology Constant to the equation:

G_{\mu \nu}=8\pi T_{\mu \nu}+ \Lambda g_{\mu \nu}

leading to the Friedmann equ:

H^{2}=\frac{\ddot{a}}{a}^2=\frac{8 \pi G}{3} \rho+ \frac{\Lambda}{3}-\frac{k}{a^{2}}

The different fluids in the Universe are matter, made of a mixture of dark matter and baryonic matter; radiation, composed of relativistic particle species such as photons and neutrinos; and dark energy, which is an unidentified component of negative pressure. In the standard cosmological model, dark matter is considered as cold, i.e. with small velocities, and dark energy is considered to be a “cosmological constant” ${\Lambda}$ with a constant density
and pressure such as $\rho_{\Lambda}=-P_{\Lambda}$ , forming the ${\Lambda}$ CDM paradigm.
The Planck Collaboration has provided the observational values of the Hubble constant and the cosmological parameters of (total) matter, cold dark matter, baryonic matter, cosmological constant and curvature, respectively:[1]

\begin{aligned} H_{0} &=67.66 \pm 0.42 \mathrm{~km} / \mathrm{s} / \mathrm{Mpc} \\ \Omega_{m} &=0.3111 \pm 0.0056 \\ \Omega_{c} h^{2} &=0.11933 \pm 0.00091 \\ \Omega_{b} h^{2} &=0.02242 \pm 0.00014 \\ \Omega_{\Lambda} &=0.6889 \pm 0.0056 \\ \Omega_{k} &=0.0007 \pm 0.0037 \end{aligned}

It is stated that the the beginning of the history of the Universe is set at $t = 0$ , which corresponds to a scale factor $a(0) = 0$ . Structure formation occurred mainly during the matter domination era, and the cosmological constant thus started playing a role only very recently.

The physics of the early Universe is indeed complicated, since the Universe went through different phases, in particular an inflation period during which the scale factor increased exponentially and also several phase transitions such as the electroweak phase transition. These periods are however not observable directly, because the photons emitted during these phases were interacting too strongly with the plasma to carry information, and only after the formation of neutral atoms during the “recombination” period at a temperature of about 1 eV, or equivalently 105 K, the photons can be considered as messengers which carry information. The photons emitted at the recombination time can still be observed today in the cosmic microwave background (CMB), on which most of the results of the Planck Collaboration are based. The energetic photons emitted during recombination have had their wavelength redshifted and reduced by the expansion, and are today observed in the microwave range corresponding to a temperature of 2.725 K. The CMB is an important vector of information, since its anisotropies, are imprints of prior phenomena and their careful examination can provide constraints on scenarios describing the beginning of the Universe. Precise analyses of the CMB have shown that the standard cosmological model is in agreement with the data[1],and that no large deviation from new phenomena is possible at the time of recombination.[2]

Figure 2: Depiction of different stages in the evolution of the Universe [6]

Another indirect information about the early Universe comes from the abundance of the chemical elements. Indeed the nuclei have been formed during the so-called Big-Bang nucleosynthesis (BBN), by the combination of the nucleons, and this epoch is well described theoretically. In spite of inconsistencies with the predictions of the lithium abundance, models of BBN are considered as successful and important tests of the Big-Bang model, and their predictions are in agreement with the observed abundances. We will discuss more thoroughly the BBN physics in the following, since BBN is the most ancient period from which one can derive constraints on the cosmological models.

The candidates for dark matter particle is listed as below

Figure 3: Candidates for dark matter

we could see its mass ranging from $10^{-22} eV$ , the fuzzy Dm, $eV ~ TeV$ , the general WIMP(weak interaction massive particle), to $M_p$ , maybe the black holes. The methods of detection also vary due to their energy. The light ones behave like wave, could detected with resonant cavity, the WIMP with Cherenkov radiation detector.

Condensed Matter Physics

The era of condensed matter physics could be divided into two times, the one before topology physics which is governed by Landau’s fermi liquid and Landau-Ginzburg-Wilson paradigm, the topology physics which revealed more intrinsic property of nature, telling us that there are more vast space behind the condensed matter.

Landau told us that states of matter are classified by the symmetries they break. But there is state of matter that goes beyond Landau’s limited conception of the world.

