Fermi Condensation Flows Induced by Ricci Flows in the String σ Model

Abstract

The Fermi condensation flows in the sine-Gordon–Thirring model with two impurities coupling are investigated in this paper; these matter flows can be induced by the Ricci flow perturbation in the two-dimensional string $\sigma$ model. The Ricci flow perturbation equations are derived according to the Gauss–Codazzi equations, and the two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, the thermodynamic quantities on the cigar soliton background are obtained by using the variational functional integrals method. Subsequently, the Fermi condensation flows varying with the momentum scale $\lambda$ are analyzed and discussed. We find that the energy density, the correlation function, and the condensation fluctuations will decrease, but the entropy will increase monotonically. The Fermi condensed matter can maintain thermodynamic stability under the Ricci flow perturbation.

1. Introduction

Studying the evolution of various geometric quantities and physical quantities is a non-trivial topic in Ricci flows theory. However, the matter flows induced by Ricci flows have not been fully studied. An entropy functional W was introduced by Perelman to prove that the Poincare conjecture, W entropy, would increase with the Ricci flow parameter t [1,2,3].The Ricci flow equation was utilized to understand the change in ADM mass m; the results showed that the mass m is flow-invariant in ≥three-dimensional manifold [4]. In addition, similar problems have been studied according to the renormalization group (R-G) flow equations [5]. The authors demonstrated that the space-time energy can decrease under world sheet R-G flows. The Ricci flow technique was also used to study the evolution of the entropy in black holes; it can be seen that both black hole entropy and entanglement entropy decrease monotonically along Ricci flows [6,7]. This means that there was a certain difference between Perelman entropy and black hole entropy [8].

On the other hand, space-time perturbations are also a noteworthy issue in the two-dimensional gravity model. Black hole solutions with time-perturbations could exist in a 2D tachyon string model, which described the changes in the event horizon and the Hawking temperature [9]. By analyzing the time-dependent growth mode in the two-dimensional heterotic string model, it can be concluded that the black hole solution is unstable under small perturbations [10]. In addition, stationary perturbations in 2D black holes could cause energy density fluctuations of the Fermi condensed matter [11,12]. According to the R-G perturbation equations of the warping factor and a dilaton field, it can be proven that the Liouville fields correspond to the growth of the warping factor in the UV region, and its flow is independent of the perturbation for the dilaton [13]. The stability properties of the cigar soliton were analyzed in reference [14], and it was found that Witten’s black hole is also unstable under Ricci flow perturbations. However, when the metric and gauge fields generate Ricci flow perturbations, in the magnetic Reissner–Nordstrom solution, there is also a stable fixed point [15].

The cigar soliton is a fundamental solution of the string $\sigma$ model and plays a key role in the study of Ricci flows. In two-dimensional string gravity, the cigar solitons are a unique and stable gradient solution of Ricci flow equations, which has attracted widespread attention in the gravity and string community [16,17,18].

The sine-Gordon–Thirring model is a widely studied theoretical model in the quantum field theory and integrable systems, which combines the characteristics of the sine-Gordon model and the fermion Thirring model and reveals the profound correspondence between the two models through bosonization techniques [19]. By using Coleman’s duality transformation, the boson field in the sine-Gordon model can be mapped to the fermion field in the Thirring model [20]. This duality allows the derivation of the properties of one model by analyzing the characteristics of another model, which is particularly suitable for solving physical quantities such as exact solutions, conservation laws, and statistical averages. In addition, the non-perturbative properties of the sine-Gordon–Thirring model can be studied through the variational functional integrals method, and it is demonstrated that Fermi condensation can have stable phase structures [21].

In 2D curved space-time, the energy density of the Fermi condensed matter will fluctuate with the perturbation the black hole [11,12]. If the gravitational perturbation changes with the momentum scale, the Ricci flow will induce the Hawking temperature flow and the energy density flow. Therefore, the Fermi condensate flows are the matter flows generated by the Ricci flow perturbation of the gravitational field. In addition, the fermions’ condensation can decay over time; a BCS-like condensation of the fermions opens an energy gap and leads to a nonsingular bounce of the universe [22]. Ricci flow equations have been used to study the expansion of anisotropic space, and the results show that the anisotropy of the universe disappears during the evolution [23].

The main aim of this article is to study the Ricci flow perturbation solution of the cigar soliton in a two-dimensional string $\sigma$ model. A perturbation equation has been derived from the Gauss–Codazzi equations, and the two-loop asymptotic perturbation solution is reduced by using a small parameter expansion method. We first briefly review these analytical techniques, and then we study the sine-Gordon–Thirring model with two impurities coupling on the cigar soliton background. Moreover, the expressions of the Fermi condensation flows are derived by means of the variational functional integrals method. The variation of the physical meaning of the thermodynamic quantities with the momentum scale is analyzed, and the thermodynamic stability of the Fermi condensed matter under Ricci flow perturbation is also discussed.

2. The Two-Loop Flow Perturbation Solution of the Cigar Soliton

The two-loop Ricci flow equations in the string $\sigma$ model can be expressed as [24,25,26,27,28,29]

$${\displaystyle \frac{\partial g_{\mu \nu }}{\partial \lambda }}=-\epsilon \left(R_{\mu \nu }+2{\nabla }_{\mu }{\nabla }_{\nu }\Phi \right)-{\displaystyle \frac{{\epsilon }^2}{2}}{R}_{\mu klm}{R}_{\nu }^{klm},$$

(1)

$${\displaystyle \frac{\partial \Phi }{\partial \lambda }}=-c+\epsilon ({\displaystyle \frac{1}{2}}{\nabla }^2\Phi -{\left(\nabla \Phi \right)}^2)-{\displaystyle \frac{{\epsilon }^2}{16}}{R}_{abcd}{R}^{abcd},$$

(2)

where $g_{\mu \nu }$ is the graviton, $\Phi$ is the dilaton, $c=(D-26)/6$ is the central charge of the conformal field, D is the space-time dimension, and $\epsilon \ll 1$ represents a string coupling coefficient. In general, the metric and curvature are functions of the time coordinate t and the space coordinate x. In this article, we consider the static-state situation, so the metric and curvature are functions of the space coordinate. The indices $\mu ,\nu$, $k,l,m$, and $a,b,c,d$ are space-time coordinate indices (0,1). The two-dimensional space-time metric is taken as $g_{00}=\alpha \left(x\right)+h(x,\lambda ),$ $g_{11}=\beta \left(x\right),$ $g_{10}=g_{01}=0$, where $h(x,\lambda )$ is the perturbation function of the flow parameter $\lambda$. The symbol x means the spatial coordinate $x_1$; it corresponds to a direction along which there are independent metric components.

