Universidade Federal do Rio Grande do Norte
Departamento de Física Teórica e Experimental
Programa de Pós-Graduação em Física
Gapless chiral spin liquid from a parton
mean-field theory on the kagome lattice
Fabrizio Giovanni Oliviero
Advisor: Prof. Dr. Rodrigo Gonçalves Pereira
Natal - RN April, 2020
Gapless chiral spin liquid
from a parton mean-field
theory on the kagome
lattice
Fabrizio Giovanni Oliviero
Master's thesis presented to the Graduate Program in Physics (PPGF) of the Federal University of Rio Grande do Norte (UFRN), as part of the requisites re-quired to obtain a Master's Degree in Physics.
Advisor: Prof. Dr. Rodrigo Gonçalves Pereira
Examination committee:
Prof. Dr. Rodrigo Gonçalves Pereira - UFRN (Advisor)
Prof. Dr. Álvaro Ferraz Filho - UFRN
Prof. Dr. Eric de Castro e Andrade - IFSC-USP
Natal - RN April, 2020
Oliviero, Fabrizio Giovanni.
Gapless chiral spin liquid from a parton mean-field theory on the kagome lattice / Fabrizio Giovanni Oliviero. - 2020.
81f.: il.
Dissertação (Mestrado) - Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Programa de Pós-Graduação em Física. Natal, 2020.
Orientador: Rodrigo Gonçalves Pereira.
1. Física Dissertação. 2. Quantum spin liquids
-Dissertação. 3. Parton construction - -Dissertação. 4. Kagome lattice - Dissertação. I. Pereira, Rodrigo Gonçalves. II. Título.
RN/UF/CCET CDU 53
Universidade Federal do Rio Grande do Norte - UFRN Sistema de Bibliotecas - SISBI
Catalogação de Publicação na Fonte. UFRN - Biblioteca Setorial Prof. Ronaldo Xavier de Arruda - CCET
Abstract
Quantum spin liquids are among the most studied phases of matter which depart from the paradigm of symmetry breaking order and Fermi-liquid theory. In this work, we investigated a sp1/2 Heisenberg model on the kagome lattice with exchange in-teraction Jd between spins connected by the diagonals of the hexagons and a staggered
chiral interaction Jχ that couples the three spins in the triangles. Applying parton
construction, we show that the model can exhibit a gapless chiral quantum spin liq-uid phase in the regime of dominant Jχ. The gapless phase is equivalent to the one
found in other mean- eld solutions based on Majorana fermions with symmetry pro-tected line Fermi surfaces. Our goal is to study the changes in the spectrum of the spinons as a function of the ratio Jd/|Jχ| for two dierent cases of a speci c ansatz
in the mean- eld theory. We expect that in the regime of relevant Jd, a non-coplanar
ordered magnetic state appears, the cuboc-2 state. The apparent phase transition is analyzed in terms of the mean- eld orders parameters and the search for insta-bilities in the generalized susceptibility. This work provides a unique example of a phase transition between a chiral spin liquid and a non-trivial ordered magnetic state.
Keywords: Quantum spin liquids, kagome lattice, parton construction, mean- eld, gapless chiral spin liquid
Natal - RN April, 2020
Resumo
Os líquidos quânticos de spin estão entre as fases da matéria mais investigadas que se afastam do paradigma da quebra simultânea de simetria e da teoria líquido de Fermi. Neste trabalho, estudamos um modelo de Heisenberg de spin 1/2 na rede kagomé, com interação de troca Jd entre os spins conectados pelas diagonais dos hexágonos e com
uma quiralidade escalar de uxos alternados Jχ que acopla os três spins dos
triângu-los. Aplicamos um campo médio baseado na representação de pártons e mostramos que o modelo apresenta um líquido de spin quiral sem gap no regime em que Jχ é a
interação dominante. A fase sem gap, é equivalente à outras fases que são obtidas a partir de soluções de campo médio utilizando representação de férmions de Majorana com superfícies de Fermi retilínia protegidas por simetria. Nosso objetivo é estudar as mudanças no espectro dos spínons em função da razão Jd/|Jχ| para dois casos de
um ansatz na teoria de campo médio. Esperamos que no limite em que Jd é o
acopla-mento relevante, temos uma fase magnética com ordenaacopla-mento não-coplanar; o estado cuboc-2. A transição de fase é analisada em termos dos parâmetros de ordem oriun-dos do campo médio e pela busca de instabilidades na susceptibilidade generalizada. Acreditamos que esse trabalho é importante porque fornece um exemplo de transição de fase entre um líquido de spin quiral e uma fase magnética ordenada não trivial.
Palavras-chave: Líquidos quânticos de spin, rede kagomé, construção de pártons, campo médio, líquido de spin chiral sem gap
Natal - RN April, 2020
Acknowledgments
First of all, I thank my advisor, Prof. Rodrigo Pereira, for his guidance, time and support provided to me during my Master's. I must say that our weekly meetings were very important to my evolution as a physicist because he always encouraged me to think from a dierent perspective, and this I think is the core of what a scientist must pursue. His professionalism will be always an example for me.
I also would like to thank the members of my examination committee, Prof. Álvaro and Prof. Eric for accepting my invitation and for the questions during my presentation, I am sure this process will improve my work and my skills as a physicist. I also thank Prof. Álvaro for the one-year course on many-body physics in 2018. The fact I took this course changed in many aspects the way of how I see condensed matter physics today.
I thank the whole faculty of the physics department in Natal at UFRN, I spent all my academic years working with these people and I am very grateful for their support. I wish to thank Prof. Luíz Felipe and Prof. Tomasso for writing recommendation letters for my P.h.D application abroad.
Here I thank not a person but a speci c place. I am very grateful for the time I lived in the 904 apartment, it was an extraordinary experience and I will never forget. I am very thankful for having a lot of friends, Pedro for being my roommate in the last six months of my Master's, The Backlands of butter which has been part of my days since the beginning of my undergraduate studies. The friends I made during my high school that still are part of my life, I am very thankfull to have you as friends. The clube do Fanfarrão" for providing me very fruitful discussions, and also for asking me out in complicated days.
There is no way that I end this list of acknowledgments, without showing how grateful I am for the immeasurable love of my mother Claudilene and my father Alberto, you are the main reason I keep myself motivated every day and never give up on my dreams.
Last but not less important, even under the hard times the science is going through in Brazil, I must acknowledge nancial support by Brazilian agency Conselho Nacional de Desenvolvimento Cientí co e Tecnológico (CNPq).
Epigraph
We should try to introduce our children to science today as a rebellion against poverty and ugliness and militarism and economic injustice. Freeman Dyson, The Scientist as Rebel
Contents
1 Introduction 1
1.1 Frustrated magnetism . . . 2
1.2 Kagome spin liquids . . . 5
1.3 Outline of this dissertation . . . 10
2 Mean- eld theory of spin liquids 13 2.1 Standard mean- eld in the Heisenberg model . . . 13
2.2 Parton construction . . . 15
2.3 U(1) gauge structure . . . 17
2.3.1 The 0- ux mean- eld state in the square lattice . . . 20
2.3.2 The π- ux mean- eld state in the square lattice . . . 21
2.4 Summarizing the general theory . . . 22
3 Gapless chiral spin liquid on the kagome lattice 23 3.1 The model Hamiltonian . . . 23
3.2 The Parton mean- eld Hamiltonian . . . 25
3.2.1 Scalar chirality operator . . . 25
3.2.2 The mean- eld ansatz . . . 27
3.2.2.1 Case I (0- ux in the trapezoids) . . . 31
3.2.2.2 Case II (π- ux in the trapezoids) . . . 37
3.2.2.3 Case III (±π/2- ux in the trapezoids) . . . 39
4 Generalized susceptibility and instability of the chiral spin liquid 42 4.1 Mean- eld spin correlation function . . . 42
4.2 The instabilities of C(q) in case I . . . 45
4.3 The instabilities of C(q) in case II . . . 49
4.4 Summarizing the chapter . . . 51
5 Self-consistency analysis 54 5.1 Self-consistency equations . . . 54 5.2 Mean- eld ground-state energy . . . 56 5.3 Other energies . . . 60
6 Conclusions and Outlook 62
List of Figures
1.1.1 Scheme of frustrated Ising spins on the triangular plaquette. . . 3 1.1.2 Kagome lattice. . . 3 1.1.3 In (a) we have the representation of the valence bond crystal on the
triangular lattice. (b) The superposition of VBC's is what we called the RVB state. We can have short-ranged RVB's states (b), or long-ranged RVB state (c). Extracted from reference [13] . . . 4 1.2.1 In (a) and (b) we have the two dierent chiralities in the q = 0 state.
