Entanglement String and Spin Liquid with Holographic Duality

We show that the quantum entanglement can be transmuted to a force using the holographic duality. First, we prove that there is an open string in the spectrum of the holographic fermion coupled with scalar. The string ends at two fermions and its tension vanishes in the limit of zero scalar condensation. We associate such string with a dimer and identify the scalar condensation as the degree of the dimerization. Together with divergently large entanglement entropy, the model is expected to describe the Spin Liquid. As a consistency check, we show that there is a Mott transition as the dimerization proceeds. We suggest that the string may be observed in an ARPES experiment of spin liquid or clean Dirac material as a tower of bands.


Introduction
Strong correlation gives many unexpected phenomena which are very interesting as well as useful. Mott insulator, Strange metal and High Tc superconductivity are some of examples. Most interesting observation is the unreasonably fast equilibration both in strange metal [1] and quark gluon plasma [2], which seems to request non-local force. In such system, the low energy spectrum is very different from that of high energy: QCD string [3,4] and fractionalized degrees in spin liquid [5][6][7] are examples. Finding such excitation often gives us a way to analyze a very difficult system in a simple way.
Recently, gravity dual description [8,9] attracted much attention as a tool [10,11] for strongly interacting system (SIS). The basic idea is to utilize the universality near the quantum critical point and its similarity with a black hole which shares the same scaling symmetry and the temperature. Since the entropy of black hole is given by its area [12] rather than the volume, we have a holographic relation [13,14] whose origin can be attributed to the equivalence principle: infinitely large number of metrics can be identified. From the SIS point of view the holographic relation comes because the configurations of SIS at different energy scales are glued together [15][16][17][18][19] by quantum entanglement to form one higher dimensional system. It is very intriguing to ask if there is a mechanism by which entanglements can give a sort of force.
Motivated with these questions, we studied the fermionic particle with a scalar coupling in the dual gravity setup. We found an exact two-point function and found that mass spectrum is that of open string excitation. Its string tension is given by the scalar condensation, therefore the string is tensionless in the zero condensation limit. The most intriguing part here is the origin of this stringy spectrum. For the hadronic physics, its appearance can be easily attributed to the QCD string, however, it is rather mysterious in the condensed matter because no confinement physics seems to be involved. Nevertheless, the appearance of massive string spectrum indicates that qualitatively similar physics is in action.
The open string should connect two fermions in the boundary which are the only object in our system, to form a dimer, the spin singlet bound state. In the absence of the scalar condensation, the string is massless so that there is no cost in forming a dimer between the far separated fermions. However, in its presence, the string has tension so that it should connect nearest two fermions to minimize its energy. In this picture, we suggest to call the new string as 'entanglement string', and the scalar condensation measures the degree of the dimerization of the system, equivalently the string condensation.
There can be a few possibilities with the condensation of the dimers: for example valence bond solid (VBS) or resonating valence bond (RVB) [5], because spin singlet condensation does not form any magnetization. See ref. [1,20] for review and references.
The key is the size of the entanglement entropy. Sometimes ago, Ryu and Takayanagi [21] showed that the presence of dual gravity request infinitely large entanglement entropy. The only system with dimerization with such large entanglement entropy is the RVB state whose fundamental aspect is also the presence of high degree of entanglement entropy [22]. Therefore it is natural to identify our system as the spin liquid rather than spin solid.
As a consistency check, we investigated the metal insulator transition, since the spin liquid is known to be an insulator. Can our holographic model show such transition? We will show that our system indeed has a Mott transition as we increase one or more of following three parameters: scalar condensation, temperature, chemical potential.
Finally we also discuss a few ways to observe the entanglement string, whose direct experimental test is the observation of the stringy spectrum.
2 Fermion in AdS 4 and string at the boundary In this paper we study the fermion dynamics with a real scalar Φ so that the fermion action is given by where D M = ∂ M + 1 4 ω abM Γ ab − iqA M is the covariant derivative in the asymptotically AdS 4 of radius L whose explicit form is given in the appendix. Φ is a scalar field whose dynamics is given by a free real scalar action with mass m 2 Φ L 2 = ∆(∆ − 3) = −2 in the same gravity background. If Φ (0) is a scalar field that couples with the boundary fermion bilinearχχ with dimension ∆ = 2, it is embedded in the bulk field Φ such that near boundary behavior of the latter is Notice that in the zero temperature zero chemical potential limit, the first two terms are two independent exact solutions of the equation of the scalar field equation in the probe limit. Since we are looking for spontaneously generated gap at zero temperature, we set the source Φ (0) = 0. Notice that the dimension of M is that of mass squared.