From Equation (1), we obtain the following Ricci flow perturbation equation

$$R_h+2[\left({\displaystyle \frac{{\Phi }^{\prime }}{2\alpha \beta }}\right){\displaystyle \frac{\partial h}{\partial x}}-\left({\displaystyle \frac{{\alpha }^{\prime }{\Phi }^{\prime }}{2{\alpha }^2\beta }}\right)h]\left(1-\epsilon R\right)+{\displaystyle \frac{\epsilon }{2}}{R}^2\left(1-\epsilon R\right)=-{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial h}{\partial \lambda }}\left(1-\epsilon \right).$$

(3)

In Equation (3), the prime symbol $\prime$ represents the differentiation of the metric factor $\alpha$ and dilaton fields $\Phi$ with respect to the space coordinate x. In order to derive the explicit form of the perturbation curvature $R_h,$ we first study the Gauss–Codazzi equations under the perturbation metric. A high-dimensional manifold is defined as a space that is locally homeomorphic to Euclidean space, and the main research object of this article is the 2D surface in the 3D Euclid space. Two-dimensional surfaces can be embedded into three-dimensional Euclidean space while still maintaining their original topological structure and local Euclidean properties. In order to understand the geometric structure of surfaces, it is necessary to introduce the concepts of the first fundamental form and the second fundamental form.

The first fundamental form, as a metric on a surface, provides intrinsic geometry information. The second fundamental form describes the extrinsic geometry characteristic (or shape of a surface). According to the second fundamental form, we can further define the Weingarten transformation on a two-dimensional vector space, which is a linear transformation between tangent planes. The eigenvalues of this linear transformation are called principal curvature. The Gauss–Codazzi equation relates the intrinsic and extrinsic geometry of a surface embedded in a three-dimensional manifold, and these equations reveal how the intrinsic curvature of a surface affects its extrinsic shape. In general relativity, the Gauss–Codazzi equation is used to describe the surface geometry of space-time, and it helps us to understand the space-time curvature of gravitational fields.

In surface differential geometry, an intuitive method of obtaining structural equations is the use of an exterior differential form and orthogonal motion frame [30,31]. The first fundamental form of the curved surface is expressed as $I={\omega }_1{\omega }_1+{\omega }_2{\omega }_2$. The second fundamental form is $II={\omega }_1{\omega }_{13}+{\omega }_2{\omega }_{23}$, where ${\omega }_{ij}=<d\overrightarrow{e_i},\overrightarrow{e_j}>$ ($i=1,2,3$) are defined by the orthogonal frame $\left\{\overrightarrow{e_i}\right\}.$ By utilizing the operation rules for the exterior differential, we further obtain the structural equations for the orthogonal frame

$$d{\omega }_{12}={\omega }_{13}\wedge {\omega }_{32},d{\omega }_{13}={\omega }_{12}\wedge {\omega }_{23},d{\omega }_{23}={\omega }_{21}\wedge {\omega }_{13}.$$

(4)

The metric of two-dimensional space-time is expressed as $d{s}^2=(\alpha +h)d{\tau }^2+\beta d{x}^2,$ where $d{s}^2=I$, $\alpha =\alpha \left(x\right),\beta =\beta \left(x\right)$ are the metric factors, and $h=h(x,\lambda )$ is a perturbation function. The orthogonal frame is chosen as $\overrightarrow{e_1}=\overrightarrow{r_{\tau }}/{(\alpha +h)}^{(1/2)}$,$\overrightarrow{e_2}=\overrightarrow{r_x}/{\left(\beta \right)}^{(1/2)}$. The 1-form is taken as ${\omega }_1={(\alpha +h)}^{(1/2)}d\tau$, ${\omega }_2={\left(\beta \right)}^{1/2}d\tau$, so we obtain differential form of the first order:

$${\omega }_{12}=-{\omega }_{21}=-{\displaystyle \frac{\partial \sqrt{\alpha +h}}{\partial x}}{\displaystyle \frac{1}{\sqrt{\beta }}}d\tau ,$$

(5)

$${\omega }_{13}={\displaystyle \frac{l}{\sqrt{\alpha +h}}}d\tau +{\displaystyle \frac{m}{\sqrt{\alpha +h}}}dx,$$

(6)

$${\omega }_{23}={\displaystyle \frac{m}{\sqrt{\beta }}}d\tau +{\displaystyle \frac{n}{\sqrt{\beta }}}dx,$$

(7)

where $l,m,n$ are the coefficients of the second fundamental form. We also obtain the exterior differential formulas

$$d{\omega }_{12}={\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{\partial \sqrt{\alpha +h}}{\partial x}}{\displaystyle \frac{1}{\sqrt{\beta }}}\right)d\tau \wedge dx,$$

(8)

$$d{\omega }_{13}={\displaystyle \frac{\partial }{\partial x}}({\displaystyle \frac{l}{\sqrt{\alpha +h}}})dx\wedge d\tau ,$$

(9)

$$d{\omega }_{23}={\displaystyle \frac{\partial }{\partial x}}({\displaystyle \frac{m}{\sqrt{\beta }}})dx\wedge d\tau .$$

