Finite temperature dynamical quantum phase transition in a non-Hermitian system

We investigate the interplay between the non-Hermiticity and finite temperature in the context of mixed state dynamical quantum phase transition (MSDQPT). We consider a $p$-wave superconductor model, encompassing complex hopping and non-Hermiticity that can lead to gapless phases in addition to gapped phases, to examine the MSDQPT and winding number via the intra-phase quench. We find that the MSDQPT is always present irrespective of the gap structure of the underlying phase, however, the profile of Fisher zeros changes between the above phases. Such occurrences of MSDQPT are in contrast to the zero-temperature case where DQPT does not take place for the gapped phase. Surprisingly, the half-integer jumps in winding number at zero-temperature are washed away for finite temperature in the gapless phase. We study the evolution of the minimum time required by the system to experience MSDQPT with the inverse temperature such that gapped and gapless phases can be differentiated. Our study indicates that the minimum time shows monotonic (non-monotonic) behavior for the gapped (gapless) phase.

Hence, it is important to study the interplay between DQPT and non-Hermiticity from the theoretical as well as experimental point of view.
The finite temperature extension of QPT has recently been examined in the context of LA [78][79][80][81]. Following the similar line of argument, the DQPT is investigated following an initial thermal distribution instead of a pure quantum state [10,[82][83][84][85][86][87]. This brings in the concept of density matrix, characterised by an inverse temperature, leading to the mixed state DQPT (MSDQPT) where the quantum coherence is lost. For the open quantum systems in contact with the thermal bath can potentially lead to such MSDQPT [88][89][90]. Given the fact that the DQPT persists in the finite temperature [10,65] and it can show anomaly in the non-Hermitian system [68], we here pose the following intriguing questions to understand the interplay between the non-Hermiticity and finite temperature: Can MSDQPT appear (disappear) when the DQPT in the the underlying non-Hermitian system at zero temperature is absent (present)? Can we differentiate various non-Hermitian phases by examining the MSDQPT? How do we understand the topology in the real-time dynamics of the non-Hermitian MSDQPT?
In this paper, we generalize the framework of DQPT and winding number for non-Hermitian finitetemperature cases such that the Hermitian and infinite temperature limits can be successfully extracted. Considering a one-dimensional (1D) p-wave superconductor with complex hopping and non-Hermiticity (see Fig. 1), we find sudden quench within the gapped phase exhibits MSDQPT unlike to the zero-temperature case [68] (see Fig. 2). Interestingly, we only observe integer jumps as a signature of MSDQPT for the sudden quenches within the gapless phases contrasting to the zero-temperature profile of DQPT as observed previously (see Figs. 3,4). However, the winding number shows non-monotonic behavior in one of the gapless phases. The threshold time, referred to as minimum critical time t cm , above which MSDQPT starts appearing can be different from the zero-as well as infinite-temperature limit (see Fig.  5). We can distinguish various gapless and the gapped phases by investigating the behavior of t cm with temperature. The above studies on lossy superconductivity are further extended to lossy chemical potential case for completeness. Our study thus indicates that the temperature can non-trivially modify the dynamics of EPs as evident from the emergence of MSDQPT.
The structure of this paper is the following. We present the framework of MSDQPT in Sec. II for finite temperature and non-Hermiticity. Next, we demonstrate the model under consideration in Sec. III. We examine the MSDQPT results for gapped phase in Sec. IV A and for gapless phases in Secs. IV B and IV C. We differentiate among these phases with respect to their temperature profile in Sec. IV D. We provide plausible explanation behind our findings in Sec. IV E. Finally, we conclude in Sec. V.

