3. Dirac model for knot on zero-lattice

#### 3.1. Dirac model on geometric space

#### 3.1.1. Dirac model in sublattice-representation on geometric space

It was known that in emergent quantum mechanics, a 3D SOC knot-crystal becomes multiknot system, of which the effective theory becomes a Dirac model in quantum field theory. In emergent quantum mechanics, the Hamiltonian for a 3D SOC knot-crystal has two terms—the kinetic term from global winding and the mass term from leapfrogging motion. Based on a representation of projected state, a 3D SOC knot-crystal is reduced into a "two-sublattice" model with discrete spatial translation symmetry, of which the knot states are described by j i <sup>L</sup> and Rj i (or the Wannier states <sup>c</sup>† L,i j i vacuum and <sup>c</sup>† R,j j i vacuum ). We call it the Dirac model in sublattice-representation.

In sublattice-representation on geometric space, the equation of motion of knots is determined by the Schrödinger equation with the Hamiltonian

$$\begin{aligned} \mathcal{H}\_{\text{krot}} &= \int \left( \psi^{\dagger} \widehat{\mathcal{H}}\_{\text{krot}} \psi \right) d^{3} \mathbf{x}, \\ \widehat{\mathcal{H}}\_{\text{krot}} &= -c\_{\text{eff}} \overrightarrow{\Gamma} \cdot \overrightarrow{p}\_{\text{krot}} + m\_{\text{krot}} c\_{\text{eff}}^{2} \Gamma^{5} \end{aligned} \tag{15}$$

where ψ† t; x ! � � is an four-component fermion field as <sup>ψ</sup>† <sup>t</sup>; <sup>x</sup> ! � � <sup>¼</sup> <sup>ψ</sup>† <sup>↑</sup><sup>L</sup> t; x ! � � <sup>ψ</sup>† <sup>↑</sup><sup>R</sup> t; x ! � � � ψ† <sup>↓</sup><sup>L</sup> t; x ! � � <sup>ψ</sup>† <sup>↓</sup><sup>R</sup> t; x ! � �Þ. Here, L, R label two chiral-degrees of freedom that denote the two possible sub-lattices, ↑, ↓ label two spin degrees of freedom that denote the two possible winding directions. We have

$$
\Gamma^5 = \vec{1} \otimes \iota\_{\text{x}} \tag{16}
$$

and

$$\begin{aligned} \Gamma^1 &= \sigma^\sharp \otimes \iota\_{\mathcal{Y}'} \\ \Gamma^2 &= \sigma^\sharp \otimes \iota\_{\mathcal{Y}'} \\ \Gamma^3 &= \sigma^z \otimes \iota\_{\mathcal{Y}} \end{aligned} \tag{17}$$

p ! knot ¼ ℏknot k ! is the momentum operator. mknotc<sup>2</sup> eff <sup>¼</sup> <sup>2</sup>ℏknotω<sup>∗</sup> plays role of the mass of knots and <sup>c</sup>eff <sup>¼</sup> <sup>a</sup>�<sup>J</sup> <sup>ℏ</sup>knot ¼ 2aω<sup>0</sup> play the role of light speed where a is a fixed length that denotes the half pitch of the windings on the knot-crystal.

In addition, the low energy effective Lagrangian of knots on 3D SOC knot-crystal is obtained as

$$\mathcal{L}\_{\text{3D}} = \overline{\psi} \left( i \gamma^{\mu} \widehat{\Theta}\_{\mu} - m\_{\text{knot}} \right) \psi \tag{18}$$

where <sup>ψ</sup> <sup>¼</sup> <sup>ψ</sup>† γ0, γμ are the reduced Gamma matrices,

$$
\gamma^1 = \gamma^0 \Gamma^1, \gamma^2 = \gamma^0 \Gamma^2, \gamma^3 = \gamma^0 \Gamma^3,\tag{19}
$$

and

Φ ! x !; t � �

> X !

3.1.1. Dirac model in sublattice-representation on geometric space

L,i

<sup>H</sup>knot <sup>¼</sup> <sup>Ð</sup> <sup>ψ</sup>†H<sup>b</sup> knot<sup>ψ</sup>

Hb knot ¼ �ceffΓ

described by the three integer numbers

38 Superfluids and Superconductors

3.1. Dirac model on geometric space

j i <sup>L</sup> and Rj i (or the Wannier states <sup>c</sup>†

by the Schrödinger equation with the Hamiltonian

sublattice-representation.

where ψ† t; x

ψ† <sup>↓</sup><sup>L</sup> t; x ! � �

and

! � �

winding directions. We have

ψ† <sup>↓</sup><sup>R</sup> t; x ! � �

3. Dirac model for knot on zero-lattice

!Φ ! x ! <sup>þ</sup>a� <sup>e</sup> !; t � �

<sup>¼</sup> ð Þ¼ <sup>X</sup>;Y;<sup>Z</sup> <sup>1</sup>

¼ Φ ! x !; t � �

Thus, on the winding space, we have a corresponding "zero-lattice" of discrete lattice sites

π Φ ! � 1 π Φ !

It was known that in emergent quantum mechanics, a 3D SOC knot-crystal becomes multiknot system, of which the effective theory becomes a Dirac model in quantum field theory. In emergent quantum mechanics, the Hamiltonian for a 3D SOC knot-crystal has two terms—the kinetic term from global winding and the mass term from leapfrogging motion. Based on a representation of projected state, a 3D SOC knot-crystal is reduced into a "two-sublattice" model with discrete spatial translation symmetry, of which the knot states are described by

j i vacuum and <sup>c</sup>†

In sublattice-representation on geometric space, the equation of motion of knots is determined

possible sub-lattices, ↑, ↓ label two spin degrees of freedom that denote the two possible

<sup>Γ</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>⊗</sup> <sup>ι</sup>y, <sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>σ</sup><sup>y</sup> <sup>⊗</sup> <sup>ι</sup>y, <sup>Γ</sup><sup>3</sup> <sup>¼</sup> <sup>σ</sup><sup>z</sup> <sup>⊗</sup> <sup>ι</sup>y:

<sup>Γ</sup><sup>5</sup> <sup>¼</sup><sup>1</sup> !

� �

! � p !

is an four-component fermion field as ψ† t; x

R,j

d3 x,

knot <sup>þ</sup> <sup>m</sup>knotc<sup>2</sup>

Þ. Here, L, R label two chiral-degrees of freedom that denote the two

effΓ<sup>5</sup> ,

! � �

See the illustration of a 1D zero-lattice in Figure 1(b) and 3D zero-lattice in Figure 1(d).

þ π: (13)

mod π: (14)

j i vacuum ). We call it the Dirac model in

<sup>¼</sup> <sup>ψ</sup>†

⊗ ιx, (16)

<sup>↑</sup><sup>L</sup> t; x ! � �

ψ† <sup>↑</sup><sup>R</sup> t; x

! � � �

(15)

(17)

$$
\gamma^0 = \Gamma^5, \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3. \tag{20}
$$

#### 3.1.2. Dirac model in vortex-representation on geometric space

In this paper, we derive the effective Dirac model for a knot-crystal based on a representation of vortex degrees of freedom. We call it vortex-representation.

In Ref. [13], it was known that a knot has four degrees of freedom, two spin degrees of freedom ↑ or ↓ from the helicity degrees of freedom, the other two vortex degrees of freedom from the vortex degrees of freedom that characterize the vortex-membranes, A or B. The basis to define the microscopic structure of a knot is given by j i ↑; A , j i ↑; B , j i ↓; A , j i ↓; B .

We define operator of knot states by the region of the phase angle of a knot: for the case of <sup>ϕ</sup><sup>0</sup> mod 2ð Þ <sup>π</sup> <sup>∈</sup>ð � �π; <sup>0</sup> , we have <sup>c</sup>†j i<sup>0</sup> ; for the case of <sup>ϕ</sup><sup>0</sup> mod 2ð Þ <sup>π</sup> <sup>∈</sup> ð � <sup>0</sup>; <sup>π</sup> , we have <sup>c</sup>†j i<sup>0</sup> � �† . As shown in Figure 2, we label the knots by Wannier state j i <sup>i</sup>; <sup>A</sup>; <sup>↑</sup> , j i <sup>i</sup> <sup>þ</sup> <sup>1</sup>; <sup>A</sup>; <sup>↑</sup> <sup>∗</sup> , j i i þ 2; A; ↑ , j i <sup>i</sup> <sup>þ</sup> <sup>3</sup>; <sup>A</sup>; <sup>↑</sup> <sup>∗</sup> ….

To characterize the energy cost from global winding, we use an effective Hamiltonian to describe the coupling between two-knot states along x<sup>I</sup> -direction on 3D SOC knot-crystal

$$J\mathbf{c}\_{\mathbf{A}/\text{Bi}}^{+}T\_{\mathbf{A}/\text{B},\mathbf{A}/\text{B}}^{I}\mathbf{c}\_{\mathbf{A}/\text{B},i+\mathbf{c}^{l}}\tag{21}$$

with the annihilation operator of knots at the site <sup>i</sup>, <sup>c</sup><sup>A</sup>=B,i <sup>¼</sup> <sup>c</sup><sup>A</sup>=B, <sup>↑</sup>,i c<sup>A</sup>=B, <sup>↓</sup>,i !. J is the coupling constant between two nearest-neighbor knots. According to the generalized translation symmetry, the transfer matrices T<sup>I</sup> <sup>A</sup>=B,A=<sup>B</sup> along xI -direction are defined by

TI <sup>A</sup>,<sup>A</sup> <sup>¼</sup> <sup>T</sup><sup>I</sup> <sup>B</sup>,<sup>B</sup> ¼ e ia b<sup>k</sup> I �σI � � (22)

Figure 2. An illustration of knot states in vortex-representation: A and B denote two 1D vortex-lines. Here B\* denotes conjugate representation of vortex-line-B. The curves with blue dots denote knots on the knot-crystal—the curves with blue dot above the line are denoted by c† <sup>i</sup> j i<sup>0</sup> and the curves with blue dot below the line are denoted by <sup>c</sup>† <sup>i</sup> j i<sup>0</sup> � �† .

and

$$T\_{\mathbf{A},\mathbf{B}}^{l} = T\_{\mathbf{B},\mathbf{A}}^{l} = \mathbf{0}.\tag{23}$$

ψð Þ¼ x

Hcoupling ¼ Hb coupling,<sup>B</sup> þ Hb coupling,<sup>A</sup>

E<sup>A</sup>=B, <sup>k</sup> ≃ceff k

! �k ! 0 � �� <sup>σ</sup>

� � and <sup>c</sup>eff <sup>¼</sup> <sup>2</sup>aJ is the velocity. In the following part we ignore <sup>k</sup>

Next, we consider the mass term from leapfrogging motion, of which the angular frequency ω<sup>∗</sup>. For leapfrogging motion obtained by [10], the function of the two entangled vortex-

> 1 CA <sup>¼</sup> <sup>r</sup><sup>0</sup> 2

� �; at <sup>t</sup> <sup>¼</sup> <sup>π</sup>

frogging knot-crystal leads to periodic varied knot states, i.e., at t ¼ 0 we have a knot on

denoted by j i σ; B . As a result, the leapfrogging motion becomes a global winding along time

state ϕAmod 2ð Þ π ∈ð � �π; 0 turns into a knot state ϕ<sup>B</sup> mod 2ð Þ π ∈ð � �π; 0 . Thus, we use the

, <sup>t</sup> <sup>þ</sup> <sup>3</sup><sup>π</sup> <sup>ω</sup><sup>∗</sup> ; <sup>B</sup> � � �

representation of knot states for knot-crystal in Figure 2(c). After a time period <sup>t</sup> <sup>¼</sup> <sup>π</sup>

<sup>¼</sup> <sup>2</sup>aJ<sup>X</sup> k ψ†

þ2aJ<sup>X</sup> k ψ†

membranes at a given point in geometric space is simplified by

0 B@

1

zA x !; t � �

0 B@

zB x !; t � �

<sup>ω</sup><sup>∗</sup> ; <sup>B</sup> � � �

vortex-membrane-A that is denoted by j i <sup>σ</sup>; <sup>A</sup> ; at <sup>t</sup> <sup>¼</sup> <sup>π</sup>

zA x !¼ <sup>0</sup>; <sup>t</sup> � �

zB x !¼ <sup>0</sup>; <sup>t</sup> � �

CA <sup>¼</sup> <sup>1</sup> 0

, <sup>t</sup> <sup>þ</sup> <sup>2</sup><sup>π</sup> <sup>ω</sup><sup>∗</sup> ; <sup>A</sup> � � �

following formulation to characterize the leapfrogging process,

denote the two possible winding directions along a given direction e

In continuum limit, we have

where the dispersion of knots is

where k ! <sup>0</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup> ; <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2

At t ¼ 0, we have

direction, j i <sup>t</sup>; <sup>A</sup> , <sup>t</sup> <sup>þ</sup> <sup>π</sup>

ψA,<sup>↑</sup> t; x ! � � 1

Topological Interplay between Knots and Entangled Vortex-Membranes

http://dx.doi.org/10.5772/intechopen.72809

CCCCCCCCA

<sup>A</sup>, <sup>k</sup> σ<sup>x</sup> cos kx þ σ<sup>y</sup> cos ky þ σ<sup>z</sup> cos kz � �ψA, <sup>k</sup>

<sup>B</sup>, <sup>k</sup> σ<sup>x</sup> cos kx þ σ<sup>y</sup> cos ky þ σ<sup>z</sup> cos kz

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>i</sup>ω∗<sup>t</sup> <sup>1</sup> � <sup>e</sup><sup>i</sup>ω∗<sup>t</sup> !

<sup>ω</sup><sup>∗</sup>, we have

!.

� �ψB, <sup>k</sup> (29)

! h i, (30)

zA x !; t � �

0 B@

zB x !; t � � ! 0.

: (31)

1

<sup>ω</sup><sup>∗</sup> we have a knot on vortex-membrane-B

, … See the illustration of vortex-

CA <sup>¼</sup> <sup>0</sup> 1

� �. The leap-

<sup>ω</sup><sup>∗</sup>, a knot

(28)

41

0

BBBBBBBB@

ψB,<sup>↑</sup> t; x ! � �

ψA,<sup>↓</sup> t; x ! � �

ψB,<sup>↓</sup> t; x ! � �

where A, B label vortex degrees of freedom and ↑, ↓ label two spin degrees of freedom that

After considering the spin rotation symmetry and the symmetry of vortex-membrane-A and vortex-membrane-B, the effective Hamiltonian from global winding energy can be described by a familiar formulation

$$
\mathcal{H}\_{\text{coupling}} = \hat{\mathcal{H}}\_{\text{coupling}, \text{B}} + \hat{\mathcal{H}}\_{\text{coupling}, \text{A}} \tag{24}
$$

where

$$\widehat{\mathcal{H}}\_{\text{coupleing},\mathcal{A}} = J \sum\_{i,\mathbf{l}} c\_{\mathbf{A},i}^{\dagger} \varepsilon^{\text{int}\left(\overset{\frown{\mathbf{\hat{\mathbf{\cdot}}}^{\mathbf{\cdot}}}{\mathbf{\hat{\mathbf{\cdot}}}^{\mathbf{\cdot}}}\right)} c\_{\mathbf{A},i+\mathbf{c}^{l}} + h.c.\tag{25}$$

and

$$
\hat{\mathcal{H}}\_{\text{coupleing,B}} = J \sum\_{i,\boldsymbol{\iota}} c\_{\mathbf{B},i}^{\dagger} \mathbf{e}^{\mathrm{id} \begin{pmatrix} \hat{\boldsymbol{\iota}}^{\boldsymbol{\iota}} & \boldsymbol{\iota}^{\boldsymbol{\iota}} \end{pmatrix}} c\_{\mathbf{B},i+\boldsymbol{\iota}^{\boldsymbol{\iota}}} + h.c. \tag{26}
$$

We then use path-integral formulation to characterize the effective Hamiltonian for a knotcrystal as

$$\int \mathcal{D}\psi^\dagger \left(t, \overrightarrow{\dot{x}}\right) \mathcal{D}\psi(t) e^{i\mathcal{S}/\hbar} \tag{27}$$

where <sup>S</sup> <sup>¼</sup> <sup>Ð</sup> Ldt and L ¼ i P <sup>i</sup> ψ† <sup>i</sup> ∂tψ<sup>i</sup> � Hcoupling. To describe the knot states on 3D knotcrystal, we have introduced a four-component fermion field to be

Topological Interplay between Knots and Entangled Vortex-Membranes http://dx.doi.org/10.5772/intechopen.72809 41

$$\psi(\mathbf{x}) = \begin{pmatrix} \psi\_{\mathbf{A},\uparrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{B},\uparrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{A},\downarrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{B},\downarrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{B},\downarrow} \left( t, \overrightarrow{\mathbf{x}} \right) \end{pmatrix} \tag{28}$$

where A, B label vortex degrees of freedom and ↑, ↓ label two spin degrees of freedom that denote the two possible winding directions along a given direction e !.

