# Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems

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## Abstract

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## 1. Introduction

## 2. NJL Model and Condensed Matter Systems

#### 2.1. Interacting Fermion

#### 2.2. Discrete Model

#### 2.3. Conducting Polymers

## 3. Inhomogeneous Solutions

#### 3.1. Kink Solution, Kink Crystal Solution, and Zero Mode

#### 3.2. Nonlinear Schrödinger Equation and Various Inhomogeneous Solutions

#### 3.3. Internal Structures behind the NJL System

#### 3.4. Inhomogeneous Solutions on a Ring

#### 3.5. Finite System and Casimir Force

## 4. Correspondence between NJL and NL$\mathit{\sigma}$ Models

#### 4.1. Model, Method, and Homogeneous Solution

#### 4.2. Inhomogeneous Solutions

#### 4.3. Finite System

#### 4.4. NJL/$\mathbb{C}{P}^{N-1}$ Correspondence in 2 + 1 Dimensions

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The energy spectrum for the normal state (left figure) and the superconducting state (right figure). In the left figure, the electron band (downward convex line) and the hole representation for the band (convex upward line) are shown. In the terminology of the electron band, the states under the Fermi level are filled (black circles), while the states above the Fermi level are filled (white circles) for the hole terminology. In the presence of the superconducting correlation, the gap in the energy spectrum appears (right figure).

**Figure 2.**The energy spectrum in realistic parameters. The top-left figure shows the energy spectrum for the normal state. The solid and the dashed lines describe the particle and hole energy dispersions, respectively. The bottom left figure shows that for the superconducting phase. The dotted and dash-dotted lines denote the positive energy and negative energy branches, respectively. The superconducting gap is much smaller than the bandwidth, and the linear approximation is justified in this case (right figure).

**Figure 3.**The energy spectrum for the tight binding model with 100 sites. The half region of the momentum space $0<k<\pi /a$ is shown. The momentum and the number n are related via ${k}_{n}=\pi n/L$ with $L=Na$. The half filling case (left figure) approximates the linear dispersion, while the small filling case (right figure) approximates the parabolic dispersion. If the characteristic energy, such as the superconducting gap, is small enough compared with the bandwidth, the discretized model approximates the continuous system well.

**Figure 4.**The schematic structure of the polyacetylene. The black dots stand for the carbon sites. For simplicity, we omit the hydrogen atoms attached to the end of the vertical lines (

**a**). The single and double bonds consist of the $\sigma $ electrons and $\sigma +\pi $ electrons, respectively. There are the right-double pattern (

**b**) and left-double pattern (

**c**), whose energy is completely degenerate. The crucial difference of left-double and right-double patterns occur when the defect is present (

**d**). In such a case, the order parameter changes its sign at the defect site.

**Figure 5.**The occupation number of the zero mode and the charge-spin separation. (

**a**–

**c**), respectively, show the empty state, the singly-occupied state, and the doubly-occupied state. Those situations yield the charge-spin separation of the conducting degrees of freedom.

**Figure 6.**The general solution of the NLSE. Both the amplitude and the phase modulate in space. Here, m is set to be one. The amplitude of the condensate does not touch zero in the presence of the current. The modulation of the phase is also not uniform in space.

**Figure 7.**The model considered in this subsection. The magnetic flux (black line) penetrating the superconducting ring. The Zeeman field is also applied on the ring. A typical configuration of the FFphase (middle figure) and that of the LOphase (right figure) with $\nu =0.99$ are shown. The horizontal and vertical axes correspond to the spatial coordinate and $\Delta $, respectively. Here, m is set to be one. The solid line represents the order parameter, and the dotted and dashed lines are the configuration of the zero modes, which are scaled so that they fit to the figure.

**Figure 8.**The phase diagram of the superconducting ring. The solid and double solid lines correspond to the constant condensation (Bardeen–Cooper–Schrieffer (BCS) phase) and the LO phase, respectively. The FF phase and FFLO phase are classified to the two classes according to the direction of the current flow (+ for clockwise and − for counterclockwise). The dotted lines represent first order phase transition lines.

**Figure 9.**The normal-like solution (top left) and the BCS-like solution (bottom left figure) for $nu=0.999$. Both solutions diverge towards the boundaries. The Casimir force for the BCS-like solution changes the sign by changing $\nu $ (right). Here, L is set to be $2K\left(\nu \right)$.

**Figure 10.**Inhomogeneous solutions for the NL$\sigma $ model. The solid lines and the dashed lines correspond to the Higgs field configuration $\sigma $ and the gap function $\lambda $, respectively. In all the figures, we set $m=1$ and normalize the Higgs fields $\sigma $ so that it fits to the figure. The upper left figure corresponds to the Higgs soliton, where the Higgs field (magnetization) has the nonzero value around $x=0$. The bottom left figure is the Higgs soliton lattice solution with $\nu =0.9$. In the upper right figure, the confining soliton solution is shown. The bottom right figure is the confining soliton lattice solution for $\nu =0.99$.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yoshii, R.; Nitta, M.
Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems. *Symmetry* **2019**, *11*, 636.
https://doi.org/10.3390/sym11050636

**AMA Style**

Yoshii R, Nitta M.
Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems. *Symmetry*. 2019; 11(5):636.
https://doi.org/10.3390/sym11050636

**Chicago/Turabian Style**

Yoshii, Ryosuke, and Muneto Nitta.
2019. "Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems" *Symmetry* 11, no. 5: 636.
https://doi.org/10.3390/sym11050636