(10)

Substituting Equations (5)–(10) into Equation (4), the Gauss–Codazzi equations can be reduced to

$$-[e^{″}-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{e^{\prime 2}}{e}}+{\displaystyle \frac{g^{\prime }{e}^{\prime }}{g}}\right)]=l.n,$$

(11)

$${\left({\displaystyle \frac{l}{e}}\right)}^{\prime }-n\left({\displaystyle \frac{{\left(\sqrt{e}\right)}^{\prime }}{g}}\right)=0,$$

(12)

where $e=\alpha +h,g=\beta$, ${\left(\sqrt{e}\right)}^{\prime }=e^{(-1/2)}{e}^{\prime }/2,{\left(\sqrt{g}\right)}^{\prime }=g^{(-1/2)}{g}^{\prime }/2$, ${\left(\sqrt{e}\right)}^{\prime }/\sqrt{g}=e^{(-1/2)}{e}^{\prime }/2g^{(1/2)}$. In Equations (11) and (12), the prime symbol $\prime$ represents the differentiation of the functions $e,g,l$ with respect to the space coordinate $x.$ The two eigenvalues of the Weingarten transformation at a specific point are called the principal curvature of the surface. To calculate this principal curvature k, we first derive the coefficient matrix M of the Weingarten transformation under the coordinate tangent vector. The principal curvature must satisfy the determinant equation $det(M-k.I)=0$. The two roots of this equation are expressed as $k_1=l/e,k_2=n/g.$ In general, the metric and curvature are functions of the time coordinate t and the space coordinate x. In the static-state case, the metric and curvature are functions of the one-dimensional space coordinate x. Thus, R is the static-state curvature in two-dimensional space-time, and $R_h$ represents the perturbation of the static-state curvature R. If the metric has a perturbation, then the Gaussian curvature $k_1{k}_2$ not only contains the unperturbed curvature R but also contains the perturbed curvature $R_h$. Therefore, the following expression for deformed curvature can be derived from the Gauss–Codazzi equations:

$$R_h=k_1{k}_2-R=-{\displaystyle \frac{h^{″}}{\alpha \beta }}+\left({\displaystyle \frac{{\alpha }^{\prime }}{{\alpha }^2\beta }}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\beta }^{\prime }}{\alpha {\beta }^2}}\right)h^{\prime }+\left({\displaystyle \frac{{\alpha }^{″}}{{\alpha }^2\beta }}-{\displaystyle \frac{{\alpha }^{\prime 2}}{{\alpha }^3\beta }}-{\displaystyle \frac{{\alpha }^{\prime }{\beta }^{\prime }}{2{\alpha }^2{\beta }^2}}\right)h.$$

(13)

Substituting Equation (13) into Equation (3), we obtain a linear perturbation equation:

$${\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}-\left({\displaystyle \frac{{\alpha }^{\prime }}{\alpha }}+{\displaystyle \frac{{\beta }^{\prime }}{2\beta }}+{\Phi }^{\prime }-\epsilon R{\Phi }^{\prime }\right){\displaystyle \frac{\partial h}{\partial x}}-\beta (1-\epsilon R){\displaystyle \frac{\partial h}{\partial \lambda }}+\left({\displaystyle \frac{{\alpha }^{\prime 2}}{{\alpha }^2}}+{\displaystyle \frac{{\alpha }^{\prime }{\beta }^{\prime }}{2\alpha \beta }}-{\displaystyle \frac{{\alpha }^{″}}{\alpha }}+{\displaystyle \frac{{\alpha }^{\prime }{\Phi }^{\prime }}{\alpha }}-\epsilon R{\displaystyle \frac{{\alpha }^{\prime }}{\alpha }}{\Phi }^{\prime }\right)h=0.$$

(14)

In Equations (13) and (14), the prime symbol $\prime$ represents the differentiation of the perturbation function $h,$ metric factors $\alpha ,\beta$ and dilaton fields $\Phi$ with respect to the space coordinate x. In the two-dimensional bosonic string $\sigma$ model, when the coupling coefficient ${\epsilon }^2=0,$ Equation (14) becomes a one-loop perturbation equation. Now, the Ricci flow Equations (1) and (2) have a cigar soliton solution, and the metric factors of the cigar soliton can be chosen as

$$\alpha \left(x\right)=\left({\displaystyle \frac{x^2}{1+x^2}}\right),\beta \left(x\right)=\left({\displaystyle \frac{1}{1+x^2}}\right),$$

(15)

The analytical solution of the dilaton field is $\Phi \left(x\right)=-ln(1+x^2)/2.$ The cigar soliton is a typical static solution in the Ricci flow theory, which can be derived from the one-loop flow equation of the string $\sigma$ model. This type of soliton holds significant importance in both geometry and physics. The cigar soliton can be used to study the eigenvalue problem of the Witten–Laplacian in two-dimensional space [32,33]. Moreover, the cigar soliton is equivalent to a two-dimensional black hole under first-order Ricci flows. A two-dimensional black hole model was examined by Witten in the conformal field theory; he found that the target space metric of the theory takes the form of a semi-infinite cigar [34]. The Ricci flow perturbation around cigar solitons’ solution can help us to understand the stability properties of a two-dimensional black hole [14], and it can also be used to study the deformation of space-time surfaces, such as the variation of two principal curvatures with the momentum scale [35].