The elipsoids indicate the zero modes present in such states. (c) We have the q =√3 ×√3state, the plus and minus corresponds to the chiralities on the triangles. Note that the unit cell size contain nine spins in this case. Extracted from [18]. . . 6 1.2.2 Spins orientation of the cuboc-1 phase in the cuboctahedron on the left,
and a representation of the unit cell on the right. Extracted from [25]. . 7 1.2.3 Cuboc-1 and cuboc-2 orientation of the spins vector. Extracted from [26] 7 1.2.4 INS from the spin excitations plotted in reciprocal space. Extracted
from [27] . . . 8 1.2.5 (a) Structure of herbertsmithite with Cu2+(brown) and Zn2+ (red). (b)
Mineral sample of herbertsmithite. Extracted from [28] . . . 8 1.2.6 Measure of the structure factor S(Q, E) using INS on kapellasite (left).
The authors found that short-correlations indicate the presence of peaks present on the cuboc-2 state (right). Extracted from [32] . . . 9 1.2.7 (a) Representantion of the main exchange interactions present in the
kagome pattern on kapellasite. (b) Image of kapellasite taken from scan-ning electron microscope (20 μm). (c) Octahedral con guration of the Cu and Zn ions. (d) In this material the kagome layers are weakly coupled. Extracted from [39] . . . 10
2.3.1 Scheme representing a loop around the elementary plaquette of the square lattice. . . 19 3.1.1 Representation of the exchange interactions in our lattice spin model on
the kagome lattice. . . 24 3.2.1 Scheme representing the Wilson loop over the triangle plaquette. . . 26 3.2.2 Mean- eld ansatz representation of the model Hamiltonian. (a) In this
case, we create 0- ux in the trapezoids formed by diagonal interactions. (b) Inverting the orientation of the arrows we create ±π- ux in the trapezoids. (c) Here we consider real hopping, thus the ux through the trapezoids is ±π/2. . . 29 3.2.3 Plot of the three bands of the spinon spectrum with tχ = 0.3and td= 0
(left). Contour plot of the E0(k) band, the dashed lines represent the
gapless lines which delimit the Fermi surface (right). The reciprocal lattice vectors are given by b1 = (2π, 2π/
√
3), b2 = (0, 4π/
√
3). . . 34 3.2.4 The dispersion relation obtained by the linear expansion of the Bloch
Hamiltonian. . . 35 3.2.5 Here we plotted the graphic of the functions that are the coe cients
which appears in Eq. (3.2.40). . . 36 3.2.6 Plot of the two bands above the apparent critical regime td/tχ' −1.45
(left). Contour plot of E−(k) with hoppings in the ratio td/tχ ' −1.45
(right). . . 36 3.2.7 Here we are able to determine the exact critical hopping td/tχ = −1.38.
Note that at this ratio the upper and lower bands touch the Fermi level E = 0. . . 37 3.2.8 Plot of the three bands above the apparent critical regime td/tχ ' 0.45
(left). Contour plot of E0(k)with hoppings above the critical ratio (right). 38
3.2.9 Expansion of the dispersion relation around Q1 for dierent values of
td/tχ with tχ = 1. We can see that at the critical ratio td/tχ = 0.44045
(red curve), the Fermi velocity (given by the slope of the curves) vanishes. 39 3.2.10Plot of the two bands at td/tχ= −1.45. . . 40
3.2.11Contour plot of E−(k) (left) and E+(k) (right) with hoppings in the
ratio td/tχ= −1.45. . . 41
4.4.1 Phase diagram of the extended Heisenberg model on the kagome lattice obtained by coupled chain construction. . . 52 4.4.2 Conjectured phase diagram for the Jd− Jχ model. . . 53
5.2.1 Contour plot of the ground-state energy as function of the order param-eters with Jd/ |Jχ| = 0.5. . . 56
5.2.2 Plot of EGS(ξd, −0.8) using Jd/ |Jχ| = 0.5. We can see that the global
minimum occurs with the order parameter given by ξd= 0. . . 57
5.2.3 Plot of EGS(ξd, ξχ) using Jd/ |Jχ| = 1.5, 2, 2.5. . . 57
5.2.4 Plot of EGS(ξd, ξχ) using Jd/ |Jχ| = 3.5, 4.5, 5.5. Clearly the
ground-state energy decreases in this interval of Jd/ |Jχ| . . . 58
5.2.5 Plot of td/tχ as function of Jd/ |Jχ| for the case I (td < 0). The dashed
line represents the critical ratio and its correspondent value of Jd/ |Jχ|. 59
5.2.6 Plot of td/tχ as function of Jd/Jχ for the case II (td> 0). . . 59
5.3.1 Classical energy of the cuboc-2 state with S = 1/2 (blue). Ground-state energy of the spin-1/2 chain (orange) . . . 61
Chapter
1
Introduction
It is important to do everything with passion, it embellishes life enor-mously. Lev Landau
For almost a half-century, the community of condensed matter physics believed that all phases of matter could be explained using two dierent theories. The rst is Landau's theory of symmetry breaking [1, 2]. This theory arose with the main goal to explain second-order phase transitions. The physical assumptions which lie behind Landau's work are the following:
The free energy must be an analytic function except at the critical temperature. The free energy must contain the symmetries of the Hamiltonian.
For example, if we are interested in the paramagnetic-ferromagnetic transition for the Ising model in three dimensions, which is one case of second-order phase transition, making the assumptions of Landau, we can expand the free energy of a magnetic system F (M, T )in the vicinity of the critical temperature (Tc), where (M) is the magnetization
and (T ) is the temperature. This approach gives us important quantities which are the critical exponents. These quantities are crucial to the study of phase transitions because they are deeply related to the concept of universality class. For example, let us consider the 3D Ising paramagnetic-ferromagnetic transition where we have the following scaling law
M ∼ (T − Tc)β.
In the picture of Landau, the correspondent critical exponent is β = 0.5. It turns out that this mean- eld critical exponent is not precisely correct. Later in 1971 with the famous work of Wilson and Fisher [3], a more accurate result was obtained β ' 0.326 for a dierent model but in the same universality class. Thus we can say that Landau's mean- eld approaches and the renormalization group methods constitute the framework of the symmetry-breaking order.
The second theory, which at that time explained phases of matter away from phase transitions, was Landau's Fermi-liquid theory [4]. This one was a theory of weakly interacting systems, and the concept of quasiparticles emerged. The Fermi-liquid the-ory presented very good results for metals if we compare theoretical predictions with experimental realizations. Together, the symmetry breaking and Fermi-liquid theory were for a long time the mechanisms of phenomena in condensed matter.
However, in 1982 a revolution in the area started with the discovery of the fractional quantum Hall eect (FQH) [5], wherein it was found that the emergent novel phase did not contain Fermi-liquid behavior. Furthermore, dierent types of fractional quantum Hall liquids can be obtained, even if they preserve the same symmetries, which goes against to the Landau symmetry breaking theory.
Then some years later, in 1989, Wen introduced the concept of topological order [6] aiming to explain the vacuum degeneracy of chiral spin states. Wen also used the concept of topological order to determine the ground-state degeneracy of the FQH [7]. Topological order is a dierent kind of order from the conventional order in Landau's symmetry breaking framework. It is highly related to ground-state degeneracy and fractional excitations that are described neither by bosonic or fermionic statistics. We know that among the systems that could present this exotic type of order are quantum spin liquids (QSL's). In this work, a speci c type of quantum spin liquid is the object of study. In the next section, we will discuss brie y what a quantum spin liquid is and in what systems this phase could appear as a possible ground-state.
