Topological order: from long-range entangled quantum matter to an unification of light and electrons

In primary school, we were told that there are four states of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four states of matter. For example, there are ferromagnetic states as revealed by the phenomenon of magnetization and superfluid states as defined by the phenomenon of zero-viscosity. The various phases in our colorful world are so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. In this paper, we will review the progress in last 20 -- 30 years, during which we discovered that there are even more interesting phases that are beyond Landau symmetry breaking theory. We discuss new"topological"phenomena, such as topological degeneracy, that reveal the existence of those new phases - topologically ordered phases. Just like zero-viscosity defines the superfluid order, the new"topological"phenomena define the topological order at macroscopic level. As a new type of order, topological order requires a new mathematical frame work, such as fusion category and group cohomology, to describe it. More recently, we find that, at microscopical level, topological order is due to long-range quantum entanglements, just like fermion superfluid is due to fermion-pair condensation. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional quantum numbers, fractional/non-Abelian statistics, and perfect conducting boundary channels. Long-range quantum entanglements can even provide a unified origin of light and electrons (or more generally, gauge interactions and Fermi statistics): light waves (gauge fields) are fluctuations of long-range entanglements, and electrons (fermions) are defects of long-range entanglements.

Symmetry is beautiful and rich.
Quantum entanglement is even more beautiful and richer. For example, particles have a random distribution in a liquid (see Fig. 1a), so a liquid remains the same as we displace it by an arbitrary distance. We say that a liquid has a "continuous translation symmetry". After a phase transition, a liquid can turn into a crystal. In a crystal, particles organize into a regular array (a lattice) (see Fig. 1b). A lattice remains unchanged only when we displace it by a particular distance (integer times of lattice constant), so a crystal has only "discrete translation symmetry". The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such change in symmetry is called "spontaneous symmetry breaking". We note that the equation of motions that govern the dynamics of the particles respect the continuous translation symmetry for both cases of liquid and crystal. However, in the case of crystal, the stronger interaction makes the particles to prefer being separated by a fixed distance and a fixed angle. This makes par- ticles to break the continuous translation symmetry down to discrete translation symmetry "spontaneously" in order to choose a low energy configuration (see Fig. 2). Therefore, the essence of the difference between liquids and crystals is that the organizations of particles have different symmetries in the two phases. spontaneously break the symmetry. The energy function ε g (φ) has a symmetry φ → −φ: ε g (φ) = ε g (−φ). However, as we change the parameter g, the minimal energy state (the ground state) may respect the symmetry (a), or may not respect the symmetry (b). This is the essence of spontaneous symmetry breaking.
superfluid is described by a U (1) symmetry breaking.
It is interesting to compare a finite-temperature phase, liquid, with a zero-temperature phase, superfluid. A liquid is described a random probability distributions of particles (such as atoms), while a superfluid is described by a quantum wave function which is the superposition of a set of random particle configurations: The superposition of many different particle positions are called quantum fluctuations in particle positions.
Since Landau symmetry-breaking theory suggests that all quantum phases are described by symmetry breaking, thus we can use group theory to classify all those symmetry breaking phases: All symmetry  We would like to point out that before the topological-order understanding of FQH states, people have tried to use the notions of off-diagonal long-range order and order parameter from Ginzburg-Landau theory to describe FQH states. [13][14][15][16] Such an effort leads to a Ginzburg-Landau Chern-Simons effective theory for FQH states. 15,16 At same time, it was also realized that the order parameter in the Ginzburg-Landau Chern-Simons is not gauge invariant and is not physical. This is consistent with the topological-order understanding of FQH states which suggests that FQH has no off-diagonal long-range order and cannot be described by local order parameters. So we can use effective theories without order parameters to describe FQH states, and such effective theories are pure Chern-Simons effective theories. 9,17-21 The pure Chern-Simons effective theories lead to a K-matrix classification 20 of all Abelian topologically ordered states (which include all Abelian FQH states).
FQH states were discovered in 1982 11 before the introduction of the concept of topological order.
But FQH states are not the first experimentally discovered topologically ordered states. The real-life superconductors, having a Z 2 topological order, [22][23][24] were first experimentally discovered topologically ordered states. 25 (Ironically, the Ginzburg-Landau symmetry breaking theory was developed to describe superconductors, despite the real-life superconductors are not symmetry breaking states, but topologically ordered states.)