Let us attempt to paraphrase the Landau Paradigm[8]

Phases of matter should be labelled by how they represent their symmetries, in particular whether they are spontaneously broken or not.
The degrees of freedom at a critical point are the fluctuations of the order parameter.

A significant corollary of assertion 1 is that gapless degrees of freedom, or groundstate degeneracy, in a phase, should be swept out by a symmetry. That is, they should arise as Goldstone modes for some spontaneously broken symmetry.

Beyond its conceptual utility, this perspective has a weaponization, in the form of Landau-Ginzburg theory, in terms of which we may find representative states, understand gross phase structure, and, when suitably augmented by the renormalization group (RG), even quantitatively describe phase transitions.
Indeed there are many apparent exceptions to the Landau Paradigm. Let us focus first on apparent exceptions to item 1. As a preview, exceptions that are only apparent include:

Topologically-ordered states. These are phases of matter distinguished from the trivial phase by something other than a local order parameter (1, 2). Symptoms include a groundstate degeneracy that depends on the topology of space, and anyons, excitations that cannot be created by any local operator. Real examples found so far include fractional quantum Hall states, as well as gapped spin liquids.
Other deconfined states of gauge theory. This category includes gapless spin liquids such as spinon Fermi surface or Dirac spin liquids (most candidate spin liquid materials are gapless). Another very visible manifestation of such a state is the photon phase of quantum electrodynamics in which our vacuum lives.
Fracton phases. Gapped fracton phases are a special case of topological order, where there are excitations that not only cannot be created by any local operator, but cannot be moved by any local operator.
Topological insulators. Here we can include both free-fermion states with topologically non-trivial bandstructure, as well as interacting symmetry-protected topological (SPT) phases.
Landau Fermi liquid.

Quantum Computation and Quantum Many-body Dynamics

The Quantum Computer has been striking a hint during this time, due to its enablity at solving various problems that are beyond the capabilities of the most powerful current supercomputers, which are based on classical technologies.

Like the digital integrated circuit nowadays, the chips are made by gate circuits from tiny, the quantum chips are also made by quantum gate. The classical gate, including OR, AND, NOT, XOR gate are made by transistors, basing on classical semiconductors physics. However, the quantum gate behave as quantum physics, differently from classical physics. The first is the axioms of unitary requires that the gate output two states when inputted two states, meaning there is no OR XOR, AND gate in quantum computation. Instead, we have single-qubit gate, including sigma-x, sigma-z, Hadamard, and T gate, and multi-qubit gate, including CNOT gate, etc. [5]

But what is the physical realization of quantum computer? It includes photonics, NMR, cavity-QED, NV Center, Joseph junction, ion trap, silicon quantum dot, Rydberg atom and the most exciting Quantum Topology.

Meanwhile, three models for quantum computation are quantum circuit model, one way quantum computation model and adiabatic quantum computation model which we will discuss later. And the three is all equivalent.

Adiabatic quantum computation model, transforms the process of quantum computation to a relatively complex Hamiltionian to be solved. Thus its ground state is the solution of our initial problem. Through adiabatic process (that is why we call it Adiabatic quantum computation model), we could control a rather simpler Hamiltionian to the Hamiltionian we want by experimental tools.

Quantum computing devices can be classified into universal and non-universal. In classical digital computers, universality means an ability perform an arbitrary sequence of operations on a bit string. The quantum counterpart of this definition is the ability to perform an arbitrary transformation of the quantum state of a set of qubits. In other words, the universal quantum computer can be understood as extension of the Turing machine into the quantum domain, as formalized by Deutsch in 1985.

In contrast, non-universal quantum computers aim to solve a specific problem or a specific class of problems. There are two important subclasses of non-universal quantum computers. The first one is analog quantum simulators: devices that simulate a process in a complex quantum system, e.g. solid matter, by another quantum system with well-known and controllable properties, e.g., an ensemble of cold atoms trapped in an optical field. The second class is special-purpose quantum computers to solve a specific restricted class of abstract mathematical problems, e.g., quantum annealing devices, which implement discrete optimization. The universality we will detail below.

In quantum computation, we have a theorem which Prof. Yang spending a long time on displaying its process of proof that night. That is any unitary operations on n qubits can be approximated to arbitrary accuracy by single-qubit gate and CNOT gate, i.e., the Hadamard gate,T gate and CNOT gate are universal gate.