1.1 Frustrated magnetism
One of the main ingredients which allow the emergence of novel phases in magnetism is frustration. Frustration can be realized in antiferromagnetic systems when mutual interactions cannot be satis ed simultaneously. For illustration, consider a triangu-lar lattice with Ising spins on each site upon antiferromagnetic interactions (see Fig. 1.1.1.). We see in this case that the ground-state is six-fold degenerate because we can align the rst two spins minimizing the classical energy, but the third spin cannot align at the same time to minimize its interactions with the other two. Moreover, this geometric frustration typical of Ising models can occur in other lattices, for example, the kagome lattice in Fig. 1.1.2 (kagome is a term from the pattern present in Japanese baskets), and all geometries that contain triangles as elementary plaquettes.
?
Figura 1.1.1: Scheme of frustrated Ising spins on the triangular plaquette.
Figure 1.1.2: Kagome lattice.
An experimental procedure for identifying frustrated magnets was proposed by Ramirez in 1994 [8]. The main idea is that we usually can t the high temperature magnetic susceptibility using the relation
χ(T ) ∝ 1
T − ΘCW
, (1.1.1)
where ΘCW is the Curie-Weiss temperature. In classical antiferromagnets where ΘCW <
0, we know that the ordering temperature is close to |ΘCW|, thus the parameter
λ = |ΘCW| /TN, with TN the Néel temperature, gives us a measure of the level of
frustration in the system. In other words, for classical antiferromagnets λ ∼ 1 whereas for frustrated systems such as spin liquids this parameter λ approaches in nity.
For the purpose of introducing the idea of spin liquids, rst let us start with the square lattice. The square lattice is an emblematic case because if we only consider nearest-neighbors for Heisenberg spin-1/2 antiferromagnet, we can minimize the energy of the ground-state considering the semiclassical con guration of the Néel state. Of course, if we add to the Heisenberg model further interactions, due to competitions of these interactions the system becomes frustrated and the Néel state is not the state that minimizes the energy. One can think a priori that the Néel state only serves to
pedagogical purposes, but we have a famous example where the Néel order can be visualized in nature, the cuprate La2CuO4 [9]. In the `50s, P. Anderson was
investigat-ing quantum models for magnetic systems such as the Heisenberg model in the square lattice. At that time he was interested in the role played by quantum uctuations and if the classical states survive upon these uctuations. One of his remarkable discover-ies was that applying spin-wave theory, and making quantum corrections to the order parameter, he veri ed that the Heisenberg model in the square lattice contains Néel order at T = 0 temperature [10]. It was really surprising that his predictions were veri ed in the La2CuO4 material.
Later, in 1973 Anderson, motivated by experiments in the TaS2 compound, which
could be modeled as a triangular lattice with spin-1/2, proposed an alternative ground-state antiferromagnetic Heisenberg model on the triangular lattice. Anderson had several in uences from Pauli's work on valence bonds, and also from the Heisenberg antiferromagnetic chain, which already had exact solution via Bethe's ansatz since 1931 [11]. The new state was a superposition of singlets in the triangular lattice that he called the resonance valence bond (RVB) [12]. This exotic state does not contain magnetic order even at T = 0, in some sense very similar to Bethe's uid present in one dimension. In Fig. 1.1.3 we have a representation of the RVB states.
Figure 1.1.3: In (a) we have the representation of the valence bond crystal on the triangular lattice. (b) The superposition of VBC's is what we called the RVB state. We can have short-ranged RVB's states (b), or long-ranged RVB state (c). Extracted from reference [13]
The RVB state is the fundamental concept behind the origin of the quantum spin liquids. Nevertheless, the idealization of RVB is not su cient to classify the true nature of spin liquids. In fact, it is a great challenge for the community to close the debate on what is exactly these exotics phases. In addition to the lack of magnetic order even at low temperatures, there are other features in the quantum spins liquids which can
give us a more precise de nition of this exotic phase. The de nition of spin liquids has changed in the last forty years after the work published by Anderson. After the publication of the famous paper from Kalmeyer and Laughlin [14], it became clear that fractionalization is one of the signatures present in QSL's [13, 15]. Fractionalization can be regarded as a consequence of the nature of the elementary excitations which are the spinons. The spinons carry spin-1/2 and neutral charge, but the eigenstates of the Hamiltonian contain integer values for the spins if we consider an even number of spins on the lattice. Another property very important to de ne QSL's is the massive quantum entanglement present in such states [16]. Here, by entanglement, we mean that the quantum state cannot be written as a product state even under local change of basis. For a system with two particles, this entanglement can be viewed as a superposition of the one-particle states, although for many-body systems this visualization of entangled states is not trivial.
There are many types of spin liquids states discussed in the literature (see table 1 in reference [16]), but loosely speaking, we can separate them in two classes; gapped or gapless. The gapped type contain short-ranged correlations and must present topolog-ical order. The gapless type has long-range correlations and the gapless behavior is not due to symmetry breaking mechanism. There are in the literature several materials that may contain spin liquid physics [13], but from now on we are going to focus on the kagome spin liquids (KSL's), which is the subject of this thesis. In the next section, we are going to discuss some of the very known states of KSL's and their experimental realizations.
1.2 Kagome spin liquids
As we discussed above, the kagome lattice is frustrated in the sense of what is the con guration of the spins that give us minimal ground-state energy. However, there are two Néel states in the spin-1/2 antiferromagnet in the Heisenberg model that are important because are selected via the mechanism of order from disorder [17]. These states are known as the q = 0 and the q = √3 ×√3. In Fig. 1.2.1 we have a representation of the orientation of the spins in both states. The q = 0 state preserves the size of the unit cell of the kagome lattice. Note that in each triangle the spins are oriented by 120◦ to each other, and we can obtain two states based on the choice of
the chirality on the triangles.
The q =√3 ×√3state, on the other hand, breaks the translation by the unit cell in the lattice. In this case, the unit cell contain three times the number of sites compared to the geometric unit cell. In both cases, we have an in nity degeneracy due to the fact that we can rotate simultaneously each spin without energy cost.
+ + + + + + + + + + -- -+ + + -+ + + + + -+ (a) (b) (c)
Figure 1.2.1: In (a) and (b) we have the two dierent chiralities in the q = 0 state. The elipsoids indicate the zero modes present in such states. (c) We have the q = √3 ×√3 state, the plus and minus corresponds to the chiralities on the triangles. Note that the unit cell size contain nine spins in this case. Extracted from [18].
In the presence of perturbations such as the Dzyaloshinski-Moriya (DM) interaction [19, 20] or further neighbors in the Heisenberg model, the ground-state degeneracy is lifted and a speci c order is selected. The type of order that appears can be visualized using inelastic neutron scattering due to formation of a sharp mode (magnon) [18]. This sharp mode has been observed in the jarosites, which are the composites where the iron ions sit on a kagome lattice, the measure which gives the intensity of the sharp modes can be viewed in reference [21]. Similar behavior has also been found in clinoatacamite [22]. In all these cases the spin is large S 1 and from the spin wave theory we expect the presence of magnons. However, in spin liquids we have low spin S = 1/2 thus quantum uctuations does not allow such type of order.
The Néel states discussed above, q = 0 and q = √3 × √3, are classical states where the direction of the spin vectors are con ned on the same plane, i.e, these states contain coplanar order. However, these states are not the only classical con gurations on the kagome lattice. It turns out, that we also can nd non-coplanar states if we consider further interactions in the Heisenberg model on the kagome lattice, which are known as the cuboc states. The cuboc states are divided into cuboc-1 and cuboc-2. The cuboc-1 was rst proposed in an antiferromagnet model for the kagome lattice with rst neighbor exchange interaction J1 and diagonal interaction across the hexagons Jd,
the study was perfomed applying exact diagonalization [23,24]. This emergent phase is characterized by a 12-sublattice non-coplanar magnetic ordering where the spins are pointing toward to the corners of the cuboctahedron, see Fig. 1.2.2.
Figure 1.2.2: Spins orientation of the cuboc-1 phase in the cuboctahedron on the left, and a representation of the unit cell on the right. Extracted from [25].
In the cuboc-2, we also have a 12-sublattice non-coplanar ordering, but here the coplanar spins live on the hexagons. In terms of physical quantities these two states dier from each other when we look for short-ranged correlations. In Fig. 1.2.3 we have a comparative of the unit cell of both phases. Now let me discuss the spin liquids that we know that may exist on the kagome lattice.
Figure 1.2.3: Cuboc-1 and cuboc-2 orientation of the spins vector. Extracted from [26]
In contrast to the sharp modes found in materials with large spin, in the kagome spin liquid material herbertsmithite, experiments using inelastic neutron scattering (INS) have been shown a broad continuum on the momentum structure (see Fig. 1.2.4) which we believe to be a signature of spinons that are the elementary excitations in spin liquids [27].
