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In this paper, based on the basic principles of thermodynamics, we explore the hydrodynamic regime of interacting Lifshitz field theories in the presence of broken rotational invariance. We compute the entropy current and discover new dissipative effects which are consistent with the principle of local entropy production in the fluid. In our analysis, we consider both the parity even and the parity odd sector upto first order in the derivative expansion. Finally, we argue that the present construction of the paper could be systematically identified as that of the hydrodynamic description associated with

Low temperature phases of various

Matter in this quantum critical domain (so called

All the above features therefore strongly suggest the breakdown of the usual Fermi liquid theory in the domain of quantum criticality and as a result the low temperature physics of strongly correlated many body systems has remained as a puzzle for the past couple of decades.

However, for various reasons, it is widely believed that the physics within this quantum critical domain is mostly governed by the obvious scaling symmetries of the underlying Quantum Critical Point (QCP). It is by now quite evident that the isometry group associated with QCPs is that of Lifshitz type; namely, the scaling symmetries are anisotropic at the fixed point [

The purpose of the present paper is however not to explore physics within this domain of quantum criticality; it is rather slightly away from it by explicitly breaking symmetries associated with the fixed point. In the following, we illustrate this in a bit detail.

It turns out that QCPs with Lifshitz isometry group explicitly break Lorentz boost invariance and on the other hand preserve rotational symmetry as well as the translational invariance. The long wavelength dynamics associated with these fixed points has been explored extensively from both the perspective of a fluid description and the AdS/CFT duality [

Two questions are quite obvious at this stage:

The anisotropy that we introduce in our analysis is in fact quite similar in spirit to that with the earlier analysis [

In our analysis, we introduce anisotropy by means of four vectors,

As far as the d.o.f. are concerned, the answer is rather tricky. However, we claim that the present analysis could be used to model the hydrodynamic description associated with

It turns out that the classical hydrodynamic description of spin waves is valid as long as spin wave fluctuations are smaller than the correlation length [

The organization of the rest of the paper is the following. In Section

The purpose of the present calculation is to explore the consequences of

We start our analysis by writing down the constitutive relations (compatible with the reduced symmetries of the system) for both the stress tensor and the charge currents associated with the underlying hydrodynamic description. We choose to work in the so called Landau frame [

We first focus on the stress tensor part. It typically consists of the following two pieces (the reader should be aware of the fact that to start with Lifshitz field theories are QFTs for which

Since the full Lifshitz isometry group is now broken down to some reduced subset of it, therefore, to start with and in principle we are allowed to incorporate more generic tensor structures in the constitutive relation which were previously disallowed due to the presence of the underlying rotational invariance. Therefore in some sense the analysis of the present paper generalizes all the previous studies in the literature [

The starting point of our analysis is quite straightforward. We write all possible distinct tensor structures which are allowed by the (reduced) symmetry of the system (consistent with the Landau frame criterion) and we express our constitutive relations as a linear combination of each of these structures. The coefficients associated with these tensor structures are finally constrained by the principle of positivity of entropy production. Those transports which do not seem to be consistent with this physical constraint are set to be zero although they are allowed by the symmetry of the system. Keeping these facts in mind, the ideal part of the stress tensor could be formally expressed as (this form of the stress tensor is unique and is indeed consistent with the Landau frame criterion, namely,

Considering the

Before we proceed further, one nontrivial yet simple check is inevitable. We consider the Lifshitz Ward identity (note that the Ward identity is always satisfied if the underlying scaling symmetry is preserved which is precisely the case for our analysis as the spin wave excitation is gapless [

We now focus on the dissipative part of the stress tensor (

Finally, the

The purpose of this section is to construct a systematic hydrodynamic description away from the quantum critical region with Lifshitz isometry group. In order to do that, in this paper we illustrate the simplest case with

As a first step of our analysis, we first construct some basic thermodynamic identities which will be required in the subsequent analysis. Considering the following identity (in order to avoid confusion, we denote chemical potential as

We would now like to interpret the entity

Using (

Following the arguments of [

Finally, using (

Before we conclude this section, it is now customary to derive the so called equation for the entropy current [

Before we proceed further, at this stage it is noteworthy to mention that the parity breaking terms would definitely modify the entropy current as expected from the earlier analysis [

We would first consider the parity

At this stage, it is also noteworthy to mention that some of the transports appearing in the isotropic part of the parity even sector also appear in its anisotropic counterpart. As a consequence of this, all the previously determined constraints [

We now focus solely on the parity

Next, using conservation equation (

Finally, using (

Following the same steps as above we note that

Finally, using (

Following the prescription of [

In the following, we enumerate each of the individual terms one by one.

Finally, collecting all the individual pieces in (

The above set of (

The above set of (

Considering the general identity (here we define

Equation (

Substituting (

From the first two equations in (

We now summarise the key findings of our analysis. The most significant outcome of our analysis is the establishment of the hydrodynamic description away from the quantum criticality. We claim that such hydrodynamic description could be identified with the corresponding hydrodynamic description of

Finally, as mentioned earlier, for realistic models of strange metal systems one should actually consider the so called

No data were used to support this study.

The author declares that there are no conflicts of interest.

The author would like to acknowledge the financial support from UGC (Project No. UGC/PHY/2014236).

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