B. Intuitive pictures of topological order
Topological order is a very new concept that describes quantum entanglements in manybody systems. Such a concept is very remote from our daily experiences and it is hard to have an intuition about it. So before we define topological order in general terms (which can be abstract), let us first introduce and explain the concept through some intuitive pictures.
We can use dancing to gain an intuitive picture of topological order. But before we do that, let us use dancing picture to describe the old symmetry breaking orders (see Fig. 4).
In the symmetry breaking orders, every particle/spin (or every pair of particles/spins) dance by itself, and they all dance in the same way. (The "same way" of dancing represents a longrange order.) For example, in a ferromagnet, every electron has a fixed position and the same spin direction. We can describe an anti-ferromagnet by saying every pair of electrons has a fixed position and the two electrons in a pair have opposite spin directions. In a boson superfluid, each boson is moving around by itself and doing the same dance, while in a fermion superfluid, fermions dance around in pairs and each pair is doing the same dance.
We can also understand topological orders through such dancing pictures. Unlike fermion superfluid where fermion dance in pairs, a topological order is described by a global dance, where every particle is dancing with every other particles in a very organized way: (a) all spins/particles dance following a set of local dancing "rules" trying to lower the energy of a local Hamiltonian. (b) If all the spins/particles follow the local dancing "rules", then they will form a global dancing pattern, which correspond to the topological order. (c) Such a global pattern of collective dancing is a pattern of quantum fluctuation which corresponds to a pattern of long range entanglements.
For example in FQH liquid, the electrons dance following the following local dancing rules: (a) electron always dances anti-clockwise which implies that the electron wave function only depend on the electron coordinates (x, y) via z = x + iy.
(b) each electron always takes exact three steps to dance around any other electron, which implies that the phase of the wave function changes by 6π as we move an electron around any other electron.
The above two local dancing rules fix a global dance pattern which correspond to the Laughlin wave function 12 Φ FQH = (z i − z j ) 3 . Such an collective dancing gives rise to the topological order (or long range entanglements) in the FQH state.
In additional to FQH states, some spin liquids also contain topological orders. 5,23,[26][27][28] (Spin liquids refer to ground states of quantum spin systems that do not break any symmetry of the spin Hamiltonian.) In those spin liquids, the spins "dance" following the follow local dancing rules: (a) Down spins form closed strings with no ends in the background of up-spins (see Fig. 6).
The global dance formed by the spins following the above dancing rules gives us a quan- (1) Topological orders produce new kind of waves (i.e. the collective excitations above the topologically ordered ground states). [31][32][33][34][35][36][37][38][39] The new kind of waves can be probed/studied in practical experiments, such as neutron scattering experiments. 35 (2) The finite-energy defects of topological order (i.e. the quasiparticles) can carry fractional statistics 40,41 (including non-Abelian statistics 42,43 ) and fractional charges 12,44 (if there is a symmetry). Such a property allows us to use topologically ordered states as a medium for topological quantum memory 45 and topological quantum computations. 29 (3) Some topological orders have topologically protected gapless boundary excitations. [46][47][48] Such gapless boundary excitations are topologically protected, which lead to perfect conducting boundary channels even with magnetic impurities. 49 This property may lead to device applications.
In the following, we will study some examples of topological orders and reveal their amazing topological properties.