Figure 1.2.4: INS from the spin excitations plotted in reciprocal space. Extracted from [27]
Figure 1.2.5: (a) Structure of herbertsmithite with Cu2+(brown) and Zn2+ (red). (b)
Mineral sample of herbertsmithite. Extracted from [28]
The most remarkble model which we believe that describes herbertsmithite is the nearest-neighbor antiferromagnetic Heisenberg model with spin-1/2 on the kagome lattice. Hastings in 2001 [29] proposed an eective low energy theory, which is described by 4 avors of massless two-component Dirac fermions coupled to a U(1) lattice gauge eld. This type of spin liquid is known as the U(1) Dirac spin liquid or algebraic spin liquid, whose correlations has algebraic decaying laws hS(ri) · S(ri)i ∼ 1/r4. The
mean- eld ansatz in this theory is characterized by gauge uxes through the elementary plaquettes (zero- ux on the triangles and π- ux on the hexagons.) for this reason, the
spin liquid is also called as SL-[0,π]. There are several pieces of evidence that this eective model shows the low energy properties of the herbertsmithite [28, 30].
The structure of herbertsmithite is ZnCu3(OH)6Cl2 (Fig. 1.2.5), and there exist
at least four polymorphs which are present in Table I of reference [31]. Here in this dissertation, we will focus on the kapellasite (see Fig. 1.2.7 for more details).
Unlike herbertsmithite, kapellasite is a candidate for spin liquid material where the best eective model that describes the low energies properties still is on debate. In Fig. 1.2.6 we have the measurements using INS exhibiting a continuum of excitations that is a strong evidence for spinons.
Figure 1.2.6: Measure of the structure factor S(Q, E) using INS on kapellasite (left). The authors found that short-correlations indicate the presence of peaks present on the cuboc-2 state (right). Extracted from [32]
The most usual approach is to study an extended Heisenberg model (J1− J2− Jd)
based on predictions from the high-temperature series analysis for the exchange inter-actions [33]. There are many numerical studies using density matrix renormalization group (DMRG), quantum variational Monte Carlo and exact diagonalization methods [24,26, 34] probing the true nature of the grounds-state of the spin liquids that could be veri ed on kapellasite. In 2015 a study of the extended Heisenberg model using large-scale Monte Carlo [35] found that the ground-state of kapellasite exhibits a gap-less chiral spin liquid phase. Based on the regime of dominant Jdwhich describes three
sets of spin chains weakly coupled, applying DMRG another study [26] found a Valence bond crystal instead of the gapless chiral spin liquid.
Moreover, a recent study [36] on the extended Heisenberg model with staggered chiral interaction Jχ (the addition of Jχ was motivated by other work [37, 38].) on
the kagome lattice used weakly coupled chain construction to probe the global phase diagram that may be relevant for kapellasite. The method consists of evaluating the limit of decoupled chains, and upon perturbations, one applies renormalization group to get the low energy properties.
Figure 1.2.7: (a) Representantion of the main exchange interactions present in the kagome pattern on kapellasite. (b) Image of kapellasite taken from scanning electron microscope (20 μm). (c) Octahedral con guration of the Cu and Zn ions. (d) In this material the kagome layers are weakly coupled. Extracted from [39]
Among the spin liquids found in these earlier works, for the purpose of this disserta-tion the relevant phase is the chiral spin liquid (CSL) [40]. A chiral spin liquid is a type of spin liquid which is not fully symmetric, speci cally, because it breaks time-reversal symmetry in the sense that the scalar chirality expectation value is non-vanishing [40]. Also, the chiral spin liquid breaks some of the re ections symmetries present in the lattice model.
More information about experimental realizations of kagome spin liquids can be found in Refs. [28, 31, 39, 41]. In the next section of this introduction, we will summarize the main content of this dissertation chapter by chapter.
1.3 Outline of this dissertation
This dissertation is structured in the following manner: In Chapter 1, we gave an introduction of what is the nature of this novel phase that is the quantum spin liquid.
Also, the materials that could support such exotic phases were presented as motivation to study spin liquids, especially the kagome type.
Chapter 2 is devoted to the theoretical framework, thus this chapter de nes our notation and gives more details about the formalism that was utilized in this work. In particular, we will present the parton mean- eld decoupling that is an alternative to the conventional mean- eld approximations for spin systems. Afterward, we will make some statements on the emergent gauge symmetries discussing a little bit about what is the importance of the projective symmetry group in order to choose a mean- eld ansatz.
Chapter 3 presents the lattice model investigated in this work. Here, we studied the Heisenberg model accounting for interactions connecting only the diagonals of the hexagons in the kagome lattice (Jd> 0), and we also considered a chiral scalar
Hamiltonian where the interactions Jχ are present on the triangles. The sign of Jχ is
alternated depending on the orientation of the triangle (Jχ < 0 for up triangles and
Jχ > 0 for down triangles) and this property is important for the gapless chiral spin
liquid [37]. The Hamiltonian is given by the following:
H = Jd X ij∈7 S(ri) · S(rj) + Jχ X ijk∈MO S(ri) · [S(rj) × S(rk)] . (1.3.1)
We did a mean- eld approximation based on the partons method, where we made three possibles choices for the gauge uxes which characterize three dierent cases for the same ansatz.
The rst step was to analyze the dispersion relation of the spinons. In each case, we varied the value of the hoppings that appeared in the mean- eld Hamiltonian, and we saw what are the main changes in the Fermi surface that may be an indicative of phase transition. In the limit where we have only the chiral interactions, we know that our model describes a pure chiral spin liquid in the kagome lattice with symmetry protected gapless lines [38]. We expect that turning on Jdleads to the chiral spin liquid
to an ordered magnetic state cuboc. In the opposite limit of our model Jd |Jχ|, the
Hamiltonian describes three sets of spin chains weakly coupled and corresponds to the same phase found in [26]. In this chapter, we based the criterion for the phase transition on the evaluation of the dispersion relation checking if there is a critical regime of the mean- eld hoppings where the Fermi velocity vanishes.
In Chapter 4 we identify a mechanism to study the instabilities of the ansatz in the cases where we have quadratic and cubic dispersion for the spinons, the idea is the following: We must investigate if the generalized susceptibility presents instabilities on the same points where a band touching occurs in the dispersion relation of the spinons. We veri ed in the case with quadratic dispersion that the generalized susceptibility
presents a logarithmic divergence that usually can be interpreted as an instability in the magnetic ordering (this type of instabilities can be found in spin density waves in metals) [42]. For the case with cubic dispersion, we found an instability in the generalized susceptibility that diverges more rapidly than a logarithmic divergence. We also compared the values of momentum which produce instabilities in the susceptibility, with the ordering vectors associated with the cuboc phases in order to distinguish which cuboc phase appeared in our model.
In Chapter 5, we show preliminary results for the evaluation of self-consistent mean- eld equations that describes two dierent cases for our ansatz (we analyzed only imaginary hopping). Based on these results we found that when the order parameter is negative de nite the ground-state energy is smaller in comparison to positive case. Finally, in Chapter 6 we will conclude and review the main results obtained in this dissertation.
Chapter
2
Mean- eld theory of spin liquids
If science ceases to be a rebellion against authority, then it does not deserve the talents of our brightest children. Freeman Dyson, The Scientist as Rebel
In this chapter, we introduce the main techniques to describe quantum spin liquids at the mean- eld level. First we explain the reason why a standard mean- eld approach cannot be applied to QSL states. Then we present the parton construction as an alternative to solve this supposed problem. We nalize the chapter discussing some of the emergent symmetries which appears within the parton construction, and based on these symmetries (we used the projective symmetry group classi cation), we explain how to search for a mean- eld ansatz.