In the appendix, it is shown that the components of the Green function of fermions in the boundary theory are given by and G R 12 = G R 21 = 0 in the standard quantization. It can be easily checked that in the limit of M → 0, our result is reduced to that in ref. [23]. The spectral function is given by the imaginary part of the Green function, which can be easily recovered from Kramers-Kronig relation or by the prescription ω → ω + i0 + .
What is remarkable in this result is that the singularities of the Green function are given by those of the Gamma function which has poles at all non-positive integer so that the spectrum of the theory is which is nothing but the spectrum of the relativistic open string α m 2 n = (n + m + 1 2 ), (2.5) whose string tension T = 1/(2πα ) with α = 1 4M . Eq. (2.4) is the Regge trajectory with slope α and intercept α 0 = −m − 1/2. This is an analogous situation of the discovery of the string from the Veneziano amplitude [3], where the origin of the string could be attributed to the confined color flux tube later. In M → 0 limit, that is, in the tensionless limit, the whole tower of string spectrum is reduced to that of a massless particle. Notice that the lowest spectrum given by n = 0 shows that even for M = 0, we can still have the massless spectrum if m = −1/2. The same formula also shows that we get tachyonic spectra for m < −1/2. In this paper we consider only −1/2 ≤ m ≤ 0.

Entanglement string and dimers
The most intriguing question here is about the origin of the string. In fact, the presence of tower of spectrum has been appeared in the numerical plots of earlier literature, although its meaning has not been discussed. From the dual gravity point of view, it is a very special type of Kaluza-Klein (KK) spectrum of the bulk fermion. Usually the KK spectrum is quadratic in n, m 2 n = c.n 2 , because one of the momentum component is quantized. Here it is linear: the AdS has a character of box drived by the quadratic gravitational potential. Such geometric picture gives an intuition why we have Regge trajectory, nevertheless, it does not help to understand its origin from the view of the original strongly interacting fermions. Let's collect some relevant facts and proceed to identify the origin of the string in a few steps.
1. We assumed the duality between the original fermion χ and the bulk fermion ψ in AdS 4 . One should notice that the spectral function is that of χ, and so is the string.
The original fermion lost its character as an elementary excitation due to the interaction and became a D0 brane [24], an object whose dynamics is given by an open string attached to it. Summarizing, the fermion particle in AdS is the holographic image of the string in the strongly interacting fermion system at the boundary. See the figure 1(a).
2. What is the physical reality of the string in the original system? Since the open string should end somewhere, our string should connect two fermions. Such fermionstring-fermion system can be identified as a dimer, which is a spin singlet Einstein-Podolski-Rosen (EPR) pair that is used as a building block of RVB (resonating valence bond) state [5,6]. See figure 1(b). Also it is natural to interpret the Φ as the low energy description of the dimer where the separation of the two fermions can be neglected. Then, the condensation of the scalar should describe the degree of dimerization of the system. is somewhat analogous to 4 He-3 He mixture. Like the 3 He atoms, the green fermions can form a Fermi surface, and extension of the Luttinger argument to the present situation shows that the Fermi volume is exactly p [45, 53, 54]. However, unlike the 4 He-3 He mixture, superfluidity is not immediate, because of the close-packing constraint on the blue+green dimers; onset of superfluidity will require pairing of the green dimers, and will not be explored here. So the state obtained by resonating motion of the dimers in Fig. 9b Fig. 7a. Numerous later studies [50, 54, 57-63] described the resulting metallic state more completely in terms of the binding of holons and spinons, similar to the discussion above. These studies also showed the presence of the emergent gauge excitations in the metal.