In continuum limit, we have

$$\begin{split} \mathcal{H}\_{\text{coupling}} &= \dot{\mathcal{H}}\_{\text{coupling,B}} + \dot{\mathcal{H}}\_{\text{coupling,A}} \\ &= 2a \text{J} \sum\_{k} \psi\_{\text{A,k}}^{\dagger} \left[ \sigma\_{\text{x}} \cos k\_{\text{x}} + \sigma\_{\text{y}} \cos k\_{\text{y}} + \sigma\_{\text{z}} \cos k\_{\text{z}} \right] \psi\_{\text{A,k}} \\ &+ 2a \text{J} \sum\_{k} \psi\_{\text{B,k}}^{\dagger} \left[ \sigma\_{\text{x}} \cos k\_{\text{x}} + \sigma\_{\text{y}} \cos k\_{\text{y}} + \sigma\_{\text{z}} \cos k\_{\text{z}} \right] \psi\_{\text{B,k}} \end{split} \tag{29}$$

where the dispersion of knots is

and

where

and

crystal as

where <sup>S</sup> <sup>¼</sup> <sup>Ð</sup>

Ldt and L ¼ i

by a familiar formulation

blue dot above the line are denoted by c†

40 Superfluids and Superconductors

TI <sup>A</sup>,<sup>B</sup> <sup>¼</sup> <sup>T</sup><sup>I</sup>

> X i, I c † A,i e ia b<sup>k</sup> I �σI � �

> X i, I c † B,i e ia b<sup>k</sup> I �σI � �

Dψ† t; x ! � �

We then use path-integral formulation to characterize the effective Hamiltonian for a knot-

Dψð Þt e

Hb coupling,<sup>A</sup> ¼ J

Hb coupling,<sup>B</sup> ¼ J

ð

P <sup>i</sup> ψ†

crystal, we have introduced a four-component fermion field to be

After considering the spin rotation symmetry and the symmetry of vortex-membrane-A and vortex-membrane-B, the effective Hamiltonian from global winding energy can be described

Figure 2. An illustration of knot states in vortex-representation: A and B denote two 1D vortex-lines. Here B\* denotes conjugate representation of vortex-line-B. The curves with blue dots denote knots on the knot-crystal—the curves with

<sup>B</sup>,<sup>A</sup> ¼ 0: (23)

<sup>i</sup> j i<sup>0</sup> � �† .

cA,iþeI þ h:c: (25)

cB,iþeI þ h:c: (26)

<sup>i</sup>S=<sup>ℏ</sup> (27)

<sup>i</sup> ∂tψ<sup>i</sup> � Hcoupling. To describe the knot states on 3D knot-

Hcoupling ¼ Hb coupling,<sup>B</sup> þ Hb coupling,<sup>A</sup> (24)

<sup>i</sup> j i<sup>0</sup> and the curves with blue dot below the line are denoted by <sup>c</sup>†

$$E\_{\mathbf{A}/\mathbf{B},k} \simeq \mathfrak{c}\_{\text{eff}} \left[ \left( \overrightarrow{k} - \overrightarrow{k}\_0 \right) \cdot \overrightarrow{\sigma} \right],\tag{30}$$

where k ! <sup>0</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup> ; <sup>π</sup> <sup>2</sup> ; <sup>π</sup> 2 � � and <sup>c</sup>eff <sup>¼</sup> <sup>2</sup>aJ is the velocity. In the following part we ignore <sup>k</sup> ! 0.

Next, we consider the mass term from leapfrogging motion, of which the angular frequency ω<sup>∗</sup>. For leapfrogging motion obtained by [10], the function of the two entangled vortexmembranes at a given point in geometric space is simplified by

$$\begin{pmatrix} \mathbf{z}\_{\mathbf{A}} \left( \overrightarrow{\mathbf{x}} = \mathbf{0}, t \right) \\\\ \mathbf{z}\_{\mathbf{B}} \left( \overrightarrow{\mathbf{x}} = \mathbf{0}, t \right) \end{pmatrix} = \frac{r\_0}{2} \begin{pmatrix} 1 + e^{i\boldsymbol{\alpha}^\* t} \\ 1 - e^{i\boldsymbol{\alpha}^\* t} \end{pmatrix}. \tag{31}$$

$$\text{At } t = 0\text{, we have } \begin{pmatrix} \mathbf{z}\_{\mathbf{A}} \left( \stackrel{\rightarrow}{\mathbf{x}}, t \right) \\\\ \mathbf{z}\_{\mathbf{B}} \left( \stackrel{\rightarrow}{\mathbf{x}}, t \right) \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; \text{ at } t = \frac{\pi}{\omega \nu}, \text{ we have } \begin{pmatrix} \mathbf{z}\_{\mathbf{A}} \left( \stackrel{\rightarrow}{\mathbf{x}}, t \right) \\\\ \mathbf{z}\_{\mathbf{B}} \left( \stackrel{\rightarrow}{\mathbf{x}}, t \right) \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \text{ The inequality holds for } \mathbf{A} \in \mathbb{R}^{m \times n} \text{ and } \mathbf{z}\_{\mathbf{A}} \in \mathbb{R}^{n \times n}.$$

frogging knot-crystal leads to periodic varied knot states, i.e., at t ¼ 0 we have a knot on vortex-membrane-A that is denoted by j i <sup>σ</sup>; <sup>A</sup> ; at <sup>t</sup> <sup>¼</sup> <sup>π</sup> <sup>ω</sup><sup>∗</sup> we have a knot on vortex-membrane-B denoted by j i σ; B . As a result, the leapfrogging motion becomes a global winding along time direction, j i <sup>t</sup>; <sup>A</sup> , <sup>t</sup> <sup>þ</sup> <sup>π</sup> <sup>ω</sup><sup>∗</sup> ; <sup>B</sup> � � � , <sup>t</sup> <sup>þ</sup> <sup>2</sup><sup>π</sup> <sup>ω</sup><sup>∗</sup> ; <sup>A</sup> � � � , <sup>t</sup> <sup>þ</sup> <sup>3</sup><sup>π</sup> <sup>ω</sup><sup>∗</sup> ; <sup>B</sup> � � � , … See the illustration of vortexrepresentation of knot states for knot-crystal in Figure 2(c). After a time period <sup>t</sup> <sup>¼</sup> <sup>π</sup> <sup>ω</sup><sup>∗</sup>, a knot state ϕAmod 2ð Þ π ∈ð � �π; 0 turns into a knot state ϕ<sup>B</sup> mod 2ð Þ π ∈ð � �π; 0 . Thus, we use the following formulation to characterize the leapfrogging process,

$$
\psi\_\mathbf{A}^\dagger \psi\_\mathbf{B}^\dagger. \tag{32}
$$

p !¼ <sup>ℏ</sup>knot <sup>k</sup>

<sup>c</sup>eff <sup>¼</sup> <sup>2</sup>a�<sup>J</sup>

j i B=A ).

where <sup>Ψ</sup> <sup>¼</sup> <sup>Ψ</sup>†

where U ¼ exp iπ

and

!

of entanglement matrices Γ

is the momentum operator. <sup>Ψ</sup>† <sup>¼</sup> <sup>ψ</sup><sup>∗</sup>

space-time, i.e., ð Þ! x; y; z ð Þ x; y; z; t . Here, we may consider Γ

!

along x/y/z/t-direction is characterize by e<sup>i</sup><sup>Γ</sup>

along arbitrary spatial direction e

tor of four-component fermions. mknotc<sup>2</sup>

<sup>A</sup>,↑;ψB, <sup>↑</sup>;ψ<sup>∗</sup>

<sup>ℏ</sup>knot play the role of light speed where a is a fixed length that denotes the half pitch of

Due to Lorentz invariance (see below discussion), the geometric space becomes geometric

matrices along spatial and tempo direction in winding space-time, respectively. A complete set

! � <sup>b</sup><sup>k</sup> � <sup>x</sup>

becomes topological defect of 3 + 1D entanglement—a knot is not only anti-phase changing

direction (along tempo direction, a knot switches a knot state Aj i =B to another knot state

<sup>¼</sup> <sup>Ψ</sup> <sup>i</sup>γμb∂<sup>μ</sup> � <sup>m</sup>knot � �<sup>Ψ</sup>

, <sup>γ</sup><sup>2</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

<sup>γ</sup><sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>τ</sup><sup>x</sup> <sup>⊗</sup> <sup>1</sup>, <sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>i</sup>γ<sup>0</sup>

j i A j i B � �

In addition, we point out that there exists intrinsic relationship between the knot states of

becomes an object with staggered R/L zeroes along x/y/z spatial directions and time direction; From the vortex-representation of knot states, the knot-crystal becomes an object with global winding along x/y/z spatial directions and time direction. See the illustration of knot states of

<sup>¼</sup> <sup>U</sup> j i <sup>L</sup> j i R

Γ2

, <sup>γ</sup><sup>3</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

γ1 γ2 γ3 :

!

� � � � . From the sublattice-representation of knot states, the knot-crystal

Γ3

∂tΨ � H3D

Finally, the low energy effective Lagrangian of 3D SOC knot-crystal is obtained as

γ0, γμ are the reduced Gamma matrices,

<sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

sublattice-representation and the knot states of vortex-representation

0 �i i 0

vortex-representation on a knot-crystal in Figure 2.

Γ1

<sup>L</sup>3D <sup>¼</sup> <sup>i</sup>Ψ†

the windings on the knot-crystal. In the following parts, we set ℏknot ¼ 1 and ceff ¼ 1.

<sup>A</sup>, <sup>↓</sup>;ψB,<sup>↓</sup>

Topological Interplay between Knots and Entangled Vortex-Membranes

� � is the annihilation opera-

http://dx.doi.org/10.5772/intechopen.72809

! and <sup>e</sup><sup>i</sup>Γ5�b<sup>ω</sup> <sup>t</sup>, respectively. Now, the knot

and Γ<sup>5</sup> to be entanglement

, (41)

� � (43)

(40)

43

(42)

eff <sup>¼</sup> <sup>2</sup>ℏknotω<sup>∗</sup> plays role of the mass of knots and

!

! but also becomes anti-phase changing along tempo

; <sup>Γ</sup><sup>5</sup> � � is called entanglement pattern. The coordinate transformation

After considering the energy from the leapfrogging process, a corresponding term is given by

$$2\hbar\_{\text{knot}}\omega^\*\psi\_\text{A}^\dagger\psi\_\text{B}^\dagger + h.c.\tag{33}$$

From the global rotating motion denoted e�iω0<sup>t</sup> , the winding states also change periodically. Because the contribution from global rotating motion e�iω0<sup>t</sup> is always canceled by shifting the chemical potential, we do not consider its effect.

From above equation, in the limit k � �! � � � � ! 0 we derive low energy effective Hamiltonian as

$$\begin{split} \mathcal{H}\_{\text{3D}} & \simeq 2a \!\!/ \sum\_{k} \psi\_{\text{A},k}^{\dagger} \left( \overrightarrow{\sigma} \cdot \overrightarrow{\bar{k}} \right) \psi\_{\text{A},k} \\ & + 2a \!\!/ \sum\_{k} \psi\_{\text{B},k}^{\dagger} \left( \overrightarrow{\sigma} \cdot \overrightarrow{\bar{k}} \right) \psi\_{\text{B},k} \\ & + 2\hbar\_{\text{knot}} \omega^{\*} \sum\_{k,\sigma} \psi\_{\text{A},\sigma,k}^{\dagger} \Psi\_{\text{B},\sigma,k}^{\dagger} \\ & = c\_{\text{eff}} \!\!/ \Psi^{\dagger} \left[ \!\!/ \!/ \!/ \!z \otimes \!\!/ \!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!\!/ \!\!\!/ \!\!\!/ \!\!\!/ \!\!\!/ \!\!\!/ \!\!\!/ \!\!\!\!/ \!\!\!\!/ \!\!\!\!\!\!\!\!/ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$$

where

$$\Psi(\mathbf{x}) = \begin{pmatrix} \psi\_{\mathbf{A},\uparrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{B},\uparrow}^{\*} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{A},\downarrow} \left( t, \overrightarrow{\mathbf{x}} \right) \\ \psi\_{\mathbf{B},\downarrow}^{\*} \left( t, \overrightarrow{\mathbf{x}} \right) \end{pmatrix}. \tag{36}$$

We then re-write the effective Hamiltonian to be

$$\mathcal{H}\_{\text{3D}} = \int \left(\Psi^{\dagger} \hat{H}\_{\text{3D}} \Psi \right) d^3 x \tag{37}$$

and

$$
\widehat{H}\_{\text{3D}} = \mathfrak{c}\_{\text{eff}} \stackrel{\cdot}{\Gamma} \cdot \vec{p}\_{\text{kroot}} + m\_{\text{knot}} \mathfrak{c}\_{\text{eff}}^2 \Gamma^5 \tag{38}
$$

where

$$
\Gamma^5 = \tau^\times \otimes 1, \Gamma^1 = \stackrel{\rightarrow}{\tau^z} \otimes \sigma^\times,\tag{39}
$$

$$
\Gamma^2 = \tau^z \otimes \sigma^y, \Gamma^3 = \tau^z \otimes \sigma^z.
$$

p !¼ <sup>ℏ</sup>knot <sup>k</sup> ! is the momentum operator. <sup>Ψ</sup>† <sup>¼</sup> <sup>ψ</sup><sup>∗</sup> <sup>A</sup>,↑;ψB, <sup>↑</sup>;ψ<sup>∗</sup> <sup>A</sup>, <sup>↓</sup>;ψB,<sup>↓</sup> � � is the annihilation operator of four-component fermions. mknotc<sup>2</sup> eff <sup>¼</sup> <sup>2</sup>ℏknotω<sup>∗</sup> plays role of the mass of knots and <sup>c</sup>eff <sup>¼</sup> <sup>2</sup>a�<sup>J</sup> <sup>ℏ</sup>knot play the role of light speed where a is a fixed length that denotes the half pitch of the windings on the knot-crystal. In the following parts, we set ℏknot ¼ 1 and ceff ¼ 1.