2.1 Standard mean- eld in the Heisenberg model
In an attempt to describe interacting electrons on a lattice, we start our discussion with the Hubbard model:
HHM = − X hiji tij c†αicαj + h.c. + UX i ni↑ni↓, (2.1.1) where cαi, c †
αi are operators of annihilation and creation of electrons respectively, thus
ni =Pαc†αicαi is the number of electrons in the site i and Ne =Pinithe total number
of electrons. The tunneling amplitude is given by tij and the one-site interaction U
represents the electron repulsion. It is well known that this Hamiltonian at half- lling in the insulating limit U/tij 1 on a two-dimensional lattice becomes a Heisenberg
model as follows [43]
H =X
hiji
where Si is the spin-1/2 operator represented by Pauli matrices and obeys the SU(2)
algebra {Si, Sj} = iijkSk. The Heisenberg antiferromagnetic exchange interaction is
given by Jij ∼ t2ij/U.
Since this Hamiltonian is not exactly solvable on general lattices, we perform a mean- eld approximation to obtain more qualitative results. Here we consider that the system has spontaneous magnetization even in the T = 0 limit. In practice, this implies a non-vanishing expectation value of Si. The rst step is to substitute the
spin operator by its average Si → hSii + δ(Si) where δ(Si) are possible quantum
uctuations. Thus the Hamiltonian (2.1.2) becomes:
H =X hiji Jij(hSii + δ(Si)) · (hSji + δ(Sj)) =X hiji JijhSii hSji + δ(Sj) hSii + δ(Si) hSji + O(δ2). (2.1.3)
If the uctuations are small we can ignore terms proportional to O(δ2). The second
step is to substitute δ(Si) = Si− hSii and we obtain
HM F =
X
hiji
Jij(SjhSii + SihSji − hSii hSji) . (2.1.4)
Now the Hamiltonian can be easily solved because it does not couple two spin operators. The mean- eld approximation must be self-consistent, therefore the expectation value hSiineeds to be chosen carefully such that it satis es the following condition:
hSii = hΦmean|Si|Φmeani , (2.1.5)
where |Φmeaniis the mean- eld wave function and also the ground-state of Eq. (2.1.4).
In general, we assume a speci c order that is represented by the form of the mean- eld state and we calculate the expectation value of the spin operator using this variational state.
Another standard procedure to describe antiferromagnetism is the semiclassical 1/S expansion, the main idea is to generalize the Heisenberg model for large spin S 1. Considering for example, the Néel state as the ground-state of a bibartite lattice, within this approximation, the spins operators are given in the Holstein-Primako representation, which is a bosonic form of the spin-S operator as follows:
SAi+ = q 2S − a†iai ai, SBj+ = b†j q 2S − b†jbj , SAi− = a†i q 2S − a†iai , SBj− = q 2S − b†jbj bj, (2.1.6)
where A, B are sublattices indices and for A we have hai, a†ii= δij. The z component
of spin Sz, we say that assume S value for the site which belongs to sublattice A and
−S for the site in the sublattice B. In the classical limit where the spin S is large, which means hSii
D
a†iaiE, we can expand the arguments of the square roots in representation (2.1.6) in powers of 1/S and the result for sublattice A for example is the following
SAiz = S − a†iai, SAi+ '√2Sai, SAi− ' a†i√2S. (2.1.7) However in spin liquids, the lack of magnetic order produces an expectation value hSii = 0, and we have in general low spin (S = 1/2). Another problem is that in some
cases, quantum corrections to the order parameter can diverge. In this case, we need another analytical method to describe phases without magnetic order of long-range. These two approaches are the core of what we call of standard mean- eld approximation for a spin system. The next section is dedicated to present an alternative mean- eld decoupling, the parton construction.
2.2 Parton construction
In 1973 P. W. Anderson proposed a new kind of insulator that he called resonating valence bond (RVB) [12]. This phase was an alternative to the Néel state as a ground state for spin-1/2 antiferromagnets. With this idea in mind, here we consider a spin liquid a system which is translationally invariant, contains rotational SU(2) symmetry, and as Anderson assumed in his work, here the system is insulating at half- lling. At that time, one of the main di culties to study QSLs was the fact that there is no magnetic ordering even at low temperatures. Hence if the starting point is a mean- eld theory, a standard approach is not appropriate, because in the case of quantum spin liquids we have hSii = 0. To solve this problem, in 1987 a dierent version of
mean- eld appears with the Parton construction1 [45,46, 47].
The parton construction is done by fractionalizing the spin operator into fermionic spinons (also called Abrikosov fermions) fiα, α = ↑, ↓, which are neutral fermions with
spin-1/2 and satisfy the following algebra nfiα† , fjβo= δijδαβ,
n
fiα, fjβo = 0. In this way, the spin operator is rewritten as
Si =
1 2f
†
iασαβfiβ, (2.2.1)
where σ = (σ1, σ2, σ3) are Pauli matrices, and i is a site index. Now we put this
representation into Hamiltonian (2.1.2) which gives us H = X αβα0β0 X ij 1 4Jij fiα† σαβfiβ ·fjα† 0σα0β0f jβ0 , = X αβα0β0 X ij 1 4Jijf † iαfiβf † jα0fjβ0(σαβ · σα0β0) . (2.2.2)
Applying the identity σαβ · σα0β0 = 2δαβ0δβα0− δαβδα0β0,and using the anticommutation
rules of the fermionics operators, we have the following fermionized Hamiltonian:
H =X αβ X ij −1 2Jijf † iαfjαf † jβfiβ + X α,β X ij Jij 1 2niα− 1 4niαnjβ . (2.2.3)
In Eq. (2.2.3) we have four states per site {|0i, f† ↑|0i, f † ↓ |0i, f † ↑f † ↓|0i}; whereas in
Eq. (2.1.2) we have two states per site {|↑i, |↓i}. This means that the new Hilbert space is larger than the original one. In order to recover physical states, that is, the states which live in the original Hilbert space, we need to impose the following constraints [45,46, 47, 48] X α fiα† fiα = 1, X αβ αβfiαfiβ = 0, (2.2.4)
where αβ is the antisymmetric tensor (↑↓ = 1).
A mean- eld theory in the scope of parton construction consists of replacing the constraint Pαf
†
iαfiα = 1 in Eq. (2.2.4) by its ground-state expectation value
X
α
D
fiα† fiαE= 1. (2.2.5)
This constraint is enforced in the Hamiltonian by inclusion of a time-independent Lagrangian multiplier a0(i). The second approximation is to replace the operator
fiα† fjα from (2.2.3) by its average X
α
D
fiα† fjαE= ξij. (2.2.6)
Within these approximations, the mean- eld Hamiltonian in this case becomes
HM F = X α X ij −1 2Jij h fiα† fjαξji + h.c. − |ξij|2 i + X α X i a0(i) fiα† fiα− 1. (2.2.7) The mean- eld value ξij and the Lagrangian multiplier a(i) specify the ansatz and must
satisfy the Eqs. (2.2.6) and (2.2.5) which are called the self-consistency equations. We can see that the Hamiltonian (2.2.7) is diagonal in the spins indices, which implies
that this mean- eld state described by the ansatz (ξij and a0(i)) is invariant under a
spin-rotation transformation. The mean- eld state is constructed such that represents a Fermi sea of the spinons, it means that we occupy the empty states creating a spinon up to the Fermi level. We must remember that the mean- eld state obtained from the parton-construction is not the eigenstate of the Heisenberg Hamiltonian, to obtain the physical spin wave function |Ψspini, we must project the mean- eld states |Ψmeani to
the subspace with one fermion per site. This is done by the action of the Gutzwiller projection operator [16, 49]
PG =
Y
i
(ni↑− ni↓)2, (2.2.8)
thus we have the following relation between the trial wave function and the real spin state:
|Ψspini = PG|Ψmeani . (2.2.9)
Untill this point, we ignored the quantum uctuations of a0(i) and ξij. In this
picture, the excitations of the Hamiltonian (2.2.7) are free spinons described by fiα.
This fact is interesting because we have a Heisenberg Hamiltonian (2.1.2) as a bosonic model with emergent fermion excitations. In the next section, we will discuss the role played by the U(1) gauge structure which is present in the parton construction. The gauge theory emerges when we consider the uctuations of the mean- eld ansatz.
