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1. Ordered Representation
- 1.1 Lattice periodicity and X-ray crystallinity
- 1.2 Long-range order
2. Quenched disorder
3. Annealed disorder
4. Topological order

In physics, the terms order and disorder denote whether there is some kind of symmetry or correlation in a system of many particles.

In condensed matter physics, systems are often ordered at low temperatures. When heated, they undergo one or several phase transitions into a less ordered state. Examples of such order-to-order transitions are:

Melting of ice: solid-liquid transformation, loss of crystallization order;
Iron is demagnetized by heating above the Curie temperature: ferromagnetic-paramagnetic transition, loss of magnetic order.

The degree of freedom of order or disorder can be translation (crystalline ordering), rotation (ferroelectric ordering) or spin state (magnetic ordering).

This ordering can be contained in the complete crystal space group symmetry or in correlation. Depending on how the correlation decays with distance, it can be said to be long-range order or short-range order.

If the disordered state is not in thermodynamic equilibrium, it is called quenched disorder. For example, glass is obtained by quenching (supercooling) a liquid. Further promotion, other quenched states are called spin glass ( spin glass ), orientation glass ( orientational glass ). In some cases, the opposite of quenched disorder is annealed disorder.

1. Ordered Representation

1.1 Lattice periodicity and X-ray crystallinity

The strictest form of order in solids is lattice periodicity: a certain pattern (the arrangement of atoms in a unit cell) repeats again and again to form a translation-invariant tiling of space. This is the defining property of crystals. Possible symmetries have been classified into 14 Bravais lattices and 230 space groups .

Lattice periodicity implies long-range order: if only one unit cell is known, all atomic positions at any distance can be accurately predicted by virtue of translational symmetry. For much of the 20th century, the opposite was also taken for granted—until the discovery of quasicrystals in 1982 showed the existence of perfectly deterministic tilings without lattice periodicity.

In addition to structural order, charge ordering , spin ordering, magnetic ordering , and compositional ordering can also be considered. Magnetic ordering can be observed in neutron diffraction.

It is a thermodynamic entropy concept usually shown by second-order phase transitions. In general, high thermal energy is associated with disorder and low thermal energy with order, but there are violations of this. In low-energy diffraction experiments, ordered peaks become apparent.

1.2 Long-range order

Long-range order characterizes physical systems in which different parts of the same sample that are far apart exhibit correlated behavior.

This can be expressed as a correlation function, the spin- spin correlation function :

$G(x,x')=\langle s(x),s(x')\rangle$

where $s$ is the spin quantum number, $x$ is a distance function within a particular system.

The function at $x = x^{'}$ is equal to an identity element, and with the distance $∣ x - x^{’} ∣$ and decrease. Increase. Typically, it decays exponentially to zero at long distances, and the system is said to be disordered. But if the correlation function decays to a large constant value $∣ x - x^{'} ∣$ then the system is said to have long-range order. If it decays to zero as a power of distance, it is called quasi-long-range. Note what constitutes $∣ x - x^{'} large value of ∣$ . is understood in the sense of asymptotics.

2. Quenched disorder

In statistical physics, a system is said to exhibit quenched disorder when some of the parameters defining its behavior are random variables that do not evolve over time, i.e. they are quenched or frozen. Spin is a typical example. It is the opposite of annealed disorder, which allows random variables to evolve on their own.

In the mathematical realm, quenched disorder is more difficult to analyze than annealed disorder because thermal averaging and noise averaging play very different roles. In fact, the problem is so hard that there are few known solutions, most of which rely on approximations. The most common methods are:

A technique based on the continuation of mathematical analysis known as the replica trick;
cavity methods; although these give results consistent with experimental results for a large number of problems, they are generally not proven to be a strictly mathematical procedure.

However, it has recently been shown by rigorous methods that replica-based solutions are indeed exact, at least in the archetypal spin-glass model (the so-called Sherrington-Kirkpatrick model).

The second most commonly used technique in this field is generating functional analysis. This method is based on path integrals and is in principle perfectly accurate, although generally more difficult to apply than the replicating process.

