{'fraction_non_alphanumeric': 0.0585515580076152, 'fraction_numerical': 0.0212914756391328, 'mean_word_length': 4.060755013763272, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Time and temperature dependent correlation functions of the 1D impenetrable electron gas

We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann-Hilbert problem we obtain formulae for the asymptotics of the correlation functions, which are valid at all finite temperatures. At low temperatures these formulae lead to explicit asymptotic expressions for the correlation functions, which describe power law behavior and exponential decay as functions of temperature, magnetic field and chemical potential.


Introduction
We consider the electron gas with delta interaction in one space and one time dimension. It can be described by canonical Fermi fields ψ α (x) with canonical equal time anticommutation relations The spin index α runs through two values, α =↑, ↓. The Hamiltonian of the model is Here µ is the chemical potential, and B is the magnetic field. σ z is a Pauli matrix, and c is the coupling constant. In this paper we shall consider the case of infinite repulsion c = ∞. The model can be solved exactly [1]. Its thermodynamics was described in [2]. The pressure serves as thermodynamic potential. T is the temperature. Note that the expression for P , eq. (3), is formally the same as for free fermions with effective chemical potential µ + T ln(2 cosh(B/T )). Thus, there are two different zero temperature phases depending on whether lim T →0+ (µ + T ln(2 cosh(B/T )) = µ + |B| is positive or negative. For µ + |B| > 0 the density D = ∂P/∂µ has a positive limit as T goes to zero. For µ + |B| < 0 the density at zero temperature vanishes. This is the phase we are interested in. We call it the gas phase. We shall assume in the following that µ + T ln(2 cosh(B/T )) < 0. The number of up-spin particles and the number of down-spin particles are separately conserved. The densities D ↑ of up-spin electrons and D ↓ of down-spin electrons are obtained as It is obvious from these expressions that they satisfy the relation The density D of the gas is Note that the density vanishes exponentially at small temperatures, At very low temperatures density and pressure are related by the ideal gas law, P = DT .
For larger temperatures there are corrections to the ideal gas law.
In this article we study the two-point correlation functions G ± α (x, t), α =↑, ↓, of local fields. We shall follow the approach of the book [3] to correlation functions of solvable models. The definitions of G ± α (x, t) are given in the next section, where we recall a determinant representation for G ± α (x, t), [4,5], which is the starting point of our considerations. The correlation functions G ± α (x, t) are not elementary functions. They are determined by a pair of coupled nonlinear partial differential equations [6] which is derived in section 3. In section 4 we shall formulate a corresponding Riemann-Hilbert problem [7,8,9]. It will fix the solution of the differential equations, which is relevant for the calculation of the correlation functions. Furthermore, the asymptotic solution of the Riemann-Hilbert problem for large x and t (x/t fixed) can be obtained by reducing it to a known form [10,11,12,13,14,15]. This is done in section 5. Using the asymptotic solution of the Riemann-Hilbert problem and the differential equations we shall obtain the asymptotics of the correlation functions G ± α (x, t) in terms of a contour integral in section 6. In section 7 we shall present explicit asymptotic expressions for G ± α (x, t) at low temperatures. The asymptotic expressions for the correlation functions G ± α (x, t) at low temperatures are products of correlation functions of free fermions and corrections caused by the interaction. The leading factors of the correction terms describe exponential decay. The next to leading factors are power law corrections. The leading factor of the asymptotics of the spinup correlation function G + ↑ (x, t), for instance, is The factor e it(µ−B) e ix 2 /4t is the exponential part of the free fermionic correlation function. The factor e −xD ↓ appears due to the interaction. The explicit form of the power law correction as a function of temperature, chemical potential and magnetic field will be given in section 7 (see eqs. (133), (132), (135), (128)). The occurrence of the density of down-spin electrons in equation (10) has a clear physical interpretation. Due to the Pauli principle and the locality of the interaction, up-spin electrons are scattered only by down-spin electrons. D ↓ appears as the reciprocal of the correlation length in eq. (10). The higher the density of down-spin electrons, the more often an up-spin electron is scattered and the smaller is the correlation length.

