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We review the progress in the last 20–30 years, during which we discovered that there are many new phases of matter that are beyond the traditional Landau symmetry breaking theory. We discuss new “topological” phenomena, such as topological degeneracy that reveals 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. More recently, we found that at the microscopical level, topological order is due to long-range quantum entanglements. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional charges and fractional statistics. Long-range quantum entanglements can even provide a unified origin of light and electrons; light is a fluctuation of long-range entanglements, and electrons are defects in long-range entanglements.

Although all matter is formed by only three kinds of particles: electrons, protons, and neutrons, matter can have many different properties and appear in many different forms, such as solid, liquid, conductor, insulator, superfluid, and magnet. According to the principle of emergence in condensed matter physics, the rich properties of materials originate from the rich ways in which the particles are organized in the materials. Those different organizations of the particles are formally called the orders in the materials.

For example, particles have a random distribution in a liquid (see Figure

(a) Particles in liquids do not have fixed relative positions. They fluctuate freely and have a random but uniform distribution. (b) Particles in solids form a fixed regular lattice.

(a) Disordered states that do not break the symmetry. (b) Ordered states that spontaneously break the symmetry. The energy function

Liquid and crystal are just two examples. In fact, particles can organize in many ways which lead to many different orders and many different types of materials. Landau symmetry-breaking theory [

Quantum phases of matter are phases of matter at zero temperature. So quantum phases correspond to the ground states of the quantum Hamiltonians that govern the systems. In this paper, we will mainly discuss those quantum phases of matter. Crystal, conductor, insulator, superfluid, and magnets can exist at zero temperature and are examples of quantum phases of matter.

Again, physicists used to believe that Landau symmetry-breaking theory also describes all possible quantum phases of matter and all possible (continuous) quantum phase transitions. (Quantum phase transitions are zero temperature phase transitions.) For example, the superfluid is described by a

It is interesting to compare a finite-temperature phase,

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-breaking quantum phases are classified by a pair of mathematical objects

However, in late 1980s, it became clear that Landau symmetry-breaking theory did not describe all possible phases. In an attempt to explain high-temperature superconductivity, the chiral spin state was introduced [

But, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry [

But experiments soon indicated that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity [

FQH states are gapped ground states of 2D electrons under strong magnetic field. FQH states have a property that a current density will induce an electric field in the transverse direction:

2D electrons in strong magnetic field may form FQH states. Each FQH state has a quantized Hall coefficient

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 [

FQH states were discovered in 1982 [

Topological order is a very new concept that describes quantum entanglements in many body 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 Figure

The dancing patterns for the symmetry-breaking orders.

We can also understand topological orders through such dancing pictures (see Figure

The dancing patterns for the topological orders.

For example, in FQH liquid, the electrons dance following the following local dancing rules.

Electron always dances anticlockwise which implies that the electron wave function only depends on the electron coordinates

Each electron always takes exact three steps to dance around any other electron, which implies that the phase of the wave function changes by

The above two local dancing rules fix a global dance pattern which corresponds to the Laughlin wave function [

In addition to FQH states, some spin liquids also contain topological orders [

down-spins form closed strings with no ends in the background of up-spins (see Figure

strings can otherwise move freely, including reconnecting freely (see Figure

The strings in a spin-1/2 model. In the background of up-spins, the down-spins form closed strings.

In the string liquid, strings can move freely, including reconnecting the strings.

The global dance formed by the spins following the above dancing rules gives us a quantum spin liquid which is a superposition of all closed-string configurations [

The above descriptions of topological order are intuitive and not concrete. It is not clear if the topological order (the global dancing pattern or the long-range entanglement) has any experimental significance. In order for the topological order to be a useful concept, it must have new experimental properties that are different from any symmetry-breaking states. Those new experimental properties should indicate the nontrivialness of the topological order. In fact, the concept of topological order should be defined by the collection of those new experimental properties.

Indeed, topological order does have new characteristic properties. Those properties of topological orders reflect the significance of topological order.

Topological orders produce new kind of waves (i.e., the collective excitations above the topologically ordered ground states) [

The finite-energy defects of topological order (i.e., the quasiparticles) can carry fractional statistics [

Some topological orders have topologically protected gapless boundary excitations [

In the following, we will study some examples of topological orders and reveal their amazing topological properties.

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”.

