# Ground state instability in nonrelativistic QFT and Euler–Heisenberg Lagrangian via holography

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## Abstract

We study the ground state instability of a strongly coupled QFT with the \(z=2\) Schrödinger symmetry in a constant electric field using probe branes holography. The system is \(N_f\) \({\mathcal {N}}=2\) hypermultiplet fermions at zero charge density in the supergravity Schrödinger background. We show that the instability occurs due to Schwinger-like effect and an insulator state will undergo a transition to a conductor state. We calculate the decay rate of instability and pair production probability by using the *gauge* / *gravity* duality. At zero temperature for massive fermions, we suggest that the instability occurs if the critical electric field is larger than the confining force between fermions, which is proportional to an effective mass. We demonstrate that, at zero temperature, the Schrödinger background simulates the role of a crystal lattice for massive particles. We also show that at finite ’t Hooft coupling for particles with a mass higher than \(\frac{\sqrt{\lambda }}{\pi \beta }\), in this background, instability does not occur, no matter how large the external electric field is, meaning that we have a *perfect insulator*. Moreover, we derive Euler–Heisenberg effective Lagrangian for the non-relativistic strongly correlated quantum theory from probe branes holography in Schrödinger spacetime.

## 1 Introduction

The decay of false vacuum to true vacuum might be considered as a pair production mechanism. The Schwinger pair production results an instability of the vacuum in QED [1, 2]. In the condensed matter systems ground state or vacuum could also experience instability in the same situation. For example, the electric breakdown of an insulator, when subjected to a high external electric field, could be considered as instability in vacuum. At the presence of an external electric field, the decay of the ground state to ground state in condensed matter physics is considered as the Zener breakdown of the Mott (or band) insulator [3]. The revisited version of Euler–Heisenberg effective Lagrangian in condensed matter physics is studied through the ground state to ground state transition amplitude or the Zener tunneling rate [4].

Generally, the Schwinger effect as a vacuum instability is a non-perturbative phenomenon. In strongly correlated systems, calculating the Schwinger effect demands great effort. The well-known toolbox for studying robust coupled systems in quantum field theories is *AdS* / *CFT* correspondence or more generally the *gauge* / *gravity* duality. The original AdS/CFT correspondence states that: The the \(AdS_5 \times S^5\), as the near extremal solution of \(N_c\) coincident \(D3-\)branes, is dual to \(3+1\) dimensional super-conformal field theory with \(SU(N_c)\) gauge degrees of freedom. The gauge degrees are shown by the adjoint representation of the \(SU(N_c)\). For adding other degrees of freedom in addition to of gauge fields, we add \(N_f\) \(D7-\)branes, see [5], and for the simplicity, consider them as a probe (\(N_f \ll N_c\)). This configuration describes a QCD-like system [5]. In general, this means that we add \({\mathcal {N}}=2\) hypermultiplet fermions (quarks) in fundamental representation to the background gauge theory. From the gravity side, these fermions are the strings with one end on *D*7 branes and the other on *D*3 branes. Dynamics of the *D*3 / *D*7 system is given by the *DBI* action of probe *D*7 branes. From the gauge/gravity dictionary, this action would be an effective action of fermions in the boundary theory. The vacuum instability of the supersymmetric QED(QCD) is studied through probe brane holography in the AdS background in the [6]. Results are in agreement with Schwinger pair production in QED after replacing ’t Hooft coupling \(\lambda \) with the QED coupling constant, \(e^2\). We aim to generalize this idea to QFT with the Schrödinger symmetry instead of the conformal symmetry. The cold atoms system known as fermions at unitarity is a famous example of a system with the Schrödinger symmetry studied using holography in the [7], see also [8, 9, 10, 11].