II. MSDQPT FRAMEWORK
Let us consider a 2-level system, described by Hamiltonian H k = h k · σ = h kĥk .⃗ σ, that is thermally attached to a heat bath at temperature T = 1/β; here Note that H k can be considered to be non-Hermitian without loss of generality where h i k can be complex with i = x, y, z. The associated density matrix takes the form ρ k = exp(−βH k )/ Tr [exp(−βH k )] = σ 0 − m(ĥ k · ⃗ σ /2, where m = tanh(βh k ) (see Appendix. A for more details). We start with this finite-temperature initial mixed state at time t = 0 i.e., ρ k (0) corresponding to H k,i and suddenly quench to the final Hamiltonian H k,f such that the LA at a later time t is given by [10] B for more details). The dynamical analog of free energy is called rate function which is given by the logarithm of LA [63] The nonanalyticities in the rate function, given by g k (t) = 0, causes the Fisher zeros to appear in the complex time plane (see Appendix. C for more details) where, z n,k = it, and n ∈ Z. Note that the zeros of partition function is referred to as Fisher zeros. As a result, MSDQPT occurs at the momentum k = k c and critical time t c = −iz n,kc where Re[z n,kc ] = 0 leading to and t c = π n + 1 2 with C k = tanh −1 (B k ) (see Appendices D and E for more details). To be precise, MSDQPT occurs when z n , k crosses the positive side of the imaginary axis such that positive t c 's can only be the meaningful solutions of Eq. (4).
On the other hand, the dynamical phase is given by (see Appendix. F for more details) .
(5) The winding number, capturing the dynamical order parameter [91], appear to be Hence the geometric phase is the net phase acquired by a non-equillibrium quantum system other than the dynamical phase.
The physical picture of MSDQPT refers to the quantum dynamics of a mixed state density matrix. The MS-DQPT essentially captures the interference between the time evolved and initial density matrices. To be precise, where there is a complete destructive interference in real time i.e., g k (t) = 0, the rate function exhibits a singular behavior. Using the concept of parallel transport it has been shown that noncyclic and unitary quantum evolutions of a pure quantum state are related to that of a mixed state [91]. Therefore, the geometric phases for a mixed state can be thoroughly investigated with real time following the analysis of MSDQPT. The non-Hermiticity can effectively mimic the effect of external bath attached to a quantum system, and/or interaction in the quantum system. In the present context, our study qualitatively tracks the evolution of geometric phase, associated with a mixed state, in an interacting system by considering a non-Hermitian system.
The Hamiltonian Eq. (7) becomes gapless for critical momentum k * when the real part of energies satisfies the following condition This allows us to chart out the phases diagram of the model Hamiltonian as shown in Fig. 1.

IV. RESULT
We focus on the gapped and gapless phases in the above model in presence of lossy superconductivity only. The MSDQPT is studied for the intra-phase quench. For completeness, we briefly discuss the fate of MSDQPT for other inter-phase quench. We also discuss the MS-DQPT in the above model with non-Hermitian chemical potential. The Hermitian counterpart of MSDQPT are demonstrated in Appendix. I.

A. Quench within gapped phase I
We first examine the MSDQPT following the quench within the phase I as shown in Fig. 1 (b). For finite temperature (β = 1), Fisher zeros profile, rate function, geometric phase, and winding number are depicted in Fig. 2 (a), (b), (c), and (d) referring to the fact that MSDQPT takes place. We notice that z n,k always cross the imaginary axis except for n = n min = 0. What we find is that n min increases from zero as β increases, but yet indicating to the emergence of MSDQPT for any temperature. Interestingly, the nonanalyticities are not visible macroscopically, however, there exists the singular micro-structures at critical time t = t c ≈ 3.22, 3.76, 4.22, 4.59, · · · over the oscillating profile. We find abrupt changes in geometric phase Φ G k (t), marked by white circles in Fig. 1(c), around the above values of t for k being close to π. The profile of Φ G k (t) looks quite different as compared to the non-Hermitian zero temperature case. The winding number shows step-like jumps at the above critical times. Since z n,k encloses a close loop by crossing the imaginary axis twice, winding number is expected to exhibit both increase and decrease with time. However, we only find decrease in winding number within 3 < t < 5 where z n,k crosses the real axis once.