Due to Lorentz invariance (see below discussion), the geometric space becomes geometric space-time, i.e., ð Þ! x; y; z ð Þ x; y; z; t . Here, we may consider Γ ! and Γ<sup>5</sup> to be entanglement matrices along spatial and tempo direction in winding space-time, respectively. A complete set of entanglement matrices Γ ! ; <sup>Γ</sup><sup>5</sup> � � is called entanglement pattern. The coordinate transformation along x/y/z/t-direction is characterize by e<sup>i</sup><sup>Γ</sup> ! � <sup>b</sup><sup>k</sup> � <sup>x</sup> ! and <sup>e</sup><sup>i</sup>Γ5�b<sup>ω</sup> <sup>t</sup>, respectively. Now, the knot becomes topological defect of 3 + 1D entanglement—a knot is not only anti-phase changing along arbitrary spatial direction e ! but also becomes anti-phase changing along tempo direction (along tempo direction, a knot switches a knot state Aj i =B to another knot state j i B=A ).

Finally, the low energy effective Lagrangian of 3D SOC knot-crystal is obtained as

$$\begin{aligned} \mathcal{L}\_{\text{3D}} &= i\Psi^{\dagger}\partial\_{\text{i}}\Psi - \mathcal{H}\_{\text{3D}} \\ &= \overline{\Psi} \Big( \mathrm{i}\gamma^{\mu}\widehat{\partial}\_{\mu} - m\_{\text{krot}} \Big) \Psi \end{aligned} \tag{40}$$

where <sup>Ψ</sup> <sup>¼</sup> <sup>Ψ</sup>† γ0, γμ are the reduced Gamma matrices,

$$
\gamma^1 = \gamma^0 \Gamma^1, \gamma^2 = \gamma^0 \Gamma^2, \gamma^3 = \gamma^0 \Gamma^3,\tag{41}
$$

and

ψ† Aψ†

2ℏknotω<sup>∗</sup>

� �! � � �

¼ ceff Ð

<sup>þ</sup>mknotc<sup>2</sup> eff Ð

Ψð Þ¼ x

H3D ¼

Hb 3D ¼ ceff Γ

<sup>Γ</sup><sup>2</sup> <sup>¼</sup> <sup>τ</sup><sup>z</sup> <sup>⊗</sup> <sup>σ</sup><sup>y</sup>

ð Ψ† Hb 3DΨ � �

! �p !

<sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>τ</sup><sup>x</sup> <sup>⊗</sup> <sup>1</sup>, <sup>Γ</sup><sup>1</sup> <sup>¼</sup> <sup>τ</sup><sup>z</sup> <sup>⊗</sup> <sup>σ</sup><sup>x</sup>

<sup>H</sup>3D <sup>≃</sup>2aJ<sup>X</sup>

k ψ† <sup>A</sup>, <sup>k</sup> σ ! � <sup>k</sup> � �!

<sup>þ</sup>2ℏknotω<sup>∗</sup><sup>P</sup>

Ψ† T <sup>z</sup> ⊗ σ

k,σψ†

! � bk h i � �

> ! ÞΨd<sup>3</sup> x:

Ψ† τ<sup>x</sup> ⊗ 1

ψA,<sup>↑</sup> t; x ! � �

ψA,<sup>↓</sup> t; x ! � �

ψ∗ <sup>B</sup>, <sup>↑</sup> t; x ! � �

0

BBBBBBBB@

ψ∗ <sup>B</sup>, <sup>↓</sup> t; x ! � �

þ2aJ<sup>X</sup> k ψ† <sup>B</sup>, <sup>k</sup> σ ! � <sup>k</sup> � �!

From the global rotating motion denoted e�iω0<sup>t</sup>

chemical potential, we do not consider its effect.

We then re-write the effective Hamiltonian to be

From above equation, in the limit k

42 Superfluids and Superconductors

where

and

where

After considering the energy from the leapfrogging process, a corresponding term is given by

ψ† Aψ†

Because the contribution from global rotating motion e�iω0<sup>t</sup> is always canceled by shifting the

<sup>B</sup>: (32)

<sup>B</sup> þ h:c: (33)

, the winding states also change periodically.

� (35)

: (36)

x (37)

effΓ<sup>5</sup> (38)

(34)

(39)

� ! 0 we derive low energy effective Hamiltonian as

ψA, <sup>k</sup>

ψB, <sup>k</sup>

<sup>A</sup>,σ, <sup>k</sup>ψ† B,σ, k

> Ψd<sup>3</sup> x

1

CCCCCCCCA

d3

2

,

:

knot þ mknotc

!

, <sup>Γ</sup><sup>3</sup> <sup>¼</sup> <sup>τ</sup><sup>z</sup> <sup>⊗</sup> <sup>σ</sup><sup>z</sup>

$$
\gamma^0 = \Gamma^5 = \tau\_x \otimes 1,\\
\gamma^5 = \dot{\imath}\dot{\gamma}^0 \gamma^1 \gamma^2 \gamma^3. \tag{42}
$$

In addition, we point out that there exists intrinsic relationship between the knot states of sublattice-representation and the knot states of vortex-representation

$$\mathcal{U}\begin{pmatrix}|\mathbf{A}\rangle\\|\mathbf{B}\rangle\end{pmatrix} = \mathcal{U}\begin{pmatrix}|\mathbf{L}\rangle\\|\mathbf{R}\rangle\end{pmatrix} \tag{43}$$

where U ¼ exp iπ 0 �i i 0 � � � � . From the sublattice-representation of knot states, the knot-crystal becomes an object with staggered R/L zeroes along x/y/z spatial directions and time direction; From the vortex-representation of knot states, the knot-crystal becomes an object with global winding along x/y/z spatial directions and time direction. See the illustration of knot states of vortex-representation on a knot-crystal in Figure 2.

#### 3.1.3. Emergent Lorentz-invariance

We discuss the emergent Lorentz-invariance for knot states on a knot-crystal.

Since the Fermi-velocity ceff only depends on the microscopic parameter J and a, we may regard ceff to be "light-velocity" and the invariance of light-velocity becomes an fundamental principle for the knot physics. The Lagrangian for massive Dirac fermions indicates emergent SO(3,1) Lorentz-invariance. The SO(3,1) Lorentz transformations SLor is defined by

$$\mathcal{S}\_{\text{Lor}} \gamma^{\mu} \mathcal{S}\_{\text{Lor}}^{-1} = \gamma^{\prime \mu} \tag{44}$$

where Eknot ≃ <sup>p</sup>

V <sup>p</sup> e�i2ω∗<sup>t</sup>

function <sup>1</sup>ffiffiffi

x

!2 knot <sup>2</sup>mknot, p !

Lorentz transformation SLor x

a given direction e

with ϕx, ϕy, ϕ<sup>z</sup> ∈ ð � 0; π .

and ϕ !

noninertial system and curved space-time.

3.2. Dirac model on winding space

determined by two kinds of values: X

denote internal winding angles

knot ≃ ω v

exp �i Eknott � p

!ÞÞ comes from the Lorentz boosting <sup>S</sup>Lor.

! knot� x

t ! t <sup>0</sup> x !; t � � <sup>¼</sup> <sup>t</sup>� <sup>x</sup>

x !! <sup>x</sup> !0 x !; t � � <sup>¼</sup> <sup>x</sup>

Ψ x !; t � � ! <sup>Ψ</sup><sup>0</sup> <sup>x</sup>

!; t

We can also do non-uniform Lorentz transformation SLor x

Noninertial system can be obtained by considering non-uniformly velocities, i.e., v

According to the linear dispersion for knots, we can do local Lorentz transformation on x

! and <sup>m</sup>knotc<sup>2</sup> <sup>¼</sup> <sup>2</sup>ω<sup>∗</sup>. As a result, we derive a new distribution of

Topological Interplay between Knots and Entangled Vortex-Membranes

V

http://dx.doi.org/10.5772/intechopen.72809

<sup>p</sup> expð � �i Eð knott p

!! <sup>Δ</sup> <sup>v</sup>

!; t � �, i.e.,

> ! . Along

! knot� 45

! x !; t � �.

!; t � � i.e.,

(50)

knot-pieces by doing Lorentz transformation, that are described by the plane-wave wave-

! � � � � . The new wave-function <sup>1</sup>ffiffiffi

! �<sup>Δ</sup> <sup>v</sup> ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � Δ v ! � �<sup>2</sup> <sup>r</sup> ,

! �<sup>Δ</sup> <sup>v</sup> ! �<sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � Δ v ! � �<sup>2</sup> <sup>r</sup> :

!0 x !; t � �; <sup>t</sup>

¼ SLor x !; t � � � <sup>Ψ</sup> <sup>x</sup>

where the new wave-functions of all quantum states change following the non-uniform

In this part, we show the effective Dirac model of knot states on winding space.

<sup>¼</sup> ð Þ¼ <sup>X</sup>;Y; <sup>Z</sup> <sup>1</sup>

¼ ϕx; ϕy; ϕ<sup>z</sup> � � <sup>¼</sup> <sup>Φ</sup>

!

X !

> ϕ !

The coordinate measurement of zero-lattice on winding space is the winding angles, Φ

are integer numbers

π Φ ! � 1 π Φ ! !; t

!; t

� �. It is obvious that there exists intrinsic relationship between

<sup>0</sup> x !; t

!, after shifting the distance a, the winding angle changes π. The position is

!

� � � �

� � on knot states <sup>Ψ</sup> <sup>x</sup>

� � (51)

modπ (52)

modπ (53)

(μ ¼ 0; 1; 2; 3) and

$$\mathcal{S}\_{\text{Lor}} \boldsymbol{\chi}^{5} \mathcal{S}\_{\text{Lor}}^{-1} = \boldsymbol{\chi}^{5}. \tag{45}$$

For a knot state with a global velocity v !, due to SO(3,1) Lorentz-invariance, we can do Lorentz boosting on x !; t � � by considering the velocity of a knot,

$$t \to t' = \frac{t - \overrightarrow{x} \cdot \overrightarrow{v}}{\sqrt{1 - \overrightarrow{v}^2}},$$

$$\overrightarrow{x} \to \overrightarrow{x}' = \frac{\overrightarrow{x} - \overrightarrow{v} \cdot t}{\sqrt{1 - \overrightarrow{v}^2}}.\tag{46}$$

We can do non-uniform Lorentz transformation SLor x !; t � � on knot states <sup>Ψ</sup> <sup>x</sup> !; t � �. The inertial reference-frame for quantum states of the knot is defined under Lorentz boost, i.e.,

$$
\Psi\left(\overrightarrow{\mathbf{x}},t\right) \to \Psi'\left(\overrightarrow{\mathbf{x}}',t'\right) = \mathbf{S}\_{\text{Lor}} \cdot \Psi\left(\overrightarrow{\mathbf{x}}',t'\right).\tag{47}
$$

For a particle-like knot, a uniform wave-function of knot states ψð Þt is

$$
\psi(t) = \frac{1}{\sqrt{V}} e^{-i2\omega^\* t}.\tag{48}
$$

Under Lorentz transformation with small velocity v ! � � � � � �, this wave-function <sup>ψ</sup>ð Þ<sup>t</sup> is transformed into

$$\begin{split} \psi(t) &= \frac{1}{\sqrt{V}} e^{-i2\omega^\*t} \\ &\to \psi' = \frac{1}{\sqrt{V}} e^{-i2\omega^\*t'} \\ &\simeq \frac{1}{\sqrt{V}} e^{-i2\omega^\*t} \exp\left(-i\left(E\_{\text{knot}}t - \overrightarrow{p}\_{\text{knot}}\cdot\overrightarrow{\chi}\right)\right) \end{split} \tag{49}$$

where Eknot ≃ <sup>p</sup> !2 knot <sup>2</sup>mknot, p ! knot ≃ ω v ! and <sup>m</sup>knotc<sup>2</sup> <sup>¼</sup> <sup>2</sup>ω<sup>∗</sup>. As a result, we derive a new distribution of knot-pieces by doing Lorentz transformation, that are described by the plane-wave wavefunction <sup>1</sup>ffiffiffi V <sup>p</sup> e�i2ω∗<sup>t</sup> exp �i Eknott � p ! knot� x ! � � � � . The new wave-function <sup>1</sup>ffiffiffi V <sup>p</sup> expð � �i Eð knott p ! knot� x !ÞÞ comes from the Lorentz boosting <sup>S</sup>Lor.

Noninertial system can be obtained by considering non-uniformly velocities, i.e., v !! <sup>Δ</sup> <sup>v</sup> ! x !; t � �. According to the linear dispersion for knots, we can do local Lorentz transformation on x !; t � � i.e.,

$$\begin{split} t &\to t'\left(\overrightarrow{\mathbf{x}}, t\right) = \frac{t - \overrightarrow{\mathbf{x}} \cdot \boldsymbol{\Delta} \cdot \overrightarrow{\boldsymbol{v}}}{\sqrt{1 - \left(\boldsymbol{\Delta} \cdot \overrightarrow{\boldsymbol{v}}\right)^2}}, \\ \overrightarrow{\mathbf{x}} &\to \overrightarrow{\mathbf{x}}'\left(\overrightarrow{\mathbf{x}}, t\right) = \frac{\overrightarrow{\boldsymbol{x}} - \boldsymbol{\Delta} \cdot \overrightarrow{\boldsymbol{v}} \cdot \boldsymbol{t}}{\sqrt{1 - \left(\boldsymbol{\Delta} \cdot \overrightarrow{\boldsymbol{v}}\right)^2}}. \end{split} \tag{50}$$

We can also do non-uniform Lorentz transformation SLor x !; t � � on knot states <sup>Ψ</sup> <sup>x</sup> !; t � �, i.e.,

$$\begin{split} \Psi \left( \overrightarrow{\mathbf{x}}, t \right) &\to \Psi' \left( \overrightarrow{\mathbf{x}}' \left( \overrightarrow{\mathbf{x}}, t \right), t' \left( \overrightarrow{\mathbf{x}}, t \right) \right) \\ &= \mathsf{S}\_{\text{Lor}} \left( \overrightarrow{\mathbf{x}}, t \right) \cdot \Psi \left( \overrightarrow{\mathbf{x}}, t \right) \end{split} \tag{51}$$

where the new wave-functions of all quantum states change following the non-uniform Lorentz transformation SLor x !; t � �. It is obvious that there exists intrinsic relationship between noninertial system and curved space-time.

#### 3.2. Dirac model on winding space

3.1.3. Emergent Lorentz-invariance

44 Superfluids and Superconductors

For a knot state with a global velocity v

!; t � �

(μ ¼ 0; 1; 2; 3) and

boosting on x

into

We discuss the emergent Lorentz-invariance for knot states on a knot-crystal.

Lorentz-invariance. The SO(3,1) Lorentz transformations SLor is defined by

by considering the velocity of a knot,

t ! t

reference-frame for quantum states of the knot is defined under Lorentz boost, i.e.,

! Ψ<sup>0</sup> x !0 ; t 0 � �

ψðÞ¼ t

ψðÞ¼ t

1 ffiffiffiffi V p e

1 ffiffiffiffi V p e

! <sup>ψ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

<sup>≃</sup> <sup>1</sup> ffiffiffiffi V p e

x !! <sup>x</sup> !0 ¼ x ! � <sup>v</sup> ! �<sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v !2

For a particle-like knot, a uniform wave-function of knot states ψð Þt is

We can do non-uniform Lorentz transformation SLor x

Under Lorentz transformation with small velocity v

Ψ x !; t � �

Since the Fermi-velocity ceff only depends on the microscopic parameter J and a, we may regard ceff to be "light-velocity" and the invariance of light-velocity becomes an fundamental principle for the knot physics. The Lagrangian for massive Dirac fermions indicates emergent SO(3,1)

Lor <sup>¼</sup> <sup>γ</sup>0<sup>μ</sup> (44)

!, due to SO(3,1) Lorentz-invariance, we can do Lorentz

<sup>q</sup> : (46)

on knot states Ψ x

!; t � �

: (47)

: (48)

�, this wave-function <sup>ψ</sup>ð Þ<sup>t</sup> is transformed

! knot� x

! � � � �

. The inertial

(49)

: (45)

SLorγ<sup>μ</sup>S�<sup>1</sup>

SLorγ<sup>5</sup> S�<sup>1</sup> Lor <sup>¼</sup> <sup>γ</sup><sup>5</sup>

<sup>0</sup> <sup>¼</sup> <sup>t</sup>� <sup>x</sup>

! � <sup>v</sup> ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v !2 q ,

> !; t � �

¼ SLor � Ψ x

�i2ω∗<sup>t</sup>

! � � � � �

�i2ω∗<sup>t</sup>

ffiffiffiffi V p e

�i2ω∗<sup>t</sup>

�i2ω∗<sup>t</sup> 0

exp �i Eknott � p

!0 ; t 0 � �

In this part, we show the effective Dirac model of knot states on winding space.