A determinant representation for correlation functions
We shall consider the following two-point correlation functions of local fields, where α =↑, ↓. These correlation functions depend not only on space and time variables x and t, but also on the temperature T , the chemical potential µ (or the density D) and the magnetic field B. Due to the invariance of the Hence, we can restrict our attention in the following to the correlation functions G ± ↑ (x, t). These correlation functions were represented by means of determinants of integral operators [4,5]. Now we shall recall these formulae: It will take us some time to define notations. First of all and V is an integral operator with kernel V (k, p). It can be represented in the "standard form" (see page 316 of the book [3]) Here and The integral operator (18) closely reminds an integral operator which appears in the theory of correlation functions of the impenetrable Bose gas (cf. page 346 of [3]). The inverse of Î + γV can be defined in the following way, The resolventR is an integral operator with kernel R(k, p). It can be represented in the "standard form", where the functions f ± (k) can be defined as solutions of the integral equations This theorem is proven on page 318 of the book [3]. We should mention that both kernels V (k, p) and R(k, p) are symmetric, At this point it is convenient to define potentials B ab and C ab , Here both indices a and b run through two values, a, b = ±. For example, the factor B −− in (15) is defined as In order to define b ++ we need the function The factor b ++ in (14) is defined as (31) In such a way we accomplished the description of the determinant representation of the correlation functions (14), (15). It reminds the determinant representation for correlation functions of the impenetrable Bose gas (see page 346 of the book [3]). Let us also mention that the symmetry of V (k, p) leads to B +− = B −+ .
Let us discuss the factor 1−cos(η) in the denominator of (14). Let us show that there is no singularity at η = 0. Indeed, e + (k), eq. (20), is proportional to η as η → 0. Hence b ++ ∼ η 2 , and there is no singularity. Let us change the integration variable to z = e iη . Then Here the contour of integration is the unit circle, and

Differential equations
Starting from a determinant representation for a quantum correlation function it is possible to derive differential equations which drive this correlation function (see [6] and chapter XIV of the book [3]). In our specific case calculations similar to the ones presented on pages 345 -353 of the book [3] lead us to the following differential equations, We should notice that a change of variables, leads to coincidence of our system of eqs. (35) with the system (5.47) on page 344 of the book. The system of equations (35) is called separated nonlinear Schrödinger equation. It is completely integrable. It has many solutions. The modern way to fix a solution of the system (35) is to formulate a corresponding Riemann-Hilbert problem (see chapter XV of the book [3]). We shall formulate this Riemann-Hilbert problem in the next section. We will be able to solve the Riemann-Hilbert problem asymptotically as x → ∞ and t → ∞ (x/t fixed). This will give us asymptotic expressions for b ++ and B −− . The solution of the Riemann-Hilbert problem will also give us asymptotic expressions for the other potentials B ab and C ab (28).
Let us now explain how to express det(Î +γV ) in terms of these potentials. The logarithmic derivatives of the determinant can be expressed in terms of B and C, We shall also mention several identities, which will be useful for further calculations: In the next section we shall introduce a Riemann-Hilbert problem and explain how one can express the potentials B ab and C ab in terms of its solution χ(k).

Riemann-Hilbert problem
The modern way to solve completely integrable differential equations is by means of an equivalent Riemann-Hilbert problem [7,8,9]. Solving a Riemann-Hilbert problem amounts to constructing a piecewise analytic matrixfunction. Let us formulate the Riemann-Hilbert problem connected to the separated nonlinear Schrödinger equation (35). Let us introduce a 2 × 2matrix χ(k). The matrix χ(k) should be analytic as a function of k in the upper half plane, Im(k) > 0, and in the lower half plane, Im(k) < 0. In general, it has different limits χ + (k) as k approaches the real axis from above and χ − (k) as k approaches the real axis from below. These limits are related by means of a conjugation matrix G(k), The conjugation matrix G(k) is of dimension two. It is defined only on the real axis. The matrix χ(k) should also satisfy the normalization condition The problem is to find such a matrix χ(k). In order to relate χ(k) to the integral operator (18) we shall use a conjugation matrix G(k) of the special form The Riemann-Hilbert problem with this form of conjugation matrix was studied in detail in chapter XV of the book [3]. Let us recall some fundamental facts: One can prove that The functions f ± (k) were defined by the integral equations (26). On the other hand, the Riemann-Hilbert problem can be solved by means of these functions In order to express the potentials B ab and C ab in terms of χ(k), let us study the asymptotics of χ(k) as k → ∞, Comparing (46), (47) and (28) we obtain In such a way we have formulated the Riemann-Hilbert problem and related it to the integral operatorV and to the correlation functions G + ↑ (x, t) and Next we shall solve the Riemann-Hilbert problem at large space time separation x → ∞, t → ∞, x/t fixed. Then we shall calculate the potentials B ab and C ab from (47). The knowledge of the potentials will permit us to evaluate det(Î + γV ), see (37). This will give us an asymptotic expression for the correlation functions (14), (15). We also want to note that det G(k) = det χ(k) = 1 . (49)

Asymptotic solution of the Riemann-Hilbert problem
In order to solve the Riemann-Hilbert problem asymptotically [7,10,11,12,13,14,15], we shall reformulate it twice. This will bring it into the canonical form, which was solved by S. V. Manakov [10,11]. First reformulation: We want to simplify the conjugation matrix G(k). Let us define a matrix Here τ (p) = ip 2 t − ipx. The matrix χ 0 (k) is analytic for Im(k) > 0 and for Im(k) < 0. At large k it approaches the unit matrix as a Taylor series in 1/k. Now, instead of the unknown matrix function χ(k), let us introduce a new unknown matrix functionχ(k), Let us formulate the Riemann-Hilbert problem forχ(k).