B. Quench within horizontal gapless phase IV
We now focus on the occurrences of MSDQPT following the quench inside the gapless phase IV as shown in Fig. 1 Figure 3(a) depicts the lines of Fisher zeroes z n,k , crossing imaginary axis twice for all values of n. This is in contrast to the previous situation for the quench inside the gapped phase I, presented in Fig. 2, where MSDQPT only takes place for n > n min . The nonanalyticities [discontinuous change] in the rate function I(t) [geometric phase] are captured at the critical times t c ≈ 1.38, 3.78, 4.71, 5.68, · · · in Fig. 3 The nonanalyticities in the rate function is more clearly visible in the present case as compared to the previous one in Fig. 2 (b). The winding number shows non-monotonic jump profile with time that is caused by the double crossing of imaginary axis by z n,k . The important point to note here is that these jumps are always of unit magnitudes unlike the previous zero-temperature case [68]. The unit jumps are a consequence of the continuous crossing of Fisher zeros through the imaginary axis that we find in the present case. The lines of Fisher zeros, z n,k with n = 0 (blue), · · · , n = 4 (red) cross imaginary axis twice leading to the nonanalyticities in rate function at critical times, tc ≈ 1.38, 3.78, 4.71, 5.68, · · · around which winding number exhibits integer jumps. The parameters are taken to be (µi,

C. Quench within vertical gapless phase V
We now demonstrate the MSDQPT following the quench within the vertical gapless phase V as presented in Fig. 1 (b). Unlike to the previous cases, we here find that z n,k crosses the imaginary axis once (see Fig. 4(a)). The nonanalyticities in the rate function are captured with time in Fig. 4 (b). The geometric phase, shown in Fig. 4 (c), exhibits a similar profile as compared to that of the gapped phase I. The oscillatory profile of geometric phase is a common characteristic of finite-temperature case. The winding number shows monotonic increase with time due to the single crossing of imaginary axis by z n,k . The unit jumps in MSDQPT, associated with finite temperature, are in contrast to the half-integer jumps of DQPT corresponding to the zero temperature case [68]. We additionally check for ϕ = 0 case where we also do not find the half-integer jumps in the winding number (not shown here). The lines of Fisher zeros, z n,k with n = 0 (blue), · · · , n = 4 (red) cross imaginary axis once leading to the nonanalyticities in rate function at critical times, tc ≈ 1.29, 2.69, 3.51, · · · around which winding number exhibits monotonic integer jumps. All the parameters are taken to be (µi, µ f , ∆i, D. Distinct temperature profile of MSDQPTs for phases I, IV and V As illustrated above, the MSDQPT takes place in all three phases irrespective of their gap structure. To be precise, for a given quench amplitude and temperature, there exist multiple critical times t c 's. We here focus on the evolution of the minimum critical time, referred to as t cm , that captures the minimum time taken by the system to witness the first occurrence of MSDQPT. We numerically study the temperature dependence of t cm such that the phase I, IV and V can be distinguished. Figure 5 (a), (b) and (c) depict the temperature profile of t cm following the large (small) intra-phase quench amplitude, denoted by red (blue) lines, within regions I, IV and V, respectively. The infinite temperature β → 0 value of t cm is found to be insensitive to the quench amplitude. This suggests that MSDQPT is anyway present in the infinite temperature case as long as the quench amplitude is finite. Connecting with the Fig. 2 (a), one can find that n min increases for smaller quench amplitude. For phase V, MSDQPT takes place early as compared to the phase I and IV in the infinite temperature limit. On the other hand, t cm saturates with increasing β above a certain value. We now find that the zero temperature β → ∞ value of t cm strongly depends on the quench amplitude (see insets of Figs. 5 (a), (b) and (c)). To be precise, MSDQPT appears quickly with time in the zero temperature limit for larger quench amplitude. Interestingly, for the present case, MSDQPT occurs more quickly with time for gapless phases IV and V as compared to the gapped phase I in the limit β → ∞.
The nonmonotonic behavior in (b) and (c) is in complete contrast to that of (a). We show the saturation of tcm over a wide range of β ≫ 1 as the insets. The insets show the saturation profile of tcm for β ≫ 1. We consider w0 = 1, ϕ = π/4, γ1 = 0, and γ2 = 1.
For the intermediate temperature with finite value of β, we find non-monotonic behavior of t cm only for the gapless phases IV and V. In the case of the gapped phase I as shown in Fig. 5 (a), t cm increases almost monotonically from β → 0. This is followed by a saturation for β → ∞. However, there exist a small dip around β ≈ 0. The non-Hermiticity induced vertical gapless phase V shows sharp dip for intermediate values of β while a broadened dip is noticed for complex hopping-induced horizontal gapless phase IV (Figs. 5 (b) and (c)). Such a dip in t cm for gapless phases refers to the fact that MSDQPT can even appear early with time as compared to the infinite temperature case. This is in contrast to the behavior of gapped phase where MSDQPT can only appear at later time for any finite temperature as compared to the infinite temperature limit. The details of dip structure of t cm is expected to depend on the quench amplitudes for a given gapless phase. The location of such dips might depend on the details of gapless phase whether it is caused by non-Hermiticity or phase of complex hopping for identical quench amplitude. For example, the relative locations of such dips on the β axis are altered between the phases IV and V. However, we emphasize that the detailed future analysis of t cm vs. β behavior is yet to be required to comment on their phase-dependent distinct characteristics.
The spectral gap profiles in the phases I, IV and V differ from each other significantly. The above analysis on MSDQPT by varying β is able to capture the interplay between the temperature and the gap profile of the eigenstates associated with these phases. In the gapless regions IV and V with finite T , the minimum time t cm at which the first destructive interference takes place is relatively less than that for gapped region I. This can be intimately connected to the distinct spectral profiles of these phases.