We assume that the perturbation solution has a form $h(x,\lambda )=e^{(q-2)\lambda }f\left(x\right),$ where q is the perturbation index. In general, the two-loop perturbation Equation (14) has no analytical solution, but it is possible to find an asymptotic solution to this equation. The two-loop perturbation equation of the cigar soliton can be derived as follows:

$$f^{″}\left(x\right)+\left[{\displaystyle \frac{2(x^2-1)}{x(x^2+1)}}-\epsilon {\displaystyle \frac{4x}{{(x^2+1)}^2}}\right]f^{\prime }\left(x\right)+[{\displaystyle \frac{2+(2-q)x^2}{x^2(x^2+1)}}+\epsilon {\displaystyle \frac{4(q-2)(x^2+1)+8}{{(x^2+1)}^3}}]f\left(x\right)=0.$$

(16)

wheretwo $\epsilon$ terms represent the quantum correction effect of the graviton. In non-commutative space-time, there exists a minimum length $l=\sqrt{\theta },$ beyond which the space-time coordinates become blurred. The $\theta$ parameter can describe the fuzziness of space-time. When $x\gg \sqrt{\theta }$, the non-commutative metric degenerates into a classical metric at large distances, and the quantum effects tend to disappear. When $x\ll \sqrt{\theta }$, the new physical phenomena will occur at very small distances, and non-commutative effects can erase the black hole hole singularities, while the space-time manifolds become smooth [36].

In this article, in order to study the perturbation solution of the cigar soliton with quantum correction, we will consider the particular region of the space coordinate, where $x\ll 1.$ The two-loop quantum correction is obvious in this domain. The quantum effect can be ignored in the long-distance domain, where $x\gg 1.$ On the other hand, the one-loop perturbation equation has an analytic solution, and this solution can be expressed by the hypergeometric functions [35,37]. When $x\ll 1$, the expansion of the hypergeometric function can then be used to obtain the one-loop asymptotic solution. The connection between this asymptotic solution and the subsequent analysis of the Fermi condensate model will be discussed in Section 4.

When $x\ll 1$, the asymptotic expression of the one-loop perturbation solution is $f_0\left(x\right)=c_{1}x+c_2{x}^2,$ where $c_1$ and $c_2$ are coefficients of the perturbation solution. The next step is to consider the contribution of quantum correction terms. The two-loop perturbation Equation (16) can be rewritten as a differential equation with a small parameter $\epsilon$:

$$f^{″}+\left(p_0\left(x\right)+\epsilon p_1\left(x\right)\right)f^{\prime }+\left(q_0\left(x\right)+\epsilon q_1\left(x\right)\right)f=0,$$

(17)

where

$$\begin{array}{c}{p}_0\left(x\right)={\displaystyle \frac{2(x^2-1)}{x(x^2+1)}},p_1\left(x\right)=-{\displaystyle \frac{4x}{{(x^2+1)}^2}},\\ q_0\left(x\right)={\displaystyle \frac{2+(2-q)x^2}{x^2(x^2+1)}},q_1\left(x\right)={\displaystyle \frac{4(q-2)(x^2+1)+8}{{(x^2+1)}^3}}.\end{array}$$

(18)

We set the perturbation solution of the parameter Equation (16) as $f=f_0+\epsilon f_1,$ where $f_0$ is the one-loop asymptotic perturbation solution. If the perturbation coefficients are taken as $c_1=0$$,c_2\ne 0,$ the differential equation of the two-loop modified solution $f_1$ has the following form:

$$f_1^{″}+\left[{\displaystyle \frac{2(x^2-1)}{x(x^2+1)}}\right]f_1^{\prime }+\left[{\displaystyle \frac{2+(2-q)x^2}{x^2(x^2+1)}}\right]f_1+c_2{x}^2\left[{\displaystyle \frac{4(q-2)x^2+4q}{{(x^2+1)}^3}}\right]+c_{2}x\left[{\displaystyle \frac{8x}{{(x^2+1)}^2}}\right]=0.$$

(19)

When $q=4$, $x\ll 1,$ the asymptotic solution of this equation is $f_1=c_2(x-2x^2);$ thus, the two-loop perturbation solution of the cigar soliton can now be written as follows:

$$f\left(x\right)=c_2\left[\left((1-2\epsilon )x^2+\epsilon x\right)\right],$$

(20)

Thus, the Ricci flow perturbation solution of the metric becomes

$$h(x,\lambda )=c_2[(1-2\epsilon )x^2+\epsilon x]e^{2\lambda }.$$

(21)

where $\epsilon c_2$ terms are the two-loop perturbation effect. If the perturbation coefficient is $c_2>0,$ then its derivative is $\partial g_{00}/\partial \lambda >0$. The metric factor increases with the increase in the momentum scale $\lambda$. In quantum electrodynamics (QED), the effective coupling constant increases with the increase in the momentum scale, and the change trend of the perturbations is similar to QED. If the perturbation coefficient is $c_2<0$, then its derivative is $\partial g_{00}/\partial \lambda <0$. The metric factor decreases with the increase in the momentum scale $\lambda .$ In quantum chromodynamics (QCD), the effective coupling constant also decreases with the increase in the momentum scale. The variation trend of the perturbations behaves as in QCD, so the perturbation function has two different trends of change.

When the perturbation index is taken as $q=4,$ Equation (19) has an analytical solution. If the perturbation index is $q=4$ and the perturbation coefficient is $c_2>$ 0, the above argument and conclusion are only valid for the special value of q. For an arbitrary perturbation index q, the perturbation Equations (16) and (19) do not have analytical solutions. To confirm the reasonableness of our argument, such as the behavior of the metric perturbation with respect to the flow parameter $\lambda$, it is necessary to solve differential Equations (16) and (19) by using numerical methods. This is a major task for our future work.

The change trend of the Fermi condensation with momentum scale will be investigated in the next section; we will see that when the perturbation coefficient is $c_2>0$, the Fermi condensation flows can exist, and the entropy of condensed matter will monotonically increase with the increase in momentum scale. The change trend of the metric factor and the entropy is the same.