IV. EXAMPLES OF TOPOLOGICAL ORDER: QUANTUM LIQUID OF ORI-ENTED STRINGS AND A UNIFICATION OF GAUGE INTERACTIONS AND FERMI STATISTICS
Our first example is a quantum liquid of oriented strings. We will discuss its new topological properties (1) and (2). We find that the new kind of waves and the emergent statistics are so profound, that they may change our view of universe. Let us start by explaining a basic notion -"principle of emergence". that a deformation in a material can propagate just like the ripple on the surface of water.
The propagating deformation corresponds to a wave traveling through the material. Since liquids can resist only compression deformation, so liquids can only support a single kind of wave -compression wave (see Fig. 8). (Compression wave is also called longitudinal wave.) Mathematically the motion of the compression wave is governed by the Euler equation where ρ is the density of the liquid.
where the vector field u i (x, t) describes the local displacement of the solid.
We would like to point out that the elasticity equation and the Euler equations not only describe the propagation of waves, they actually describe all small deformations in solids  A recent study provides an positive answer to the above questions. 30,37,38 We find that if bosons/spins form large oriented strings and if those strings form a quantum liquid state, then the collective motion of the such organized bosons/spins will correspond to waves described by Maxwell equation and Dirac equation. The strings in the string liquid are free to join and cross each other. As a result, the strings look more like a network (see Fig. 12).
For this reason, the string liquid is actually a liquid of string-nets, which is called string-net condensed state.
But why the waving of strings produces waves described by the Maxwell equation? We know that the particles in a liquid have a random but uniform distribution. A deformation of such a distribution corresponds a density fluctuation, which can be described by a scaler field ρ(x, t). Thus the waves in a liquid is described by the scaler field ρ(x, t) which satisfy the Euler equation (2). Similarly, the strings in a string-net liquid also have a random but uniform distribution (see Fig. 13). A deformation of string-net liquid corresponds to a change of the density of the strings (see Fig. 14). However, since strings have an orientation, the "density" fluctuations are described by a vector field E(x, t), which indicates there are more strings in the E direction on average. The oriented strings can be regarded as flux lines.
The vector field E(x, t) describes the smeared average flux. Since strings are continuous (i.e. they cannot end), the flux is conserved: of strings cannot change in the direction along the strings (i.e. along the E(x, t) direction).
E(x, t) can change only in the direction perpendicular to E(x, t). Since the direction of the propagation is the same as the direction in which E(x, t) varies, thus the waves described by E(x, t) must be transverse waves: E(x, t) is always perpendicular to the direction of the propagation. Therefore, the waves in the string liquid have a very special property: the waves have only transverse modes and no longitudinal mode. This is exactly the property of the light waves described by the Maxwell equation. We see that "density" fluctuations of strings (which are described be a transverse vector field) naturally give rise to the light (or electromagnetic) waves and the Maxwell equation. [33][34][35][36][37][38] It is interesting to compare solid, liquid, and string-net liquid. We know that the particles in a solid organized into a regular lattice pattern. The waving of such organized particles produces a compression wave and two transverse waves. The particles in a liquid have a more random organization. As a result, the waves in liquids lost two transverse modes and contain only a single compression mode. The particles in a string-net liquid also have a random organization, but in a different way. The particles first form string-nets and stringnets then form a random liquid state. Due to this different kind of randomness, the waves in string-net condensed state lost the compression mode and contain two transverse modes.
Such a wave (having only two transverse modes) is exactly the electromagnetic wave.
To understand how electrons appear from string-nets, we would like to point out that if we only want photons and no other particles, the strings must be closed strings with no ends. How do we know that ends of strings behave like electrons? First, since the waving of string-nets is an electromagnetic wave, a deformation of string-nets correspond to an electromagnetic field. So we can study how an end of a string interacts with a deformation of string-nets. We find that such an interaction is just like the interaction between a charged electron and an electromagnetic field. Also electrons have a subtle but very important property -Fermi statistics, which is a property that exists only quantum theory. Amazingly, the ends of strings can reproduce this subtle quantum property of Fermi statistics. 30,51 Actually, string-net liquids explain why Fermi statistics should exist.
We see that string-nets naturally explain both light and electrons (gauge interactions and Fermi statistics). In other words, string-net theory provides a way to unify light and electrons. 37,38 So, the fact that our vacuum contains both light and electrons may not be a mere accident. It may actually suggest that the vacuum is indeed a string-net liquid.
C. More general string-net liquid and emergence of non-Abelian gauge theory Here, we would like to point out that there are many different kinds of string-net liquids.
The strings in different liquids may have different numbers of types. The strings may also join in different ways. For a general string-net liquid, the waving of the strings may not correspond to light and the ends of strings may not be electrons. Only one kind of string-net liquids give rise to light and electrons. On the other hand, the fact that there are many different kinds of string-net liquids allows us to explain more than just light and electrons.
We can design a particular type of string-net liquids which not only gives rise to electrons and photons, but also gives rise to quarks and gluons. 30,34 The waving of such type of string-nets corresponds to photons (light) and gluons. Our attempt to understand light has long and evolving history. We first thought light to be a beam of particles. After Maxwell, we understand light as electromagnetic waves. After Einstein's theory of general relativity, where gravity is viewed as curvature in space-time, Weyl and others try to view electromagnetic field as curvatures in the "unit system" that we used to measure complex phases. It Later, people in high-energy physics and in condensed matter physics have found another way in which gauge field can emerge: 54-57 one first cut a particle (such as an electron) into two partons by writing the field of the particle as the product of the two fields of the two partons. Then one introduces a gauge field to glue the two partons back to the original particle. Such a "glue-picture" of gauge fields (instead of the fiber bundle picture of gauge fields) allow us to understand the emergence of gauge fields in models that originally contain no gauge field at the cut-off scale.
A string picture represent the third way to understand gauge theory. String operators appear in particles. Such a picture has an experimental prediction that is described in the next section IV D.
We like to point out that the string-net unification of gauge bosons and fermions is very different from the superstring theory for gauge bosons and fermions. In the string-net theory, gauge bosons and fermions come from the qubits that form the space, and "string-net" is simply the name that describe how qubits are organized in the ground state. So string-net is not a thing, but a pattern of qubits. In gauge theories will lead to new cosmic strings which will appear in very early universe.