The coordinate measurement of zero-lattice on winding space is the winding angles, Φ ! . Along a given direction e !, after shifting the distance a, the winding angle changes π. The position is determined by two kinds of values: X ! are integer numbers

$$\vec{X} = (X, Y, Z) = \frac{1}{\pi}\vec{\Phi} - \frac{1}{\pi}\vec{\Phi}\text{ mod }\pi\tag{52}$$

and ϕ ! denote internal winding angles

$$
\overrightarrow{\phi} = \left(\phi\_x, \phi\_y, \phi\_z\right) = \overrightarrow{\Phi} \bmod \pi \tag{53}
$$

with ϕx, ϕy, ϕ<sup>z</sup> ∈ ð � 0; π .

Therefore, on winding space, the effective Hamiltonian turns into

$$
\hat{H}\_{\text{3D}} = \stackrel{\rightarrow}{\Gamma} \cdot \vec{p}\_{\text{knot}} + m\_{\text{knot}} \Gamma^5 = \stackrel{\rightarrow}{\Gamma} \cdot \vec{p}\_{X,\text{knot}} + \stackrel{\rightarrow}{\Gamma} \cdot \vec{p}\_{\phi,\text{knot}} + m\_{\text{knot}} \Gamma^5 \tag{54}
$$

where p ! <sup>X</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup> i <sup>d</sup> dX ! and p ! <sup>ϕ</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup> i <sup>d</sup> dϕ !. Because of ϕ<sup>j</sup> ∈ ð � 0; π , quantum number of p ! <sup>ϕ</sup> is angular momentum L ! <sup>ϕ</sup> and the energy spectra are <sup>1</sup> <sup>a</sup> L ! ϕ � � � � � �. If we focus on the low energy physics <sup>E</sup> <sup>≪</sup> <sup>1</sup> a (or L ! <sup>ϕ</sup> ¼ 0), we may get the low energy effective Hamiltonian as

$$
\widehat{H}\_{\text{3D}} \simeq \overrightarrow{\Gamma} \cdot \overrightarrow{p}\_{\text{X, knot}} + m\_{\text{knot}} \Gamma^5. \tag{55}
$$

4.1. Entanglement transformation

global entanglement transformation.

� � and <sup>δ</sup>Φ<sup>t</sup> <sup>x</sup>

!; t

entanglement transformation Ub ET x

coordination transformation, i.e.,

!; t

Φ ! x !; t � � ) <sup>Φ</sup>

Φ<sup>t</sup> x !; t � � ) <sup>Φ</sup><sup>0</sup>

For zero-lattice, Ub ET x

where

Here, δ Φ !

where δ Φ ! x !; t

zero-lattice.

(δ Φ ! x !; t � �, <sup>δ</sup>Φ<sup>t</sup> <sup>x</sup>

duce the concept of "entanglement transformation (ET)". Under global entanglement transformation, we have

> Ψ x !; t � � ! <sup>Ψ</sup><sup>0</sup> <sup>x</sup>

Firstly, based on a uniform 3D knot-crystal (uniform entangled vortex-membranes), we intro-

� � <sup>¼</sup> <sup>U</sup><sup>b</sup> ET <sup>x</sup>

iδΦ ! �Γ ! � e <sup>i</sup>δΦt�Γ<sup>5</sup>

direction on geometric space-time, respectively. The dispersion of the excitation changes under

iδΦ ! x ! ð Þ:<sup>t</sup> �<sup>Γ</sup> ! � e iδΦ<sup>t</sup> x ! ð Þ:<sup>t</sup> �Γ<sup>5</sup>

For knots on a deformed zero-lattice, there exists an intrinsic correspondence between an

These equations also imply a curved space-time: the lattice constants of the 3 + 1D zero-lattice

2a ! 2aeff x

!; t � � � <sup>Ψ</sup> <sup>x</sup>

!; t

Topological Interplay between Knots and Entangled Vortex-Membranes

!

� � are not constant. We call a system with smoothly varied-

� � and a local coordinate transformation that becomes a

� �) deformed knot-crystal and its projected zero-lattice deformed (3 + 1D)

� � changes the winding degrees of freedom that is denoted by the local

!; t � � <sup>þ</sup> <sup>δ</sup>Φ<sup>t</sup> <sup>x</sup>

!; t

! x !; t � �,

> !; t � �:

� � (63)

� � (59)

http://dx.doi.org/10.5772/intechopen.72809

: (60)


(61)

47

(62)

!; t

Ub ET x !; t � � <sup>¼</sup> <sup>e</sup>

and δΦ<sup>t</sup> are constant winding angles along spatial Φ

In general, we may define (local) entanglement transformation, i.e.,

Ub ET x !; t � � <sup>¼</sup> <sup>e</sup>

4.2. Geometric description for deformed zero-lattice: curved space-time

!; t

!0 x !; t � � <sup>¼</sup><sup>Φ</sup> ! x !; t � � <sup>þ</sup> <sup>δ</sup> <sup>Φ</sup>

<sup>t</sup> x !; t � � <sup>¼</sup> <sup>Φ</sup><sup>t</sup> <sup>x</sup>

(the size of a lattice constant with 2π angle changing) are not fixed to be 2a, i.e.,

fundamental principle for emergent gravity theory in knot physics.

!; t

We introduce 3 + 1D winding space-time by defining four coordinates on winding space, Φ ¼ Φ ! ; Φ<sup>t</sup> � � where <sup>Φ</sup><sup>t</sup> is phase changing under time evolution. For a fixed entanglement pattern Γ ! ; <sup>Γ</sup><sup>5</sup> � �, the coordinate transformation along <sup>x</sup>/y/z/t-direction on winding space-time is given by ei<sup>Γ</sup> ! � b <sup>Φ</sup> and e<sup>i</sup>Γ5� b <sup>Φ</sup> <sup>t</sup> , respectively.

For low energy physics, the position in 3 + 1D winding space-time is 3 + 1D zero-lattice of winding space-time labeled by four integer numbers, X ¼ X ! ; X<sup>0</sup> � � where

$$\begin{aligned} \overrightarrow{X} &= \frac{1}{\pi} \overrightarrow{\Phi} - \frac{1}{\pi} \overrightarrow{\Phi} \bmod \pi, \\ X\_0 &= \frac{1}{\pi} \Phi\_t - \frac{1}{\pi} \Phi\_t \bmod \pi. \end{aligned} \tag{56}$$

The lattice constant of the winding space-time is always π that will never be changed. As a result, the winding space-time becomes an effective quantized space-time. Because of x<sup>μ</sup> ¼ a � Xμ, the effective action on 3 + 1D winding space-time becomes

$$\mathcal{S}\_{\text{3D}} \simeq (a)^4 \sum\_{\mathbf{X}\_{\prime} \mathbf{Y}\_{\prime} \mathbf{Z}\_{\prime} \mathbf{X}\_0} \mathcal{L}\_{\text{3D}} \tag{57}$$

where

$$\mathcal{L}\_{\text{3D}} = \overline{\Psi} \left[ i \frac{1}{a} (\gamma^{\mu}) \widehat{\Theta}\_{\mu} - m\_{\text{knot}} \right] \Psi. \tag{58}$$

## 4. Deformed zero-lattice as curved space-time

In this section, we discuss the knot dynamics on smoothly deformed knot-crystal (or deformed zero-lattice). We point out that to characterize the entanglement evolution, the corresponding Biot-Savart mechanics for a knot on smoothly deformed zero-lattice is mapped to that in quantum mechanics on a curved space-time.

#### 4.1. Entanglement transformation

Firstly, based on a uniform 3D knot-crystal (uniform entangled vortex-membranes), we introduce the concept of "entanglement transformation (ET)".

Under global entanglement transformation, we have

$$
\Psi\left(\vec{\mathbf{x}},t\right) \to \Psi'\left(\vec{\mathbf{x}},t\right) = \widehat{\mathcal{U}}\_{\text{ET}}\left(\vec{\mathbf{x}},t\right) \cdot \Psi\left(\vec{\mathbf{x}},t\right) \tag{59}
$$

where

Therefore, on winding space, the effective Hamiltonian turns into

<sup>ϕ</sup> and the energy spectra are <sup>1</sup>

b

<sup>Φ</sup> <sup>t</sup> , respectively.

winding space-time labeled by four integer numbers, X ¼ X

effective action on 3 + 1D winding space-time becomes

4. Deformed zero-lattice as curved space-time

quantum mechanics on a curved space-time.

X ! ¼ 1 π Φ ! � 1 π Φ !

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> π <sup>Φ</sup><sup>t</sup> � <sup>1</sup> π

S3D ≃ð Þa

1

L3D ¼ Ψ i

knot <sup>þ</sup> <sup>m</sup>knotΓ<sup>5</sup> <sup>¼</sup><sup>Γ</sup>

<sup>ϕ</sup> ¼ 0), we may get the low energy effective Hamiltonian as

Hb 3D ≃ Γ ! �p !

! �p !

<sup>a</sup> L ! ϕ � � � � �

We introduce 3 + 1D winding space-time by defining four coordinates on winding space,

For low energy physics, the position in 3 + 1D winding space-time is 3 + 1D zero-lattice of

The lattice constant of the winding space-time is always π that will never be changed. As a result, the winding space-time becomes an effective quantized space-time. Because of x<sup>μ</sup> ¼ a � Xμ, the

> <sup>4</sup> X X, Y, Z, <sup>X</sup><sup>0</sup>

<sup>a</sup> <sup>γ</sup><sup>μ</sup> ð Þb∂<sup>μ</sup> � <sup>m</sup>knot � �

In this section, we discuss the knot dynamics on smoothly deformed knot-crystal (or deformed zero-lattice). We point out that to characterize the entanglement evolution, the corresponding Biot-Savart mechanics for a knot on smoothly deformed zero-lattice is mapped to that in

X,knotþ Γ ! �p !

X,knot <sup>þ</sup> <sup>m</sup>knotΓ<sup>5</sup>

where Φ<sup>t</sup> is phase changing under time evolution. For a fixed entanglement

, the coordinate transformation along x/y/z/t-direction on winding space-time

modπ,

Φ<sup>t</sup> modπ:

! ; X<sup>0</sup> � �

!. Because of ϕ<sup>j</sup> ∈ ð � 0; π , quantum number of p

<sup>ϕ</sup>,knot <sup>þ</sup> <sup>m</sup>knotΓ<sup>5</sup> (54)

: (55)

�. If we focus on the low energy physics <sup>E</sup> <sup>≪</sup> <sup>1</sup>

where

L3D (57)

Ψ: (58)

!

<sup>ϕ</sup> is angular

a

(56)

Hb 3D ¼Γ ! �p !

where p ! <sup>X</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup> i <sup>d</sup> dX ! and p ! <sup>ϕ</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup> i <sup>d</sup> dϕ

(or L !

Φ ¼ Φ ! ; Φ<sup>t</sup> � �

where

pattern Γ

is given by ei<sup>Γ</sup>

! ; <sup>Γ</sup><sup>5</sup> � �

> ! � b <sup>Φ</sup> and e<sup>i</sup>Γ5�

momentum L

!

46 Superfluids and Superconductors

$$
\widehat{\mathcal{U}}\_{\rm ET}\left(\overrightarrow{\mathbf{x}},t\right) = e^{i\boldsymbol{\delta\Phi}\cdot\overrightarrow{\Gamma}} \cdot e^{i\boldsymbol{\delta\Phi}\_{\rm l}\cdot\overrightarrow{\Gamma}^{\delta}}.\tag{60}
$$

Here, δ Φ ! and δΦ<sup>t</sup> are constant winding angles along spatial Φ ! -direction and that along tempo direction on geometric space-time, respectively. The dispersion of the excitation changes under global entanglement transformation.

In general, we may define (local) entanglement transformation, i.e.,

$$
\widehat{\mathcal{U}}\_{\rm ET} \left( \overrightarrow{\mathbf{x}}, t \right) = e^{i\delta \overrightarrow{\Phi} \left( \overrightarrow{\mathbf{x}}, t \right) \cdot \overrightarrow{\Gamma}} \cdot e^{i\delta \Phi\_t \left( \overrightarrow{\mathbf{x}}, t \right) \cdot \Gamma^5} \tag{61}
$$

where δ Φ ! x !; t � � and <sup>δ</sup>Φ<sup>t</sup> <sup>x</sup> !; t � � are not constant. We call a system with smoothly varied- (δ Φ ! x !; t � �, <sup>δ</sup>Φ<sup>t</sup> <sup>x</sup> !; t � �) deformed knot-crystal and its projected zero-lattice deformed (3 + 1D) zero-lattice.

#### 4.2. Geometric description for deformed zero-lattice: curved space-time

For knots on a deformed zero-lattice, there exists an intrinsic correspondence between an entanglement transformation Ub ET x !; t � � and a local coordinate transformation that becomes a fundamental principle for emergent gravity theory in knot physics.

