BCS thermal vacuum of fermionic superfluids and its perturbation theory

The thermal field theory is applied to fermionic superfluids by doubling the degrees of freedom of the BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon, where the pairing persists when the order parameter vanishes, emerges from the perturbation theory. The correspondence between the thermal vacuum and purification of the density matrix allows a unitary transformation, and we found the geometric phase associated with the transformation in the parameter space.

Quantum many-body systems can be described by quantum field theories [1][2][3][4] . Some available frameworks for systems at finite temperatures include the Matsubara formalism using the imaginary time for equilibrium systems 1,5 and the Keldysh formalism of time-contour path integrals 3,6 for non-equilibrium systems. There are also alternative formalisms. For instance, the thermal field theory [7][8][9] is built on the concept of thermal vacuum. The idea of thermal vacuum is to construct temperature-dependent augmented states and rewrite the statistical average of observables as quantum-mechanical expectation values. Thermal field theory was introduced a while ago 7,10 , and more recently it has found applications beyond high-energy physics 9 .
The thermal vacuum of a non-interacting bosonic or fermionic system has been obtained by a Bogoliubov transformation of the corresponding two-mode vacuum 11 , where an auxiliary system, called the tilde system, is introduced to satisfy the statistical weight. The concepts of Bogoliubov transformation and unitary inequivalent representations are closely related to the development of thermal vacuum theory 9 , and they are also related to the quantum Hall effect 12 and orthogonality catastrophe 4 . The thermal vacuum of an interacting system can be constructed if a Bogoliubov transformation of the corresponding two-mode vacuum is found. In the following, we will use the BCS theory of fermionic superfluids as a concrete example. By construction, the thermal vacuum provides an alternative interpretation of the statistical average. Nevertheless, we will show that the introduction of the thermal vacuum simplifies certain calculations from the level of quantum field theory to the level of quantum mechanics. As an example, we will apply the thermal field theory to develop a perturbation theory where the BCS thermal vacuum serves as the unperturbed state, and the fermion-fermion interaction ignored in the BCS approximation is the perturbation. Because the particle-hole channel, or the induced interaction 13 , is not included in the BCS theory, the BCS thermal vacuum inherits the same property and its perturbation theory does not produce the Gorkov-Melik-Barkhudarov effect 14 predicting the transition temperature is more than halved.
The thermal vacuum of a given Hamiltonian can alternatively be viewed as a purification of the density matrix of the corresponding mixed state 15,16 . Since the thermal vacuum is a pure state, the statistical average of a physical quantity at finite temperatures is the expectation value obtained in the quantum mechanical manner. Therefore, one can find some physical quantities not easily evaluated in conventional methods. This is the reason behind the perturbation theory using the BCS thermal vacuum as the unperturbed state. The corrections from the full fermion-fermion interaction can be evaluated order-by-order following the standard time-independent quantum-mechanical perturbation theory 17 . In principle, the corrections to any physical quantities can be expressed as a perturbation series. In this work, the first-order corrections to the order parameter will be evaluated and analyzed.
Furthermore, the BCS thermal vacuum allows us to gain insights into the BCS superfluids. We will illustrate the applications of the BCS thermal vacuum by calculating the pairing correlation, which may be viewed as the quantum correlation matrix 18 of the pairing operator. We found that the corrected order parameter vanishes at a lower temperature compared to the unperturbed one, and the pairing correlation persists above the corrected transition temperature. Thus, the perturbation theory offers an algebraic foundation for the pseudogap effect 19,20 , where the pairing effect persists above the transition temperature. Moreover, the purification of the density matrix of a mixed state allows a unitary transformation 15,16 . For the BCS thermal vacuum, this translates to a U(1) phase in its construction. By evaluating the analogue of the Berry phase 21 along the U(1)-manifold of the unitary transformation, we found a thermal phase characterizing the thermal excitations of the BCS superfluid.
Throughout this paper, we choose ℏ = k B = 1 and set |q e | = 1, where q e is the electron charge. In quantum statistics, the expectation value of an operator  in a canonical ensemble is evaluated by the statistical (thermal) average SCieNtifiC REPORTS | (2018) 8:11995 | DOI:10.1038/s41598-018-30438-1 Results BCS Thermal Vacuum. We first give a brief review of the BCS theory. The fermion field operators ψ σ and ψ σ †  where m and μ are the fermion mass and chemical potential, respectively, g is the coupling constant, and 3 3 with  being the volume of the system. In the BCS theory, the pairing field leads to the gap func- which is also the order parameter in the broken-symmetry phase. Physically, pairing between fermions in the time-reversal states ↑ k ( ) and (− ↓ k ) can make the Fermi sea unstable if the inter-particle interaction is attractive 23 .