3. Variational Functional Integrals Method and Fermi Condensate Model

The partition function on the cigar soliton background is defined as [38]

$${\displaystyle \frac{Z}{Z_0}}={\displaystyle \frac{\int D_{\mu }exp[S_0+S_1+S_c]}{\int D_{\mu }exp\left[S_0\right]}}={\displaystyle \frac{\int {\prod }_{i,k}d{a}_i^{*}\left(k\right)d{a}_i\left(k\right)exp\left[S\right]}{\int {\prod }_{i,k}d{a}_i^{*}\left(k\right)d{a}_i\left(k\right)exp\left[S_0\right]}},$$

(22)

where $D_{\mu }$ is an integration measure of the Fermi fields, $S_0=S_0(\alpha ,\beta )$ is a free action, $S_1=S_1(\alpha ,\beta )$ is the coupling term of the fermions, $S_c=S_c(\alpha ,\beta )$ is the condensate term of the fermions, and $S_0$ and S are abbreviated as

$$S_0=-\sum _k{A}^{*}\left(k\right)S_{0}A\left(k\right),$$

(23)

$$S=-\sum _k{A}^{*}\left(k\right)(S_0+S_1)A\left(k\right)+S_c.$$

(24)

We notice that $A^{*}\left(k\right)=(a_1^{*}\left(k\right),a_2^{*}\left(k\right))$, $A\left(k\right)$ is its adjoint matrix, and $a_i^{*}\left(k\right)$ and $a_i\left(k\right)$ are the generators of the Grassmann algebra. Let us introduce a trial action $S_J$ with variational parameter J; then, the partition function (22) is given by

$$ln{\displaystyle \frac{Z}{Z_0}}=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n!}}<{(S_1-S_J)}^n{>}_c+lndet(I+S_J{S}_0^{-1})+S_c,$$

(25)

where

$$ln<e^{\xi (S_1-S_J)}{>}_{0,J}=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n!}}<{(S_1-S_J)}^n{>}_c=lndet(I+\xi (S_1-S_J){(S_0+S_J)}^{-1}),$$

(26)

and $\xi$ is a free parameter. From Equations (25) and (26), we obtain the following free energy:

$$F\left(\overline{\beta }\right)=-{\overline{\beta }}^{-1}ln{\displaystyle \frac{Z}{Z_0}}=-{\overline{\beta }}^{-1}ln<e^{S_1-S_J}{>}_{0,J}-{\overline{\beta }}^{-1}lndet(I+S_J{S}_0^{-1})-{\overline{\beta }}^{-1}{S}_c.$$

(27)

where $\overline{\beta }=1/T$ is a finite absolute temperature. By using the convexity inequality <$e^x$> $⩾exp\left(x\right)$, we obtain the relation

$$F\left(\overline{\beta }\right)\le F_J\left(\overline{\beta }\right)=F_{0,J}\left(\overline{\beta }\right)-{\overline{\beta }}^{-1}<S_1-S_J{>}_{0,J}-{\overline{\beta }}^{-1}{S}_c,$$

(28)

The first variational parameter J is determined by $\delta F_J\left(\overline{\beta }\right)/\delta J=0.$ The statistical correlation between two matrices is defined as

$$<e^{JA+KB}>=<e^{JA-S_J+KB-S_K}{>}_{0,J,K}.<e^{J{S}_J+S_K}{>}_0,$$

(29)

where A and B are two matrix elements in the action $S_1$. A statistical correlation with respect to the trial action $S_{J,K}$ is denoted by

$$<e^{JA-S_J+KB-S_K}{>}_{0,J,K}={\displaystyle \frac{\int {\prod }_{i,k}d{a}_i^{*}\left(k\right)d{a}_i\left(k\right)exp[JA-S_J+KB-S_K]exp[S_0+S_J+S_K]}{\int {\prod }_{i,k}d{a}_i^{*}\left(k\right)d{a}_i\left(k\right)exp[S_0+S_J+S_K]}}.$$

(30)

Now, we choose $S_J=J{S}_0,S_K=K{S}_0$; then, we obtain the cumulant expansion for the correlation function:

$$<AB>=2+2<A-S_0{>}_c+2<B-S_0{>}_c+<A-S_0{>}_c.<B-S_0{>}_c+<(A-S_0).(B-S_0){>}_c,$$

(31)

where

$$<A-S_0{>}_c={(1+J+K)}^{-1}Tr(A{S}_0^{-1}-1),<B-S_0{>}_c={(1+J+K)}^{-1}Tr(B{S}_0^{-1}-1),$$

(32)

$$<(A-S_0)(B-S_0){>}_c=-{\displaystyle \frac{{(1+J+K)}^{-2}}{2}}Tr{\left[(A{S}_0^{-1}-1)(B-S_0)\right]}_{+}.$$

(33)

The convexity inequality tells us that

$$ln<e^{JA+KB}>⩾W=J<A-S_0{>}_c+K<B-S_0{>}_c+2ln(1+J+K),$$

(34)

and the second variational parameter W is given by the equation $\delta W/\delta K=0.$

The Hamiltonian of the sine-Gordon–Thirring model on the cigar soliton background is written as [12]

$$\begin{array}{c}H={\int }_0^L\sqrt{-g}dx[i\overline{\psi }{\gamma }^{\mu }{\nabla }_{\mu }\psi +m\overline{\psi }\psi +\mu \overline{\psi }{\gamma }^0\psi +u_0\delta (x-x_0)\overline{\psi }{\gamma }^0\psi \\ +v_0\delta (x-x_0)\overline{\psi }\psi -{\displaystyle \frac{\omega }{2}}\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\left(\overline{\psi }{\gamma }_{\mu }\psi \right)],\end{array}$$