Transition of graph from disordered (left) to ordered (right) state

3. Annealed disorder

When some of the parameters in the system definition are random variables, but their evolution is related to the evolution of the degrees of freedom defining the system, the system is said to exhibit annealed disorder. It is by definition the opposite of quenched disorder, where random variables may not change their values.

Systems with annealed disorder are generally considered easier to treat mathematically, since averages of disorder and thermal averages can be treated on the same basis.

4. Topological order

Transition of graph from disordered (left) to ordered (right) state

In physics, topological order is an ordering of phases of matter (also known as quantum matter) at zero temperature. Macroscopically, the topological order is defined and described by the robust ground state degeneracy and the quantized non-Abelian geometric phase of the degenerated ground states. On a microscopic level, topological order corresponds to patterns of long-range quantum entanglement. States with different topological orders (or different modes of long-range entanglement) cannot change each other without a phase transition.

Various topologically ordered states have interesting properties, such as (1) topological degeneracy and fractional statistics or non-abelian statistics can be used to implement topological quantum computers; (2) Perfect conducting edge states that may have important device applications; (3) Emerging gauge fields and Fermi statistics indicate the origin of quantum information of elementary particles; (4) Topological entanglement entropy (topological entanglement entropy) reveals the topological order Entanglement Origins, etc. Topological order is important in the study of several physical systems such as spin liquids and the quantum Hall effect, as well as for potential applications in fault-tolerant quantum computing.

Topological insulators and topological superconductors (more than one dimension) do not have the topological order defined above, and their entanglement is only short-range.

4.1 Background

Although all matter is made up of atoms, matter can have different properties and appear in different forms such as solid, liquid, superfluid, etc. These different forms of matter are often referred to as states or phases of matter. According to condensed matter physics and emergence principles, the different properties of materials arise from the different ways in which atoms are organized in the material. These different organizations of atoms (or other particles) are formally known as order in the material.

Atoms can be organized in many ways, leading to many different orders and many different types of materials. Landau's symmetry breaking theory provides a general understanding of these different orders. It states that different orders actually correspond to different symmetries in the organization of the constituent atoms. What happens when a material changes from one order to another (that is, when a material undergoes a phase transition) is a change in the symmetry of the atomic organization.

For example, atoms have a random distribution in a liquid, so when we replace the atoms by any distance, the liquid remains the same. We say that liquids have continuous translational symmetry. After a phase change, a liquid can become a crystal. In a crystal, atoms are organized into a regular array (lattice). The lattice stays the same only if we shift it by a certain distance (an integer times the lattice constant), so the crystal has only discrete translational symmetries. A phase transition between a liquid and a crystal is a transition that reduces the continuous translational symmetry of a liquid to the discrete symmetry of a crystal. This change in symmetry is called symmetry breaking. Thus, the essence of the difference between liquids and crystals is that the organization of atoms has a different symmetry in the two phases.

Landau's symmetry breaking theory has been a very successful theory. Physicists have long believed that Landau's theory describes all possible orders in materials, as well as all possible (continuous) phase transitions.

4.2 Discovery and characterization

However, since the late 1980s it has gradually become apparent that Landau's symmetry breaking theory may not describe all possible orders. To explain high-temperature superconductivity, a chiral spin state is introduced. At first, physicists also wanted to use Landau's symmetry breaking theory to describe chiral spin states. They identify chiral spin states as those that break time-reversal and even-even symmetry, rather than spin-rotation symmetry. According to Landau's description of the symmetry breaking, this should be the end of the story. However, it was quickly realized that there are many different chiral spin states with exactly the same symmetry, so symmetry alone is not enough to characterize the different chiral spin states. This means that the chiral spin state contains a new order beyond the usual symmetry description. The proposed new order is named "topological order". The name "topological order" is inspired by the low energy effective theory of chiral spin states, which is a type of topological quantum field theory (TQFT ) . New quantum numbers, such as ground-state degeneracy (which can be defined on closed spaces or open spaces with interstitial boundaries, including Abelian and non-Abelian topological orders) and non-Abelian geometric phases of degenerate ground states, are introduced in Characterize and define different topological orders of chiral spin states. Recently, it was shown that topological order can also be characterized by topological entropy.