If one takes only the k = 0 contribution in Eq. (7), the Hamiltonian of a homogeneous system can be approximated by the BCS form 24,25 with which can be diagonalized as This is achieved by implementing the Bogoliubov transformation, which can also be cast in the form of a similarity transformation: and all other anti-commutators vanish. We remark that the BCS theory only considers the particle-particle channel (pairing) contribution. As pointed out in ref. 26 , the BCS theory is not compatible with a split, density contribution to the chemical potential. It is, however, possible to add the particle-hole (density) diagrams to the Feynman diagrams describing the scattering process 13 and obtain a modified effective interaction, which then leads to the Gorkov-Melik-Barkhudarov effect 14 of suppressed superfluid transition temperature. Here we base the theory on the BCS theory, so the resulting thermal field theory also does not explicitly exhibit the particle-hole channel effects.
Rewriting the Bogoliubov transformation as a similarity transformation leads to a connection between the Fock-space vacuum |0〉 of the ψ σ quanta and the BCS ground state |g〉, which can be viewed as the vacuum of the α k and β −k quasi-particles because α k |g〉 = 0 = β −k |g〉. Explicitly, We remark the relation between the similarity transformation of the fields and the unitary transformation of the Fock-space vacuum resembles the connection between the Schrodinger picture and the Heisenberg picture in quantum dynamics (see ref. 17 for example).
The thermal vacuum of the BCS theory is constructed by introducing the tilde partners of the α k and β −k quanta, α  k and β −  k . They satisfy the algebra and all other anti-commutators vanish. Moreover, the tilde fields anti-commute with the quasi-particle quanta α k and β k . Next, the two-mode BCS ground state is constructed as follows.
The occupation number of each fermion quasi-particle state can only be 0 or 1. According to Eq. (3), the two-mode BCS thermal vacuum can be expressed as . The coefficients f 0k and f 1k can be deduced from Eq. (4). For each k, we define = + β − Z 1 e E k k and then the partition function is = ∏ Z Z k k . Comparing with Eq. (4), we get According to Eq. (5), one may choose a relative phase between the different two-mode states. Here, we choose χ 0 = 0 and χ 1 = −χ. Thus, the coefficients are parametrized by The phase χ parametrizes the U(1) transformation allowed by the BCS thermal vacuum, and we will show its consequence later.
The BCS thermal vacuum can be obtained by a unitary transformation of the two-mode BCS ground state. Explicitly, Following a similarity transformation using the unitary operator e −Q , the BCS thermal vacuum is the Fock-space vacuum of the thermal quasi-quanta One can construct similar relations for the tilde operators. Moreover, |0(β)〉 is the ground state of the thermal BCS Hamiltonian Incidentally, the similarity transformation does not change the eigenvalues. At zero temperature, | f 0k | = 1 and f 1k = 0. Hence, the BCS thermal vacuum reduces to the ground state of the conventional BCS theory, but in the augmented two-mode form. It is important to notice that the two-mode BCS ground state differs from the BCS thermal vacuum at finite temperatures in their structures: The former is the Fock-space vacuum of the quasi-particles, i.e., α β ; the latter is the Fock-space vacuum of the thermal quasi-particles, i.e., α k (T)|0(β)〉 = β k (T)|0(β)〉 = 0. We mention that the BCS thermal vacuum is a generalized coherent squeezed state (see the Supplemental Information for details.) By construction, the statistical average of any physical observable can be obtained by taking the expectation value with respect to the thermal vacuum. In the following we show how this procedure reproduces the BCS number and gap equations. One can show that Here we have used Eq. (20) and Applying these identities and the inverse transformation of Eq. (10), the total particle number is given by the expectation value of the number operator with respect to the state |0(β)〉: When compared to the Green's function approach 2 , the thermal-vacuum approach is formally at the quantum mechanical level. To emphasize this feature, we will use some techniques from quantum mechanics to perform calculations instead of using the framework of quantum field theory.