For zero-lattice, Ub ET x !; t � � changes the winding degrees of freedom that is denoted by the local coordination transformation, i.e.,

$$\begin{split} \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right) &\Rightarrow \overrightarrow{\Phi}' \left( \overrightarrow{\boldsymbol{x}}, t \right) = \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right) + \delta \ \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right), \\ \Phi\_t \left( \overrightarrow{\boldsymbol{x}}, t \right) &\Rightarrow \Phi'\_t \left( \overrightarrow{\boldsymbol{x}}, t \right) = \Phi\_t \left( \overrightarrow{\boldsymbol{x}}, t \right) + \delta \Phi\_t \left( \overrightarrow{\boldsymbol{x}}, t \right). \end{split} \tag{62}$$

These equations also imply a curved space-time: the lattice constants of the 3 + 1D zero-lattice (the size of a lattice constant with 2π angle changing) are not fixed to be 2a, i.e.,

$$2a \to 2a\_{\rm eff} \left( \vec{\times}, t \right) \tag{63}$$

The distance between two nearest-neighbor "lattice sites" on the spatial/tempo coordinate changes, i.e.,

$$\begin{aligned} \Delta \overrightarrow{\mathbf{x}} &= \left( \overrightarrow{\mathbf{x}}' + \overrightarrow{\mathbf{e}}\_x \right) - \overrightarrow{\mathbf{x}} = \overrightarrow{\mathbf{e}}\_{x} \\ \Delta \overrightarrow{\mathbf{x}}' &= \left( \overrightarrow{\mathbf{x}}' + \overrightarrow{\mathbf{e}}\_x' \right) - \overrightarrow{\mathbf{x}}' = \overrightarrow{\mathbf{e}}\_x' \left( \overrightarrow{\mathbf{x}}, t \right) \end{aligned} \tag{64}$$

winding space-time. Because one may smoothly deform the zero-lattice and get the same low energy effective model for knots on winding space-time, there exists diffeomorphism invariance, i.e.,

Knot–invariance on winding space–time

Therefore, from the view of mathematics, the physics on winding space-time is never changed! The invariance of the effective model for knots on winding space-time indicates the diffeomorphism

On the other hand, the condition of very smoothly entanglement transformation guarantees a (local) Lorentz invariance in long wave-length limit. Under local Lorentz invariance, the knot-

According to the local coordinate transformation, the deformed zero-lattice becomes a curved space-time for the knots. In continuum limit <sup>Δ</sup><sup>k</sup> <sup>≪</sup> ð Þ<sup>a</sup> �<sup>1</sup> and <sup>Δ</sup><sup>ω</sup> <sup>≪</sup> <sup>ω</sup>0, the diffeomorphism invariance and (local) Lorentz invariance emerge together. E. Witten had made a strong claim about emergent gravity, "whatever we do, we are not going to start with a conventional theory of nongravitational fields in Minkowski space-time and generate Einstein gravity as an emergent phenomenon." He pointed out that gravity could be emergent only if the notion on the space-time on which diffeomorphism invariance is simultaneously emergent. For the emergent quantum gravity in knot physics, diffeomorphism invariance and Lorentz invariance are simultaneously emergent. In particular, the diffeomorphism invariance comes from information invariance of knots on winding space-time—when the lattice-distance of zero-lattice changes, the size of the

> !0 x !; t � �; <sup>t</sup>

<sup>β</sup> ¼ gαβ where η is the internal space metric tensor. The geometry fields (vierbein fields

<sup>b</sup> <sup>þ</sup> <sup>ω</sup><sup>a</sup> <sup>c</sup> ∧ ω<sup>c</sup> b

<sup>b</sup>μνdx<sup>μ</sup> <sup>∧</sup> dx<sup>ν</sup>

βμν are the components of the usual Riemann tensor projection on the tangent

description. In addition to the existence of a set of vierbein fields ea, the space metric is defined

<sup>0</sup> x !; t

� �) are determined by the non-uniform local coordinates

� � � � , we introduce a geometric

!; t � � and the

, (68)

Ψ i 1 a <sup>γ</sup><sup>μ</sup>b<sup>∂</sup> X μ

<sup>4</sup> X X, Y, Z, <sup>X</sup><sup>0</sup>

Szero�lattice � ð Þa

pieces of a given knot are determined by local Lorentz transformations.

!; t

� � � � . Furthermore, one needs to introduce spin connections <sup>ω</sup>ab <sup>x</sup>

¼ 1 2 Ra

Ra <sup>b</sup> <sup>¼</sup> <sup>d</sup>ω<sup>a</sup>

space. The deformation of the zero-lattice is characterized by

invariance

knots correspondingly changes.

� � and spin connections <sup>ω</sup>ab <sup>x</sup>

<sup>0</sup> x !; t

<sup>b</sup>μν � <sup>e</sup><sup>a</sup> αe β bR<sup>α</sup>

Riemann curvature two-form as

by ηabe<sup>a</sup> αeb

e<sup>a</sup> x !; t

> x !0 x !; t � �; <sup>t</sup>

where Ra

To characterize the deformed 3 + 1D zero-lattice x

) Diffeomorphism invariance: (66)

Topological Interplay between Knots and Entangled Vortex-Membranes

http://dx.doi.org/10.5772/intechopen.72809

49

� <sup>m</sup>knot � �Ψ: (67)

and

$$\begin{aligned} \Delta t &= (t + \varepsilon\_0) - t = \varepsilon\_0 \\ \Delta t' &= \left(t' + \varepsilon\_0'\right) - t' = \varepsilon\_0' \left(\overrightarrow{\bf x}, t\right) \end{aligned} \tag{65}$$

where eað Þ a ¼ 0; 1; 2; 3 and e<sup>0</sup> <sup>a</sup> x !; t are the unit-vectors of the original frame and the deformed frame, respectively. See the illustration of a 1 + 1D deformed zero-lattice on winding spacetime with a non-uniform distribution of zeroes in Figure 3(d).

However, for deformed zero-lattice, the information of knots in projected space is invariant: when the lattice-distance of zero-lattice changes a ! aeff x !; t , the size of the knots correspondingly changes a ! aeff x !; t . Therefore, due to the invariance of a knot, the deformation of zero-lattice does not change the formula of the low energy effective model for knots on

Figure 3. (a) An illustration of deformed knot-crystal; (b) an illustration of smoothly deformed relationship between winding angle Φ and spatial coordinate x. The zero-lattice in winding space is still uniform; while the zero-lattice in geometric space is deformed; (c) an illustration of a uniform 1 + 1D zero-lattice in geometric space-time; and (d) an illustration of a deformed 1 + 1D zero-lattice in geometric space-time.

winding space-time. Because one may smoothly deform the zero-lattice and get the same low energy effective model for knots on winding space-time, there exists diffeomorphism invariance, i.e.,

The distance between two nearest-neighbor "lattice sites" on the spatial/tempo coordinate

� x !0 ¼ e !0 <sup>x</sup> x !; t

Δt ¼ ð Þ� t þ e<sup>0</sup> t ¼ e0,

frame, respectively. See the illustration of a 1 + 1D deformed zero-lattice on winding space-

However, for deformed zero-lattice, the information of knots in projected space is invariant:

of zero-lattice does not change the formula of the low energy effective model for knots on

Figure 3. (a) An illustration of deformed knot-crystal; (b) an illustration of smoothly deformed relationship between winding angle Φ and spatial coordinate x. The zero-lattice in winding space is still uniform; while the zero-lattice in geometric space is deformed; (c) an illustration of a uniform 1 + 1D zero-lattice in geometric space-time; and (d) an

<sup>0</sup> þ e<sup>0</sup> 0 � <sup>t</sup> � x ! <sup>¼</sup> <sup>e</sup> ! x,

<sup>0</sup> ¼ e<sup>0</sup> <sup>0</sup> x !; t

are the unit-vectors of the original frame and the deformed

!; t 

. Therefore, due to the invariance of a knot, the deformation

(64)

(65)

, the size of the knots corre-

Δ x ! <sup>¼</sup> <sup>x</sup> ! <sup>þ</sup><sup>e</sup> ! x 

Δt <sup>0</sup> ¼ t

Δx !0 ¼ x !0 þ e !0 x 

<sup>a</sup> x !; t 

time with a non-uniform distribution of zeroes in Figure 3(d).

when the lattice-distance of zero-lattice changes a ! aeff x

!; t 

illustration of a deformed 1 + 1D zero-lattice in geometric space-time.

changes, i.e.,

48 Superfluids and Superconductors

where eað Þ a ¼ 0; 1; 2; 3 and e<sup>0</sup>

spondingly changes a ! aeff x

and

$$\begin{aligned} & \text{Knot-invariance on winding space-time} \\ & \Rightarrow \text{Diffeomorphisms invariance.} \end{aligned} \tag{66}$$

Therefore, from the view of mathematics, the physics on winding space-time is never changed! The invariance of the effective model for knots on winding space-time indicates the diffeomorphism invariance

$$\mathcal{S}\_{\text{zero-lattice}} \equiv (a)^4 \sum\_{\mathcal{X}\_\tau \mathcal{Y}\_\tau Z\_\tau \mathcal{X}\_0} \overline{\Psi} \left[ i \frac{1}{a} \gamma^\mu \widehat{\otimes}^\times\_\mu - m\_{\text{krot}} \right] \Psi. \tag{67}$$

On the other hand, the condition of very smoothly entanglement transformation guarantees a (local) Lorentz invariance in long wave-length limit. Under local Lorentz invariance, the knotpieces of a given knot are determined by local Lorentz transformations.

According to the local coordinate transformation, the deformed zero-lattice becomes a curved space-time for the knots. In continuum limit <sup>Δ</sup><sup>k</sup> <sup>≪</sup> ð Þ<sup>a</sup> �<sup>1</sup> and <sup>Δ</sup><sup>ω</sup> <sup>≪</sup> <sup>ω</sup>0, the diffeomorphism invariance and (local) Lorentz invariance emerge together. E. Witten had made a strong claim about emergent gravity, "whatever we do, we are not going to start with a conventional theory of nongravitational fields in Minkowski space-time and generate Einstein gravity as an emergent phenomenon." He pointed out that gravity could be emergent only if the notion on the space-time on which diffeomorphism invariance is simultaneously emergent. For the emergent quantum gravity in knot physics, diffeomorphism invariance and Lorentz invariance are simultaneously emergent. In particular, the diffeomorphism invariance comes from information invariance of knots on winding space-time—when the lattice-distance of zero-lattice changes, the size of the knots correspondingly changes.

To characterize the deformed 3 + 1D zero-lattice x !0 x !; t � �; <sup>t</sup> <sup>0</sup> x !; t � � � � , we introduce a geometric description. In addition to the existence of a set of vierbein fields ea, the space metric is defined by ηabe<sup>a</sup> αeb <sup>β</sup> ¼ gαβ where η is the internal space metric tensor. The geometry fields (vierbein fields e<sup>a</sup> x !; t � � and spin connections <sup>ω</sup>ab <sup>x</sup> !; t � �) are determined by the non-uniform local coordinates x !0 x !; t � �; <sup>t</sup> <sup>0</sup> x !; t � � � � . Furthermore, one needs to introduce spin connections <sup>ω</sup>ab <sup>x</sup> !; t � � and the Riemann curvature two-form as

$$\begin{split} R^{a}\_{b} &= d\omega^{a}\_{b} + \omega^{a}\_{c} \wedge \omega^{c}\_{b} \\ &= \frac{1}{2} R^{a}\_{b\mu\nu} d\mathbf{x}^{\mu} \wedge d\mathbf{x}^{\nu} \end{split} \tag{68}$$

where Ra <sup>b</sup>μν � <sup>e</sup><sup>a</sup> αe β bR<sup>α</sup> βμν are the components of the usual Riemann tensor projection on the tangent space. The deformation of the zero-lattice is characterized by

$$R^{ab} = d\omega^{ab} + \omega^{ac} \wedge \omega^{cb}.\tag{69}$$

Γ !

!

x denotes the space-time position of a site of zero-lattice, x

ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>5</sup> � �<sup>0</sup>

4.3.2. Gauge description for deformed tempo entanglement matrix

where

Γ !0

entanglement matrices) Γ

To characterize Γ<sup>5</sup> � �<sup>0</sup>

projected space.

ð Þ¼ x Ub ETð Þx Γ

!0

; <sup>Γ</sup><sup>5</sup> � � ! <sup>Γ</sup>

<sup>U</sup><sup>b</sup> ETð Þ<sup>x</sup> �<sup>1</sup>

ð Þx

Figure 4(d), in which the arrows denote deformed entanglement matrix Γ<sup>5</sup> � �<sup>0</sup>

!0

, Γ<sup>5</sup> � �<sup>0</sup>

becomes a unit SO(4) vector-field on each lattice site. The deformed zero-lattice induced by local entanglement transformation Ub ETð Þx is characterized by four SO(4) vector-fields (four

Firstly, we study the unit SO(4) vector-field of deformed tempo entanglement matrix Γ<sup>5</sup> � �<sup>0</sup>

ð Þ<sup>x</sup> , the reduced Gamma matrices γμ is defined as

Figure 4. (a) An illustration of the effect of an extra knot on a 1D knot-crystal along spatial direction; (b) an illustration of the effect of an extra knot on a 1D knot-crystal along tempo direction. Here A<sup>∗</sup>/B<sup>∗</sup> denotes conjugate representation of vortex-line-A/B; (c) the entanglement pattern for a uniform knot-crystal. The arrows denote the directions of entanglement matrices; and (d) the entanglement pattern for a knot-crystal with an extra knot at center. The purple spot denotes the knot. The red arrows that denote local tangential entanglement matrices have vortex-like configuration on 2D

ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>5</sup> � �<sup>0</sup>

ð Þx

ð Þ¼ x Ub ET x

!; t � �Γ<sup>5</sup>

!; t

� �. See the illustration of a 2D deformed zero-lattice in

� � (75)

Topological Interplay between Knots and Entangled Vortex-Membranes

<sup>U</sup><sup>b</sup> ETð Þ<sup>x</sup> �<sup>1</sup>

� �. Each entanglement matrix

http://dx.doi.org/10.5772/intechopen.72809

ð Þx .

: (76)

51

ð Þx .

So the low energy physics for knots on the deformed zero-lattice turns into that for Dirac fermions on curved space-time

$$\mathcal{S}\_{\text{cavred-ST}} = \int \sqrt{-g} \overline{\Psi} \left( e\_a^{\mu} \gamma^{\mu} \left( i \hat{\partial}\_{\mu} + i \omega\_{\mu} \right) - m\_{\text{knot}} \right) \Psi d^4 x \tag{70}$$

where ωμ <sup>¼</sup> <sup>ω</sup><sup>0</sup><sup>i</sup> <sup>μ</sup> γ<sup>0</sup><sup>i</sup> =2; ωij μγij=2 � � ð Þ <sup>i</sup>; <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup> and <sup>γ</sup>ab ¼ � <sup>1</sup> <sup>4</sup> γ<sup>a</sup> ; ; <sup>γ</sup><sup>b</sup> � � ð Þ <sup>a</sup>; <sup>b</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup> [15]. This model described by Scurved�ST is invariant under local (non-compact) SO(3,1) Lorentz transformation S x!; t � � <sup>¼</sup> <sup>e</sup> θab x ! ð Þ;<sup>t</sup> <sup>γ</sup>ab as

$$\begin{split} \Psi\left(\overrightarrow{\boldsymbol{x}},t\right) &\to \Psi'\left(\overrightarrow{\boldsymbol{x}},t\right) = \mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\Psi\left(\overrightarrow{\boldsymbol{x}},t\right), \\ \gamma^{\mu} &\to \left(\gamma^{\mu}\left(\overrightarrow{\boldsymbol{x}},t\right)\right)' = \mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\gamma^{\mu}\left(\mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\right)^{-1}, \\ \omega\_{\mu} &\to \omega\_{\mu}'\left(\overrightarrow{\boldsymbol{x}},t\right) = \mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\omega\_{\mu}\left(\overrightarrow{\boldsymbol{x}},t\right)\left(\mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\right)^{-1} \\ &\to \mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\partial\_{\mu}\left(\mathcal{S}\left(\overrightarrow{\boldsymbol{x}},t\right)\right)^{-1}. \end{split} \tag{71}$$

γ<sup>5</sup> is invariant under local SO(3,1) Lorentz symmetry as

$$\begin{split} \gamma^5 \rightarrow \left( \gamma^5 \right)' &= \mathcal{S} \left( \overrightarrow{\mathbf{x}}, t \right) \gamma^5 \left( \mathcal{S} \left( \overrightarrow{\mathbf{x}}, t \right) \right)^{-1} \\ &= \gamma^5. \end{split} \tag{72}$$

In general, an SO(3,1) Lorentz transformation S x!; t � � is a combination of spin rotation transformation R x <sup>b</sup> !; <sup>t</sup> � � <sup>¼</sup> <sup>R</sup>bspin <sup>x</sup> !; t � � � <sup>R</sup>bspace <sup>x</sup> !; t � � and Lorentz boosting <sup>S</sup>Lor <sup>x</sup> !; t � �.