Typically, one thinks that the properties of a material should be determined by the components that form the material. However, this simple intuition is incorrect, since all the materials are made of same components: electrons, protons, and neutrons. So, we cannot use the richness of the components to understand the richness of the materials. In fact, the various properties of different materials originate from various ways in which the particles are organized. Different orders (the organizations of particles) give rise to different physical properties of a material. It is the richness of the orders that gives rise to the richness of material world.

Let us use the origin of mechanical properties and the origin of waves to explain, in a more concrete way, how orders determine the physics properties of a material. We know 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 Figure

Liquids only have a compression wave—a wave of density fluctuations.

Solid can resist both compression and shear deformations. As a result, solids can support both compression wave and transverse wave. The transverse wave corresponds to the propagation of shear deformations. In fact, there are two transverse waves corresponding to two directions of shear deformations. The propagation of the compression wave and the two transverse waves in solids are described by the elasticity equation

Drawing a grid on a sold helps us to see the deformation of the solid. The vector

We would like to point out that the elasticity equation and the Euler equations not only describe the propagation of waves, but also they actually describe all small deformations in solids and liquids. Thus, the two equations represent a complete mathematical description of the mechanical properties of solids and liquids.

But why do solids and liquids behave so differently? What makes a solid to have a shape and a liquid to have no shape? What are the origins of elasticity equation and Euler equations? The answer to those questions has to wait until the discovery of atoms in 19th century. Since then, we realized that both solids and liquids are formed by collections of atoms. The main difference between the solids and liquids is that the atoms are organized very differently. In liquids, the positions of atoms fluctuate randomly (see Figure

How can different organizations of atoms affect mechanical properties of materials? In solids, both the compression deformation (see Figure

The atomic picture of (a) the compression wave and (b) the transverse wave in a crystal.

In contrast, a shear deformation of atoms in liquids does not result in a new configuration since the atoms still have uniformly random positions. So, the shear deformation is a do-nothing operation for liquids. Only the compression deformation which changes the density of the atoms results in a new atomic configuration and costs energies (see Figure

The atomic picture of the compression wave in liquids.

We see that the properties of the propagating wave are entirely determined by how the atoms are organized in the materials. Different organizations lead to different kinds of waves and different kinds of mechanical laws. Such a point of view of different kinds of waves/laws originated from different organizations of particles is a central theme in condensed matter physics. This point of view is called the principle of emergence.

The elasticity equation and the Euler equation are two very important equations. They lay the foundation of many branches of science, such as mechanical engineering and aerodynamic engineering. But, we have a more important equation, Maxwell equation, that describes light waves in vacuum. When Maxwell equation was first introduced, people firmly believed that any wave must correspond to motion of something. So, people want to find out what is the origin of the Maxwell equation? The motion of what gives rise to electromagnetic wave?

First, one may wonder, can Maxwell equation comes from a certain symmetry-breaking order? Based on Landau symmetry-breaking theory, the different symmetry-breaking orders can indeed lead to different waves satisfying different wave equations. So, maybe a certain symmetry-breaking order can give rise to a wave that satisfies Maxwell equation. But people have been searching for ether—a medium that supports light wave—for over 100 years and could not find any symmetry-breaking states that can give rise to waves satisfying the Maxwell equation. This is one of the reasons why people give up the idea of ether as the origin of light and Maxwell equation.

However, the discovery of topological order [

In addition to the Maxwell equation, there is an even stranger equation, Dirac equation, that describes wave of electrons (and other fermions). Electrons have Fermi statistics. They are fundamentally different from the quanta of other familiar waves, such as photons and phonons, since those quanta all have Bose statistics. To describe the electron wave, the amplitude of the wave must be anticommuting Grassmann numbers, so that the wave quanta will have Fermi statistics. Since electrons are so strange, few people regard electrons and the electron waves as collective motions of something. People accept without questioning that electrons are fundamental particles, one of the building blocks of all that exist.

However, from a condensed matter physics point of view, all low-energy excitations are collective motion of something. If we try to regard photons as collective modes, why cannot we regard electrons as collective modes as well? So, maybe Dirac equation and the associated fermions can also arise from a new kind of organizations of bosons/spins that have nontrivial topological orders.

A recent study provides a positive answer to the above questions [

A quantum ether: the fluctuation of oriented strings gives rise to electromagnetic waves (or light). The ends of strings give rise to electrons. Note that oriented strings have directions which should be described by curves with arrow. For ease of drawing, the arrows on the curves are omitted in the above plot.

But why does the waving of strings produce 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 to a density fluctuation, which can be described by a scaler field

The fluctuating strings in a string liquid.