(35)

where $g=det\left(g_{\mu \nu }\right)$, $g_{\mu \nu }=e_{\mu }^{\left(a\right)}{e}_{\nu }^{\left(b\right)}{\eta }_{ab}$ is the metric of two-dimensional space-time, and ${\psi }_{i=1,2}$ are the Fermi fields. L is a distribution length of the Fermi matter, m is a mass term, $\mu$ is the chemical potential, and $u_0$ and $v_0$ are the coupling constants of two-impurities. $x_0$ is the common coordinate of two impurities, where 0$<x_0<L\ll 1,\omega$ is the coupling constant of the fermions, and ${\gamma }^{\mu }$ are the Dirac matrices in the curved space-time. The covariant derivative is defined as [37]

$${\gamma }^{\mu }{\nabla }_{\mu }={\gamma }^a{e}_{\left(a\right)}^{\mu }{\partial }_{\mu }+{\displaystyle \frac{1}{2}}{\gamma }^a{\displaystyle \frac{1}{\sqrt{g}}}{\partial }_{\mu }\left[\right(\sqrt{g}{e}_{\left(a\right)}^{\mu }].$$

(36)

We take the tetrad fields as follows: $e_{\left(0\right)}^0=1/\sqrt{\alpha +h},e_{\left(0\right)}^1=e_{\left(1\right)}^0=0,e_{\left(1\right)}^1=1/\sqrt{\beta }.$ The Fourier series of the Fermi fields can be expressed as [39,40]

$${\psi }_i(x,\tau )={\left(\overline{\beta }L\right)}^{-1/2}\sum _k{e}^{ikx}{a}_i\left(k\right),\overline{{\psi }_i}(x,\tau )={\left(\overline{\beta }L\right)}^{-1/2}\sum _k{e}^{-ikx}{a}_i^{*}\left(k\right),$$

(37)

where $\tau$ is the imaginary time, and k is the wave vector. Let us introduce the auxiliary Bose fields, ${\varphi }_i(x,\tau )={\psi }_i(x,\tau ){\psi }_i(x,\tau ),i=1,2.$ We consider the case ${\varphi }_i(x,\tau )=$ ${\varphi }_0$. The momentum space representation of the Bose fields is $b\left(0\right)={\left(\overline{\beta }L\right)}^{1/2}{\varphi }_0,$ and then the interacting part of the Fermi fields can be expressed by the auxiliary Bose fields ${\varphi }_0.$ In momentum space, the functional action of the Hamiltonian (35) is written as

$$S_0=-\sum _k{A}^{*}\left(k\right)(\tilde{k}{\sigma }_3+\tilde{m}{\sigma }_1)A\left(k\right),$$

(38)

$$S_1=-\tilde{\mu }\sum _k{A}^{*}\left(k\right)A\left(k\right)-{\tilde{u}}_0\sum _k{A}^{*}\left(k\right)A\left(k\right)-{\tilde{v}}_0\sum _k{A}^{*}\left(k\right){\sigma }_{1}A\left(k\right)-{\displaystyle \frac{\tilde{\omega }}{2}}{\varphi }_0\sum _k{A}^{*}\left(k\right)A\left(k\right)-\tilde{\omega }\overline{\beta }L{\varphi }_0^2,$$

(39)

where $S_0$ is the free action, $S_1$ is the coupling action, ${\sigma }_3$ and ${\sigma }_1$ are the Pauli matrices, and $S_0^{-1}=-(\tilde{k}{\sigma }_3+\tilde{m}{\sigma }_1)$ is a free propagator. $\tilde{k}=B_{0}k=k,$$\tilde{m}=G_{0}m,\tilde{\mu }=D_0\mu ,{\tilde{u}}_0=f_1{u}_0, {\tilde{v}}_0=f_2{v}_0,\tilde{\omega }=D_0\omega .$ The gravitational correction factors of the physical parameters and coupling coefficients are derived as follows:

$$\begin{array}{c}{D}_0={\displaystyle \frac{1}{L}}{\int }_0^L\sqrt{{\displaystyle \frac{\beta }{\alpha +h}}}dx,G_0={\displaystyle \frac{1}{L}}{\int }_0^L\sqrt{\beta }dx,\\ f_1={\displaystyle \frac{1}{L}}\sqrt{{\displaystyle \frac{\beta \left(x_0\right)}{\alpha \left(x_0\right)+h\left(x_0\right)}}},f_2={\displaystyle \frac{1}{L}}\sqrt{\beta \left(x_0\right)}.\end{array}$$

(40)

From the equations $\delta F_J\left(\overline{\beta }\right)/\delta J=0$ and $\delta W/\delta K=0,$ we obtain two variational parameters:

$$J={\displaystyle \frac{{\tilde{v}}_0}{\tilde{m}}},K={\displaystyle \frac{J}{2}}[Tr(A{S}_0^{-1}-1)-Tr(B{S}_0^{-1}-1)-2]-[1+{\displaystyle \frac{Tr(B{S}_0^{-1}-1)}{2}}],$$

(41)

In the long-wave approximation $k\to 0,$ the effective action of the fermions becomes

$$S_{eff}\left({\varphi }_0\right)=-\tilde{\omega }\overline{\beta }L{\varphi }_0^2+2ln(1+{\displaystyle \frac{{\tilde{v}}_0}{\tilde{m}}})+2\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^{2n-1}}{2n}}{\left({\displaystyle \frac{\tilde{\mu }+{\tilde{u}}_0+\frac{\tilde{\omega }}{2}\varphi }{\tilde{m}+{\tilde{v}}_0}}\right)}^{2n},$$

(42)

where the convergence condition satisfies this relation $(\tilde{\mu }+{\tilde{u}}_0+{\displaystyle \frac{\tilde{\omega }}{2}}\varphi )/\widetilde{(m}+{\tilde{v}}_0)<1$. If the wave vector $k\ll 1$, the convergence properties of the partition function have been proven by using the variational functional integrals technique and mathematical induction [38]. The free energy of the system has an alternative form, $F\left(\overline{\beta }\right)$=${\overline{\beta }}^{-1}{S}_{eff}\left({\varphi }_0\right),$ when the summation index is $n=1$. From the condition equation $\partial F\left(\overline{\beta }\right)$/$\partial {\varphi }_0$ = 0, we obtain the first-order condensation density

$${\varphi }_0={\displaystyle \frac{-2(\tilde{\mu }+{\tilde{u}}_0)}{\tilde{\omega }+4\overline{\beta }L{(\tilde{m}+{\tilde{v}}_0)}^2}}.$$

(43)

Moreover, the correlation function can be calculated from the Equation (31):

$$<AB>=-\tilde{\omega }{\displaystyle \frac{{\tilde{u}}_0}{{\tilde{m}}^2}}{\varphi }_0-4,$$

(44)

where $A={\tilde{u}}_{0}I$ is the impurity matrix and $B=(\tilde{\omega }{\varphi }_0/2)I$ is the condensation matrix.

