Holographic fermionic system with dipole coupling on Q-lattice

Journal of High Energy Physics, Dec 2014

We construct a holographic model for a fermionic system on Q-lattice and compute the spectral function in the presence of a dipole coupling. Both key features of doped Mott insulators, the dynamical generation of a gap and spectral weight transfer, are observed when adjusting the value of the coupling parameter p. Of particular interest is that when the background is in a deep insulating phase, the Mott gap opens much easier with a smaller coupling parameter in comparison with a metallic background. The effects of lattice parameters on the width of the gap Δ/μ are studied and a turning point is observed near the critical regime of metal-insulator transitions of the background. Furthermore, the temperature dependence of the spectral function is studied. Finally, we also observe that the anisotropic Q-lattice generates anisotropic peaks with different magnitudes, indicating that insulating and metallic phases arise in different directions.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP12%282014%29149.pdf

Holographic fermionic system with dipole coupling on Q-lattice

Q-lattice 0 1 5 6 7 Open Access 0 1 5 6 7 c The Authors. 0 1 5 6 7 0 Jinzhou 121013 , China 1 Beijing 100190 , China 2 State Key Laboratory of Theoretical Physics 3 Department of Physics, School of Mathematics and Physics, Bohai University 4 Institute of High Energy Physics, Chinese Academy of Sciences 5 Institute of Theoretical Physics, Chinese Academy of Sciences 6 Beijing 100049 , China 7 [46] I. Bah , A. Faraggi, J.I. Jottar, R.G. Leigh and L.A. Pando Zayas, Fermions and D = 11 We construct a holographic model for a fermionic system on Q-lattice and compute the spectral function in the presence of a dipole coupling. Both key features of doped Mott insulators, the dynamical generation of a gap and spectral weight transfer, are observed when adjusting the value of the coupling parameter p. Of particular interest is that when the background is in a deep insulating phase, the Mott gap opens much easier with a smaller coupling parameter in comparison with a metallic background. The effects of lattice parameters on the width of the gap / are studied and a turning point is observed near the critical regime of metal-insulator transitions of the background. Furthermore, the temperature dependence of the spectral function is studied. Finally, we also observe that the anisotropic Q-lattice generates anisotropic peaks with different magnitudes, indicating that insulating and metallic phases arise in different directions. 1 Introduction 2 3 4 Holographic Q-lattice geometry Dirac equation Mott transition on Q-lattice Free fermionic spectral function The Mott transition The dynamics at different temperatures Fermi surface and gap on anisotropic Q-lattice Conclusion and discussion Introduction To understand and describe Mott metal-insulator transition (MIT) is a long-standing and widely known difficult problem in condensed matter physics because it involves a strongly correlated electron system, in which the conventional theoretical tools prove of little help and non-perturbative techniques are called for. At this stage, holography may provide insights into the associated mechanisms of these strongly correlated electron systems by building a gravitational dual model which is usually solvable in the large N limit. Some excellent examples are the holographic superconductor [13] and holographic (non)-Fermi As early as 1930s, it was reported that many transition-metal oxides (such as NiO) with partially filled bands show insulating behavior. Peierls ascribed that to the strong electron-electron correlation. And then, Mott made a series of pioneering work towards understanding how electron-electron correlations could explain the insulating state [15 19]. Roughly speaking, the main idea of Mott is that the transition from metal to insulator occurs as lattice constant increases. Subsequently, this idea of Mott is formalized in the Hubbard model in which the Mott transition depends on the competition between the prevails if the kinetic energy t overcomes the Coulomb energy U while insulating phase is favored for U/t 1 and a gap opens in the single-particle excitation spectrum, resulting in a Mott transition at a critical ratio of U/t. By adding probe fermions with dipole coupling in RN-AdS black hole [20, 21], a Mott gap opens dynamically, which exhibits two key features of doped Mott insulator, i.e., the dynamical generation of a gap and spectral weight transfer. And then, the dipole coupling effects have also been studied in more general geometries in [2229]. Along this line, here we shall study the holographic fermionic system with dipole coupling in Q-lattice geometry. Holographic Q-lattice model is firstly proposed in [30], which is similar to the construction of Q-balls [31]. Some extensive studies have been presented in [3235]. In this framework, the translational symmetry is