Using (50) and (51) we can also evaluate the coefficients of the expansion (52) ofχ(k) for large k,χ Let us recall that In order to prepare for the second step let us introduce a function α(k), This function solves the scalar Riemann-Hilbert problem The conjugation function here is the elementG 11 (k) of the conjugation matrix G(k), eq. (55). Later we shall need the coefficients α 1 , α 2 of the expansion of ln(α(k)) for large k, Second reformulation: We want to bring the Riemann-Hilbert problem to the canonical form. Instead ofχ(k) let us define another 2 × 2-matrix This matrix function is analytic for Im(k) > 0 and Im(k) < 0. It approaches the unit matrix at large k, It has a discontinuity on the real axis, which can be described by a conjuga- Here The matrix elements of G Φ (k) are We also note that Let us introduce the notations Then we can write the conjugation matrix G Φ (k) in the canonical form Here τ (k) = ik 2 t − ikx. In the following the stationary point of the phase τ (k), ∂ k τ (k 0 ) = 0, will play an important role. Let us express the potentials B ab in terms of Φ 1 , eq. (64), Here is the first coefficient of the expansion (62) of ln(α(k)) for large k. Using (37) and (73) we obtain The time derivative of ln det Î + γV can be expressed in terms of Φ 2 , eq. (64), where α 2 is the second coefficient in the expansion (62), This is the end of the second reformulation. The asymptotic solution of the Riemann-Hilbert problem with conjugation matrix G Φ (k), eq. (71), is known. It is given by Manakov's Ansatz (see page 452 of the book [3]). In order to describe Manakov's Ansatz we first solve a scalar Riemann-Hilbert problem for a function δ(k), Here θ is the Heavyside step function, The solution is Now we can define functions I p (k) and I q (k), Using these functions we can write the solution of the Riemann-Hilbert problem (65), (71) as This solution is an asymptotic solution which is valid only as x → ∞, t → ∞ for fixed ratio x/t.
In order to evaluate the correlation functions (14), (15) we also need b ++ and see (73). Using the explicit estimate for the error (see page 453 of the book [3]) of the asymptotic solution Φ(k), eq. (83), of the Riemann-Hilbert problem we find which shows that b ++ and B −− decay with time. In order to get a more precise estimate of the asymptotics, we shall turn to the differential equations (35). We shall use some information from the book [3]. So we shall rewrite the differential equations (35) in the variablesx = −x/2,t = t/2. Then The asymptotics of the decaying solutions of this system is well known [16,17,18]. The leading terms are Hereũ 0 andṽ 0 only depend on k 0 , T , µ and B. Substitution of these asymptotic solutions into the differential equations (101) gives In order to calculate ν we use the identity (38), Let us substitute (91) here, From (102) and (103) we have Using (104), (105) we get So we have calculated the asymptotics of b ++ and B −− , see (102). We can use the differential equations (101) even more. We can improve the asymptotic expression (97) for ln det(Î + γV ). Following the steps from pages 455-457 of the book [3] we obtain ln det(Î + γV ) = 1 2π ∞ −∞ dp |x − 2pt| ln 1 + γϑ(p) e −iηsign(p−k 0 ) − 1 + ν 2 2 ln t . (108) I order to get the correlation functions (14), (15) let us calculate We did not calculate the coefficients u 0 and v 0 in these expressions. We calculated the leading exponential factor and power (of time) corrections. Further down, while substituting we shall not pay attention to numerical coefficients. Let us get all notations together, and let us present our asymptotic expressions for the correlation functions (11), (12), Here the contour of integration is the unit circle. At the end of section 2 we showed that b ++ cancels the second order pole at z = 1. This means that u 0 will cancel the second order pole at z = 1 in (111). The remaining contour integrals in (111), (112) can be evaluated by the method of steepest descent (see next section). Eqs. (111), (112) are integral formulae for the large time, long distance asymptotics of correlation functions of local fields for the delta-interacting electron gas. These formulae are valid for arbitrary finite temperatures. They are the main result of the present article. In the next section we shall simplify our asymptotic expressions (111), (112) at low temperatures. Let us note that S(z) can be written as

Low temperature asymptotics
We obtained the asymptotic formulae (111), (112). The remaining contour integral should be taken by means of the steepest descent (saddle point) method. Let us consider here only the leading asymptotic factor. Then we can rewrite the simplified expression for the asymptotics as The saddle point equation ∂S/∂z = 0 can be represented in the form k 0 = x/2t, γ = 1 + e 2B/T . We prove in appendix A that there is only one solution of this equation in the interval 0 < z ≤ 1. Considering the pure time direction, k 0 = 0, we find the solution which leads to Ω ± ∼ t −1 .