Here, we can see the presence of emergent gauge fields, and associated low energy states sensitive to the topology of the manifold, in our simplified description here of the FL* metal. Indeed, such low energy states are required to evade the Luttinger theorem on the Fermi surface volume [45,46]. The FL* metal shares its topological features with corresponding insulating spin liquids, and we can transfer all of the arguments of Section 2 practically unchanged, merely by applying them to wavefunctions like Fig. 9b in a 'color blind' manner. So the arguments in Fig. 3 on the conservation of the number of valence bond across the cut modulo 2, and associated near-degeneracies on the torus, apply equally to the FL* wavefunction after counting the numbers of both blue and green dimers (see Fig. 10). The presence of these near-degenerate topological states is also crucial for the Luttinger-volume-violating Fermi surface. Oshikawa [64] presented a proof of the Luttinger volume in a Fermi liquid by considering the consequences of adiabatically inserting a fluxoid = h/e through a cycle of the torus, while assuming that the only low energy excitations on the torus are the quasiparticles around the Fermi surface. However, with the availibility of the low energy topological states discussed in Fig. 10, which are not related to quasiparticle excitations, it is possible to modify Oshikawa's proof, and obtain a Fermi volume different from the Luttinger volume [45, 46]; indeed a Luttinger volume of p holes appears naturally in many models, including the simple models discussed here. For the fermion bilinear, ∆ = d − 1.
6. There still are a few options for configuration of dimers: either it can have some order in their alignment called valence bond solid (VBS), or it can be symmetry unbroken state called resonating valence bond (RVB). Their main difference is the amount of the entanglement entropy [22]. The VBS can not be described by a scalar condensation because the rotation symmetry is broken. See figure 2(a,b).
According to Ryu-Takayanagi [26], any system with holographic description has divergently large entanglement entropy proportional to the interface area. Therefore we can identify the system of holographic fermions with scalar condensation as spin liquid or resonating valence bond (RVB), the system of dimer condensation with maximal entanglement. See figure 2(b)  appears as a low temperature instability of the pseudogap, so a theory of the high value of Tc can only appear after a theory of the PG metal. We begin our discussion by describing the simpler phases at the extremes of p in Fig. 1b. At (and near) p = 0, we have the antiferromagnet (AF) which is sketched in Fig. 2a. The Coulomb repulsion between the electrons keeps their charges immobile on the Cu lattice sites, so that each site has exactly one electron. The Coulomb interaction is insensitive to the spin of the electron, and so it would appear that each electron spin is free to rotate independently on each site. However, there are virtual 'superexchange' processes which induce terms in the effective Hamiltonian which prefer opposite orientations of nearest-neighbor spins, and the optimal state turns out to be the antiferromagnet (AF) sketched in Fig. 2a. In this state, the spins are arranged in a checkerboard pattern, so that all the spins in one sublattice are parallel to each other, and antiparallel to spins on the other sublattice. Two key features of this AF state deserve attention here: (i) The state breaks a global spin rotation symmetry, and essentially all of its low energy properties can be described by well-known quantum field theory methods associated with spontaneously broken symmetries. (ii) The wavefunction does not have long-range entanglement, and the exact many-electron wavefunction can be obtained by a series of local unitary transformations on the simple product state sketched in Fig. 2a.
At the other end of larger values of p, we have the Fermi liquid (FL) phase. This is a metallic state, in which the electronic properties are most similar to those of simple monoatomic metals like sodium or gold. This is also a quantum state without long-range entanglement, and the manyelectron wavefunction can be well-approximated by a product over single electron momentum eigenstates (Bloch waves); note the contrast from the AF state, where the relevant single-particle states were localized on single sites in position space. We will discuss some further important properties of the FL state in Section 3. Section 2 will describe possible insulating states on the square lattice, other than the simple AF state found in the cuprate compounds at p = 0. The objective here will be to introduce states with long-range quantum entanglement in a simple setting, and highlight their connection to emergent gauge fields. Then Section 4 will combine the descriptions of Sections 3 and 2 to propose a metallic state with long-range quantum entanglement and emergent gauge fields: the fractionalized Fermi liquid (FL*). Finally, in Section 5, we will review the evidence from recent experiments that the pseudogap (PG) regime of Fig. 1b is described by a FL* phase.
I also note here another recent review article [9], which discusses similar issues at a more specialized level aimed at condensed matter physicists. The gauge theories of the insulators discussed in Section 2 were reviewed in earlier lectures [10,11]. The message is that the presence of the holography which by itself is a consequence of entanglement, gives a set up where the quantum entanglement (QE) acts on a system as a force, the string tension, enforcing the system to realize the dimerization by nearest fermions. It also suggests that even without a strong correlation, a string is associated with EPR pair which resembles the statement 'ER=EPR' [16,29,30]. This may be useful to explain the non-local forces that seem to be responsible for the rapid thermalization of the strange metal [1] and quark gluon plasma [2].