2.3 U(1) gauge structure
The parton construction produces an enlarged Hilbert space. This leads to additional symmetries that are not present in the original spin system. One of these symmetries is the U(1) phase. To see the eects of this phase, we start the discussion by studying what happens if we consider the quantum uctuations. The rst of this uctuations is inserted by a time dependence of a0(i) → a0(i, t). To analyze these uctuations, let us
consider the imaginary-time path integral formulation. If we consider that the partition function is dominated by saddle points [49], the correspondent partition function after a Hubbard-Stratonovich transformation is given by
Z = ˆ D[f, ¯f ]D[a0(i)]Dξije− ´β 0 dτ(L− P
αa0(i,t)(f¯iα(t)fiα(t)−1)), (2.3.1)
where the Langrangian is
L =X
αi
¯
We can see from Eq. (2.3.1) that an integration over the time-dependent a0(i, t)
La-grangian multiplier leads to products of delta functions as follows:
Y i δ X α ¯ fiα(t)fiα(t) − 1 ! . (2.3.3)
The above result is essentially the constraints in Eq. (2.2.4). An integration over the auxiliary eld ξij will result in the exact spin Hamiltonian (2.2.3). Therefore in the
parton construction, the excitations above the mean- eld ground-state are described by these quantum uctuations of ξij and a0(i). Now if we take into account U(1)
gauge uctuations in the mean- eld ansatz 2 such as ξ
ij → e−iaijξij, we will obtain the
rst-order mean- eld Hamiltonian
HM F = X αβ X ij −Jij h fiα† fjαξjie−iaji + h.c. − |ξij|2 i − X α X i a0(i) fiα† fiα− 1. (2.3.4) Notice that if we make a gauge transformation such as
fiα → fiαeiθ(i), aij → aij+ θ(i) − θ(j), (2.3.5)
the mean- eld Hamiltonian remains unchanged. Instead of the free spinons of Eq. (2.2.7), in the rst-order case, the excitations are described by spinons coupled to the U (1)gauge eld. In the literature this gauge symmetry is also called gauge redundancy. This is because after the Gutzwiller projection the physical states do not contain the gauge symmetry, i.e., the two gauge equivalent states in the larger Hilbert space can-not be distinguished physically [50]. Moreover, the lattice gauge eld mediates the spinon-spinon interaction, but at the men- eld level these interactions are neglected. However, the U(1) gauge redundancy is not the only symmetry which appears in the parton construction. It turns out, that the spinon representation also contains another particle-hole redundancy, namely the SU(2) gauge symmetry [51].
The mean- eld theory is a self-consistent procedure. Even if we follow correctly all the steps explained above, such as checking self-consistency or analyze the quantum uctuations, still it is important to construct an ansatz which contains more physical motivation. In this section, we will elucidate the main idea and why the PSG is im-portant to choose a mean- eld ansatz. The projective symmetry group was introduced rst by Wen [50], the simplest de nition is that a PSG is a property of an ansatz. An element of a PSG we say that is composed by a symmetry transformation let us call U, and each element U is followed by a gauge transformation GU. In the U(1) gauge
2We neglected the amplitude uctuations because they are gapped. While gauge uctuations are
redundancy the ansatz in is speci ed by a complex number ξij. In that case the ansatz
invariance in the PSG is expressed as follows:
GUU (ξij) = ξij, GU ∈ U (1). (2.3.6)
The PSG contains a special subgroup which Wen called the invariant gauge group (IGG) denoted by G. This group is formed by all the pure gauge transformations that leaves ξij unchanged.
G = {Wi|WiξijW †
j = ξij, Wi ∈ U (1)}. (2.3.7)
The invariant gauge group G is responsible for the most general classi cation of spin liquids, basically dividing the spin liquids in three classes (within the fermionic formal-ism), U(1), SU(2), and Z2. The identi cation of the IGG for the U(1) case (when the
ansatz is speci ed by a complex number ξij ), can be done by the evaluation of the
gauge uxes through the elementary plaquettes of the lattice model. For example see the scheme for a square plaquette in Fig. 2.3.1.
i
j
l
k
ϕ
Figure 2.3.1: Scheme representing a loop around the elementary plaquette of the square lattice.
If we introduce a loop variable Pi on the path C clockwise oriented around the
plaquette, where the i index indicates the starting point of the loop, we have the following de nition for such variable
Pi = ξijξjkξklξli. (2.3.8)
Now let us assume that link variables ξij are phases, which in Fig. 2.3.1 are indicated
by the direction of the arrows in the links. We say that Pi relates to the ϕ- ux of the
plaquette by the following equation:
The relation above holds for ansätze that only contain hopping terms, the U(1) case, and this type of loop variable can be interpretated as the analogue of Wilson loops in a lattice gauge theory [52]. We can consider the ux per plaquette a gauge invariant, and into PSG the ux can be used to classify the mean- eld state. For the other gauge uxes, more information can be found in Refs. [49, 50,52].
2.3.1 The 0- ux mean- eld state in the square lattice
Let us discuss two mean- eld states in the square lattice as an application of the theoretical framework presented in this chapter. The rst case we are going to study is the 0- ux state. Consider the Heisenberg nearest-neighbor model on the square lattice, into parton construction the mean- eld Hamiltonian for this case is the following:
HM F = X α X <ij> −1 2Jij h fiα† fjαξji + h.c. − |ξij|2 i , (2.3.10) Jij =
J, if (i, j) are rst neighbors
0, otherwise. (2.3.11)
The 0- ux state is given by considering that the quantity ξij is a constant real number
that we are going to denote ξij = ξ. Hence the Hamiltonian describes a problem that
is an analogue of an electron hopping in the lattice. The diagonal Hamiltonian is given by: HM F = B.Z X k ε(k)fk†fk + J |ξ|2Ns, (2.3.12)
with the dispersion relation given by Eq. (2.3.13).
ε(k) = −J ξ [cos(kx) + cos(ky)] . (2.3.13)
Based on Eq. (2.3.13) we can see that the dispersion relation satis es the perfect nesting condition
ε(k + Q) = −ε(k), (2.3.14)
with Q = (π, π). Due to the presence of nesting in the Fermi surface, the density of states of the 0- ux ansatz diverges. This is an indicative that this mean- eld state is unstable against Néel order. The instability can be veri ed in the generalized suscep-tibility C(0)(Q). Where for this ordering vector Q = (π, π), C(0)(Q) diverges
logarith-mically. This criterion of instability for the ansatz is based on Stoner criterion for spin density waves [42]. In order to propose a stable ansatz, instead of real hopping, one can consider imaginary hopping.
2.3.2 The π- ux mean- eld state in the square lattice
The second mean- eld state we are going to discuss is the π- ux [47]. This ansatz produces π- ux through the square plaquettes, because around a plaquette the ux is given by Q ξij ∝ eiπ. The π- ux ansatz pattern doubles the size of the square lattice
unit cell. The dispersion relation for the π- ux is the following:
ε(k) = ±J ξ q
sin2(kx) + sin2(ky), (2.3.15)
The self-consistent order parameter ξ can be determined by minimizing the mean- eld ground-state energy. We consider that the system is at half- lling, thus the ground-state is obtained by occupying all the negatives states (ε = 0 is the Fermi level). Minimizing the ground-state energy we obtain
ξ = 1 4N X k q sin2(kx) + sin2(ky), (2.3.16) with −π 2 < kx < π 2 and −π < ky < π.
The Fermi surface at half- lling in the π- ux are points (kx, ky)at (0, 0) and (0, π).
An interesting feature of the π- ux is that at the Fermi surface, the elementary excita-tions are gapless spinons which correspond to particle-hole excitaexcita-tions across the Fermi points. In contrast with the 0- ux state, where the instability occurs due to nesting conditions in the Fermi surface, the instability of the π- ux occurs if we go beyond the mean- eld approximation and we consider the interaction of the spinons with the U(1) gauge eld [52]. However, the π- ux state is among the mean- eld ansätze that gives rise to translational symmetric spin liquids for the nearest-neighbor Heisenberg model with the lowest ground-state energy.
Another important spin liquid in the square lattice can be obtained analyzing the Heisenberg model with the addition of second-neighbors, the Heisenberg J1−J2. In this
case, the mean- eld ground-state breaks spontaneously the time-reversal symmetry and is called chiral spin liquid. The chiral spin liquid contains a non-vanishing expectation value of the scalar chirality operator, and also breaks some of the re exion symmetries present in the Heisenberg model. In addition to chiral spin liquids, there are many discussions on the stability of the mean- eld state against uctuations of the U(1) gauge eld, but in general we believe that chiral spin liquids are stable in such cases [53].