4. Fermi Condensation Flows Induced from the Ricci Flows

Thermodynamic quantities on the cigar soliton background are expressed as

$$\rho =-{\displaystyle \frac{1}{L}}{\displaystyle \frac{\partial S_{eff}}{\partial \overline{\beta }}},S=S_{eff}-\overline{\beta }{\displaystyle \frac{\partial S_{eff}}{\partial \overline{\beta }}},\sigma ={\displaystyle \frac{{\partial }^2{S}_{eff}}{{\partial }^2\overline{\beta }}},$$

(45)

where $\rho$ is the energy density, S is the entropy, $\sigma$ is the fluctuations, and the effective action of the condensed fermions is

$$S_{eff}=2ln(1+{\displaystyle \frac{{\tilde{v}}_0}{\tilde{m}}})-{\displaystyle \frac{4\overline{\beta }L{(\tilde{\mu }+{\tilde{u}}_0)}^2}{\tilde{\omega }+4\overline{\beta }L{(\tilde{m}+{\tilde{v}}_0)}^2}}.$$

(46)

From Equations (43), (45), and (46), we obtain the Fermi condensation flows for the thermodynamic quantities

$${\displaystyle \frac{\partial {\varphi }_0}{\partial \lambda }}={\displaystyle \frac{{(\tilde{\mu }+{\tilde{u}}_0)}_{\lambda }}{\tilde{\mu }+{\tilde{u}}_0}}+{\displaystyle \frac{{\tilde{\omega }}_{\lambda }}{\tilde{\mu }+{\tilde{u}}_0}}{\varphi }_0,$$

(47)

$${\displaystyle \frac{\partial \rho }{\partial \lambda }}=2\tilde{\omega }{\varphi }_0{\displaystyle \frac{\partial {\varphi }_0}{\partial \lambda }},$$

(48)

$${\displaystyle \frac{\partial S}{\partial \lambda }}=-8\overline{\beta }L{(\tilde{m}+{\tilde{v}}_0)}^2{\varphi }_0{\displaystyle \frac{\partial {\varphi }_0}{\partial \lambda }},$$

(49)

$${\displaystyle \frac{\partial \sigma }{\partial \lambda }}=-{\displaystyle \frac{4L{(\tilde{m}+{\tilde{v}}_0)}^2}{\tilde{\mu }+{\tilde{u}}_0}}[2\tilde{\omega }{\varphi }_0^2{\displaystyle \frac{\partial {\varphi }_0}{\partial \lambda }}+{\tilde{\omega }}_{\lambda }(1+{\displaystyle \frac{\tilde{\omega }}{2}}{\displaystyle \frac{{\varphi }_0}{\tilde{\mu }+{\tilde{u}}_0}}).$$

(50)

From Equations (44) and (47), we obtain the correlation function flow:

$${\displaystyle \frac{\partial <AB>}{\partial \lambda }}=-{\displaystyle \frac{\tilde{\omega }{\tilde{u}}_0}{{\tilde{m}}^2}}[{\displaystyle \frac{\partial {\varphi }_0}{\partial \lambda }}+({\displaystyle \frac{{\tilde{\omega }}_{\lambda }}{\tilde{\omega }}}+{\displaystyle \frac{{\tilde{u}}_{0\lambda }}{{\tilde{u}}_0}})],$$

(51)

where ${\left(\right)}_{\lambda }=\partial \left(\right)/\partial \lambda ,$ the gravitational correction factors of the physical parameters and the coupling coefficients are

$$\begin{array}{c}{D}_0={\displaystyle \frac{1}{L\sqrt{a}}}ln(1+{\displaystyle \frac{2a}{b}}L+2\sqrt{{\displaystyle \frac{a}{b}}L}),G_0={\displaystyle \frac{1}{L}}ln(L+\sqrt{L^2+1}),\\ f_1={\displaystyle \frac{1}{L\sqrt{a{x}_0^2+b{x}_0}}},f_2={\displaystyle \frac{1}{L\sqrt{x_0^2+1}}},\end{array}$$

(52)

where $a=1+c_2\left[\left((1-6\epsilon )\right)\right]e^{2\lambda },b=c_2\epsilon e^{2\lambda },$$c_2$> $0,\epsilon >0,$ the derivative form for the chemical potential and the coupling coefficients is

$${\tilde{\mu }}_{\lambda }\sim -\left({\displaystyle \frac{\tilde{\mu }}{c_2\epsilon }}\right)e^{-2\lambda },{\tilde{u}}_{0\lambda }\sim -\left({\displaystyle \frac{\tilde{\mu }}{\sqrt{c_2\epsilon }}}\right)e^{-\lambda },{\tilde{\omega }}_{\lambda }\sim -\left({\displaystyle \frac{\tilde{\omega }}{c_2\epsilon }}\right)e^{-2\lambda }.$$

(53)

On the background of a two-dimensional point matter black hole, the perturbation of the gravitational field can cause the energy density fluctuations of the Fermi condensed matter [11]. In the string $\sigma$ model, the Ricci flow perturbation also induces the Fermi condensate flows, and the thermodynamic quantities are set on the cigar soliton background, which will change with the flow parameter $\lambda$. In the sine-Gordon–Thirring model, the perturbation function h in the gravitational correction factors of physical parameters and coupling coefficients will affect the change trend of Fermi condensate flows. The derivation process of the gravitational correction factors is provided in Appendix A. For example, substituting the metric functions (15) and asymptotic perturbation solution (21) into Equation (40), the specific expression (52) for the gravitational correction factors can be derived from the cigar soliton solution. The sign of the flows’ derivative for the physical parameters and coupling coefficients can determine the variation law of the Fermi condensate flows.