In physics, under a Lorentz transformation, a distribution of knot-pieces changes into another distribution of knot-pieces. For this reason, the velocity ceff and the total number of zeroes Nknot are invariant,

$$
\mathcal{c}\_{\rm eff} \to \mathcal{c}'\_{\rm eff} \equiv \mathcal{c}\_{\rm eff} \tag{73}
$$

and

$$N\_{\rm knot} \to N\_{\rm knot}' \equiv N\_{\rm knot}.\tag{74}$$

#### 4.3. Gauge description for deformed zero-lattice

#### 4.3.1. Deformed entanglement matrices and deformed entanglement pattern

The deformation of the zero-lattice leads to deformation of entanglement pattern, i.e.,

Topological Interplay between Knots and Entangled Vortex-Membranes http://dx.doi.org/10.5772/intechopen.72809 51

$$\left(\overrightarrow{\Gamma}, \Gamma^{5}\right) \rightarrow \left(\overrightarrow{\Gamma}'(\mathbf{x}), \left(\Gamma^{5}\right)'(\mathbf{x})\right) \tag{75}$$

where

<sup>R</sup>ab <sup>¼</sup> <sup>d</sup>ωab <sup>þ</sup> <sup>ω</sup>ac <sup>∧</sup> <sup>ω</sup>cb: (69)

� mknot

<sup>4</sup> γ<sup>a</sup>

γ<sup>μ</sup> S x!; t � � � � �<sup>1</sup>

Ψ x !; t � � ,

ωμ x !; t � �

:

and Lorentz boosting SLor x

Ψd<sup>4</sup>

x (70)

(71)

(72)

; ; <sup>γ</sup><sup>b</sup> � � ð Þ <sup>a</sup>; <sup>b</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup> [15].

,

is a combination of spin rotation trans-

!; t � � .

eff � ceff (73)

knot � Nknot: (74)

S x!; t � � � � �<sup>1</sup>

So the low energy physics for knots on the deformed zero-lattice turns into that for Dirac

ð Þ <sup>i</sup>; <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup> and <sup>γ</sup>ab ¼ � <sup>1</sup>

This model described by Scurved�ST is invariant under local (non-compact) SO(3,1) Lorentz

!; t � � � � <sup>0</sup>

> <sup>¼</sup> S x!; <sup>t</sup> � �

<sup>¼</sup> <sup>γ</sup><sup>5</sup> :

!; t � �

ceff ! c 0

Nknot ! N<sup>0</sup>

The deformation of the zero-lattice leads to deformation of entanglement pattern, i.e.,

4.3.1. Deformed entanglement matrices and deformed entanglement pattern

In physics, under a Lorentz transformation, a distribution of knot-pieces changes into another distribution of knot-pieces. For this reason, the velocity ceff and the total number of zeroes

<sup>μ</sup> x !; t � �

b∂<sup>μ</sup> þ iωμ � �

<sup>¼</sup> S x!; <sup>t</sup> � �

<sup>¼</sup> S x!; <sup>t</sup> � �

∂<sup>μ</sup> S x!; t � � � � �<sup>1</sup>

� �

<sup>¼</sup> S x!; <sup>t</sup> � �

γ<sup>5</sup> S x!; t � � � � �<sup>1</sup>

� �

μ <sup>a</sup> γ<sup>a</sup> i

fermions on curved space-time

50 Superfluids and Superconductors

<sup>μ</sup> γ<sup>0</sup><sup>i</sup>

� �

where ωμ <sup>¼</sup> <sup>ω</sup><sup>0</sup><sup>i</sup>

formation R x

Nknot are invariant,

and

<sup>b</sup> !; <sup>t</sup> � �

transformation S x!; t

Scurved�ST ¼

μγij=2

Ψ x !; t � �

γ<sup>5</sup> is invariant under local SO(3,1) Lorentz symmetry as

In general, an SO(3,1) Lorentz transformation S x!; t

!; t � �

¼ Rbspin x

4.3. Gauge description for deformed zero-lattice

=2; ωij

¼ e θab x ! ð Þ;<sup>t</sup> <sup>γ</sup>ab

� �

<sup>ð</sup> ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> <sup>Ψ</sup> <sup>e</sup>

as

! Ψ<sup>0</sup> x !; t � �

<sup>γ</sup><sup>μ</sup> ! <sup>γ</sup><sup>μ</sup> <sup>x</sup>

<sup>þ</sup>S x!; <sup>t</sup> � �

ωμ ! ω<sup>0</sup>

<sup>γ</sup><sup>5</sup> ! <sup>γ</sup><sup>5</sup> � �<sup>0</sup>

� Rbspace x

$$
\overrightarrow{\Gamma}'(\mathbf{x}) = \widehat{\mathcal{U}}\_{\text{ET}}(\mathbf{x}) \, \overrightarrow{\Gamma}^{\prime} \widehat{\mathcal{U}}\_{\text{ET}}(\mathbf{x})^{-1}, \{\Gamma^{5}\}'(\mathbf{x}) = \widehat{\mathcal{U}}\_{\text{ET}}(\overrightarrow{\mathbf{x}},t) \Gamma^{5} \widehat{\mathcal{U}}\_{\text{ET}}(\mathbf{x})^{-1}. \tag{76}
$$

x denotes the space-time position of a site of zero-lattice, x !; t � �. Each entanglement matrix becomes a unit SO(4) vector-field on each lattice site. The deformed zero-lattice induced by local entanglement transformation Ub ETð Þx is characterized by four SO(4) vector-fields (four entanglement matrices) Γ !0 ð Þ<sup>x</sup> ; <sup>Γ</sup><sup>5</sup> � �<sup>0</sup> ð Þx � �. See the illustration of a 2D deformed zero-lattice in Figure 4(d), in which the arrows denote deformed entanglement matrix Γ<sup>5</sup> � �<sup>0</sup> ð Þx .

#### 4.3.2. Gauge description for deformed tempo entanglement matrix

Firstly, we study the unit SO(4) vector-field of deformed tempo entanglement matrix Γ<sup>5</sup> � �<sup>0</sup> ð Þx . To characterize Γ<sup>5</sup> � �<sup>0</sup> ð Þ<sup>x</sup> , the reduced Gamma matrices γμ is defined as

Figure 4. (a) An illustration of the effect of an extra knot on a 1D knot-crystal along spatial direction; (b) an illustration of the effect of an extra knot on a 1D knot-crystal along tempo direction. Here A<sup>∗</sup>/B<sup>∗</sup> denotes conjugate representation of vortex-line-A/B; (c) the entanglement pattern for a uniform knot-crystal. The arrows denote the directions of entanglement matrices; and (d) the entanglement pattern for a knot-crystal with an extra knot at center. The purple spot denotes the knot. The red arrows that denote local tangential entanglement matrices have vortex-like configuration on 2D projected space.

$$
\gamma^1 = \gamma^0 \Gamma^1, \gamma^2 = \gamma^0 \Gamma^2, \gamma^3 = \gamma^0 \Gamma^3,\tag{77}
$$

<sup>γ</sup><sup>5</sup> ! <sup>γ</sup><sup>5</sup> � �<sup>0</sup>

The correspondence between index of γ<sup>a</sup> and index of space-time xa is

As a result, we can introduce an auxiliary gauge field Aab

A<sup>i</sup><sup>0</sup>

characterize the deformation of the zero-lattice. The auxiliary gauge field Aab

<sup>A</sup>ijð Þ¼ <sup>x</sup> tr <sup>γ</sup>ij <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

ð Þ¼ <sup>x</sup> tr <sup>γ</sup><sup>i</sup><sup>0</sup>U<sup>b</sup> ð Þ<sup>x</sup>

ð Þ<sup>x</sup> � �<sup>0</sup>

The total field strength <sup>F</sup>ijð Þ<sup>x</sup> of i, j <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3 components can be divided into two parts

<sup>F</sup>ijð Þ¼ <sup>x</sup> <sup>F</sup>ij <sup>þ</sup> Ai<sup>0</sup> <sup>∧</sup> <sup>A</sup><sup>j</sup><sup>0</sup>

<sup>F</sup>ij <sup>¼</sup> dAij <sup>þ</sup> <sup>A</sup>ik <sup>∧</sup> Akj

Next, we study the unit SO(4) vector-field of deformed spatial entanglement matrix Γ<sup>i</sup> � �<sup>0</sup>

ð Þ<sup>x</sup> , the reduced Gamma matrices <sup>γ</sup><sup>μ</sup> is defined as

� �A<sup>i</sup><sup>0</sup> <sup>∧</sup> <sup>A</sup><sup>j</sup><sup>0</sup>

:

<sup>¼</sup> <sup>γ</sup><sup>0</sup><sup>d</sup> <sup>γ</sup><sup>i</sup>

According to pure gauge condition, we have Maurer-Cartan equation,

Finally, we emphasize the equivalence between γ<sup>0</sup><sup>i</sup> and Γ<sup>i</sup>

4.3.3. Gauge description for deformed spatial entanglement matrix

We denote this correspondence to be

space-time x<sup>a</sup>

.

two parts [15]: SO(3) parts

and SO(4)/SO(3) parts

or

characterize Γ<sup>i</sup> � �<sup>0</sup>

<sup>¼</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup> <sup>γ</sup><sup>5</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

γ<sup>1</sup> ⇔ x, γ<sup>2</sup> ⇔ y,

where 1ð Þ ; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> ET denotes the index order of <sup>γ</sup><sup>a</sup> and 1ð Þ ; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> ST denotes the index order of

� �<sup>d</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

� �<sup>d</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

¼ �γ<sup>i</sup>

� ��<sup>1</sup>

d γ<sup>0</sup> ð Þ<sup>x</sup> � �<sup>0</sup> :

Þ

<sup>F</sup>ijð Þ¼ <sup>x</sup> <sup>F</sup>ij <sup>þ</sup> Ai<sup>0</sup> <sup>∧</sup> <sup>A</sup><sup>j</sup><sup>0</sup> � <sup>0</sup> (90)

, i.e., γ<sup>0</sup><sup>i</sup> ⇔ Γ<sup>i</sup>

.

� ��<sup>1</sup>

<sup>¼</sup> <sup>γ</sup><sup>5</sup>: (84)

http://dx.doi.org/10.5772/intechopen.72809

<sup>μ</sup> ð Þx and use a gauge description to

: (89)

<sup>μ</sup> ð Þx is written into

(88)

53

(91)

ð Þx . To

<sup>γ</sup><sup>3</sup> <sup>⇔</sup> z, <sup>γ</sup><sup>0</sup> <sup>⇔</sup> <sup>t</sup>: (85)

Topological Interplay between Knots and Entangled Vortex-Membranes

� ��<sup>1</sup> � � (87)

ð Þ 1; 2; 3; 0 ET ⇔ ð Þ 1; 2; 3; 0 ST (86)

and

$$
\gamma^0 = \Gamma^5 = \tau^x \otimes 1,\\
\gamma^5 = \stackrel{\cdot}{\dot{\gamma}^0} \gamma^1 \gamma^2 \gamma^3. \tag{78}
$$

Under this definition (γ<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup> ), the effect of deformed zero-lattice from spatial entanglement transformation e<sup>i</sup>Γ1�ΔΦ<sup>x</sup>, e<sup>i</sup>Γ2�ΔΦ<sup>y</sup> , e<sup>i</sup>Γ3�ΔΦ<sup>z</sup> can be studied due to

$$
\Gamma^5 \to \left(\Gamma^5\right)'(\mathbf{x}) = \widehat{\mathcal{U}}\_{\text{ET}}^{x/y/z}\left(\overrightarrow{\mathbf{x}},t\right)\Gamma^5\widehat{\mathcal{U}}\_{\text{ET}}^{x/y/z}(\mathbf{x})^{-1} \neq \Gamma^5. \tag{79}
$$

However, the effect of deformed zero-lattice from tempo entanglement transformation e<sup>i</sup>δΦt�Γ<sup>5</sup> cannot be well defined due to

$$
\Gamma^5 \to \left(\Gamma^5\right)'(\mathbf{x}) = \hat{\mathcal{U}}\_{\text{ET}}^t\left(\vec{\mathbf{x}}, t\right) \Gamma^5 \hat{\mathcal{U}}\_{\text{ET}}^t(\mathbf{x})^{-1} = \Gamma^5. \tag{80}
$$

We introduce an SO(4) transformation U x b !; t � � that is a combination of spin rotation transformation Rbð Þx and spatial entanglement transformation (entanglement transformation along x/y/ <sup>z</sup>-direction) <sup>U</sup><sup>b</sup> <sup>x</sup>=y=<sup>z</sup> ET ð Þ¼ <sup>x</sup> <sup>e</sup><sup>i</sup>δ<sup>Φ</sup> ! ð Þ� x Γ ! , i.e.,

$$
\widehat{\boldsymbol{\mathcal{U}}}(\mathbf{x}) = \widehat{\boldsymbol{\mathcal{R}}}(\mathbf{x}) \oplus \widehat{\boldsymbol{\mathcal{U}}}\_{\text{ET}}^{\mathbf{x}/y/z}(\mathbf{x}).\tag{81}
$$

Here, ⊕ denotes operation combination. Under a non-uniform SO(4) transformation Ub ð Þx , we have

$$\gamma^0 \to \hat{\mathcal{U}}\left(\mathbf{x}\right)\gamma^0 \left(\hat{\mathcal{U}}(\mathbf{x})\right)^{-1} = \left(\gamma^0(\mathbf{x})\right)' = \sum\_{a} \gamma^a n^a(\mathbf{x})\tag{82}$$

where <sup>n</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>; <sup>n</sup><sup>2</sup>; <sup>n</sup><sup>3</sup>ϕ<sup>0</sup> 0 � � <sup>¼</sup> <sup>n</sup> !; ϕ<sup>0</sup> 0 � � is a unit SO(4) vector-field. For the deformed zero-lattice, according to <sup>γ</sup><sup>0</sup>ð Þ<sup>x</sup> � �<sup>0</sup> 6¼ <sup>γ</sup>0, the entanglement matrix <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> along tempo direction is varied, <sup>Γ</sup><sup>5</sup> ! <sup>Γ</sup><sup>5</sup> � �<sup>0</sup> ð Þ<sup>x</sup> 6¼ <sup>Γ</sup><sup>5</sup> .

In general, the SO(4) transformation is defined by <sup>U</sup><sup>b</sup> ð Þ¼ <sup>x</sup> <sup>e</sup><sup>Φ</sup>ab ð Þ<sup>x</sup> <sup>γ</sup>ab <sup>γ</sup>ab ¼ � <sup>1</sup> <sup>4</sup> γ<sup>a</sup> ; γ<sup>b</sup> � � � � . Under the SO(4) transformation, we have

$$\gamma^{\mu} \to \left(\gamma^{\mu}(\mathbf{x})\right)' = \hat{\mathcal{U}}(\mathbf{x})\gamma^{\mu}\left(\hat{\mathcal{U}}(\mathbf{x})\right)^{-1},$$

$$A\_{\mu} \to A\_{\mu}'\left(\vec{\mathbf{x}},t\right) = \hat{\mathcal{U}}\left(\vec{\mathbf{x}},t\right)A\_{\mu}(\mathbf{x})\left(\hat{\mathcal{U}}(\mathbf{x})\right)^{-1} \tag{83}$$

$$+ \hat{\mathcal{U}}(\mathbf{x})\partial\_{\mu}\left(\hat{\mathcal{U}}(\mathbf{x})\right)^{-1}.$$

In particular, γ<sup>5</sup> is invariant under the SO(4) transformation as

Topological Interplay between Knots and Entangled Vortex-Membranes http://dx.doi.org/10.5772/intechopen.72809 53

$$\left(\gamma^5 \to \left(\gamma^5\right)' = \hat{\mathcal{U}}\left(\mathbf{x}\right)\gamma^5 \left(\hat{\mathcal{U}}\left(\mathbf{x}\right)\right)^{-1} = \gamma^5. \tag{84}$$

The correspondence between index of γ<sup>a</sup> and index of space-time xa is

$$\begin{aligned} \gamma^1 &\Leftrightarrow \chi^2 \Leftrightarrow y, \\ \gamma^3 &\Leftrightarrow z, \gamma^0 \Leftrightarrow t. \end{aligned} \tag{85}$$

We denote this correspondence to be

$$(1,2,3,0)\_{\text{ET}} \Leftrightarrow (1,2,3,0)\_{\text{ST}} \tag{86}$$

where 1ð Þ ; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> ET denotes the index order of <sup>γ</sup><sup>a</sup> and 1ð Þ ; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> ST denotes the index order of space-time x<sup>a</sup> .