broken and MIT is observed through the study of optical conductivity. Different from the holographic scalar lattice and ionic lattice constructed in [3641], which involve solving the PDEs and need a hard numerical work, but here we only need solve ODEs for Q-lattice. In particular, MIT is not observed in scalar lattice or ionic lattice background yet. The difficulties of numerical calculation prevent one from dropping the temperature down to an extremely low level in this context.1 Here, we are interested in understanding if there is a relationship between the phase (metal/insulator) the background geometry with Q-lattice is in and that the fermionic excitations on Q-lattice are in. Therefore, in this paper we shall study the fermionic excitations by adding a probe fermion with dipole coupling in Q-lattice geometry. This paper is organized as follows. In section 2, based on the holographic model originally presented in [30], which introduces Q-lattice structure in one spatial direction, we generalize it to a two dimensional lattice model which in general can be anisotropic in in section 3. In section 4, we present our numerical results for the fermionic spectral function. We conclude in section 5 with a summary of main results and suggestions for future research. Holographic Q-lattice geometry Here we are interested in a holographic Q-lattice geometry with no translational symmetry in both of the spatial directions. For this purpose, we consider a system containing two be seen in [32]. The simplest action may take the form as, S = 4 g R + 6 2 1 F 2 |1|2 m12|1|2 |2|2 m22|2|2 , 3+ +(1)1 +(2)2 + FF 4 g F 2 , Consider the following ansatz A = At(z)dt, 1MIT is also observed in helical lattice model [42] and holographic charge density waves [43]. In addition, a simpler construction which breaks the translational invariance can be found in [44, 45], in which the lattice amplitude is absent, and thus no MIT happens in these models. gtt(z) = gxx(z) = (1 z)P (z)Q(z) P (z) = 1 + z + z2 2 gzz(z) = gyy(z) = z2(1 z)P (z)Q(z) and substitute them into (2.2), one has five second order ODEs for V1, V2, a, 1, 2 and one first order ODE for Q. Note that in above ansatz, k1 and k2 are two waveaddition, in our holographic setup, the dual CFT involves two complex scalar operators scalar field, possibly leading to a different ground state. To avoid this possibility, we will To solve the ODEs numerically, we impose a regular boundary condition at the horizon In this paper, we only focus on the standard quantisation of the scalar field, in which the The UV behavior of the scalar field corresponds to a Q-lattice deformation with lattice T = denote the above five dimensionless quantities in what follows. Dirac equation To explore the properties of fermionic spectral function in the Q-lattice geometry, we consider the following action including the dipole coupling with strength p between the fermion and gauge field [20, 21, 46, 47] SD = i 4 a set of orthogonal normal vector bases and the spin connection 1-forms, respectively. Here, gzz 0(i iqAt)F (z, k) + gxx Choose the following gamma matrices + gyy 2ikyF (z, k) mF (z, k) + gzzgtt the relevant quantities are the products gF q and gF p. The Dirac equation can be deduced from the above action the same time, by the Fourier expansion, gzz gzz F = kx gxx gzz z + gyy + gzzgtt can be expressed as + gzzgtt (zAt) + gzzgtt + gyy (zAt) + gyy Z ddkxdky F (z, k)eit+ikxx+ikyy, gxx + gxx = 0. = 0, = 0. At the horizon, we can find that z = 0. In order to obtain the retarded Green function on the boundary by holography, the independent ingoing boundary condition should be imposed at the horizon, i.e., Near the AdS boundary, the Dirac field reduces to And so by holography, the retarded Green function can be read off i Note that since the four components of the Dirac fields couple to one another, we need Green function. We are mainly interested in the measurable spectral function, which is Mott transition on Q-lattice In this section, we study the properties of spectral function on Q-lattice. We shall firstly where we give a brief discussion on anisotropic spectral function. For definiteness, we work non-zero temperature of T ' 0.00398 for this paper except the subsection 4.3, in which we explore the dynamics at different temperatures. Free fermionic spectral function In this subsection, we present the results for free fermionic spectral function on Q-lattice. Because the notion of Fermi surface is only well defined at zero temperature, for our Qlattice system of low but non-zero temperature, we need an operational definition, which has been proposed in [48] and adopted in holographic models [40, 49]. It is argued that Fermi surface is a circle, which is in agreement with the fact that our holographic Q-lattice and kF ' 1.359, indicating a Fermi surface. The Q-lattice effects on the evolution of the free fermionic spectral function can be seen as follows. On one hand, one can fix the wave-number and see the changes of spectrum of the Fermi