This result is valid for all finite temperatures. Let us consider the other asymptotic regions, k 0 > 0, x → ∞ and t → ∞. In these regions the saddle point equation (121) can be solved explicitly only for small temperatures. There are two solutions z ± = ±z c , If we deform the contour of integration from a circle to a line through z + parallel to the imaginary axis, we pass the saddle point in the right direction (see appendix A).
In appendix A we obtain the following low temperature expansion for the phase tS(z), Here D is the density (8), Eq. (125) shows that at low temperatures the relevant parameter for the calculation of the asymptotics of Ω ± is 2k 0 Dt = xD rather than t. The parameter xD has a simple interpretation. It is the average number of particles in the interval [0, x]. We have to distinguish two limiting cases.
(a) xD → 0, the number of electrons in the interval [0, x] vanishes. In this regime the interaction of the electrons is negligible. An electron propagates freely from 0 to x. Ω ± cannot be calculated by the method of steepest descent. We have to use (111), (112) to calculate the asymptotics of G + ↑ (x, t) and G − ↑ (x, t). Now tS(z) and ν(z) tend to zero on the contour of integration. Thus G ± ↑ (x, t) ∼ t − 1 2 e ±it(µ−B)±ix 2 /4t , which, as expected, is the same as for free fermions. The proportionality factor is different for G + ↑ (x, t) and G − ↑ (x, t). To calculate it from (111), (112) we would have to know u 0 and v 0 . By comparison with free fermions (see below) we expect it to be a number for G + ↑ (x, t) and proportional to e (µ−B−k 2 0 )/T for G − ↑ (x, t).
(b) xD ≫ 1, the average number of electrons in the interval [0, x] is large. Now the interaction becomes important, and we can use the method of steepest descent to calculate Ω ± . This is the most interesting case. In the following we will investigate it in detail.
The contribution of the saddle point to Ω ± , eq. (120), is where D ↓ is the low temperature expression for the density of down-spin electrons, eq. (5).
We further have to take into account the pole of the function F at z = γ −1 . While deforming the contour, we could have crossed it. It turns out that the pole contributes to Ω ± , when the magnetic field B is below a critical positive value (see appendix A). The pole contribution always dominates the contribution of the saddle point. It is obtained as Finally, we have the following low temperature asymptotic expressions for the leading factors of the correlation functions. For B < B c : where ν(γ −1 ) is given by eq. (132), and for B > B c : where ν(z + ) is given by eq. (128). Recall that we have assumed the average number of electrons in the interval [0, x] to be large, xD ≫ 1. The corresponding expressions for G ± ↓ (x, t) follow from equation (13). The leading low temperature expressions for D and D ↓ agree for positive magnetic field. Hence, above the critical field B c the correlation functions depend on the magnetic field only through the trivial factors e ±it(µ−B) . The system is saturated. Another way of looking at the condition (130), which would be appropriate for experiments with fixed magnetic field B, is the following. For positive magnetic field there are two different asymptotic regions. One is the space like region, where k 2 0 > 4B, the other one is the time like region, where k 2 0 < 4B. Correlation functions of local fields have asymptotics (133), (134) in the space like region and asymptotics (135), (136) in the time like region. For non-positive magnetic field there is no distinction between space like and time like region.