V. EXAMPLES OF TOPOLOGICAL ORDER: QUANTUM LIQUID OF UNORI-ENTED STRINGS AND EMERGENCE OF STATISTICS
In the above, we discussed how light and electrons may emerge from a quantum liquid of orientable strings. We like to point out that quantum liquids of orientable strings are not the simplest topologically ordered state. Quantum liquids of unoriented strings are simpler topologically ordered states. In this section, we will discuss quantum liquids of unoriented strings and their topological properties. Using those simpler examples, we will discuss in detail how can ends of strings become fermions, or even anyons.
A. Quantum liquids of unoriented strings and the local "dancing" rules The strings in quantum liquids of unoriented strings can be realized in a spin-1/2 model.
We can view up-spins as background and lines of down-spins as the strings (see Fig. 6).
Clearly, such string is unoriented. The simplest topologically ordered state in such spin-1/2 system is given by the equal-weight superposition of all closed strings: 29 Then the second rule relates the amplitudes of close strings in the ground state as we change the strings locally: In other words, if we locally deform/reconnect the strings as in Fig. 7, the amplitude (or the ground state wave function) does not change.
The first rule tells us that the amplitude of a string configuration only depend on the topology of the string configuration. Starting from a single loop, using the local deformation and the local reconnection in Fig. 7, we can generate all closed string configurations with any number of loops. So all those closed string configurations have the same amplitude.
Therefore, the local dancing rule fixes the wave function to be the equal-weight superposition of all closed strings: |Φ Z 2 = all closed strings . In other words, the local dancing rule fixes the global dancing pattern. If we choose another local dancing rule, then we will get a different global dancing pattern that corresponds to a different topological order. One of the new choices is obtained by just modifying the sign in eqn. (5): We note that each local reconnection operation changes the number of loops by 1. Thus the new local dancing rules gives rise to a wave function which has a form |Φ Semi = all closed strings (−) N loops , where N loops is the number of loops. The wave function |Φ Semi corresponds to a different global dance and a different topological order.
In the above, we constructed two quantum liquids of unoriented strings in a spin-1/2 model. Using a similar construction, we can also obtain a quantum liquid of orientable strings which gives rise to waves satisfying Maxwell equation as discussed before. To obtain quantum liquid of orientable strings, we need to start with a spin-1 model, where spins live on the links of honeycomb lattice (see Fig.   15). Since the honeycomb lattice is bipartite, each link has an orientation from the A-sublattice to the B-sublattice (see Fig. 15). The orientable strings is formed by alternating S z = ±1 spins on the background of S z = 0 spins. The string orientation is given be the orientation of the links under the S z = 1 spins (see Fig. 15). The superposition of the orientable strings gives rise to quantum liquid of orientable strings.

B. Topological properties of quantum liquids of unoriented strings
Why the two wave functions of unoriented strings, |Φ Z 2 and |Φ Semi , have non-trivial topological orders? This is because the two wave functions give rise to non-trivial topological properties. The two wave functions correspond to different topological orders since they give rise to different topological properties. In this section, we will discuss two topological properties: emergence of fractional statistics and topological degeneracy on compact spaces.