surface. On the other hand, one can fix the lattice amplitude and adjust the wave-number k to see its impact on the shape and location of Fermi surface. From the right plot in figure 2, we see that with the augmentation of the wave-number k, the Fermi peak shifts to positions with larger momenta. Moreover, we notice that the height of peak Finally, we would like to point out that no matter how to tune the lattice amplitude though the Q-lattice background is dual to a deep insulating phase. It implies that in the absence of coupling, the holographic Q-lattice background itself is not able to drive fermionic probes to undergo a Mott transition. To model Mott physics, we shall introduce the dipole coupling term as proposed in [20]. The Mott transition geometry [20, 21] and other backgrounds [2227]. In addition, two bands are located in the positive frequency and negative frequency regions, respectively. Furthermore, we also show certain critical point pc, a gap opens and exists for all kx. The above observation strongly manifests that a transition occurs from a (non)-Fermi like phase to an insulating phase. To gain a better understanding of the spectral measure with dipole coupling on Q(black) and p = 4.5 (red). generation of a gap with the increase of the diploe coupling p. More importantly, as p increases, the spectral weight switches gradually from the positive frequency region to the negative frequency region. The dynamical generation and spectral weight transfer are two key features of doped Mott physics. Such observations are confirmed by checking other lattice fermionic system can be implemented by introducing a dipole coupling, which is supposed to play double roles of doping as well as on-site interaction like U in Hubbard model, as argued in RN-AdS geometry [20, 21]. Now, we turn to study the critical value pc of Mott transition, which reflects how easy it is to open the gap dynamically. In numerical calculations, the onset of gap can be phase than metallic phase. This observation can be further confirmed by evaluating the k1 = k2 = 0.8. value pc diminishes slightly with the increase of k, as opposite to that for k < kc. Subsequently, we shall study the features of Mott insulating phase over a Q-lattice the width of the gap becomes larger. It is just the effects of on-site interaction strength U of Hubbard model, which p plays. We also study the effects on the width of the gap of wave-number k and lattice ampliis shown in figure 7. Specifically, as k increases from small one, the width of the gap electron-electron correlations increases with the decrease of lattice constant [1519]. Interestingly enough, we find the values of kc falls into the critical regime of MIT in the phase diagram of the background, which is quite general. We expect to understand this anomaly with an analytical treatment in near future. The critical value kc is almost independent of the dipole coupling p but becomes larger with the increase of the lattice amplitude, as shown in figure 7 and table 3. In addition, it is interesting to notice that in the right plot of figure 7 all the lines almost intersect at one point of k ' 0.38. rises up at first and then tends to a constant or very slightly decreases, depending on the dipole coupling p. But for k > 0.38 the cases are just opposite. As a consistent check, one may find that in the right plot in figure 8, all the lines black hole background. The dynamics at different temperatures So far, we have only focused on the system at a very low temperature T ' 0.00398. Now, we turn to study the evolution of the spectral function with temperature. The temperature dynamics is one of important aspects of Mott insulators. A characteristic quantity is the approximately 20. Compared with the case of superconductor, in which the U(1) symmetry one of the unresolved puzzles with V O2. It indicates that the driven force to form the gap in Mott transition is strongly correlated. By holography, the authors in [21] show the holography can provide a good description on strong correlation induced transition. Here we shall address this issue on Q-lattice by holography. Figure 9 shows a 3D plot of the spectral function at T ' 0.115 and the DOS for different system with dipole coupling on Q-lattice, the temperature dynamics possesses non-trivial behavior, as revealed in RN-AdS black hole [21]. T 0.00398 instead of zero temperature gap. Fermi surface and gap on anisotropic Q-lattice In this subsection, we briefly discuss the Fermi surface and gap on anisotropic Q-lattice. we find that the Fermi peaks along kx direction develop into some bumps. It exhibits an anisotropic peaks with different magnitudes, which suggests that insulating phase arises in one direction while metallic in the other. Furthermore, when we switch on the dipole coupling p, the gap along kx direction gradually opens but the bump still remain along ky direction (figure 11). Finally, we would like to comment that the same statement is true for the conductivity on Q-lattice. For