Complete and precise theoretical descriptions of complex quantum systems, such as solid state, which involves interaction of multiple microscopic quantum objects, commonly call the quantum many-body dynamics. The curse of dimensionality makes the analysis exponentially hard for classical computers. As discussed above, this is one of main motivations behind quantum computing.
Each of the above described universal quantum computing models can in principle be used to simulate arbitrary complex quantum systems. For example, a digital superconducting quantum computer was used to simulate the interaction of two fermions, whose states are
encoded in four qubitsc[3]. This rather simple simulation required as many as 300 single-qubit and two-qubit gates.

QCD, Perturbated QCD and Lattice QCD

Dating back to 1970s, the physicsts had been used QED to probe the inner structure of nuclei, like belowed Feynman Diagram, electron-proton scattering:

Figure 4: The Feynmann diagram of electron-proton scattering proved that nuclei has its inner structure, distributing with spin and magnetic pole.

In the theory of strong interactions quarks, fermions, interact via coupling to gluons, vector (quantum spin 1) bosons, the quanta of the strong interaction fields, color replaces the electric charge in QED, which is why it is called Quantum Chromodynamics or QCD.
The QCD Lagrangian is [7]

\begin{aligned} \mathcal{L}_{\mathcal{Q C D}} &=-\frac{1}{2} \operatorname{tr}\left[G_{\mu \nu} G^{\mu \nu}\right]+\sum_{k} \bar{q}_{k}\left(i \gamma^{\mu}\left(\partial_{\mu}-i g A_{\mu}\right)-m_{k}\right) q_{k} \\ G_{\mu \nu} &=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}-i g\left[A_{\mu} A_{\nu}-A_{\nu} A_{\mu}\right] \\ A_{\mu} &=\sum_{1}^{8} A_{\mu}^{a} \lambda^{a} / 2 \end{aligned}

Under different energy level nature shows different kind of physics, as the figure show:

Figure 5: asymptotic freedom of qcd

This is called the Asymptotic freedom of QCD, meaning that we can use perturbation in high energy zone, non-perturbation in low energy zone. And it is just because of this we develop different kind of methods including the perturbated QCD at the high energy zone, the Lattice QCD at the low energy zone, and effective field theory to describe the Quark-Gluon Plasma(QGP).

Figure 6: The phase diagram of QGP

Figure 7: A summary of particle physics discoveries made with different machines

Figure 8: The cross sections for various processes expected at the LHC

Reference

[1] Nabila Aghanim, Yashar Akrami, Mark Ashdown, J Aumont, C Baccigalupi, M Ballardini, AJ Banday, RB Barreiro, N Bartolo, S Basak, et al. Planck 2018 results-vi. cosmological parameters. Astronomy & Astrophysics, 641:A6, 2020.

[2] A. Arbey and F. Mahmoudi. Dark matter and the early universe: A review. Progress in Particle and Nuclear Physics, page 103865, apr 2021.

[3] R. Barends, L. Lamata, J. Kelly, L. Garcia-Alvarez, A. G. Fowler, A Megrant, E Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, E. Solano, and John M. Martinis. Digital quantum simulation of fermionic models with a superconducting circuit. Nature Communications, 6(1), jul 2015.

[4] Jorge L. Cervantes-Cota, George Smoot, Luis Arturo Urena-Lopez, Hugo Aurelio Morales-Tecotl, Roman Linares-Romero, Eli SantosRodriguez, and Sendic Estrada-Jimenez. Cosmology today—a brief review. In AIP Conference Proceedings. AIP, 2011.

[5] A. K. Fedorov, N. Gisin, S. M. Beloussov, and A. I. Lvovsky. Quantum computing at the quantum advantage threshold: a down-to-business review, 2022.

[6] Rohini M. Godbole. Story of a journey: Rutherford to the large hadron collider and onwards, 2010.

[7] Leonard S. Kisslinger and Debasish Das. Review of QCD, quark–gluon plasma, heavy quark hybrids, and heavy quark state production in p–p and a–a collisions. International Journal of Modern Physics A, 31(07):1630010, mar 2016

[8] John McGreevy. Generalized symmetries in condensed matter, 2022.