Emergence of Fermi and fractional statistics
The two topological states in two dimensions contain only closed strings, which represent Only its two ends carry energies and correspond to two point-like particles.
We note that such a point-like particle from an end of string cannot be created alone.
Thus an end of string correspond to a topological point defect, which may carry fractional quantum numbers. This is because an open string as a whole always carry non-fractionalized quantum numbers. But an open string corresponds to two topological point defects from the two ends. So we cannot say that each end of string carries non-fractionalized quantum numbers. Some times, they do carry fractionalized quantum numbers.
Let us first consider the defects in the |Φ Z 2 state. To understand the fractionalization, let us first consider the spin of such a defect to see if the spin is fractionalized or not. 73,74 An end of string can be represented by which is an equal-weight superposition of all string states obtained from the deformations and the reconnections of .
Under a 360 • rotation, the end of string is changed to def , which is an equal weight superposition of all string states obtained from the deformations and the reconnections of . Since def and def are alway different, def is not an eigenstate of 360 • rotation and does not carry a definite spin.
To construct the eigenstates of 360 • rotation, let us make a 360 • rotation to def . To do that, we first use the string reconnection move in Fig. 7  We see that the 360 • rotation exchanges def and def . Thus the eigenstates of 360 • rotation are given by def + def with eigenvalue 1, and by def − def with eigenvalue −1. So the particle def + def has a spin 0 (mod 1), and the particle def − def has a spin 1/2 (mod 1).
If one believes in the spin-statistics theorem, one may guess that the particle def + def is a boson and the particle def − def is a fermion. This guess is indeed correct. Form Fig.   16, we see that we can use deformation of strings and two reconnection moves to generate an exchange of two ends of strings and a 360 • rotation of one of the end of string. Such operations allow us to show that Fig. 16a and Fig. 16e have the same amplitude, which means that an exchange of two ends of strings followed by a 360 • rotation of one of the end of string do not generate any phase. This is nothing but the spin-statistics theorem.
The emergence of Fermi statistics in the |Φ Z 2 state of a purely bosonic spin-1/2 model indicates that the state is a topologically ordered state. We also see that the |Φ Z 2 state has a bosonic quasi-particle def + def , and a fermionic quasi-particle def − def . The bound state of the above two particles is a boson (not a fermion) due to their mutual semion statistics. Such quasi-particle content agrees exactly with the Z 2 gauge theory which also has three type of non-trivial quasiparticles excitations, two bosons and one fermion. In fact, the low energy effective theory of the topologically ordered state |Φ Z 2 is the Z 2 gauge theory and we will call |Φ Z 2 a Z 2 topologically ordered state.
Next, let us consider the defects in the |Φ Semi state. Now and a similar expression for def , due to a change of the local dancing rule for reconnecting the strings (see eqn. (6)). Using the string reconnection move in Fig. 7, we find that def = − def . So a 360 • rotation, changes ( def , def ) to ( def , − def ). We find that def + i def is the eigenstate of the 360 • rotation with eigenvalue −i, and def − i def is the other eigenstate of the 360 • rotation with eigenvalue i. So the particle def + i def has a spin −1/4, and the particle def − i def has a spin 1/4. The spin-statistics theorem is still valid for |Φ Semi def state, as one can see form Fig. 16. So, the particle def + i def and particle def − i def have fractional statistics with statistical angles of semion: ±π/2.
Thus the |Φ Semi state contains a non-trivial topological order. We will call such a topological order a double-semion topological order.
It is amazing to see that the long range quantum entanglements in string liquid can give rise to fractional spin and fractional statistics, even from a purely bosonic model. Fractional spin and Fermi statistics are two of most mysterious phenomena in natural. Now, we can understand them as merely a phenomenon of long-range quantum entanglements. They are no longer mysterious.

Topological degeneracy
The Z 2 and the double-semion topological states (as well as many other topological states) have another important topological property: topological degeneracy. 6,7 Topological degeneracy is the ground state degeneracy of a gapped many-body system that is robust against any local perturbations as long as the system size is large.
Topological degeneracy can be used as protected qubits which allows us to perform topological quantum computation. 29 It is believed that the appearance of topological degeneracy implies the topological order (or long-range entanglements) in the ground state. 6,7 Manybody states with topological degeneracy are described by topological quantum field theory at low energies. 8 The simplest topological degeneracy appears when we put topologically ordered states on compact spaces with no boundary. We can use the global dancing pattern to understand the  For the Z 2 topological state on torus, the local dancing rule relate the amplitudes of the string configurations that differ by a string reconnection operation in Fig. 7. On a torus, the closed string configurations can be divided into four sectors (see Fig. 17), depending on even or odd number of strings crossing the x-or y-axises. The string reconnection move only connect the string configurations among each sector. So the superposition of the string configurations in each sector represents a different global dancing pattern and a different degenerate ground state. Therefore, the local dancing rule for the Z 2 topological order gives rise to four fold degenerate ground state on torus. 23 Similarly, the double-semion topological order also gives rise to four fold degenerate ground state on torus.