By using null Melvin twist(*NMT*) transformations [12], or *TsT* [13] or the *AdS* geometry solution of the type *IIB* supergravity, which has a dual QFT with conformal symmetry, the spacetime can be generated with the Schrödinger symmetry which has a dual non-relativistic QFT. Following [14], we study the DBI action of the probe *D*7 branes in the Schrödinger background. From Legendre transformation of the DBI action, we propose the effective action and Euler–Heisenberg effective Lagrangian of the systems with strong interaction with the \(z=2\) Schrödinger symmetry. The electric field on the probe branes will distance the two ends of a string on the same D7-brane, and if it is larger than a critical value, it will tear the string apart. In other words, meson dissociates, instability occurs, and non-zero current produces. In the gravity side, this means that the probe branes fall in the background black holes. At non-zero charge density, this always occurs in the AdS background see [15]. We check this statement in the supergravity Schrödinger background. On the other hand, the presence of the electric field on the branes introduce world-volume horizon, to which we could assign a temperature. This temperature is different from the background Hawking temperature. Thus, we deal with a non-equilibrium situation. Consequently, the occurrence of instability means that we switch from an equilibrium state to a non-equilibrium one. We study the decay rate of this instability that might produce the fermion and anti-fermion pairs through the Schwinger effect in the Schrödinger background. Generally, the instability at the presence of a constant electric field transforms us from an insulator state into a conductor state. We study the breakdown of the vacuum (ground state) of strongly coupled systems with the \(z=2\) Schrödinger symmetry at the presence of an external electric field via probe branes holography.

## 2 Review on probe branes in Schrödinger background

*D*7 branes as the probe in the background of

*D*3 branes. Due to the probe limit, the dynamics of the system is given by the DBI action [14].

*D*7 worldvolume coordinates and

*D*7 branes tension. The \(g_{ab}\) and \(B_{ab}\) are the induced metric and induced

*B*field from the background on the probe branes, respectively. Let embed the

*D*7 branes in 10 dimension space–time as follows :

*D*3 and

*D*7 branes at \(r=0\) which is denoted by the \((x^+,x^-,\mathbf {x})\). As it clear, there is a

*O*(2) symmetry in \((\chi ,\theta )\) direction which clarify the shape of

*D*7 branes relative to the background. Without loss of generality, we assume that \(\chi =0\) and \(\theta =\theta (r)\). We introduce following gauge field on the probe branes

^{1}

*m*is representing the mass of the fermions. It was shown that at the zero electric fields on the

*D*7 branes in the

*AdS*background, there are two allowed embedding for probe branes which could be classified by the ratio of the flavors mass and the background temperature, \((\frac{m}{T})\). The DBI action as the free energy of dual theory would tell us which embedding is thermodynamically favorable [17, 18]. For the large value of \(\frac{m}{T}\) the Minkowski embedding (ME) and for small \(\frac{m}{T}\) the black hole embedding (BE) is favorable. From a geometry point of view, the ME will happen if the compact dimension of the probe

*D*7 branes shrinks to zero outside of the background event horizon. The BE embedding, as its name is, will happen if the compact coordinates fall into the background Blackhole. For the non-zero electric field on the probe branes, we also have another class of embedding [17, 20] which known as Minkowski embedding with the horizon (MEH). At the non-zero electric field, the probe

*D*-branes would have the world-volume horizon which, in general, differs from the background event horizon. The same embeddings is allow in the Schrödinger space–time. For example see Fig. 1.

*J*with the respect to the one

*m*.

^{2}These two different current are illustrating the MEH and BE, for more detail see [19]. To explain these embedding let us solve Euler–Lagrange equation for \(a_x(r)\):

^{3}by the Legendre transformation of the on-shell action [6], which is

*U*(

*r*) and

*V*(

*r*) could be negative or positive. The real condition of the action force

*U*and

*V*to change their signs at the same point where we call it \(r_*\)(\(0<r_*<r_H) \). This point extracts from the following equations:

*T*. This situation shows that we deal with a non-equilibrium condition. In other words, the matter sector, which realize by the probe

*D*7 branes, has different temperature relative to the background plasma,

*D*3 branes. Therefore, in the dual theory we deal with non-equilibrium steady state, [24, 25].