A “density” wave of oriented strings in a string liquid. The wave propagates in

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 contained 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 string-nets 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 contained 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. The fluctuations of closed strings produce only photons. If strings have open ends, those open ends can move around and just behave like independent particles. Those particles are not photons. In fact, the ends of strings are nothing but electrons.

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 corresponds 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 in quantum theory. Amazingly, the ends of strings can reproduce this subtle quantum property of Fermi statistics [

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 [

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 gives 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 [

We like to stress that the string-nets are formed by qubits. So, in the string-net picture, both the Maxwell equation and Dirac equation emerge from

The electric field and the magnetic field in the Maxwell equation are called gauge fields. The fields in the Dirac equation are Grassmann-number valued field. (Grassmann numbers are anticommuting numbers. For a long time, we thought that we have to use gauge fields to describe light waves that have only two transverse modes, and we thought that we have to use Grassmann-number valued fields to describe electrons and quarks that have Fermi statistics. So, gauge fields and Grassmann-number valued fields became the fundamental build blocks of quantum field theory that describes our world. The string-net liquids demonstrate that we do not have to introduce gauge fields and Grassmann-number valued fields to describe photons, gluons, electrons, and quarks. It demonstrates how gauge fields and Grassmann fields emerge from local qubit models that contain only complex scaler fields at the cut-off scale.

Our attempt to understand light has a 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 leads to the notion of gauge theory. The general relativity and the gauge theory are two corner stones of modern physics. They provide a unified understanding of all four interactions in terms of a beautiful mathematical framework; all interactions can be understood geometrically as curvatures in space-time and in “unit systems” (or more precisely, as curvatures in the tangent bundle and other vector bundles in space-time).

Later, people in high-energy physics and in condensed matter physics have found another way in which gauge field can emerge [

A string picture represents the third way to understand gauge theory. String operators appear in the Wilson-loop characterization [

Viewing gauge field (and the associated gauge bosons) as fluctuations of long-range entanglements has an added bonus; we can understand the origin of Fermi statistics in the same way; fermions emerge as defects of long-range entanglements, even though the original model is purely bosonic. Previously, there are two ways to obtain emergent fermions from purely bosonic model: by binding gauge charge and gauge flux in

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 describes how qubits are organized in the ground state. So string-net is not a thing, but a pattern of qubits. In the string-net theory, the gauge bosons are waves of collective fluctuations of the string-nets, and a fermion corresponds to one end of string. In contrast, gauge bosons and fermions come from strings in the superstring theory. Both gauge bosons and fermions correspond to small pieces of strings. Different vibrations of the small pieces of strings give rise to different kind of particles. The fermions in the superstring theory are put in by hand through the introduction of Grassmann fields.

In the string-net unification of light and electrons [

The

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.

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 Figure

As we have mentioned before, the global dancing pattern is determined by local dancing rules. What are those local rules that give rise to the global dancing pattern

The first rule tells us that the amplitude of a string configuration only depends on the topology of the string configuration. Starting from a single loop, using the local deformation and the local reconnection in Figure

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 (

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 Figure

The orientable strings in a spin-1 model. In the background of

Why do the two wave functions of unoriented strings,

The two topological states in two dimensions contain only closed strings, which represent the ground states. If the wave functions contain open strings (i.e., have nonzero amplitudes for open string states), then the ends of the open strings will correspond to point-like topological excitations above the ground states. Although an open string is an extended object, its middle part merges with the strings already in the ground states and is unobservable. 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 corresponds to a topological point defect, which may carry fractional quantum numbers. This is because an open string as a whole always carries nonfractionalized quantum numbers. But an open string corresponds to

Let us first consider the defects in the

Under a

To construct the eigenstates of

We see that the

If one believes in the spin-statistics theorem, one may guess that the particle

Deformation of strings and two reconnection moves, plus an exchange of two ends of strings and a

The emergence of Fermi statistics in the

Next, let us consider the defects in the

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 the most mysterious phenomena in nature. Now, we can understand them as merely a phenomenon of long-range quantum entanglements. They are no longer mysterious.

The

Topological degeneracy can be used as protected qubits which allow us to perform topological quantum computation [

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 topological degeneracy. We know that the local dancing rules determine the global dancing pattern. On a sphere, the local dancing rules determine a unique global dancing pattern. So, the ground state is nondegenerate. However, on other compact spaces, there can be several global dancing patterns that all satisfy the local dancing rules. In this case, the ground state is degenerate.

For the

On a torus, the closed string configurations can be divided into four sectors, depending on even or odd number of strings crossing the

Similarly, the double-semion topological order also gives rise to fourfold degenerate ground state on torus.