VI. A MACROSCOPIC DEFINITION AND THE CHARACTERIZATION OF TOPOLOGICAL ORDER
So far in this paper, we discussed topological order using an intuitive dancing picture.
Then we discussed a few simple examples. In the rest of this paper, we will give a more rigorous description and a systematic understanding of topological order and its essence. 6,7 Historically, the more rigorous description of topological order was obtained before the intuitive dancing picture and the simple examples of topological order discussed in the  FIG. 18: A X-ray diffraction pattern defines/probes the crystal order.

Order Experiment
Crystal order X-ray diffraction

Ferromagnetic order Magnetization
Anti-ferromagnetic order Neutron scattering

Superfluid order Zero-viscosity & vorticity quantization
Topological order Topological degeneracy, (Global dancing pattern) non-Abelian geometric phase previous part of the paper.
First, we would like to give a physical definition of topological order (at least in 2+1 dimensions). Here, we like to point out that to define a physical concept is to design experiments or numerical calculations that allow us probe and characterize the concept. For example, the concept of superfluid order, is defined by zero viscosity and the quantization of vorticity, and the concept of crystal order is defined by X-ray diffraction experiment (see Fig. 18).
The experiments that we use to define/characterize superfluid order and crystal order are linear responses. Linear responses are easily accessible in experiments and the symmetry breaking order that they define are easy to understand (see Table I). However, topological order is such a new and elusive order that it cannot be probed/defined by any linear responses. To probe/define topological order we need to use very unusual "topological" probes.
It was through such topological probes that allowed us to introduce the concept of topological order. Just like zero viscosity and the quantization of vorticity define the concept of superfluid order, the topological degeneracy and the non-Abelian geometric phases of the degenerate ground states define the concept of topological order.
A. What is "topological ground state degeneracy" Topological ground state degeneracy, or simply, topological degeneracy is a phenomenon of quantum many-body systems, that the ground state of a gapped many-body system become degenerate in the large system size limit, and that such a degeneracy cannot be lifted by any local perturbations as long as the system size is large. 6,9,63,76 The topological degeneracy for a given system usually is different for different topologies of space. 77 For example, for the Z 2 topologically ordered state in two dimensions, 5 the topological degeneracy is D g = 4 g on genus g Riemann surface (see Fig. 19).
People usually attribute the ground state degeneracy to symmetry. But topological degeneracy, being robust against any local perturbations, is not due to symmetry. So the very existence of topological degeneracy is a surprising and amazing phenomenon. Such an amazing phenomenon defines the notion of topological order. As a comparison, we know that the existence of zero-viscosity is also an amazing phenomenon, and such an amazing shear-deformed torus is the same as the original torus after a coordinate transformation: x → x + y, y → y. phenomenon defines the notion of superfluid order. So topological degeneracy, playing the role of zero-viscosity in superfluid order, implies the existence of a new kind of quantum phase -topologically ordered phases.
B. What is "non-Abelian geometric phase of topologically degenerate states" However, the ground state degeneracy is not enough to completely characterize/define topological order. Two different topological orders may have exactly the same topological degeneracy on space of any topology. We would like to find, as many as possible, quantum numbers associated with the degenerate ground states, so that by measuring these quantum numbers we can completely characterize/define topological order. The non-Abelian geometric phases of topologically degenerate states are such quantum numbers. 7,10 The non-Abelian geometric phase is a unitary matrix U that can be calculated from an one parameter Similarly, we may ask: Laughlin's theory for FQH effect capture the essence of the FQH effect, but what is this essence? Our answer is that the topological order defined by the topological ground state degeneracy and the non-Abelian geometric phases of those degenerate ground states is the essence of FQH effect.
One may disagree with the above statement by pointing out that the essence of FQH effect should be the quantized Hall conductance. However, such an opinion is not quite correct, since even after we break the particle number conservation (which breaks the quantized Hall conductance), a FQH state is still a non-trivial state with a quantized thermal Hall conductance. 81 The non-trivialness of FQH state does not rely on any symmetry (except the conservation of energy). In fact, the topological degeneracy and the non-Abelian geometric phases discussed above are the essence of FQH states which can be defined even without any symmetry. They provide a characterization and definition of topological order that does not rely on any symmetry. We would like to point out that the topological entanglement entropy is another way to characterize the topological order without any symmetry. 82

VII. THE MICROSCOPIC DESCRIPTION OF TOPOLOGICAL ORDER
After the experimental discovery of superconducting order via zero-resistance and Meissner effect, 84 it took 40 years to obtain the microscopic understanding of superconducting order through the condensation of fermion pairs. 85 However, we are luckier for topological orders. After the theoretical discovery of topological order via the topological degeneracy and the non-Abelian geometric phases of the degenerate ground states, 7 it took only 20 years to obtain the microscopic understanding of topological order: topological order is due to longrange entanglements and topological order is simply pattern of long-range entanglements. 86 In this section, we will explain such a microscopic understanding.