instance, since the lattice is introduced in just one direction, the Q-lattice geometries are highly anisotropic in [30], in which the calculation of the conductivity reveals that they can be insulators in the direction where the lattice is placed whilst remaining a pure metal in the other direction where the lattice is absence and the translational invariance is reserved. Therefore, the anisotropic geometry with insulating phase in one direction but metallic in the other provides us more space to model real materials with anisotropy by holography. Conclusion and discussion Our main results in this paper are: Two key features of doped Mott physics, the dynamical emergence of a gap and the spectral weight transfer, are observed in the Q-lattice background with dipole coupling. Such features have been observed in many of holographic fermionic systems including dipole coupling, e.g., RN-AdS geometry and other geometry. It confirms the robustness of the generations of Mott gap induced by dipole coupling, which play the double roles of doping as well as the interaction strength U in the Hubbard model. The most important thing is that the fermionic system with dipole coupling on Qlattice can exhibit abundant Mott physics due to the introduction of wave-number k The evolution of the spectral function as a function of temperature reveals the nontrivial temperature dynamical behavior, which indicates the Mott transition induced by dipole coupling on Q-lattice is due to strong correlations but not the spontaneous breaking of some symmetry. not generate the Mott gap. When dipole coupling p exceeds some critical value, the gap opens and, usually, the gap opens much easily in deep insulating phase. The anisotropic peaks with different magnitudes occurs in our holographic fermionic system on Q-lattice. It indicates that insulating and metallic phases arise in different directions, respectively. Still, a number of problems deserve further exploration. Rather than the dipole coupling, we may introduce other sorts of couplings to induce h.c may have a manifest periodic structure and generate Brillouin zones. Our work on this subject is under progress. By the poles and zeros duality through the detGR, the pseudo-gap phase can be observed in the holographic fermions with dipole coupling in RN-AdS geometry [50]. It is valuable to address this problem in the holographic Q-lattice model. It is interesting to study the Fermi arc phenomenology in the anisotropic Q-lattice Acknowledgments We are grateful to the anonymous referee for valuable comments. This work is supported by the Natural Science Foundation of China under Grant Nos.11275208, 11305018 and 11178002. Y.L. also acknowledges the support from Jiangxi young scientists (JingGang Star) program and 555 talent project of Jiangxi Province. J.P. Wu is also supported by Program for Liaoning Excellent Talents in University (No. LJQ2014123). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE]. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE]. [3] S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE]. Rev. D 79 (2009) 086006 [arXiv:0809.3402] [INSPIRE]. (2011) 065029 [arXiv:0903.2477] [INSPIRE]. [5] H. Liu, J. McGreevy and D. Vegh, Non-Fermi liquids from holography, Phys. Rev. D 83 (2014) 086 [arXiv:1409.2945] [INSPIRE]. [6] M. Cubrovic, J. Zaanen and K. Schalm, String Theory, Quantum Phase Transitions and the Emergent Fermi-Liquid, Science 325 (2009) 439 [arXiv:0904.1993] [INSPIRE]. [arXiv:1103.3982] [INSPIRE]. hole, Phys. Rev. D 84 (2011) 064008 [arXiv:1108.6134] [INSPIRE]. 867 (2013) 810 [arXiv:1203.0674] [INSPIRE]. the appearance of liquid phases in holographic theories with hyperscaling violation, JHEP 11 A 49 (1937) 72. charged dilatonic black hole, JHEP 01 (2012) 153 [arXiv:1111.3783] [INSPIRE]. Transition Metals, Proc. Phys. Soc. A 62 (1949) 416. 04 (2013) 073 [INSPIRE]. B 728 (2014) 450 [INSPIRE]. branes, JHEP 04 (2012) 068 [arXiv:1201.2485] [INSPIRE]. [23] J.-P. Wu, Emergence of gap from holographic fermions on charged Lifshitz background, JHEP [arXiv:1205.6674] [INSPIRE]. black hole, Class. Quant. Grav. 30 (2013) 145011 [arXiv:1210.5735] [INSPIRE]. potential and dipole coupling, Nucl. Phys. B 877 (2013) 807 [arXiv:1304.7431] [INSPIRE]. coupling, JHEP 11 (2011) 018 [arXiv:1110.4559] [INSPIRE]. [arXiv:1311.3292] [INSPIRE]. 007 [arXiv:1401.5077] [INSPIRE]. lattices, arXiv:1409.6875 [INSPIRE]. (2013) 649 [arXiv:1212.2998] [INSPIRE]. (2014) 101 [arXiv:1311.5157] [INSPIRE]. 181 [arXiv:1401.5436] [INSPIRE]. JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE]. arXiv:1410.6761 [INSPIRE]. JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE]. holographic systems with massless fermions and a dipole coupling, arXiv:1404.4010


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP12%282014%29149.pdf

Yi Ling, Peng Liu, Chao Niu, Jian-Pin Wu, Zhuo-Yu Xian. Holographic fermionic system with dipole coupling on Q-lattice, Journal of High Energy Physics, 2014, 149, DOI: 10.1007/JHEP12(2014)149