As a result, we can introduce an auxiliary gauge field Aab <sup>μ</sup> ð Þx and use a gauge description to characterize the deformation of the zero-lattice. The auxiliary gauge field Aab <sup>μ</sup> ð Þx is written into two parts [15]: SO(3) parts

$$A^{\vec{\eta}}(\mathbf{x}) = \text{tr}\left(\gamma^{\vec{\eta}} \Big(\widehat{\mathcal{U}}(\mathbf{x})\Big) d\Big(\widehat{\mathcal{U}}(\mathbf{x})\Big)^{-1}\right) \tag{87}$$

and SO(4)/SO(3) parts

$$\begin{split} A^{i0}(\mathbf{x}) &= \text{tr}\left(\boldsymbol{\gamma}^{i0}\widehat{\boldsymbol{U}}(\mathbf{x})\right) d\left(\widehat{\boldsymbol{U}}(\mathbf{x})\right)^{-1}) \\ &= \boldsymbol{\gamma}^{0} d\left(\boldsymbol{\gamma}^{i}(\mathbf{x})\right)' = -\boldsymbol{\gamma}^{i} d\left(\boldsymbol{\gamma}^{0}(\mathbf{x})\right)'. \end{split} \tag{88}$$

The total field strength <sup>F</sup>ijð Þ<sup>x</sup> of i, j <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; 3 components can be divided into two parts

$$\mathcal{F}^{\vec{\gamma}}(\mathbf{x}) = F^{\vec{\gamma}} + A^{i0} \wedge A^{j0}. \tag{89}$$

According to pure gauge condition, we have Maurer-Cartan equation,

$$\mathcal{F}^{\vec{\eta}}(\mathbf{x}) = F^{\vec{\eta}} + A^{i0} \wedge A^{j0} \equiv \mathbf{0} \tag{90}$$

or

<sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

transformation e<sup>i</sup>Γ1�ΔΦ<sup>x</sup>, e<sup>i</sup>Γ2�ΔΦ<sup>y</sup> , e<sup>i</sup>Γ3�ΔΦ<sup>z</sup> can be studied due to

<sup>Γ</sup><sup>5</sup> ! <sup>Γ</sup><sup>5</sup> � �<sup>0</sup>

<sup>γ</sup><sup>0</sup> ! <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup> <sup>γ</sup><sup>0</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

In general, the SO(4) transformation is defined by <sup>U</sup><sup>b</sup> ð Þ¼ <sup>x</sup> <sup>e</sup><sup>Φ</sup>ab ð Þ<sup>x</sup> <sup>γ</sup>ab

A<sup>μ</sup> ! A<sup>0</sup>

In particular, γ<sup>5</sup> is invariant under the SO(4) transformation as

<sup>μ</sup> x !; t � �

!; ϕ<sup>0</sup> 0 � �

<sup>Γ</sup><sup>5</sup> ! <sup>Γ</sup><sup>5</sup> � �<sup>0</sup>

and

Under this definition (γ<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup>

52 Superfluids and Superconductors

cannot be well defined due to

<sup>z</sup>-direction) <sup>U</sup><sup>b</sup> <sup>x</sup>=y=<sup>z</sup>

where <sup>n</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup>; <sup>n</sup><sup>2</sup>; <sup>n</sup><sup>3</sup>ϕ<sup>0</sup>

ð Þ<sup>x</sup> 6¼ <sup>Γ</sup><sup>5</sup> .

the SO(4) transformation, we have

according to <sup>γ</sup><sup>0</sup>ð Þ<sup>x</sup> � �<sup>0</sup>

<sup>Γ</sup><sup>5</sup> ! <sup>Γ</sup><sup>5</sup> � �<sup>0</sup>

have

We introduce an SO(4) transformation U x

! ð Þ� x Γ ! , i.e.,

ET ð Þ¼ <sup>x</sup> <sup>e</sup><sup>i</sup>δ<sup>Φ</sup>

0 � � <sup>¼</sup> <sup>n</sup> Γ1

ð Þ¼ <sup>x</sup> <sup>U</sup><sup>b</sup> <sup>x</sup>=y=<sup>z</sup>

ð Þ¼ <sup>x</sup> <sup>U</sup><sup>b</sup> <sup>t</sup>

b !; t � �

ET x !; t � � Γ5 <sup>U</sup><sup>b</sup> <sup>x</sup>=y=<sup>z</sup>

However, the effect of deformed zero-lattice from tempo entanglement transformation e<sup>i</sup>δΦt�Γ<sup>5</sup>

ET x !; t � � Γ5 Ub t

mation Rbð Þx and spatial entanglement transformation (entanglement transformation along x/y/

<sup>U</sup><sup>b</sup> ð Þ¼ <sup>x</sup> <sup>R</sup>bð Þ<sup>x</sup> <sup>⊕</sup> <sup>U</sup><sup>b</sup> <sup>x</sup>=y=<sup>z</sup>

� ��<sup>1</sup>

Here, ⊕ denotes operation combination. Under a non-uniform SO(4) transformation Ub ð Þx , we

<sup>¼</sup> <sup>γ</sup><sup>0</sup> ð Þ<sup>x</sup> � �<sup>0</sup>

<sup>γ</sup><sup>μ</sup> ! γμ ð Þ ð Þ<sup>x</sup> <sup>0</sup> <sup>¼</sup> <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup> γμ <sup>U</sup><sup>b</sup> ð Þ<sup>x</sup>

¼ U x b !; t � �

, <sup>γ</sup><sup>2</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

Γ2

, <sup>γ</sup><sup>3</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

Γ3

<sup>γ</sup><sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>τ</sup><sup>x</sup> <sup>⊗</sup> <sup>1</sup>, <sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>i</sup>γ<sup>0</sup>γ<sup>1</sup>γ<sup>2</sup>γ<sup>3</sup>: ! (78)

), the effect of deformed zero-lattice from spatial entanglement

ET ð Þ<sup>x</sup> �<sup>1</sup> 6¼ <sup>Γ</sup><sup>5</sup>

ETð Þ<sup>x</sup> �<sup>1</sup> <sup>¼</sup> <sup>Γ</sup><sup>5</sup>

<sup>¼</sup> <sup>X</sup> a γa na

6¼ <sup>γ</sup>0, the entanglement matrix <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> along tempo direction is varied,

� ��<sup>1</sup>

� ��<sup>1</sup>

� ��<sup>1</sup>

Aμð Þx Ub ð Þx

þUb ð Þx ∂<sup>μ</sup> Ub ð Þx

,

:

is a unit SO(4) vector-field. For the deformed zero-lattice,

that is a combination of spin rotation transfor-

ET ð Þx : (81)

<sup>γ</sup>ab ¼ � <sup>1</sup>

<sup>4</sup> γ<sup>a</sup> ; γ<sup>b</sup> � � � � . Under

(83)

, (77)

: (79)

: (80)

ð Þx (82)

$$\begin{aligned} F^{\ddagger} &= d A^{\ddagger} + A^{\ddagger} \wedge A^{kj} \\ \equiv &-A^{\ddagger0} \wedge A^{\ddagger0}. \end{aligned} \tag{91}$$

Finally, we emphasize the equivalence between γ<sup>0</sup><sup>i</sup> and Γ<sup>i</sup> , i.e., γ<sup>0</sup><sup>i</sup> ⇔ Γ<sup>i</sup> .

#### 4.3.3. Gauge description for deformed spatial entanglement matrix

Next, we study the unit SO(4) vector-field of deformed spatial entanglement matrix Γ<sup>i</sup> � �<sup>0</sup> ð Þx . To characterize Γ<sup>i</sup> � �<sup>0</sup> ð Þ<sup>x</sup> , the reduced Gamma matrices <sup>γ</sup><sup>μ</sup> is defined as

$$
\gamma^1 = \gamma^0 \Gamma^j, \gamma^2 = \gamma^0 \Gamma^k, \gamma^3 = \gamma^0 \Gamma^5,\tag{92}
$$

and

$$\begin{aligned} \gamma^0 &= \Gamma' = \tau^z \otimes \sigma^i, \\ \gamma^5 &= i \gamma^0 \gamma^1 \gamma^2 \gamma^3. \end{aligned} \tag{93}$$

<sup>U</sup><sup>~</sup> ð Þ<sup>x</sup> <sup>γ</sup>~<sup>0</sup> <sup>U</sup><sup>~</sup> ð ÞÞ <sup>x</sup> �<sup>1</sup>

�

0

Finally, we emphasize the equivalence between γ~<sup>0</sup><sup>i</sup> and Γ<sup>a</sup>

4.3.4. Hidden SO(4) invariant for gauge description

different directions in 3 + 1D (winding) space-time Γ

with <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> <sup>¼</sup> 1. Here, <sup>α</sup>, <sup>β</sup>, <sup>γ</sup>, <sup>δ</sup> are constant.

curved space-time by using geometric description, x ¼ x

hand, we need to consider a varied vector-field

tempo directions to x<sup>0</sup> ¼ αx þ βy þ γz þ δt.

zero-lattice

lattice.

SO(4) invariant, we define the reduced Gamma matrices γ~<sup>μ</sup> as

<sup>γ</sup>~<sup>1</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

<sup>γ</sup>~<sup>5</sup> <sup>¼</sup> <sup>i</sup>γ~<sup>0</sup>

Γ2

γ~1 γ~2 γ~3

where <sup>n</sup><sup>~</sup> <sup>¼</sup> <sup>n</sup>~<sup>1</sup>; ; <sup>n</sup>~<sup>2</sup>; ; <sup>n</sup>~<sup>3</sup>; <sup>ϕ</sup>~<sup>0</sup>

by

<sup>¼</sup> <sup>γ</sup>~<sup>0</sup>ð Þ<sup>x</sup> � �<sup>0</sup>

<sup>A</sup><sup>~</sup> abð Þ¼ <sup>x</sup> tr <sup>γ</sup>~ij <sup>U</sup><sup>~</sup> ð ÞÞ <sup>x</sup> <sup>d</sup> <sup>U</sup><sup>~</sup> ð ÞÞ <sup>x</sup> �<sup>1</sup> � �: � �

<sup>F</sup>~ij <sup>¼</sup> dA<sup>~</sup> ij <sup>þ</sup> <sup>A</sup><sup>~</sup> ik <sup>∧</sup> <sup>A</sup><sup>~</sup> kj � �A<sup>~</sup> <sup>i</sup><sup>0</sup> <sup>∧</sup> <sup>A</sup><sup>~</sup> <sup>j</sup><sup>0</sup>

In addition, there exists a hidden global SO(4) invariant for entanglement matrices along

, <sup>γ</sup>~<sup>2</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

<sup>γ</sup>~<sup>0</sup> <sup>¼</sup> <sup>α</sup>Γ<sup>1</sup> <sup>þ</sup> <sup>β</sup>Γ<sup>2</sup> <sup>þ</sup> <sup>γ</sup>Γ<sup>3</sup> <sup>þ</sup> <sup>δ</sup>Γ<sup>5</sup>

Under this description, we can study the entanglement deformation along orthotropic spatial/

Due to the generalized spatial translation symmetry there exists an intrinsic relationship between gauge description for entanglement deformation between two vortex-membranes and geometric description for global coordinate transformation of the same deformed zero-

On the one hand, to characterize the changes of the positions of zeroes, we must consider a

4.4. Relationship between geometric description and gauge description for deformed

!

Γ3

; <sup>Γ</sup><sup>5</sup> � � ! <sup>Γ</sup>

, <sup>γ</sup>~<sup>3</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

,

According to pure gauge condition, we also have the following Maurer-Cartan equation,

<sup>¼</sup> <sup>X</sup> a γ~a n~a ð Þx

, i.e., γ~<sup>01</sup> ⇔ Γ<sup>2</sup>

Topological Interplay between Knots and Entangled Vortex-Membranes

!0 ; Γ<sup>5</sup> � �<sup>0</sup>

Γ5 ,

!; t

� � ! <sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup>

!0 ; t 0

� �. On the other

� � is a unit vector-field. The auxiliary gauge field <sup>A</sup><sup>~</sup> abð Þ<sup>x</sup> are defined

(100)

55

(101)

: (102)

� �. To show the hidden

, γ~<sup>03</sup> ⇔ Γ<sup>5</sup>

.

(103)

, γ~<sup>02</sup> ⇔ Γ<sup>3</sup>

http://dx.doi.org/10.5772/intechopen.72809

Here, Γ<sup>i</sup> , Γ<sup>j</sup> , and Γ<sup>k</sup> are three orthotropic spatial entanglement matrices. Under this definition (γ<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>i</sup> ), the effect of deformed zero-lattice from partial spatial/tempo entanglement transformation e<sup>i</sup>Γ<sup>j</sup> �ΔΦ<sup>j</sup> , e<sup>i</sup>Γ<sup>k</sup> �ΔΦ<sup>k</sup> , e<sup>i</sup>Γ5�ΔΦ<sup>t</sup> can be studied due to

$$
\Gamma^i \to \left(\Gamma^i\right)'(\mathbf{x}) = \widehat{\mathcal{U}}\_{\text{ET}}^{\mathbf{x}/\mathbf{x}\_k/t}\left(\overrightarrow{\mathbf{x}},t\right)\Gamma^i \widehat{\mathcal{U}}\_{\text{ET}}^{\mathbf{x}/\mathbf{x}\_k/t}(\mathbf{x})^{-1} \neq \Gamma^i. \tag{94}
$$

However, the effect of deformed zero-lattice from spatial entanglement transformation e<sup>i</sup>δΦt�Γ<sup>5</sup> cannot be well defined due to

$$
\Gamma^i \to \left(\Gamma^i\right)'(\mathbf{x}) = \widehat{\mathcal{U}}\_{\text{ET}}^{\mathbf{x}\_i}\left(\overrightarrow{\mathbf{x}},t\right)\Gamma^i \widehat{\mathcal{U}}\_{\text{ET}}^{\mathbf{x}\_i}(\mathbf{x})^{-1} = \Gamma^i. \tag{95}
$$

We use similar approach to introduce the gauge description. We can also define the reduced Gamma matrices γ~<sup>μ</sup> as

$$
\hat{\gamma}^1 = \hat{\gamma}^0 \Gamma^2, \hat{\gamma}^2 = \hat{\gamma}^0 \Gamma^3, \hat{\gamma}^3 = \hat{\gamma}^0 \Gamma^5,\tag{96}
$$

and

$$\begin{aligned} \tilde{\gamma}^0 &= \Gamma^i = \tau^z \otimes \sigma^\imath , \\ \tilde{\gamma}^5 &= \dot{\imath} \tilde{\gamma}^0 \dot{\gamma}^1 \tilde{\gamma}^2 \tilde{\gamma}^3 . \end{aligned} \tag{97}$$

The correspondence between index of γ~<sup>a</sup> and index of space-time xa is

$$\begin{array}{l} \tilde{\gamma}^1 \Leftrightarrow y, \tilde{\gamma}^2 \Leftrightarrow z, \\ \tilde{\gamma}^3 \Leftrightarrow t, \tilde{\gamma}^0 \Leftrightarrow x. \end{array} \tag{98}$$