More on experimental implementation
The most direct prediction of the theory is the existence of the string. Finding string means observing its spectrum given in figure 3. We expect that such tower of string spectrum may be found in the ARPES experiment of some Dirac materials with strong correlation.
Any Dirac material with small fermi surface, which is realized when the material is clean enough, is strongly interacting. Therefore we expect that the stringy spectrum can be observed in the experiment with clean Dirac material. Notice that the higher spectral tower with large n is suppressed only by 1/πn instead of 1/n!, which is naively  Finding such stringy spectrum would be extremely interesting, since it is an evidence of the holographic dual space as well as the entanglement string in real matter.

Mott transition
Now lets do some consistency check. The spin liquid is known to be a Mott insulator. Is  Fermi sea ω = 0 and gapped phase does not have any, pseudo gap means the depletion of DOS at the central peak. We choose the onset of pseudo gap as the 10% depletion.
We remark that a similar model with the Pauli interaction instead of scalar coupling [31] it was shown that the model exhibits the Mott transition [32] connecting the free fermion point [33][34][35] and gapped phase [36,37] under increasing the coupling strength.
To compare with our phase diagram with that of the Pauli interaction, we divide the gapless region into 4 subclasses: Fermi Liquid like (FL), Bad Metal (BM), Bad Metal prime (BM') and half Metal (hM). The characters of these phases are qualitatively similar to those of Pauli case and we refer the interested readers to the ref. [31]. From the fact that m = −1/2, M = 0 is the free fermion point and there is a large gapped phase region, any line connecting these two can realize the Mott transition. Furthermore, increasing M induces the transfer of the degree of freedom from the central peak to shoulder peak.
These features are common with the model with Pauli interaction studied in [31].
There is an important difference, however: while there is a Mott transition for any value of m as M varies for the scalar interaction, there is none for fixed m if |m| > 0.35 in the Pauli interaction case. Now we study the temperature and chemical potential dependence of the phase diagram. The figure 5(a,b) show how the m-M phase diagram(PD) changes as we change those parameters respectively. We can easily sea that as we increase temperature, the boundary of PG and Gapped phases is moving to the higher M region significantly, while the boundary of Gapless and PG phase does not move much. The chemical potential evolution shows similar behavior qualitatively. Phase boundaries move to the higher µ region as we increase µ. These are consistent with our expectations that Gap formation is harder at higher temperature or higher chemical potentials. We also draw µ-T phase diagram explicit in figure 5(c). It would be interesting if these diagrams can be compared with experimental data.

Summary and Discussion
In this paper we investigated the scalar coupling of the fermion. First, it is shown that two entangled fermions are connected by a string and the string tension is proportional to the scalar condensation, therefore for vanishing condensation, the string becomes tensionless and all spectrum becomes massless so that the string is invisible. Secondly, the scalar condensation is identified as the dimerization and the system is suggested to be describing the spin liquid. Thirdly, we found that there is a Mott transition as we change the scalar coupling.
The scalar field condensation in this paper is just one possible mechanism of gap generation and it is identified as dimerization of the system. It would be interesting if we can similarly identify other gap generating interactions as physical phenomena. In this regard, we mention that in ref. [31], we suggested that when the mass generation is associated with the Pauli term, such instability can be related to the "nesting" for the charge density wave. This means that the gap generated by the Pauli term is associated with the CDW order just as the gap generated by the Yukawa term is associated with dimerization.
We expect that for all gap generating mechanism, similar trajectory should appear although it may not be strictly a linear one. Also, notice that in this paper the scalar field is considered as a probe. It would be interesting to couple it with the gravity so that the condensation is determined including the gravity back-reaction.
The appearance of the stringy spectrum can be more universal than we expect because matter can become a SIS in many ways: first, by having a small Fermi surface [38][39][40] as in the Dirac materials which form rather wide class, second by having small Fermi velocity as in the case of all transition metal oxides. This is because the effective coupling inside matter is ∼ α/ v F , where is a screening constant. We wish that the entanglement string can be found in the spectrum of diverse material.