*DC*conductivity, \(\langle J^x \rangle =\sigma (E_{\beta }) E_{\beta }\) :

For the zero electric fields, we do not have any world-volume horizon or the singularity for heat capacity (16). One can conclude that: At non zero temperature, the electric field breaks the bond state of neutral charge pairs, which are binding as mesons or Cooper pairs. Also at the zero background temperature, this phenomenon happens because of the external electric field and existence of MEH. A transition from the state with\( {\langle J^x \rangle =0}\) in the dual theory to the states with \({\langle J^x \rangle \ne 0}\) can be considered as a change from false or metastable ground state (or vacuum) to a true ground state. Consequently, at the presence of the electric field an insulator state \(\langle J^x \rangle =0\) will suffer from the instability due to the electrical breakdown. Although we will see that, on the Schrödinger background, there is a situation to have a perfect insulator. In the next section we investigate the instability which causes the phase transitions in schrödinger geometry from type IIB supergravity.

## 3 Ground state instability

^{4}which means that the electric field is turned on and we have our probe branes with Minkowski embedding yet. In other words we are studying electric field effects on the insulator state which is a pair of fermions bound together. In the condensed matter environment the dielectric breakdown of band(or Mott) insulator from the ground state to ground state transition was studied in [3]. As repeatedly mentioned, living in an insulator state means we have \(\langle J^x \rangle =0\) so from Eq. (13) we will have:

*U*(

*t*) is unitary time evolution operator of the system and

*V*is a volume of the space and \(|0\rangle \) stands for a ground state.In general we have

*Schwinger-like*pair production in the Schrödinger geometry.

^{5}In the following sections, we study the imaginary and the real part of the effective action for the massless and massive charge carriers in the Schrödinger background and compare our result with the relativistic one in the AdS background.

## 4 Ground state instability for gapless systems

^{6}this configuration resembles the gapless systems in the condensed matter systems.

^{7}For the \(\theta (r)=0\) Eq. (7) reduce to

*AdS*background, see [6].

### 4.1 Decay rate of ground state for the gapless systems

*AdS*result in [6], if we replace \(E_{\beta }=E/2\beta \) and \(N_f=1\). The reason for the similarity result between relativistic and non-relativistic Schwinger instability is that they have the same bulk mechanism of instability; the electric field will tear apart strings with both ends on the same probe brane. So finally we could say that for the \(E_{\beta }\ne 0\) the system always will decay from \(\langle J^x \rangle =0\), an insulator state, to \(\langle J^x \rangle \ne 0\) or a conductor state. This result, as Eq. (27) shows it, is independent of the background temperature. Therefore, at zero background temperature, we also will have the ground state instability, and the Minkowski embedding will switch to the other embedding with non-zero current which is MEH. Moreover, it is evident from Eq. (27) that

*it does not matter how small the electric field is*, the ground state of massless fermions is always unstable because of the electric field. At the next section, we will see that the ground state in the electric field for the massive fermions behaves entirely different to the one for massless fermions.

### 4.2 Euler–Heisenberg action for the gapless systems

*C*is a positive numeric constant. Thus we could presume that the energy difference is also finite so the real part of the effective action for zero current is meaningful, see also [6].

## 5 Ground state instability for the gapped systems

^{8}With this assumptions the induced metric on the probe

*D*7 branes, with Table 1 embedding, is

^{9}For \(m=0\), the metric Eq. (31) would have \(z=2\) Schrödinger isometry. With a little bit of work we recover the Eq. (17) with

**Note:**For \(( 2\pi \alpha ' m)^2 \beta ^2<1 \) we could always define a real effective electric field \({\tilde{E}}\) such that \({\tilde{E}}^2=4\beta ^2E_{\beta }^2 (1-( 2\pi \alpha ' m)^2 \beta ^{2})\); therefore, we could rewrite the Eq. (33) as

*AdS*background for QCD like systems:

^{10}So the critical electric field \(E_c\) is

\(D3-D7\) embedding

\(x^+\) | \(x^-\) | x, y | r | \(\alpha _1\) | \(\alpha _2\) | \(\alpha _3\) | \(\theta \) | \(\chi \) | |
---|---|---|---|---|---|---|---|---|---|

D3 | \(\times \) | \(\times \) | \(\times \) | ||||||

D7 | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) |

^{11}

^{12}By defining the

*effective mass*\(m_{*}\) as :