When the chemical potential is $\tilde{\mu }<0,$ the coupling constants are ${\tilde{u}}_0<0$ and $\tilde{\omega }>0.$ Then, the condensation density can exist, and it is ${\varphi }_0>0.$ The derivative of the free energy becomes ${\partial }^{2}F\left(\overline{\beta }\right)$/${\partial }^2{\varphi }_0>0.$ The Ricci flow perturbation is a small quantity and it does not change the minimum value of the free energy, so we believe that the Fermi condensation can reach a stable state. In addition, from Equation (47), it can be seen that $\partial {\varphi }_0/\partial \lambda <0,$ and the thermodynamic quantities varying with the momentum scale $\lambda$ are

$${\displaystyle \frac{\partial \rho }{\partial \lambda }}<0,{\displaystyle \frac{\partial S}{\partial \lambda }}>0,{\displaystyle \frac{\partial \sigma }{\partial \lambda }}<0,$$

(54)

The correlation function varying with the momentum scale $\lambda$ is

$${\displaystyle \frac{\partial <AB>}{\partial \lambda }}<0.$$

(55)

The entropy of matter in a gravitational field refers to the thermodynamic properties of matter under the action of gravity. The interaction between matter and gravitational field can cause changes in the microscopic state of the system, thereby affecting the entropy of matter. Specifically, the entropy of matter in a gravitational field is influenced by various factors, including the gravitational potential, the temperature changes, and the interactions with the surrounding environment. In the present article, which mainly studies the entropy of the Fermi condensed matter against the background of the cigar soliton, this entropy should be viewed as an effective quantity derived from a variational ansatz. It contains information about the soliton gravitational field. Since the perturbation function h varies with the momentum scale, the Ricci flow perturbation induces the Fermi condensate flow through gravitational correction factors, and this leads to a change in the effective entropy of matter.

Looking to the above calculation results, we can see the evolutionary trend of the thermodynamic quantities under the Ricci flow perturbation. The results indicate that the energy density, the correlation function, and the condensation fluctuations will decrease with the increase in the momentum scale, but the effective entropy will monotonically increase with the increase in the momentum scale. According to string theory, the Perelman entropy recovers the low-energy structure of space-time, which has the feature of increasing under the action of the Ricci flow. Through the study of the fixed points of the flow, it can be found that the Perelman entropy is not related to the Bekenstein–Hawking entropy [8]. In addition, the entanglement entropy in a two-dimensional black hole monotonically decreases under the Ricci flow [7]. The application of Zamolodchikov’s c theorem could prove that the energy of the UV fixed point is greater than that of the IR fixed point, so the space-time energy would decrease under the world-sheet Ricci flow [5]. Therefore, the effective entropy of the Fermi condensation is similar to Perelman’s geometric entropy, but it is not like the entanglement entropy of a black hole, and the change in the condensation density is similar to the change in the space-time energy. Vacaru et al. studied geometric thermodynamic quantities corresponding to the Ricci flow equation, and they derived expressions for the energy, the entropy, and the fluctuations by using Perelman’s functional [41,42]. This article mainly analyzes the evolution trend of the thermodynamic quantities in condensed fermions and compares the thermodynamic quantities of matter with the geometric thermodynamic quantities of space-time.

5. Conclusions and Discussion

In this article, the Ricci flow perturbation equation is derived based on the exterior differential form and the structural equations; we obtain a two-loop asymptotic perturbation solution for the cigar soliton by using a small parameter expansion method. In addition, we use the variational functional integrals technique to calculate the Fermi condensation flows in the sine-Gordon–Thirring model with coupling of two impurities. When the perturbation coefficient is $c_2>0,$ and its derivative is $\partial g_{00}/\partial \lambda >0$, the metric factor increases with the increase in the momentum scale. Thus, the Fermi condensation flow can exist under the Ricci flow perturbation, and the sign of the Fermi condensation flows is determined by the gravitational correction factors in the background of the cigar soliton. The results show that the entropy increases with the increase in the momentum scale; the energy density, the correlation function, and the fluctuations decrease with the increase in the momentum scale. From the evolution trend of the Fermi condensation flows, it can be seen that the entropy of the fermions is similar to the geometric entropy of Perelman, which is different from the entanglement entropy of a two-dimensional black hole. The evolution of the matter density is similar to the change in the space-time energy. Although gravitational perturbations can alter the stability of the 2D black holes, the Ricci flow perturbation cannot affect the thermodynamic stability of the Fermi condensation matter.

Recently, Hoare et al. calculated the two-loop $\beta$ function in a two-dimensional “sausage” model, and they obtained the R-G flow equation for the metric and $\eta$ deformation parameter [43]. By calculating the single-loop $\beta$ function in the Bose–Thirring model, it can be shown that the deformation parameter $s=e^{2\lambda }$ is an analytical solution of the Ricci flow equation [44]. When the $\eta$ deformation parameter was equal to the Ricci flow parameter $\lambda$, the $\eta$ deformation metric also satisfied the Ricci flow equation [45]. Perturbation and deformation would cause the deviation of the space-time structure from the equilibrium state, which would lead to a series of physical effects in the two-dimensional black holes, such as the fluctuations of the condensation density with the perturbation parameter [11], the variation of the phonon mass, and the variation of the Hawking temperature with the deformation parameter [46]. In this article, we mainly study the evolution trend of the Fermi condensation flows under the Ricci flow perturbation; our method can also be used to study the evolution of the matter flows in deformed space-time, and these interesting topics are worth considering in future work.