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 [

First, we would like to give a physical definition of topological order (at least in

A X-ray diffraction pattern defines/probes the crystal order.

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

topological ground-state degeneracies on closed spaces of various topologies (see Figure

non-Abelian geometric phases [

Symmetry-breaking orders can be probed/defined through linear responses. But topological order cannot be probed/defined through linear responses. We need topological probes to define topological orders.

Order | Experiment |
---|---|

Crystal order | X-ray diffraction |

Ferromagnetic order | Magnetization |

Antiferromagnetic order | Neutron scattering |

Superuid order | Zero viscosity and vorticity quantization |

Topological order | Topological degeneracy |

(Global dancing pattern) | non-Abelian geometric phase |

The topological ground state degeneracies of topologically ordered states depend on the topology of the space, such as the genus

(a) The shear deformation of a torus generates a (projective) non-Abelian geometric phase

It was through such topological probes that we introduce the concept of topological order.

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

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 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.

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 [

The non-Abelian geometric phase is a unitary matrix

To use non-Abelian geometric phases to characterize/define topological order, let us put the many-body state on a torus [

Yang once asked; the microscopic theory of fermionic superfluid and superconductor, BCS theory, captures the essence of the superfluid and superconductor, but what is this essence? This question led him to develop the theory of off-diagonal long-range order, [

Similarly, we may ask; Laughlin’s theory for FQH effect captures 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), an FQH state is still a nontrivial state with a quantized thermal Hall conductance [

After the experimental discovery of superconducting order via zero resistance and Meissner effect [

The long-range entanglements are defined through local unitary (LU) transformations. LU transformation is an important concept which is directly related to the definition of quantum phases [

Let us first introduce local unitary evolution. An LU evolution is defined as the following unitary operator that acts on the degrees of freedom in a quantum system:

The LU evolution is closely related to

(a) A graphic representation of a quantum circuit, which is formed by (b) unitary operations on blocks of finite size

The notion of LU transformations leads to the following more general and more systematic picture of phases and phase transitions (see Figure

The possible gapped phases for a class of Hamiltonians

SRE states are states that can be transformed into direct product states via LU transformations. All SRE states can be transformed into each other via LU transformations. So, all SRE states belong to the same phase (see Figure

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 a pattern of long-range entanglements.

Such understanding of topological order in terms of long-range entanglements leads to a systematic description of boundary-gapped (BG) topological orders in

In

By applying the local rules in (

However, an arbitrary choice of

We know that group theory is the mathematical foundation of symmetry-breaking theory of phases and phase transitions. The above systematic description of

In this paper, 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 framework (such as group theory and Ginzburg-Landau theory). The main goal of this paper 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

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, nontrivial topological orders—FQH states—can be realized by 2D electron gas under very strong magnetic fields and very low temperatures [

Apart from the FQH effects, nontrivial topological order may also appear in quantum spin systems. In fact, the concept of topological order was first introduced [

More recently, extensive new numerical calculations indicated that the Heisenberg model on Kagome lattice [

The Heisenberg model on Kagome lattice can be realized in Herbertsmithite

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, and perfect conducting boundary. 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!

Our world is rich and complex. When we discover the inner working of our world and try to describe it, we often find that we need to invent new mathematical language to 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 the 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, Riemannian 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 world based on quantum field theory, and we tried to understand everything in our world in terms 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. They greatly expand our understanding of possible quantum phases and bring the research of quantum matter to a whole new level. To gain a systematic understanding of new quantum phases and long-range entanglements, we like to know what 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 framework that describes long-range entanglements. The further progresses in this direction will lead to a comprehensive understanding of long-range entanglements and topological quantum matter.

However, what is really exciting in the study of quantum matter is that it might lead to a whole new point of view of our world. This is because 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 we should not use geometric pictures, based on fields and fiber bundles, to understand our world. Maybe we should use entanglement pictures to understand our world. This way, we can get both gauge interactions and fermions from a single origin—qubits. We may live in a truly quantum world. So, quantum entanglements represent a new chapter in physics.

_{c}superconductors

_{1}-

_{2}Heisenberg model

_{1}-

_{2}Heisenberg model: a tensor product state approach

_{2}topological order

_{2}spin liquid and chiral antiferromagnetic phase in the Hubbard model on a honeycomb lattice

_{3}(OH)

_{6}Cl

_{2}

_{3}(OH)

_{6}Cl

_{2}

_{2}spin liquids in the