But experiments soon showed that chiral spin states cannot describe high-temperature superconductors, and the theory of topological order became a theory without experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use topological order theory to describe different quantum Hall states. Like chiral spin states, the different quantum Hall states all share the same symmetry and fall outside the Landau symmetry-breaking description. It is found that the different orders of different quantum Hall states can indeed be described by topological order, so topology does have experimental realization.

Fractional quantum Hall ( FQH) states were discovered in 1982, before the concept of topological order was introduced in 1989. But the FQH state is not the first topologically ordered state discovered experimentally. The superconductor, discovered in 1911, was the first state of topological order discovered experimentally. It has $\mathsf{Z}_{2}$ topological order.

While topological order usually occurs in strongly interacting boson/fermion systems, a simple topological order can also occur in free fermion systems. This topological order corresponds to integral quantum Hall states, which can be characterized by band-filling Chern numbers if we consider integer quantum Hall states on lattices. Theoretical calculations show that this Chern number can be measured experimentally for free fermion systems. It is well known that such Chern numbers can be (possibly indirectly) measured by edge states.

The most important features of topological order are the underlying fractional excitations (such as anyons ) and their fusion statistics and braiding statistics (quantum statistics that can go beyond bosons or fermions). The current research work shows that there are ring and string-like excitations in the topological order in the 3+1-dimensional space-time, and their multi-ring/chord braiding statistics are the key features to identify the 3+1-dimensional topological order. The multi-loop/string-braiding statistics of the 3+1-dimensional topological order can be captured by the link invariants of the specific topological quantum field theory in the 4 space-time dimensions.

4.3 Mechanism

A large class of 2+1D topological orders is achieved through a mechanism called string-net condensation . Such topological orders can have a gap edge and are classified by the unitary fusion category (or monoidal category) theory. It has been found that string-net condensation can generate an infinite variety of different types of topological order, which may indicate that there are still many different new materials to be discovered.

Collective motion of condensed strings produces excitations higher than string-net condensation. These excitations turn out to be gauge bosons. The end of the string is a defect corresponding to another excitation. These excitations are gauge charges and can carry Fermi or fractional statistics.

Condensation of other extended objects such as 'membranes', 'brane-nets' and fractals also leads to topologically ordered phases and 'quantum galssiness'.

4.4 Mathematical expressions

We know that group theory is the mathematical foundation of symmetry-breaking orders. What is the mathematical basis of topological order? Discovered that a subclass of 2+1D topological order - Abelian topological order - can be obtained by $K-$ matrix method for classification. string-net condensation shows that tensor categories (such as fusion category or monoidal category) are part of the mathematical basis for 2+1D topological orders. Recent studies have shown (up to reversible topological orders without graded excitations):

The 2+1D boson topological order is classified by unitary modular tensor categories.
The topological order of 2+1D bosons with symmetry G is classified in G-crossed tensor categories.
2+1D boson/fermion topological orders with symmetry G are classified in unitary braided fusion categories rather than symmetric fusion category, which has modular extensions. symmetric fusion category Rep(G) for boson systems and sRep(G) for fermion systems.

Higher dimensional topological order may be related to n-Category theory. Quantum operator algebra is a very important mathematical tool for studying topological order.

Some also argue that topological order is described mathematically by extended quantum symmetry.

4.5 Applications

The materials described by Landau's symmetry-breaking theory have had a major impact on technology. For example, ferromagnetic materials that break spin-rotational symmetry can be used as media for digital information storage. Hard drives made of ferromagnetic materials can store gigabytes of information. Liquid crystals that break molecular rotational symmetry are widely used in display technology. Crystals that break translational symmetry lead to well-defined electronic energy bands, allowing us to fabricate semiconductor devices such as transistors. Different types of topological orders are even more abundant than different types of symmetry-breaking orders. This suggests their potential for exciting novel applications.