E. Discussion
The estimation of critical momenta, obtained from Eq. (3), is hard as far as a closed form is concerned. In the non-Hermitian case, one can use non-Bloch form of momentum to explain the topological properties [99][100][101][102][103]. We can use the same non-Bloch notion to qualitatively predict the critical momenta k c using the DQPT framework for Hermitian system. The above effective approach is unable to predict the exact values of the critical momenta, however, one obtains a closed form expression of k c (see Appendix. G). Importantly, it can explain the emergence of multiple k c 's which is consistently visible for non-Hermitian cases [68]. The multi-valued nature of k c also exists for Hermitian system. However, this nature persists more strongly as the non-Hermiticity allows for additional solutions for k c as evident from Eq. (3).
Having demonstrated the results for quenching inside the phases I, IV and V extensively, we comment that MS-DQPT is also present for quench inside phases II as well as III. Therefore, all the intra-phase quench leads to MS-DQPT in finite temperature when the non-Hermiticity is associated with the superconductivity. By contrast, the DQPT is not always present for all of the above cases in the zero temperature [68]. On the other hand, for any inter-phase quench, the MSDQPT is present that we do not show here explicitly. The emergence of DQPT is not intrinsically connected with the crossing of QCP and EP for Hermitian and non-Hermitian systems, respectively [64,68] at zero temperature. As shown above, DQPT always present as long as the critical momenta k c exist. For finite temperature case, obtaining the critical momenta is even more probable due to the presence of thermal density matrix instead of the pure quantum state. The finite temperature broadening of the quantum energy levels yields further scope to interact with neighbouring energy levels in addition to the non-Hermiticity. This might lead to the rapid variation of geometric phases for different momentum modes at a given time. As a result, one can expect to see MSDQPT for all the cases. However, we find an exception when we study the intraphase III quench in presence of non-Hermitian chemical potential only at finite temperature (see Appendix. H). We emphasize that the half-integer jumps in winding number for zero temperature are rounded off by the finite temperature where the Fisher zeros do not show any discontinuity over the imaginary axis.
The temperature profiles of MSDQPT in the different phases clearly signal the distinct characteristics of the post quench evolution of a mixed quantum state (see Fig. 5). The unitary evolution of a pure quantum state results in revival with time referring to the fact that there exist quantum interferences between the initial and time evolved state [104,105]. The non-analytic divergences in the rate functions are connected with the complete destructive interferences. The absence of such interferences results in disappearance of the DQPT. The evolution of mixed state LA shows qualitatively similar features as far as the constructive and destructive interferences are concerned while compared with the pure state LA. However, temperature smoothens the interference patterns and the destructive interferences sustain [84,106,107]. In our case, MSDQPT always present means that complete destructive interferences are bound to happen post quench out of the mixed quantum state. For non-Hermitian case, due to the presence of EPs the destructive interferences are more probable [55]. As a result, the non-unitary evolution of mixed state at finite temperature deviates from the initial configuration more often than the zero temperature unitary counterpart. This can be connected to the thermal behavior of interacting systems where the driven systems traverses the entire phase space. Our findings on the occurence of MSDQPT essentially refer to the fact that the non-Hermitian system might not localize in the phase space as opposed to integrable Hermitian system leading to finite temperature thermal phase. A detailed investigation is required in future to study these aspects of DQPTs.
We now discuss the possible experimental connection as far as the model and MSDQPT are concerned. We know p-wave superconductor is naturally unavailable but can be engineered using proximity effect in Rashba nanowire with s-wave superconductivity [108]. On the other hand, the non-Hermiticity is more easily realizable in meta-materials as compared to the solid state systems.
The non-Hermitian dynamics in ultracold atoms are theoretically proposed to obtain new regime of quantum critical phenomena [109][110][111]. The PT-symmetric dimerized photonic lattice are experimentally engineered to study the non-Hermitian topological systems [75]. Moving onto the experimental detection of MSDQPT, we can comment that the geometric phase for mixed state can be captured by using NMR spectrospy [112]. The DQPT for fermionic many-body states has also been experimentally captured following time-resolved state tomography in a system of ultracold atoms in optical lattices [71]. The quantum logic gates in optical lattices [113], where NMR technique can be blended with ultracold atoms, might be instrumental in probing MSDQPT such that the geometric phase is measured for a time evolved mixed state on a Bloch sphere. In short, we believe that meta-material perspective of quantum phenomena could be realized in future to test the theoretical findings. However, predicting an exact experimental setup is beyond the scope of the present manuscript.