We denote this correspondence to be

$$(1,2,3,0)\_{\text{ET}} \Leftrightarrow (2,3,0,1)\_{\text{ST}}.\tag{99}$$

Now, the SO(4) transformation U x ~ !; t � � <sup>¼</sup> <sup>e</sup> Φab x ! ð Þ;<sup>t</sup> <sup>γ</sup>~ab <sup>γ</sup>~ab ¼ � <sup>1</sup> <sup>4</sup> <sup>γ</sup>~<sup>a</sup>; ; <sup>γ</sup>~<sup>b</sup> � � � � is not a combination of spin rotation symmetry and entanglement transformation along x/y/z-direction. However, for the case of a or b to be 0, U x ~ !; t � � <sup>¼</sup> <sup>e</sup> Φa<sup>0</sup> x ! ð Þ;<sup>t</sup> <sup>γ</sup>~a<sup>0</sup> denotes the entanglement transformation along y/z/t-direction. The unit SO(4) vector-field on each lattice site becomes

Topological Interplay between Knots and Entangled Vortex-Membranes http://dx.doi.org/10.5772/intechopen.72809 55

$$\left(\tilde{\mathcal{U}}(\mathbf{x})\right)^{0}\left(\tilde{\mathcal{U}}(\mathbf{x})\right)^{-1} = \left(\tilde{\mathcal{V}}^{0}(\mathbf{x})\right)' = \sum\_{a} \tilde{\mathcal{V}}^{a}\tilde{n}^{a}(\mathbf{x})\tag{100}$$

where <sup>n</sup><sup>~</sup> <sup>¼</sup> <sup>n</sup>~<sup>1</sup>; ; <sup>n</sup>~<sup>2</sup>; ; <sup>n</sup>~<sup>3</sup>; <sup>ϕ</sup>~<sup>0</sup> 0 � � is a unit vector-field. The auxiliary gauge field <sup>A</sup><sup>~</sup> abð Þ<sup>x</sup> are defined by

$$\tilde{A}^{\text{ab}}(\mathbf{x}) = \text{tr}\Big(\tilde{\gamma}^{\neq \dagger} \Big(\tilde{\mathcal{U}}(\mathbf{x})) d\Big(\tilde{\mathcal{U}}(\mathbf{x})\big)^{-1}\Big). \tag{101}$$

According to pure gauge condition, we also have the following Maurer-Cartan equation,

$$
\tilde{F}^{\dagger j} = d\tilde{A}^{\dagger j} + \tilde{A}^{\text{ik}} \wedge \tilde{A}^{k j} \equiv -\tilde{A}^{\dagger 0} \wedge \tilde{A}^{j0}.\tag{102}
$$

Finally, we emphasize the equivalence between γ~<sup>0</sup><sup>i</sup> and Γ<sup>a</sup> , i.e., γ~<sup>01</sup> ⇔ Γ<sup>2</sup> , γ~<sup>02</sup> ⇔ Γ<sup>3</sup> , γ~<sup>03</sup> ⇔ Γ<sup>5</sup> .

#### 4.3.4. Hidden SO(4) invariant for gauge description

<sup>γ</sup><sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> Γj

�ΔΦ<sup>k</sup> , e<sup>i</sup>Γ5�ΔΦ<sup>t</sup> can be studied due to

ð Þ¼ <sup>x</sup> <sup>U</sup><sup>b</sup> xj=xk=<sup>t</sup>

ð Þ¼ <sup>x</sup> <sup>U</sup><sup>b</sup> xi

Γ2

ET x !; t � � Γi <sup>U</sup><sup>b</sup> xj=xk=<sup>t</sup>

However, the effect of deformed zero-lattice from spatial entanglement transformation e<sup>i</sup>δΦt�Γ<sup>5</sup>

ET x !; t � � Γi Ub xi

We use similar approach to introduce the gauge description. We can also define the reduced

<sup>γ</sup>~<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>τ</sup><sup>z</sup> <sup>⊗</sup> <sup>σ</sup>x,

γ~<sup>1</sup> ⇔ y, γ~<sup>2</sup> ⇔ z,

of spin rotation symmetry and entanglement transformation along x/y/z-direction. However,

Γ3

, <sup>γ</sup>~<sup>3</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

, <sup>γ</sup>~<sup>2</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

<sup>Γ</sup><sup>i</sup> ! <sup>Γ</sup><sup>i</sup> � �<sup>0</sup>

<sup>Γ</sup><sup>i</sup> ! <sup>Γ</sup><sup>i</sup> � �<sup>0</sup>

<sup>γ</sup>~<sup>1</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup>

The correspondence between index of γ~<sup>a</sup> and index of space-time xa is

~ !; t � �

~ !; t � � ¼ e Φab x ! ð Þ;<sup>t</sup> <sup>γ</sup>~ab

¼ e Φa<sup>0</sup> x ! ð Þ;<sup>t</sup> <sup>γ</sup>~a<sup>0</sup>

along y/z/t-direction. The unit SO(4) vector-field on each lattice site becomes

and

Here, Γ<sup>i</sup>

(γ<sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>i</sup>

and

mation e<sup>i</sup>Γ<sup>j</sup>

, Γ<sup>j</sup>

54 Superfluids and Superconductors

�ΔΦ<sup>j</sup> , e<sup>i</sup>Γ<sup>k</sup>

cannot be well defined due to

We denote this correspondence to be

Now, the SO(4) transformation U x

for the case of a or b to be 0, U x

Gamma matrices γ~<sup>μ</sup> as

, <sup>γ</sup><sup>2</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

<sup>γ</sup><sup>0</sup> <sup>¼</sup> <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>τ</sup><sup>z</sup> <sup>⊗</sup> <sup>σ</sup><sup>i</sup>

Γk

, <sup>γ</sup><sup>3</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup>

,

, and Γ<sup>k</sup> are three orthotropic spatial entanglement matrices. Under this definition

), the effect of deformed zero-lattice from partial spatial/tempo entanglement transfor-

Γ5

<sup>γ</sup><sup>5</sup> <sup>¼</sup> <sup>i</sup>γ<sup>0</sup>γ<sup>1</sup>γ<sup>2</sup>γ<sup>3</sup>: (93)

ET ð Þ<sup>x</sup> �<sup>1</sup> 6¼ <sup>Γ</sup><sup>i</sup>

ETð Þ<sup>x</sup> �<sup>1</sup> <sup>¼</sup> <sup>Γ</sup><sup>i</sup>

Γ5

<sup>γ</sup>~<sup>5</sup> <sup>¼</sup> <sup>i</sup>γ~<sup>0</sup>γ~<sup>1</sup>γ~<sup>2</sup>γ~<sup>3</sup>: (97)

<sup>γ</sup>~<sup>3</sup> <sup>⇔</sup> t, <sup>γ</sup>~<sup>0</sup> <sup>⇔</sup> <sup>x</sup>: (98)

<sup>4</sup> <sup>γ</sup>~<sup>a</sup>; ; <sup>γ</sup>~<sup>b</sup> � � � � is not a combination

denotes the entanglement transformation

ð Þ 1; 2; 3; 0 ET ⇔ ð Þ 2; 3; 0; 1 ST: (99)

<sup>γ</sup>~ab ¼ � <sup>1</sup>

, (92)

: (94)

: (95)

, (96)

In addition, there exists a hidden global SO(4) invariant for entanglement matrices along different directions in 3 + 1D (winding) space-time Γ ! ; <sup>Γ</sup><sup>5</sup> � � ! <sup>Γ</sup> !0 ; Γ<sup>5</sup> � �<sup>0</sup> � �. To show the hidden SO(4) invariant, we define the reduced Gamma matrices γ~<sup>μ</sup> as

$$\begin{aligned} \tilde{\gamma}^1 &= \tilde{\gamma}^0 \Gamma^2, \tilde{\gamma}^2 = \tilde{\gamma}^0 \Gamma^3, \tilde{\gamma}^3 = \tilde{\gamma}^0 \Gamma^5, \\ \tilde{\gamma}^0 &= \alpha \Gamma^1 + \beta \Gamma^2 + \gamma \Gamma^3 + \delta \Gamma^5, \\ \tilde{\gamma}^5 &= i \tilde{\gamma}^0 \tilde{\gamma}^1 \tilde{\gamma}^2 \tilde{\gamma}^3 \end{aligned} \tag{103}$$

with <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> <sup>¼</sup> 1. Here, <sup>α</sup>, <sup>β</sup>, <sup>γ</sup>, <sup>δ</sup> are constant.

Under this description, we can study the entanglement deformation along orthotropic spatial/ tempo directions to x<sup>0</sup> ¼ αx þ βy þ γz þ δt.

## 4.4. Relationship between geometric description and gauge description for deformed zero-lattice

Due to the generalized spatial translation symmetry there exists an intrinsic relationship between gauge description for entanglement deformation between two vortex-membranes and geometric description for global coordinate transformation of the same deformed zerolattice.

On the one hand, to characterize the changes of the positions of zeroes, we must consider a curved space-time by using geometric description, x ¼ x !; t � � ! <sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup> !0 ; t 0 � �. On the other hand, we need to consider a varied vector-field

$$\left(\boldsymbol{\gamma}^{0}(\mathbf{x})\right)' = \widehat{\boldsymbol{\mathcal{U}}}\left(\mathbf{x}\right)\boldsymbol{\upgamma}^{0}\left(\widehat{\boldsymbol{\mathcal{U}}}(\mathbf{x})\right)^{-1} = \sum\_{a} \boldsymbol{\upgamma}^{a} n^{a}(\mathbf{x})\tag{104}$$

In addition, we point out that for different representation of reduced Gamma matrix, there exists intrinsic relationships between the gauge fields <sup>A</sup>ð Þ<sup>x</sup> and <sup>A</sup><sup>~</sup> ð Þ<sup>x</sup> . After considering these relationships, we have a complete description of the deformed zero-lattice on the geometric

Gravity is a natural phenomenon by which all objects attract one another including galaxies, stars, human-being and even elementary particles. Hundreds of years ago, Newton discovered

the distance, and M and m are the masses for two objects. One hundred years ago, the establishment of general relativity by Einstein is a milestone to learn the underlying physics of gravity that provides a unified description of gravity as a geometric property of space-time.

curved space-time. Here Rμν is the 2nd rank Ricci tensor, R is the curvature scalar, gμν is the

A knot corresponds to an elementary object of a knot-crystal; a knot-crystal can be regarded as composite system of multi-knot. For example, for 1D knot, people divide the knot-crystal into

From point view of information, each knot corresponds to a zero between two vortexmembranes along the given direction. For a knot, there must exist a zero point, at which ξAð Þx is equal to ξBð Þx . The position of the zero is determined by a local solution of the zero-

From point view of geometry, a knot (an anti-knot) removes (or adds) a projected zero of zerolattice that corresponds to removes (or adds) half of "lattice unit" on the zero-lattice according to

As a result, a knot looks like a special type of edge dislocation on 3 + 1D zero-lattice. The zero-

From point view of entanglement, a knot becomes topological defect of 3 + 1D winding space-

!; t

Δxi ¼ �aeff x

lattice is deformed and becomes mismatch with an additional knot.

y-direction, knot is anti-phase changing denoted by eiΓ2�ΔΦ<sup>y</sup>

time: along x-direction, knot is anti-phase changing denoted by e<sup>i</sup>Γ1�ΔΦ<sup>x</sup>

In this section, we point out that there exists emergent gravity for knots on zero-lattice.

<sup>r</sup><sup>2</sup> where G is the Newton constant, r is

http://dx.doi.org/10.5772/intechopen.72809

57

<sup>≃</sup> � <sup>a</sup>: (111)

, ΔΦx ¼ π; along

, ΔΦ<sup>y</sup> ¼ π; along z-direction, knot

<sup>2</sup> Rgμν ¼ 8πGTμν, the gravitational force is really an effect of

Topological Interplay between Knots and Entangled Vortex-Membranes

space-time,

5. Emergent gravity

From Einstein's equations <sup>R</sup>μν � <sup>1</sup>

5.1. Knots as topological defects

the inverse-square law of universal gravitation, <sup>F</sup> <sup>¼</sup> GMm

metric tensor, and Tμν is the energy-momentum tensor of matter.

5.1.1. Knot as SO(4)/SO(3) topological defect in 3 + 1D space-time

N identical pieces, each of which is just a knot.

equation, Fθð Þ¼ x 0 or ξA,θð Þ¼ x ξB,θð Þx .

by using gauge description. There exists intrinsic relationship between the geometry fields ea ð Þ<sup>x</sup> ð Þ <sup>a</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>0</sup> and the auxiliary gauge fields <sup>A</sup><sup>a</sup><sup>0</sup> ð Þx .

For a non-uniform zero-lattice, we have

$$\begin{split} \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right) &\Rightarrow \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right) = \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right) + \delta \ \overrightarrow{\Phi} \left( \overrightarrow{\boldsymbol{x}}, t \right), \\ \Phi\_{t} \left( \overrightarrow{\boldsymbol{x}}, t \right) &\Rightarrow \Phi\_{t}^{\prime} \left( \overrightarrow{\boldsymbol{x}}, t \right) = \Phi\_{t} \left( \overrightarrow{\boldsymbol{x}}, t \right) + \delta \Phi\_{t} \left( \overrightarrow{\boldsymbol{x}}, t \right). \end{split} \tag{105}$$

On deformed zero-lattice, the "lattice distances" become dynamic vector fields. We define the vierbein fields e<sup>a</sup> ð Þx that are supposed to transform homogeneously under the local symmetry, and to behave as ordinary vectors under local entanglement transformation along xa-direction,

$$
\varepsilon^a(\mathbf{x}) = d\mathbf{x}^a(\mathbf{x}) = \frac{a}{\pi} d\Phi^a(\mathbf{x}).\tag{106}
$$

For the smoothly deformed vector-fields n<sup>i</sup> ð Þ<sup>x</sup> <sup>≪</sup> 1, within the representation of <sup>Γ</sup><sup>5</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> we have

$$\frac{d\Phi^i(\mathbf{x})}{2\pi} = n^i(\mathbf{x}) = \text{tr}\left[\gamma^0 d\gamma^i(\mathbf{x})\right] \tag{107}$$

$$= A^{i0}(\mathbf{x}) / i = 1, 2, 3.$$

Thus, the relationship between ei ð Þ<sup>x</sup> and <sup>A</sup><sup>i</sup><sup>0</sup> ð Þx is obtained as

$$e^i(\mathbf{x}) \equiv (2a)A^{i0}(\mathbf{x}).\tag{108}$$

According to this relationship, the changing of entanglement of the vortex-membranes curves the 3D space.

On the other hand, within the representation of <sup>Γ</sup><sup>i</sup> <sup>¼</sup> <sup>γ</sup>~<sup>0</sup> we have

$$\begin{split} \frac{d\Phi^a(\mathbf{x})}{2\pi} = \check{n}^a(\mathbf{x}) &= \text{tr}\left[\check{\boldsymbol{\gamma}}^0 d\boldsymbol{\hat{\gamma}}^a(\mathbf{x})\right] \\ &= \check{A}^{i0}(\mathbf{x}), i = j, k, 0, \end{split} \tag{109}$$

and

$$e^0(\mathbf{x}) = dt(\mathbf{x}) = \frac{a}{\pi} d\Phi\_t(\mathbf{x}) = (2a)\tilde{A}^{30}(\mathbf{x}).\tag{110}$$

According to this relationship, the changing of entanglement of the vortex-membranes curves the 4D space-time.

In addition, we point out that for different representation of reduced Gamma matrix, there exists intrinsic relationships between the gauge fields <sup>A</sup>ð Þ<sup>x</sup> and <sup>A</sup><sup>~</sup> ð Þ<sup>x</sup> . After considering these relationships, we have a complete description of the deformed zero-lattice on the geometric space-time,