One theoretical application is the use of topological order states as a medium for quantum computing in a technique called topological quantum computing. A topologically ordered state is a state with complex nonlocal quantum entanglement. Nonlocality means that quantum entanglement in a state of topological order is distributed among many different particles. Therefore, the pattern of quantum entanglement cannot be disrupted by local perturbations. This significantly reduces the effect of decoherence. This suggests that if we encode quantum information using different quantum entanglements in topologically ordered states, the information could persist for longer periods of time. The quantum information encoded by topological quantum entanglement can also be manipulated by dragging topological defects towards each other. This process may provide a physical device for performing quantum computations. Therefore, topologically ordered states can provide a natural medium for quantum memory and quantum computing. Such implementations of quantum memory and quantum computing could be fault-tolerant.

Topologically ordered states often have a special property that they contain non-trivial boundary states. In many cases, these boundary states become perfect conductive pathways that conduct electricity without generating heat. This could be another potential application of topological order in electronic devices.

Similar to topological order, topological insulators also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and application of topological insulators. This observation naturally leads to the question: are topological insulators examples of topologically ordered states? In fact, topological insulators are different from the topologically ordered states defined in this paper. Topological insulators have only short-range entanglement and no topological order, while the topological order defined in this paper is a mode of long-range entanglement. The topological order is robust to any perturbation. It features emergent gauge theory, emergent fractional charge, and fractional statistics. In contrast, topological insulators only respect time reversal and $The perturbation of U (1)$ symmetry is robust. Their quasiparticle excitations have no fractional charge and fractional statistics. Strictly speaking, a topological insulator is an example of a symmetry-protected topological (SPT) order, where the first example of an SPT order is the Haldane phase of the spin-1 chain. But the Haldane phase of spin-2 chain has no SPT sequence.

4.6 Potential Impact

Landau's symmetry breaking theory is the cornerstone of condensed matter physics. It is used to define the field of condensed matter research. The existence of topological order seems to indicate that nature is much richer than Landau's symmetry-breaking theories have shown so far. So topological order opens up a new direction in condensed matter physics—a new direction for highly entangled quantum matter. We realize that quantum phases of matter (i.e., zero-temperature phases of matter) can be divided into two categories: long-range entangled states and short-range entangled states. Topological order is a concept to describe the state of long-range entanglement: topological order = long-range entanglement mode. Short-range entangled states are trivial in the sense that they all belong to one phase. In the presence of symmetry, however, even short-range entangled states are nontrivial and can belong to different phases. These phases are said to contain the SPT order. The SPT order extends the concept of topological insulators to interacting systems.

Some have argued that topological order (or, more precisely, string-net condensation) in the local boson (spin) model has the potential to provide a unified origin for photons, electrons, and other fundamental particles in our universe.

4.7 Topological entropy

Topological entanglement entropy or topological entropy, usually $\gamma$ means that it is a number that characterizes many-body states with topological order (many-body states).

Non-zero topological entanglement entropy (non-zero topological entanglement entropy) reflects the existence of long-range quantum entanglement in many-body quantum states. Thus, topological entanglement entropy links topological order to patterns of long-range quantum entanglement.

Given a topologically ordered state, topological entropy can be extracted from the asymptotic behavior of von Neumann entropy, which measures the quantum entanglement between a block of space and the rest of the system. In an infinite two-dimensional topological order state, the boundary length is Entanglement entropy of simply connected regions of $L$ $L$ has the following form:

$S_{L}\;\longrightarrow \;\alpha L-\gamma +{\mathcal {O}}(L^{- \nu })\;,\qquad \nu >0\,\!$

where $-\gamma$ is the topological entanglement entropy.

The entropy of topological entanglement is equal to the logarithm of the total quantum dimensionality of quasiparticle excitations of a state.

For example, the simplest fractional quantum Hall state, which fills the fraction Laughlin state at $1/$ $m$ $\gamma=1/2\log(m)$ 。 $\mathsf{Z}_{2}$ Fractionalized states, such as $\mathsf{Z}_{2}$ Topologically ordered states of spin liquids, quantum dimer models ( quantum dimer models ) on non-bipartite lattices , and Kitaev's toric code state are all represented by $\gamma = \log(2)$ $the$ feature.