2.4 Summarizing the general theory
In this chapter, we saw how the Heisenberg model is decoupled at the mean- eld level in terms of spinons, and what are the consequences of this new representation to the Hilbert space. The emergent symmetries which appeared in the parton construction were also presented, and we explained the basic physical ideas which lie behind these symmetries. The projective symmetry group was the last tool of the general theory that we showed in the chapter. Basically, the most important idea is that in the projective symmetry group every symmetry transformation is followed by a gauge transformation, and that we can create an universality class with the ansätze sharing the same PSG. In addition to the general theory, we presented two mean- eld states as an example of how to look for ansätze based on the gauge uxes. The 0- ux state and π- ux state were brie y discussed, but we saw how the dispersion relation changes in each case due to the choice of gauge ux in the elemenatary plaquettes of the square lattice.
Therefore, in this introductory chapter, we introduced the fundamental concepts that will be used for the kagome spin liquid which will be presented in chapter 3. The brief review on parton construction and the PSG, it might be su cient to make the reader who is not familiarized with these topics to have a general point of view of the basic technics in this intriguing study of spin liquids. But yet, there are at least two reviews papers [16, 49] that were cited in this chapter, which may clarify technical details that are not present in this introduction. We also recommend the reading of chapter 9 from [52] and reference [50] for more details on Wen's work on the classi cation of fully symmetric spin liquids applying projective symmetry group. In chapter 3, we will introduce the model for the chiral kagome spin liquid, where this general formalism will be utilized in our mean- eld approximation.
Chapter
3
Gapless chiral spin liquid on the kagome
lattice
Há que ter o máximo de cuidado com aquilo que se julga saber, porque por detrás se encontra escondida uma cadeia interminável de incógnitas, a última das quais, provavelmente, não terá solução. José Saramago, Ensaio sobre a lucidez
This chapter is devoted to exploring three cases of the mean- eld ansatz which may describe the gapless chiral spin liquid phase on the Jd − Jχ kagome lattice model.
First, we show the model Hamiltonian which we are going to investigate at the mean- eld level. In this thesis, we studied a Heisenberg model on the diagonals across the hexagons with coupling Jdand we added a chiral three-spin interaction Jχwhich breaks
time-reversal symmetry simultaneously. Afterward, we present the main symmetries of our model and explain how they compare with the symmetries of other models studied in the literature. The general techniques we are going to apply were reviewed in the previous chapter. For completeness, we show some modi cations that are needed to include the chiral three-spin interaction in the mean- eld Hamiltonian. At the end of this chapter, we are going to comment on the main results in terms of the spinon dispersion relation.
3.1 The model Hamiltonian
Since we are considering a SU(2) symmetric spin system, the exchange interactions are the same for the three spin axis components. Thus the full Hamiltonian that describes the interactions in the model is the following:
H = Jd X ij∈7 S(ri) · S(rj) + Jχ X ij∈MO S(ri) · [S(rj) × S(rk)] . (3.1.1)
In Eq. (3.1.1) S(ri) is a spin-1/2 local operator and M O indicates a sum over all
triangles. Where Jd > 0 is motivated by kapellasite, Jχ < 0 for the up triangles and
Jχ > 0 for the down triangles. The chirality in the up triangles is clockwise oriented
and in the down triangles is anticlockwise. For that reason, this interaction is also called staggered interaction. In Fig. 3.1.1 we have a representation of the interactions that are present in our spin model.
Figure 3.1.1: Representation of the exchange interactions in our lattice spin model on the kagome lattice.
A natural question at this point is: Why not consider rst and second neighbors in the Heisenberg part of Eq. (3.1.1)? The answer is because in this dissertation we are interested in a minimal model which may explain the possible CSL in the kapellasite material, and it turns out that in this material are several pieces of evidence that Jd > J1 ∼ J2 [32, 33]. In the limit |Jχ| Jd, the Hamiltonian describes three sets of
decoupled spin chains oriented 120° with each other, and for that reason, must present a nearby one-dimensional behavior which agrees with the results obtained by [26]. Meanwhile, in the opposite limit where Jχis the dominant interaction, the Hamiltonian
describes a chiral spin liquid equivalent to the found in [37, 38]. In 2018, Pereira and Bieri [36] based on the extended Heisenberg model (J1− J2− Jd) with a staggered
chiral interaction Jχ found a gapless CSL within a weakly coupled-chain construction.
The authors argued that the novel phase was equivalent to the found by Bauer et al. [37] for another model of staggered chirality.
Let us discuss in more details the symmetries which are present in this Hamiltonian Eq. (3.1.1). First of all, we have a global SU(2) rotation symmetry, because both terms of the Hamiltonian are scalar quantities. We also know that time-reversal symmetry is explicitily broken when Jχ 6= 0. Moreover, this chiral term also breaks lattice rotation
of R(θ = π/3), this can be veri ed noting that this rotation changes up triangles into down triangles, and as the triangles contain dierent chiralities this rotation is not a symmetry of the Hamiltonian. This lack of rotation symmetry can be viewed from another perspective, we have a lattice re ection symmetry along the diagonals of the hexagons (σ), which is the same symmetry in references [36, 54], and in perpendicular directions to the diagonals the re exion symmetry (σ0
) is broken. The composition of the these two mirror symmetries is equivalent to a π/3-rotation (R(π/3) = σσ0), and
since one of them is broken it follows that R is broken. The rotation symmetry which remains is a 2π/3-rotation.
3.2 The Parton mean- eld Hamiltonian
In this section, we are going to decouple the lattice Hamiltonian Eq. (3.1.1) using the mean- eld approximation presented in chapter 2. We also introduce the ansatz that we argued to present a gapless CSL phase at the mean- eld level. At this point, we already know how the Heisenberg Hamiltonian is fermionized in the parton construction, but before we discuss the mean- eld procedure for our ansätze, we make some statements about the physical interpretation of the scalar chirality operator. After that, we must obtain the fermionized form of the three-spin interaction and nally to discuss the main results of this chapter.
3.2.1 Scalar chirality operator
The physical origin of the scalar chirality interaction Jχ also cames from the
one-band Hubbard model at half- lling. We know that performing pertubation theory in the insulating limit t U, at rst order in 1/U the Hubbard model resumes to the antiferromagnetic Heisenberg model. If we consider an external magnetic eld, and expanding up to second order in 1/U2, rather the Heisenberg interaction the Hubbard
model produces the scalar chirality with Jχ ∼ t
3sin(φ)
U2 [55]. Where φ is the magnetic
ux through the elementary plaquettes of the lattice. Moreover, this chiral interaction has been used to explain an enhanced thermal Hall response in the pseudogap phase in cuprates [56]. In this section, we are going to discuss the exact eigenstates of the three spin problem in the triangle.
Let us consider the chiral operator for a three spin system ˆ
E123= S1 · (S2× S3) . (3.2.1)
This operator was rst introduced by Wen, Wilczek, and Zee [40]. The authors pro-posed that such operator is an order parameter for spin states which breaks parity (P )
and time-reversal (T ) symmetries. This type of states is what they called chiral states. Notice that under time-reversal (T ) the spin operator changes as
T−1ST = −S, (3.2.2)
and under parity (P ), let us say we change the spin in site 2 by the spin in site 3, ˆE123
transform as
P−1Eˆ123P = S1· (S3× S2) ,
= − ˆE123. (3.2.3)
However, the combination of the two transformations (P T ) leaves the chiral operator unchanged.
Now in terms of the link operators ξ(l, m) = Pαf †
α(l)fα(m), with l, m = 1, 2, 3 site
indices, and assuming a triangle plaquete for a three spin-1/2 system, see Fig. 3.2.1, we have the following identity for the chiral operator [40, 57]
ˆ E123 = i 4[ξ(1, 2)ξ(2, 3)ξ(3, 1) − ξ(1, 3)ξ(3, 2)ξ(2, 1)] . (3.2.4)
1
2
3
ϕ
Figure 3.2.1: Scheme representing the Wilson loop over the triangle plaquette.
If we take the expectation value over a loop which passes through the three sites of the triangle, and assuming that the hoppings are such as ξ(l, m) = eiϕ/3ξ for all l, m,
we obtain D ˆE123E = i 4h[ξ(1, 2)ξ(2, 3)ξ(3, 1) − ξ(1, 3)ξ(3, 2)ξ(2, 1)]i , ∼ −i 4ξ 3(eiϕ− e−iϕ ) = −1 2ξ 3sin(ϕ), (3.2.5)
where ϕ is the total ux through the triangle. We can see that for two special uxes, 0 and π, the expectation value of the order parameter vanishes, hence these two cases are symmetric under time-reversal transformation.