A. Local unitary transformations
The long-range entanglements is defined through local unitary (LU) transformations. LU transformation is an important concept which is directly related to the definition of quantum phases. 86 In this section, we will give a short review of LU transformation. 30,[86][87][88] Let us first introduce local unitary evolution. A LU evolution is defined as the following unitary operator that act on the degrees of freedom in a quantum system: where T is the path-ordering operator andH(g) = i O i (g) is a sum of local Hermitian operators. Two gapped quantum states belong to the same phase if and only if they are related by a LU evolution. 63,86,89 The LU evolutions is closely related to quantum circuits with finite depth.
We will call U M circ a LU transformation. In quantum information theory, it is known that finite time unitary evolution with local Hamiltonian (LU evolution defined above) can be simulated with constant depth quantum circuit (i.e. a LU transformation) and vice-verse: So two gapped quantum states belong to the same phase if and only if they are related by a LU transformation.

B. Topological orders and long-range entanglements
The notion of LU transformations leads to the following more general and more systematic picture of phases and phase transitions (see Fig. 22). 86 For gapped quantum systems without 2 g g 1  Fig. 22a).
LRE states are states that cannot be transformed into direct product states via LU transformations. It turns out that, many LRE states also cannot be transformed into each other.
The LRE states that are not connected via LU transformations belong to different classes and represent different quantum phases. Those different quantum phases are nothing but the topologically ordered phases. So, topological order is pattern of long-range entanglements.
Such a understanding of topological order in terms of long-range entanglements lead to a systematic description of boundary-gapped (BG) topological orders in 2+1 dimensions, 30,86,90,91 in terms of spherical fusion category. 74 (Here, an BG topological order is a long-range entangled phase which can have an gapped edge or gapped entanglement spectrum. 92 where the shaded areas represent other parts of string-nets that are not changed. Here, the type-0 string is interpreted as the no-string state. We would like to mention that we have drawn the first local rule somewhat schematically. The more precise statement of this rule is that any two string-net configurations that can be continuously deformed into each other have the same amplitude. In other words, the string-net wave function Φ only depends on the topologies of the graphs; it only depends on how the strings are connected (see Fig. 12).
By applying the local rules in eqn. (11) multiple times, one can compute the amplitude of any string-net configuration in terms of the amplitude of the no-string configuration. Thus (11) determines the string-net wave function Φ.
However, an arbitrary choice of (d i , F ijk lmn ) does not lead to a well defined Φ. This is because two string-net configurations may be related by more than one sequence of local rules. We need to choose the (d i , F ijk lmn ) carefully so that different sequences of local rules produce the same results. That is, we need to choose (d i , F ijk lmn ) so that the rules are selfconsistent. Finding these special tensors is the subject of tensor category theory 93,94 . It has been shown that only those that satisfy 30 F mlq kpn F jip mns F jsn lkr = F jip qkr F riq mls (12) will result in self-consistent rules and a well defined string-net wave function Φ. Such a wave function describes a string-net condensed state. Here, we have introduced some new The solutions (d i , F ijk lmn ) give us a quantitative description of topological orders (or pattern of long-range entanglements), in terms of local dancing rules. From the data (d i , F ijk lmn ), we can compute the topological properties of the corresponding topological phases, such as ground state degeneracy, quasi-particle statistics, etc. 30,73,86,90,94,95 The above approach can also be used to systematically describe BG topological orders in 3+1 dimensions. 30,37,96 We know that group theory is the mathematical foundation of symmetry breaking theory of phases and phase transitions. The above systematic description of (2+1)D BG topological order strongly suggests that tensor category theory is the mathematical foundation of topological order and long-range entanglements. Because of symmetry, group theory becomes very important in physics. Because of quantum entanglements, tensor category theory will becomes very important in physics.