*m*has been replaced by the effective mass \(m_{*}\). Consequently, the geometric distinction between AdS and Schrödinger space–time has been changed to the difference in masses of charge carriers in the dual theory. Therefore, we could propose that the potential between two fermions or quarks with a distance of

*l*from each other, in the Schrödinger space–time Eq. (A.5), would be:

^{13}In the solid-state physics, moving an electron inside a crystal lattice would be the same as its motion in the vacuum if we use effective mass for the electron instead of the electron’s mass; the similar behavior is observed in here. Due to the effective mass, we could assume that the compactification along the \(x^-\) brings the same physics as a periodic potential brings to the study of the band structure in a crystal lattice at solid-state physics. It should be clear that from Eq. (39), this is significant effect at the finite ’t Hooft coupling \(\lambda \) and for large ’t Hooft coupling, \(m_*= m\); therefore, we are reduced to the result in the AdS background.

### 5.1 Decay rate: imaginary part of the effective action

*AdS*background in which

*E*is replaced by \({\tilde{E}}\). With subsequent changes

*AdS*background, see [6]. Expanding imaginiary term of the effective action relative to the small \(\frac{\pi ^2 m^2}{|\mu | \lambda }\), and also replacing \(E=2\beta E_{\beta }\), the decay rate or pair production probability will be

*DBI*action for the probe

*D*7 branes does not change under NMT transformations, see [14], but the

*DBI*action for the massive particles will change under the NMT transformations, so the result in Eq. (45) make sense. It is clear that if the particle’s mass goes to zero, we will get back to

*AdS*result, due to the same effective action. The dependence of the imaginary part of the effective action to ’t Hooft coupling also is shown in [29] for chiral mesons at the presence of external electromagnetic fields, see also [30, 31]. Other terms in Eq. (45), which differ from

*AdS*, might be considered as a dipole interaction that inherently exists in the dual theory of this Schrödinger spacetime which is originated from type IIB supergravity, see [26] and references therein.

### 5.2 Euler–Heisenberg action: real part of the effective action for gaped systems

*AdS*background [6]. For small \(\epsilon \) i.e., Strong electric field \({\tilde{E}}\) relative to \(\tilde{E_c}\), the critical electric field, the finite part

^{14}of the (47) would be

*AdS*spacetime, in term of \(\frac{\pi ^2 m^2}{|\mu | \lambda }\) which are relevant at the finite ’t Hooft coupling. However, at the zero mass, we will have the same Euler–Heisenberg effective Lagrangian which is found in the AdS background. At zero temperature for the massive neutral charge carriers, there are also two different situations that the effective action is not a complex quantity, and always remains real at the presence of an external electric field. In other words, the Minkowski embedding \(\langle J^x \rangle =0\) is a stable solution, and the electric field is not strong enough to break the bond between neutral charge pairs. One of them will exist if we have an electric field below the critical electric field, i.e., \({\tilde{E}}\le {\tilde{E}}_c\). This real effective action lives in both AdS background and Schrödinger background.

**Stable bound state:**From the Eq. (33), we will have real effective action regardless of the electric field if we suppose that

*AdS*counterpart. Interestingly this condition does not depend on the electric field; therefore

*it does not matter how large the electric field is*, at the regime of Eq. (49), the effective action is always a real quantity. If we replace ’t Hooft coupling \(\lambda \) with \(e^2\), to extract QED-like results following [6], one could say that for the neutral charge pairs with the mass-to-charge \(\frac{m_e}{e}\) ratio larger than \(\frac{1}{\sqrt{2}\pi \, \beta }\) we have

*permanent or perfect insulator*. In this case, the effective action is an addition of the two real quantity, Eqs. (42) and (47). If one does not accept the existence of perfect insulator at any circumstances, one can put the upper limit for the fermion’s mass, and say that the insulator ground state will decay to the conductor ground state. For the massive particles at non-zero temperature, due to the analytic difficulties, we need numerical calculations, but naively from Eq. (15) we could argue that the conductivity might always be zero, so current \(\langle J^x \rangle \) could be zero if