V. CONCLUSIONS
Considering p-wave superconductor with complex hopping and non-Hermiticity (see Fig. 1), we examine the occurrences of MSDQPT in various gapped and gapless phases. We find MSDQPT always exists irrespective of the gap profile of the underlying phases as long as the temperature is non-zero (see Figs. 2, 3, and 4). The phase boundaries are modified by the particular choice of γ 1,2 in the non-Hermitian case, however, the qualitative findings whether the MSDQPT appears remain unaltered. This is in contrast to the absence of DQPT in the gapped phases at zero temperature. The halfinteger jumps of winding number in the zero temperature for gapless phase are washed away at finite temperature. However, in the gapped phase with finite temperature, there exist a notion of minimum integer number above which the Fisher zeros crosses the imaginary axis. We do not find any such finite integer number for Fisher zeros under finite temperature in the case of gapless phases. We analyze the minimum time t cm required by the system to experience MSDQPT as a function of the inverse temperature β such that we can distinguish the above phases (see Fig. 5). The non-monotonic (monotonic) nature of t cm with β is noticed for gapless (gapped) phases. There exist finer details in the behavior of t cm with regard to the quench amplitudes through which non-Hermiticity induced and complex hopping induced gapless phases can be differentiated. Our study can successfully bridge between the zero and infinite temperature limits. We provide an effective theoretical framework to qualitatively understand the occurrences of MSDQPT. However, we stress that the exact closed form expression of the critical time for any finite temperature MSDQPT is yet to be examined as a future study. The effect of long-range hopping, various types of disorder can be studied in this con-text of MSDQPT. Provided the experimental advancement on lossy systems [72][73][74][75][76][77], we believe that the present study is experimentally viable. Acknowledgments D.M. acknowledges SAMKHYA: High-Performance Computing Facility provided by Institute of Physics, Bhubaneswar, for numerical computations. We thank to Arijit Saha for useful discussions.

Appendix A: Initial density matrix
The initial Hamiltonian is given by (A1) For temperature T = β −1 , using Eq. (A1) the initial (t = 0) density matrix is given by where, σ 0 is 2 × 2 identity matrix, m = tanh(βh k,i ) and For Hermitian Hamiltonian, h k,i is the positive eigenvalue of H k,i that is always real. On the other hand, for non-Hermitian Hamiltonian, the eigenvalue can be imaginary as well and h k,i corresponds to positive real part of the energy eigenvalue.
For infinite temperature with T → ∞ limit, i.e., β → 0, leads to m = 0. This results in The above expression exactly matches with previous studies on DQPT with an infinite temperature mixed density matrix at initial time [65].