VIII. WHERE TO FIND LONG-RANGE ENTANGLED QUANTUM MATTER?
In this article, we described the world of quantum phases. We pointed out that there are symmetry breaking quantum phases, and there are topologically ordered quantum phases.
The topologically ordered quantum phases are a totally new kind of phases which cannot be understood using the conventional concepts (such as symmetry breaking, long range order, and order parameter) and conventional mathematical frame work (such as group theory and Ginzburg-Landau theory). The main goal of this article is to introduce new concepts and pictures to describe the new topologically ordered quantum phases.
In particular, we described how to use global dancing pattern to gain an intuitive picture of topological order (which is a pattern of long range entanglements). We further point out that we can use local dancing rules to quantitatively describe the global dancing pattern (or topological order). Such an approach leads to a systematic description of BG topological order in terms of string-net (or spherical fusion category theory), 30,86,90,91 and systematic description of 2D chiral topological order in terms of pattern of zeros [97][98][99][100][101][102][103][104][105] (which is a generalization of "CDW" description of FQH states [106][107][108][109][110][111][112][113][114] ).
The local-dancing-rule approach also leads to concrete and explicit Hamiltonians, that allow us to realize each string-net state and each FQH state described by pattern of zeros.
However, those Hamiltonians usually contain three-body or more complicated interactions, and are hard to realize in real materials. So here we would like to ask: can topological order be realized by some simple Hamiltonians and real materials?
Of cause, non-trivial topological orders -FQH states -can be realized by 2D electron gas under very strong magnetic fields and very low temperatures. 11,12 Recently, it was proposed that FQH states might appear even at room temperatures with no magnetic field in flat-band materials with spin-orbital coupling and spin polarization. [115][116][117][118][119] Finding such materials and realizing FQH states at high temperatures will be an amazing discovery. Using flat-band materials, we may even realize non-Abelian fractional quantum Hall states 42,43,120,121 at high temperatures.
Apart from the FQH effects, non-trivial topological order may also appear in quantum spin systems. In fact, the concept of topological order was first introduced 6 to describe a chiral spin liquid, 4,5 which breaks time reversal and parity symmetry. Soon after, time reversal and parity symmetric topological order was proposed in 1991, 23,26-28 which has spincharge separation and emergent fermions. The new topological spin liquid is called Z 2 spin liquid or Z 2 topological order since the low energy effective theory is a Z 2 gauge theory.
In 1997, an exactly soluble model 29 (that breaks the spin rotation symmetry) was obtained that realizes the Z 2 topological order. Since then, the Z 2 topological order become widely J 2 S i · S j , J 2 /J 1 ∼ 0.5, may have gapped spin liquid ground states, and such spin liquids are very likely to be Z 2 spin liquids. However, with spin rotation, time reversal, and lattice symmetry, there are many To summarize, topological order and long-range entanglements give rise to new states of quantum matter. Topological order has many new emergent phenomena, such as emergent gauge theory, fractional charge, fractional statistics, non-Abelian statistics, perfect conducting boundary, etc. In particular, if we can realize a quantum liquid of oriented strings in certain materials, it will allow us to make artificial elementary particles (such as artificial photons and artificial electrons). So we can actually create an artificial vacuum, and an artificial world for that matter, by making an oriented string-net liquid. This would be a fun experiment to do!

IX. A NEW CHAPTER IN PHYSICS
Our world is rich and complex. When we discover the inner working of our world and try to describe it, we ofter find that we need to invent new mathematical language describe our understanding and insight. For example, when Newton discovered his law of mechanics, the proper mathematical language was not invented yet. Newton (and Leibniz) had to develop calculus in order to formulate the law of mechanics. For a long time, we tried to use theory of mechanics and calculus to understand everything in our world.
As another example, when Einstein discovered the general equivalence principle to describe gravity, he needed a mathematical language to describe his theory. In this case, the needed mathematics, Riemann geometry, had been developed, which leaded to the theory of general relativity. Following the idea of general relativity, we developed the gauge theory.
Both general relativity and gauge theory can be described by the mathematics of fiber bundles. Those advances led to a beautiful geometric understanding of our would based quantum field theory, and we tried to understand everything in our world in term of quantum field theory.
Now, I feel that we are at another turning point. In a study of quantum matter, we find that long-range entanglements can give rise to many new quantum phases. So long-range entanglements are natural phenomena that can happen in our world. But mathematical language should we use to describe long-range entanglements? The answer is not totally clear. But early studies suggest that tensor category and group cohomology should be a part of the mathematical frame work that describes long-range entanglements. What is surprising is that such a study of quantum matter might lead to a whole new point of view of our world, since long-range entanglements can give rise to both gauge interactions and Fermi statistics.
(In contrast, the geometric point of view can only lead to gauge interactions.) So maybe