It is instructive to compare the asymptotic expressions (133) -(136) for the correlation functions with the corresponding asymptotics for free fermions. The two-point correlation functions for free fermions have the low temperature asymptotics Here H 0 is the free Hamiltonian ((2) with c = 0). The leading low temperature contributions to the constants in (137) and (138) are C + = e −i π 4 /2 √ π and C − = e (µ−B−k 2 0 )/T e i π 4 /2 √ π. Comparing the free fermionic expressions with (133), (134) or (135), (136), respectively, we can interpret the factors t ±iν(γ −1 ) e −xD ↓ and t − 1 2 ±iν(z + ) e −xD ↓ as low temperature corrections, which appear due to the interaction. The occurrence of the density of down-spin electrons in the exponential factors in eqs. (133) -(136) has a natural interpretation. Correlations decay due to interaction. Because of the Pauli principle and the locality of the interaction in our specific model, eq. (2), up-spin electrons interact only with down-spin electrons. Therefore the correlation length is expected to be a decreasing function of the density of down-spin electrons, which diverges as the density of down spin electrons goes to zero. We see from (133) -(136) that the low temperature expression for the correlation length is just 1/D ↓ and thus meets these expectations.
In the limit B → −∞, µ → −∞, µ−B fixed there are no ↓-spin electrons left in the system, D ↓ → 0, D ↑ → D. This is the free fermion limit. In the free fermion limit B < B c , and the asymptotics of G + ↑ (x, t) and G − ↑ (x, t) are given by eqs. (133), (134), which then turn into the expressions (137), (138) for free fermions.
So we have explicitly evaluated the asymptotics of correlation functions of local fields G ± ↑ (x, t), (11), (12). Our main results are formulae (111), (112). In this section we considered the saddle point calculation of the remaining contour integral, and we discussed the physical meaning of our asymptotics in the low temperature regime.
all sub-leading factors. Then we are left with the calculation of the contour integral (120), The contour of integration is the unit circle.
Let us cut the complex plane from −∞ to −γ −1 e (B−µ)/T and from −γe (µ−B)/T to zero. The integrand in eq. (A.1) is analytic in the cut plane except for the two simple poles of F (z) at z = γ ±1 . We may therefore deform the contour of integration as long as we never cross the cuts and take into account the contributions from the residua, if we cross z = γ or z = γ −1 . Let us comment on the somewhat unusual fact that t in eq. (A.1) appears not only in the exponent. Calculating an integral of the form (A.1) by the method of steepest descent amounts to deforming the contour of integration in such a way that we are left with a Laplace type integral. Let us therefore consider the large t asymptotics of the integral Here ε = ln(t)/t becomes small as t → ∞. We assume that f (x) has a unique minimum x 0 . Then, first calculating the large t asymptotics of the right hand side of (A.5) by Laplace's method and expanding for small ε afterwards, we obtain the leading asymptotics This means that the factor t − 1 2 (1±iν(z)) 2 in equation (A.1) can be treated the same way as a t-independent function. Thus, to leading order, a saddle point z c contributes to Ω ± as Ω ± ∼ t − 1 2 − 1 2 (1±iν(zc)) 2 e −tS(zc) . (A.7) The saddle point equation ∂S/∂z = 0 can be represented in the form . This is still a transcendental equation, which cannot be solved explicitly for z except at low or high temperatures. Before turning to the case of low temperatures, let us show that (A.8) restricted to the positive real axis, z > 0, has a unique solution, which is located in the interval 0 < z ≤ 1. Let us define two functions We will further use the abbreviation Note that Λ > 0. Using the above conventions the saddle point equation where the primes denote derivatives with respect to the arguments. Since and J ± (x) > 0 for all positive x, we conclude that f (z) is monotonically decreasing for positive z. We can decompose J − (x) into where (A. 15) It follows that We can further show that Then The low temperature asymptotics of I + and I − are easily calculated. We find It follows that Here we suppressed ν 2 (z + ), which is small compared to 1. The function F (z), eq. (A.2), has two simple poles at z = γ ±1 , which may contribute to the integral (A.1), if they are crossed in the process of deformation of the path of integration. Let us investigate this question in the low temperature limit. The cases B > 0, B = 0 and B < 0 have to be studied separately.
(a) B > 0: In this case γ = e 2B/T , γ −1 = e −2B/T . Since z + becomes small at low temperatures, we may have crossed the pole at γ −1 . The condition for having no contribution from the pole is e −2B/T < z + , or, using (A.31), k 2 0 < 4B. Let us assume that k 2 0 > 4B . Here we used (A.39) once more to estimate that γz 2 c ≪ 1. We see that the contribution of the pole to the exponential decay is exactly the same as the contribution of the saddle point. This is no longer true for the power law. We still have to discuss the range of validity of the above low temperature calculations. The low temperature expression for the saddle point z c is sufficiently precise, as long as z c satisfies (A.24). Yet, there is another condition to be satisfied. Note that the phase S(z) in low temperature approximation, eq. (A.33), is multiplied by a factor which vanishes as the temperature approaches zero. We have Thus the large parameter, which we need for the saddle point approximation to be valid, is xD = 2k 0 Dt rather than t.