Appendix B: Loschmidt amplitude
The final Hamiltonian is given by where, h k,f represents the positive real part of the energy eigenvalue for final Hamiltonian without loss of generality. The initial thermal density matrix is evolved with the final Hamiltonian H k,f in time following the sudden quench from H k,i . The time evolution operator is given by The LA at zero temperature is given by g k (t) = initial pure quantum state |Ψ k,i ⟩, associated with H k,i , is evolved with H k,f . In the case of finite temperature, the LA thus can be written with the initial thermal density matrix (Eq. (A2) and time evolution operator (Eq. (B2)) as follows [10] g where, Note that in DQPT, LA plays the same role as the partition function for an equilibrium phase transition. For the infinite temperature case with T → ∞, β vanishes yielding m = 0. This leads to the following The above expression exactly matches with previous studies on MSDQPT [65].

Appendix C: Lines of Fisher zeros
Similar to the vanishing of the partition function in equilibrium phase transition, here also the lines of Fisher zeros are given by the suppression of LA i.e. g k (t) = 0. This represents a complete destructive interference between the initial and time evolved states. From Eq. (B3), one can obtain the Fisher zeros as follows Hence the general expression for Fisher zeros z n,k is given by where, z n,k = it, and n ∈ Z. Note that z n,k in Eq. (C2) is a complex function, and MSDQPT happens when the lines of Fisher zeros cut the imaginary axis i.e. Re[z n,k ] = 0. This refers to the fact tanh −1 (B k )/h k,f = 0 with B k = 0.
We again compare with the infinite temperature case i.e, T → ∞ limit. Here, β becomes zero giving rise to m = 0, B k = 0. As a result, we find Note that the above expression exactly matches with the previous findings [65].

Appendix D: Critical momenta
Let us define a new quantity as C k = tanh −1 (B k ). Hence Eq. (C2) becomes The critical momenta k c is then obtained by solving the equation below for k c We now reduce the above expression in the case of infinite temperature. For T → ∞ limit, i.e., β → 0, one can obtain B k = 0, and C k = 0. As a result, Eq. (D1) takes the following form This is consistent with the previous findings [65]. We would like to point out an important observation here for γ 2 , ∆ ̸ = 0, and γ 1 = 0. The above Eq. (D2) can only yield k c = nπ with n = 0, 1, 2, · · · as the valid solution. This further indicates that the critical momenta k c are independent of all the model parameters. In order to derive the above solution, we use z 1/2 = |z| 1/4 exp(iϕ/2) with z = x+iy, ϕ = arctan(y/x). A complete calculation suggests that y = 0 is the only solution possible provided |z| ̸ = 0.

Appendix E: Critical time
We here derive the critical time t c = −iz n,kc corresponding to k c as given below The above general expression in the case of infinite temperature can be reduced further. Considering T → ∞, β → 0, i.e., B k = 0 i.e. C k = 0, we find This is consistent with the previous findings [65]. Using the above lines of argument, discussed after Eq. (D2), one can find that for a fixed µ f and γ 2 , ∆ ̸ = 0, t cm | T →∞ is independent of quench amplitude. Interestingly, t cm | T →∞ can only depend on µ f while ∆ dependence is completely absent for the critical momentum k c = nπ.