Still on the three-spin problem in the triangle, we can nd the exact eigenstates of the Hamiltonian given by the chiral operator changing to the total spin basis. The total spin is given by
S = S1+ S2+ S3, (3.2.6)
and S commutes with the z-projection of the total spin Sz where its eigenvalue we
denote by Mz.Thus we can represent the eigenstates of the chiral operator as |S, Mz; χi,
where χ is the chirality eigenvalue. The total spin S can assume two values, 1/2 and 3/2. Let us consider the highest-weight state (|↑↑↑i). This state is invariant under any permutation of the spins, and since under permutation the chiral operator ˆE123
changes to − ˆE123, the eigenvalue χ for the state with maximum z-projection 32,32; 0 is zero. The multiplet with spin-3/2 can be determined by applying the lowering operator S− to
32,32; 0
. It turns out, that all the states in the same multiplet de ned by S, Mz the chirality eigenvalue is the same for all set of states. The sector with
spin S = 1/2, contains two orthogonal sectors with Mz = ±1/2 which dier by their
chirality eigenvalue. Consider the eigenstates of ˆE123: |χ+i , |χ−i which are given by:
|χ+i =
1 √ 3
|↑↑↓i + ei2π3 |↑↓↑i + e−i2π 3 |↓↑↑i , (3.2.7) |χ−i = 1 √ 3
|↑↑↓i + e−i2π3 |↑↓↑i + ei 2π 3 |↓↑↑i , (3.2.8) with eigenvalues χ± = ∓ √ 3 2 respectively.
Therefore, the nal message here is that the chiral interaction allow us to make two dierent choices of chirality. If we are interested in the positive chirality for example, we must set Jχ < 0 in order to obtain the lowest ground-state energy in our lattice
Hamiltonian. Henceforth this is going to be our convention, Jd > 0 and Jχ < 0 (up
triangles), Jχ > 0 (down triangles). Now we have discussed the main features of the
chiral operator, we are able to continue to the next step of the chapter which is the mean- eld ansatz.
3.2.2 The mean- eld ansatz
At this moment we introduce for the rst time the mean- eld ansatz. Here we are going to put in more details the representation of the parts of the Hamiltonian into parton construction. Then we will show the diagonalization procedure and nally we discuss what are the main results.
Let us consider the Heisenberg part of the Hamiltonian Eq.( 3.1.1)
Hd= Jd
X
ij∈7
S(ri) · S(rj). (3.2.9)
In the parton construction this Hamiltonian at the mean- eld level becomes (Eq. 2.2.7):
Hd= − Jd 2 X ij∈7 f†(ri)f (rj) ξji+ h.c − |ξij|2 , (3.2.10)
with f (rj) a spinon eld de ned by:
f (rj) =
f↑(rj)
f↓(rj)
!
, (3.2.11)
and ξji is the expectation value in the variational ground-state
ξij =f†(ri)f (rj) . (3.2.12)
The chiral Hamiltonian is given by:
Hχ = Jχ
X
ijk∈MO
S(ri) · [S(rj) × S(rk)] . (3.2.13)
The three-spin chiral Hamiltonian can be written using the identity Eq. (3.2.4) in terms of the spinons as follows
S1(ri) · (S2(rj) × S3(rk)) = i 4f † 3(rk)f1(ri)f † 1(ri)f2(rj)f † 2(rj)f3(rk) + h.c. (3.2.14)
Where the 1, 2, 3 represents the sublattices indices. Thus the chiral Hamiltonian in the fermionic representation is given by
Hχ= Jχ 4 X ijk∈MO if3†(rk)f1(ri)f1†(ri)f2(rj)f2†(rj)f3(rk) + h.c. . (3.2.15)
To perform the mean- eld decoupling of the Hamiltonian above, we need to substitute the link operators by their expectation values
fl†(ri)fm(rj) → ξ(l, m)ij+ δ(l, m)ij, (3.2.16)
Hamiltonian that will support our ansatz HM F = − Jd 2 X ij∈7 f†(ri)f (rj) ξji+ h.c. − |ξij|2 + Jχ 4 X ijk∈MO h if3†(rk)f1(ri)ξ(1, 2)ijξ(2, 3)jk+ if † 1(ri)f2(rj)ξ(2, 3)jkξ(3, 1)ki +if2†(rj)f3(rk)ξ(3, 1)jkξ(1, 2)ij − 2i (ξ31ξ12ξ23) 3 + h.c.i (3.2.17)
Let me now give a motivation for the ansatz we chose. As mentioned in Chapter 2, the projective symmetry group is an important tool on the searching of mean- eld ansätze for spin liquids. Here we do not have a fully symmetric spin liquid since the Hamiltonian breaks the P and T symmetries. Apparently, here we are not allowed to apply exactly the same method introduced by Wen in 2002 [50]. It turns out, in 2016, Bieri et al. [54] generalized the PSG proposed by Wen for systems which break such symmetries, in other words, chiral spin liquids. The authors constructed the possible choices of ansätze for the triangular and kagome lattice. They also gave a detailed prescription on how to construct mean- eld Hamiltonians and the associated wave functions for each representation class on these lattices. Our ansatz can be regarded as the No. 11 (imaginary diagonal hopping) and No. 9 (real diagonal hopping) in Table IX of reference [54]. All the symmetries present in our ansatz are basically the same. The three cases that we are going to investigate in our ansatz are represented in Fig. 3.2.2.
Figure 3.2.2: Mean- eld ansatz representation of the model Hamiltonian. (a) In this case, we create 0- ux in the trapezoids formed by diagonal interactions. (b) Inverting the orientation of the arrows we create ±π- ux in the trapezoids. (c) Here we consider real hopping, thus the ux through the trapezoids is ±π/2.
Now let us de ne the convention for the sign of ξij in order to distinguish the
with ξχ < 0. Where making a loop in the direction of the arrows, we assume positive
sign, and negative sign otherwise. Following our convention for the hoppings signs and orienting the loops clockwise, we can see that the ux through the up triangles is π/2 while in the down triangles we have a −π/2- ux per plaquette. Into this picture, this ansatz is usualy denominated as a staggered π/2- ux state. The ux through the hexagons is kept zero. To do so, we considered three dierent choices for the uxes through the trapezoids. The rst case we are going to study the spectrum is the orientation given in Fig. 3.2.2 (a) which creates a 0- ux per trapezoid. The second 3.2.2 (b), can be obtained basically inverting the direction of the arrows connecting the diagonals and gives us ±π- ux per trapezoid. The last, represented in Fig. 3.2.2 (c), we exclude the arrows in the links connecting the diagonals of the hexagons. This choice creates ±π/2- ux per trapezoid and we assume a real hopping ξij = ξd with ξd
a real number.
Case (a): In the case represented in Fig. 3.2.2 (a), making loops in the direction of the arrows ξij = iξd, otherwise ξij = −iξd. Here the mean- eld order parameter
is negative de nite ξd< 0.
Case (b): This case is represented in Fig. 3.2.2 (b), making loops in the direction of the arrows ξij = iξd, otherwise ξij = −iξd. Here the mean- eld order parameter
is positive de nite ξd> 0.
Case (c): We represent the ansatz in Fig. 3.2.2 (c), here we consider that ξij is
a real number thus ξij = ξd.
Since cases (a) and (b) preserve the particle-hole symmetry (inversion of the sign by conjugate complex transformation), the Lagrangian multipliers that enforce the single occupancy constraint in this ansatz are identically zero. Although we discuss the results of the spectrum in case (c), we will not focus in this case in the next chapters.
The kagome lattice can be interpreted as a triangular lattice with triangles at the corners. Considering that the side of the triangles is a = 1, we can use the following direct lattice vectors:
a1 = 2 (1, 0) , a2 = 2 −1/2,
√
3/2 . (3.2.18)
Given the direct lattice vectors we can de ne the nearest-neighbor vectors
δ1 = (1, 0) , δ2 = − 1 2, √ 3 2 ! , δ3 = − 1 2, − √ 3 2 ! . (3.2.19)