Perturbation Theory Based on BCS Thermal Vacuum. The BCS equations of state can be derived from a formalism formally identical to quantum mechanics by the BCS thermal vacuum. We generalize the procedure to more complicated calculations such as evaluating higher-order corrections to the BCS mean-field theory by developing a perturbation theory like the one in quantum mechanics. The idea is to take the BCS thermal vacuum as the unperturbed state and follow the standard time-independent perturbation formalism 17 to build the corrections order by order. To develop a perturbation theory at the quantum mechanical level based on the BCS thermal vacuum, we first identify the omitted interaction term in the BCS approximation as the perturbation. By comparing the total Hamiltonian (7) and the BCS Hamiltonian (8), one finds  . However, the inclusion of V invalidates the Bogoliubov transformation and H is not diagonalized by the quasi-particles α k and β −k . As a consequence, and the evaluation of |0(β)〉 c can be difficult. Formally, the perturbed thermal vacuum may be expressed as β , c denotes the perturbed two-mode ground state. Since the thermal Hamiltonian (26) involves Q c , a full treatment of the perturbed thermal vacuum would be quite challenging.
Here we adopt the approximation  Q Q c in the thermal Hamiltonian (26) and obtain the corresponding thermal vacuum by a perturbation theory. When V = 0, |0(β)〉 is the ground state of the unperturbed thermal BCS Hamiltonian H BCS (T). We will take it as the starting point and evaluate |0(β)〉 c by applying the quantum perturbation theory 17 . Explicitly, the perturbed thermal vacuum is approximated by where V k0 = 〈k (0) |V(T)|0(β)〉 is the matrix element of the perturbation with respect to the unperturbed states, V(T) = e Q Ve −Q , |k (0) 〉 includes all possible unperturbed excited states given by α β , etc., and E n (0) is the corresponding unperturbed energy. After obtaining the perturbed BCS thermal vacuum order by order, the corrections to physical quantities such as the order parameter can be found by taking the expectation values of the corresponding operators with respect to the perturbed thermal vacuum. We caution that the thermal vacuum (27) is the ground state of the thermal Hamiltonian with the unperturbed Q in Eq. (26).
The perturbation theory requires the evaluation of the matrix elements of the perturbation, V k0 , = … k 1, 2, , which are determined as follows.
The matrix element associated with the two-particle excited states, V 20 , can be evaluated with the method shown in the Supplemental Information. Thus, the nonvanishing elements involve one αand one βquanta and are given by To evaluate the matrix element associated with the four-particle excited states, V 40 , we need to consider the following matrix elements It can be shown that only the term β β is nonzero (see the SI). Therefore, the nonvanishing matrix element is  Finally, since the perturbation V contains at most four quasi-particle operators, all matrix elements associated with higher order (k > 4) excited states vanish (see Eq. (28)).
According to Eq. (27), the perturbed BCS thermal vacuum up to the first order is given by 2 By solving the full chemical potential μ c from the equation, one can obtain the correction to μ. The expansion of the order parameter can be obtained in a similar fashion: Here we emphasize that Δ c (Δ) denotes the full (unperturbed) order parameter. The first-order correction to the order parameter can be found numerically for different coupling strengths. The fermion density is with k F being the Fermi momentum of a noninteracting Fermi gas with the same particle density. The Fermi energy is defined by . We first solve the unperturbed equations of states, Eqs (23) and (24), at different temperatures to obtain Δ and μ. Then, we substitute the unperturbed solution to Eq. (35) and get the first-order correction to Δ c (T). The critical temperature T c can be found by checking where the full order parameter Δ c (T) vanishes. Eq. (35) indicates the full order parameter is lowered by the first-order correction. We found that the critical temperature T c is also lowered when compared to the unperturbed value. However, our numerical results show that the correction to the critical temperature is small if the particle-particle interaction is weak. The strong correction to T c due to the particle-hole channel (induced interaction) 13,14 is not included in the BCS theory. Since the perturbation V considered here carries non-zero momentum, the calculation here shows the correction from finite-momentum effects to the Cooper pairs. Figure 1 shows the unperturbed and perturbed (up to the first-order correction) order parameters as functions of temperature. In Fig. 1(a), a relatively small coupling constant gn/E F = 0.385 is chosen, which corresponds to the conventional BCS case where the interaction energy is much smaller compared to the Fermi energy. We first order correction of the order parameter. Hence, δΔ is indeed small in the BCS limit. We also found the ratio .