Appendix F: Dynamical phase
Dynamical phase is nothing but the phase acquired by a quantum state due to the time evolution of the underlying Hamiltonian. We here illustrate the dynamical phase for the non-Hermitian system such that H † ̸ = H [65]. Let us denote the right and left eigenvectors as |ψ r s (k)⟩, and |ψ l s (k)⟩ respectively. Here s = ± denotes two energy bands for the 2-level systems. These eigenvectors satisfy the following equations In this representation, the Hamiltonian H k can be expressed as In the space of these left and right eigenvectors, right and left time evolution operators can be expressed as respectively. The bi-orthogonality conditions s |ψ r s (k)⟩⟨ψ l s (k)| = σ 0 and ⟨ψ l s (k)|ψ r s ′ (k)⟩ = δ ss ′ are required to further simplify the expressions. The time evolved density matrix is written as ρ k (t) = U l † k (t)ρ k (0)U r k (t). The dynamical phase is expressed as follows [10] Φ dyn Now, using Eq (A2), we obtain Now using Eqs. (F3), (F4), (F5) Similarly, Now combining all the above expression, the dynamical phase for non-Hermitian system is found to be In the infinite temperature T → ∞ limit, the dynamical phase reads as The above expression is consistent with earlier findings [65].
Appendix G: Effective theory for non-Hermitian DQPT We here discuss the effective theory for the MSDQPT. We rewrite the non-Hermitian Hamiltonian under consideration Replacing 'k' by 'k + iκ', and saying e k+iκ ≡ x, we can write the above Hamiltonian as The eigenvalues are given by Now, E ± → 0 limit, we get Solutions of the above equation are Now, x 1 x 2 x 3 x 4 = 1, x 1 x 2 = −∆+w0 cos ϕ ∆+w0 cos ϕ , and x 3 x 4 = ∆+w0 cos ϕ −∆+w0 cos ϕ . Therefore, x can be written as, none of the above is a good choice as their final expressions are independent of γ 2 . We hence use x = √ x 1 x 4 as it depends on γ 2 . This allows us to write the Eq. (G2) as follows In order to get the critical momentum, k c we use the known DQPT framework for the Hermitian system using ⃗ h kc,i . ⃗ h kc,f [10] for our case such that Solving the above equation for a fixed quench amplitude, one can get multiple critical momenta k c unlike single critical momentum for Hermitian case [10]. This can qualitatively explain the emergence of multiple k c 's for the non-Hermitian case. We here consider the non-Hermiticity only in the chemical potential i.e., γ 1 = 1 and γ 2 = 0. Note that the phase diagram changes for lossy chemical potential, however, the phases I, II, III, IV, and V are present similar to the lossy superconductivity (see Fig. 1). We study the behavior of the Fisher zeros for quench within the phases I, III, IV and V, as shown in Fig. 6 (a), (b), (c), and (d), respectively. We find that MSDQPT can exist for phases I, IV and V except for phase III as the Fisher zeros crosses the imaginary axis in the prior phases but not in the later phase. This is in contrast to the non-Hermitian superconductor case where the MSDQPT always persists irrespective of the phases as long as the temperature is non-zero. On the other hand, for phases I and IV, one can respectively find n min and n max for the lines of Fisher zeros, indicating to the fact that MSDQPT is absent below (above) a certain time scale. This time scale is directly related to n min and n max for phases I and IV, respectively. For the case of non-Hermitian superconductor, we do not find any such n max in phase IV, however, we do find n min for phase I. Therefore, the gapped and gapless phases for non-Hermitian superconductor and non-Hermitian chemical potential do not show identical properties as far as the MSDQPT is concerned. Interestingly, for quench within region III, we find the absence of MSDQPT for the finite temperature non-Hermitian chemical potential similar to the zero temperature case [68]. However, likewise the zero temperature case, we do not find any discontinuity in the Fisher zeros in any of the above cases.
Here, we are interested only in the lines of Fisher zeroes that are enough to confirm the occurrences of MSDQPT. We investigate the Fisher zero profiles for quenches within regions III and I which are shown in Fig. 7 (a) and (b) respectively. The non-crossing nature conveys the absence of the MSDQPT for an intra-phase quench inside region I. Therefore, for intra-phase quench within region I, Fig. 7 (b) suggests that MSDQPT does not happen. By contrast, (γ 1 , γ 2 ) = (̸ = 0, 0) and (0, ̸ = 0) as shown in Figs. 6 (a) and 4 respectively, we find that MS-DQPT exists which is a marked difference as compared to the finite temperature Hermitian case. The MSDQPT is absent for intra-phase quench in region III irrespective of the Hermicity of the problem (see Figs. 7 (a) and 6 (b)). The Fisher zeros are depicted in Fig. 7 (c) ((d)) for large (short) quench mertic within region IV. Interestingly, the lines of Fisher zeroes cross imaginary axis twice indicating MSDQPT for two types of critical momenta in Fig. 7 (c). On the other hand, the lines of Fisher zeroes do not cross imaginary axis referring to the absence of MSDQPT as demonstrated in Fig. 7 (d). Based on the above analysis on the Hermitian case at finite temperature (β = 1), we can comment that the results are similar as obtained for the zero temperature case [68].