## 6 Conclusion and summary

We study the breakdown of the vacuum (ground state) of strongly coupled systems with the \(z=2\) Schrödinger symmetry at the presence of an external electric field via probe branes holography. By using the holographic argument, the decay rate of the ground state to the stable ground states, which causes the Schwinger pair production, is calculated in a finite ’t Hooft coupling. From the gravity side of the duality, there are three embedding classes: Minkowski embedding (ME) which exists if the probe brane closes off the background event horizon, black hole embedding (BE) which exists if the probe branes fall into the background black hole, and Minkowski embedding with the horizon (MEH) which exists at the presence of the non-zero electric field. For both BE and MEH, we have the non-zero current \(\langle J^x \rangle \ne 0\), and thus the system in the boundary quantum theory lives in the conductor state. For ME, the current is zero, and at the boundary theory, we have an insulator state. We might consider the string with both ends at the same brane as mesons, Cooper pairs or a bonded electron–hole. In this study, we investigate instability at the presence of the constant external electric field when we have \(\langle J^x \rangle =0\) or the insulator state in the Schrödinger background. For the massless particles or gapless systems in the Schrödinger spacetime, both the real and imaginary parts of the effective action look similar to the effective action in the AdS background. For the massive fermions, the decay rate from the insulator to the conductor would be the same as the AdS results if we recall the effective mass \(m_*\) instead of the mass *m* and electric field with \(E (1-\frac{\pi ^2 m^2}{|\mu | \lambda })^{1/2}\). For an electric field greater than the critical electric field, which is proportional to the square of the effective mass, the ground state or insulator will decay to other ground states or conductors. We show that the false vacuum would be faded out if there is an upper bound for the mass of the massive particles. In other words, the bond between fermions–anti-fermions or quarks-anti-quarks will not break by the electric field if we have particles with the mass larger than \(\frac{\sqrt{\lambda \, |\mu |}}{\pi }\). Therefore, the system lives in the insulator phase forever. The effective action in here looks similar to the relativistic one [6], if the electric field is replaced by \(E (1-\frac{\pi ^2 m^2}{|\mu | \lambda })^{1/2}\).

## Footnotes

- 1.
Which we normalized it with volume of boundary theory, i.e., \(Vol_{x^+,x^-,x,y}\).

- 2.
For example, see

*A*and*B*in Fig. 1. - 3.
- 4.
- 5.
Instead of

*vacuum*we repeatedly use*ground state*due to the non-zero chemical potential \(\mu \) which in the dual theory related to the number operator of a Schrödinger algebra. - 6.
At the zero temperature.

- 7.
In condensed matter physics it was suggested that Kondo insulators are gapless, see [38].

- 8.
We consider the distance between D3 and D7 branes such that \(\sin \theta (r)=2\pi \alpha ' m\).

- 9.
This is quit similar to \(ds_{S^3}^2\).

- 10.
We should note that \(E_c =2\pi \alpha ' m^2\) is the critical electric field in the AdS background, see [6].

- 11.
Note that we considered \(( 2 \pi \alpha ' m )^2 \beta ^2 <1\) or in term of \(\mu \) and \(\lambda \), \(\frac{m^2}{\lambda }<\frac{|\mu |}{\pi ^2}\).

- 12.
We must notice that electric field in the relativistic theory has a different scale dimension in comparison to theory with Schrödinger symmetry, so we compare \( 2\beta E_{\beta }^c= E_c^{sch}\) and \(E_c^{AdS}\).

- 13.
- 14.
After regularization.

- 15.
there is also five form

*RR*field. - 16.
At zero temperature or extremal limit we have supersymmetry. At non zero temperature, we have a thermal state in a dual field theory where supersymmetry is broken.

- 17.
- 18.
\(ds_{S^5}^2=\left( d\chi + \mathcal {A} \right) ^2 + ds^2_{\mathbb {CP}^2}\).

## Notes

### Acknowledgements

We want to thank S. F. Taghavi, K. Bitaghsir, R. Mohammadi and J. Khodagholizade and M. B Fathi and Mobin. Shakeri for useful discussions. Special thank to D. Allahbakhshi for reading the manuscript and fruitful comments.

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