, which is close to the mean-field BCS result of 1.76 2,23 . In Fig. 1(b), a relatively large coupling constant gn/E F = 1.19 is chosen. The first-order correction δΔ/Δ is more visible, but the ratio is still less than 8% at any temperature below T c . We also found .
, which is more distinct from the unperturbed BCS value. According to the value of Δ(T = 0)/E F indicated by Fig. 1, the system with gn/E F = 0.385 is in the BCS regime while the one with gn/E F = 1.19 is beyond the BCS limit because of its relatively large gap. The perturbation calculations allow us to improve the results order by order, but the complexity of the calculations increases rapidly. The insets of Fig. 1 show the detailed behavior close to T c and indeed the critical temperature is lowered by the perturbation.
We remark that the BCS theory is usually viewed as a variational theory 23,24,27 . The introduction of the perturbation calculation using the BCS thermal vacuum reproduces the thermodynamics of the BCS theory at the lowest order, and it introduces a quantum-mechanical style perturbation theory. The corrections to the BCS theory thus can be obtained by the perturbation theory, albeit the exact results require a full treatment of the transformation from the two-mode vacuum to the thermal vacuum. After some algebra, we obtain k k which is proportional to the total quasi-particle number N α (β) + N β (β) at temperature = β T k 1 B . The quasi-particle number is nonzero only when T > 0.
The origin of the thermal phase can be understood as follows. The thermal vacuum can be thought of as a purification of the mixed state at finite temperatures. This can be clarified by noting that Eq. (5) leads to is the density matrix of the non-tilde system at temperature T, and the phase nχ is because each excitation of a quasi-particle contributes e −iχ indicated by Eq. (18). The unitary operator  has the following matrix representation in the basis formed by {|n〉}: There is another way of purifying the density matrix 16 (42) and (44), one can find a one-to-one mapping between the thermal vacuum |0(β)〉 and the amplitude w β .
However, the purification of a mixed state is not unique. For instance, Eq. (17) allows a relative phase χ between f 0k and f 1k . A U(1) transformation corresponding to a change of the parameter χ leads to another thermal vacuum. Hence, the BCS thermal vacuum can be parametrized in the space 0 ≤ χ < 2π, i.e. one may recognize the collection of BCS thermal vacua as a U(1) manifold parametrized by χ. The thermal phase (41) from the BCS thermal vacuum may be understood as follows. If χ is transported along the U(1) manifold along a loop, every excited quasi-particle acquires a phase 2π. Therefore, the thermal phase indicates the number of thermal excitations in the system.
When T → 0, the statistical average becomes the expectation value with respect to the ground state. In the present case, the BCS thermal vacuum reduces to the (two-mode) BCS ground state. As a consequence, the U(1) manifold of the unitary transformation of the BCS thermal vacua is no longer defined at T = 0 because there is no thermal excitation and f 1k = 0 in Eq. (16). Importantly, the BCS ground state is already a pure state at T = 0, so there is no need to introduce χ for parametrizing the manifold of the unitary transformation in the purification. Hence, the thermal phase should only be defined at finite temperatures when the system is thermal.

Discussion
By introducing the two-mode BCS vacuum and the corresponding unitary transformation, we have shown how to construct the BCS thermal vacuum. A perturbation theory is then developed based on the BCS thermal vacuum. In principle, one can evaluate the corrections from the original fermion interactions ignored in the BCS approximation. Importantly, the perturbation calculations are at the quantum-mechanical level even though the BCS theory and the BCS thermal vacuum are based on quantum field theory. The interaction, however, renders the transformation from the two-mode ground state to the thermal vacuum nonlinear. We took an approximation in developing the perturbation theory, and a full treatment awaits future investigations.
The BCS thermal vacuum is expected to offer insights into interacting quantum many-body systems. We have shown that the pairing correlation from the perturbation theory persists when the corrected order parameter vanishes, providing evidence of the pseudogap phenomenon. The thermal phases associated with the BCS thermal vacuum elucidates the internal geometry of its construction. The BCS thermal vacuum and its perturbation theory thus offers an alternative way for investigating superconductivity and superfluidity. Data availability. All data generated or analysed during this study are included in this published article (and its Supplementary Information files).