diff --git a/_thesis/0_Preface/0.1_Aknowledgements.html b/_thesis/0_Preface/0.1_Aknowledgements.html index 13cca4a..9e7e6f8 100644 --- a/_thesis/0_Preface/0.1_Aknowledgements.html +++ b/_thesis/0_Preface/0.1_Aknowledgements.html @@ -39,34 +39,21 @@ image: -

I would like to thank my supervisor, Professor Johannes Knolle and -co-supervisor Professor Derek Lee for guidance and support during this -long process.

-

Dan Hdidouan for being an example of how to weather the stress of a -PhD with grace and kindness.

+

I would like to thank my supervisor, Professor Johannes Knolle and co-supervisor Professor Derek Lee for guidance and support during this long process.

+

Dan Hdidouan for being an example of how to weather the stress of a PhD with grace and kindness.

Arnaud for help and guidance…

-

Carolyn, Juraci, Ievgeniia and Loli for their patience and -support.

+

Carolyn, Juraci, Ievgeniia and Loli for their patience and support.

Nina del Ser

-

Brian Tam for his endless energy on our many many calls while we -served as joint Postgraduate reps for the department.

-

All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, -Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM -in Munich, Alex and Rohit.

-

Gino, Peru and Willian for their collaboration on the Amorphous -Kitaev Model.

+

Brian Tam for his endless energy on our many many calls while we served as joint Postgraduate reps for the department.

+

All the students in CMTH, Halvard, Tom, Chris, Krishnan, David, Tonny, Emanuele … and particularly to Thank you to the CMT group at TUM in Munich, Alex and Rohit.

+

Gino, Peru and Willian for their collaboration on the Amorphous Kitaev Model.

Mr Jeffries who encouraged me to pursue physics

All the gang from Munich, Toni, Mine, Mike, Claudi.

-

Dan Simpson, the poet in residence at Imperial and one of my -favourite collaborators during my time at Imperial.

-

Lou Khalfaoui for keeping me sane during the lockdown of March 2022. -Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design -expertise and patience.

+

Dan Simpson, the poet in residence at Imperial and one of my favourite collaborators during my time at Imperial.

+

Lou Khalfaoui for keeping me sane during the lockdown of March 2022. Sophie Nadel, Julie Ketcher and Kim ??? for their graphic design expertise and patience.

All the I-Stemm team, Katerina, Jeremey, John, ….

-

And finally, I’d like the thank the staff of the Camberwell Public -Library where the majority of this thesis was written.

-

We thank Angus MacKinnon for helpful discussions, Sophie Nadel for -input when preparing the figures.

+

And finally, I’d like the thank the staff of the Camberwell Public Library where the majority of this thesis was written.

+

We thank Angus MacKinnon for helpful discussions, Sophie Nadel for input when preparing the figures.

diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index 2f5a923..6dc4b9c 100644 --- a/_thesis/1_Introduction/1_Intro.html +++ b/_thesis/1_Introduction/1_Intro.html @@ -28,13 +28,10 @@ image:
@@ -48,13 +45,10 @@ id="toc-quantum-spin-liquids">Quantum Spin Liquids -
- - +Figure 3: From  [44]. +
-
-

Spin-orbit coupling is a relativistic effect, that very roughly -corresponds to the fact that in the frame of reference of a moving -electron, the electric field of nearby nuclei look like magnetic fields -to which the electron spin couples. This effectively couples the spatial -and spin parts of the electron wavefunction, meaning that the lattice -structure can influence the form of the spin-spin interactions leading -to spatial anisotropy. This anisotropy will be how we frustrate the Mott -insulators  [48,49]. As we saw with the Hubbard model, -interaction effects are only strong or weak in comparison to the -bandwidth or hopping integral \(t\) so -what we need to see strong frustration is a material with strong -spin-orbit coupling \(\lambda\) -relative to its bandwidth \(t\).

-

In certain transition metal based compounds, such as those based on -Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling -and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with -strongly anisotropic spin-spin couplings. These transition metal -compounds, known Kitaev Materials, draw their name from the celebrated -Kitaev Honeycomb Model which is expected to model their low temperature -behaviour  [44,50–53].

-

At this point we can sketch out a phase diagram like that of fig. 3. When both electron-electron -interactions \(U\) and spin-orbit -couplings \(\lambda\) are small -relative to the bandwidth \(t\) we -recover standard band theory of band insulators and metals. In the upper -left we have the simple Mott insulating state as described by the -Hubbard model. In the lower right, strong spin-orbit coupling gives rise -to Topological insulators (TIs) characterised by symmetry protected edge -modes and non-zero Chern number. Kitaev materials occur in the region -where strong electron-electron interaction and spin-orbit coupling -interact. See  [54] for a much more expansive version of -this diagram.

-

The Kitaev Honeycomb model  [55] was the first concrete spin model -with a QSL ground state. It is defined on the two dimensional honeycomb -lattice and provides an exactly solvable model that can be reduced to a -free fermion problem via a mapping to Majorana fermions. This yields an -extensive number of static \(\mathbb -Z_2\) fluxes tied to an emergent gauge field. The model is -remarkable not only for its QSL ground state but also for its -fractionalised excitations with non-trivial braiding statistics. It has -a rich phase diagram hosting gapless, Abelian and non-Abelian -phases  [56] and a finite temperature phase -transition to a thermal metal state  [57]. It been proposed that its -non-Abelian excitations could be used to support robust topological -quantum computing  [58–60].

-

As Kitaev points out in his original paper, the model remains -solvable on any tri-coordinated \(z=3\) -graph which can be 3-edge-coloured. Indeed many generalisations of the -model to   [61–65]. -Notably, the Yao-Kivelson model  [66] -introduces triangular plaquettes to the honeycomb lattice leading to -spontaneous chiral symmetry breaking. These extensions all retain -translation symmetry, likely because edge-colouring and finding the -ground state become much harder without it. Finding the ground state -flux sector and understanding the QSL properties can still be -challenging  [67,68]. Undeterred, -this gap lead us to wonder what might happen if we remove translation -symmetry from the Kitaev Model. This might would be a model of a -tri-coordinated, highly bond anisotropic but otherwise amorphous -material.

-

Amorphous materials do no have long-range lattice regularities but -covalent compounds can induce short-range regularities in the lattice -structure such as fixed coordination number \(z\). The best examples being amorphous -Silicon and Germanium with \(z=4\) -which are used to make thin-film solar cells  [69,70]. -Recently is has been shown that topological insulating (TI) phases can -exist in amorphous systems. Amorphous TIs are characterized by similar -protected edge states to their translation invariant cousins and -generalised topological bulk invariants  [71–77]. However, research on amorphous -electronic systems has been mostly focused on non-interacting systems -with a few exceptions, for example, to account for the observation of -superconductivity  [78–82] in amorphous materials or very -recently to understand the effect of strong electron repulsion in -TIs  [83].

-

Amorphous magnetic systems has been investigated since the -1960s, mostly through the adaptation of theoretical tools developed for -disordered systems  [84–87] and with -numerical methods  [88,89]. -Research on classical Heisenberg and Ising models has been shown to -account for observed behaviour of ferromagnetism, disordered -antiferromagnetism and widely observed spin glass behaviour  [90]. -However, the role of spin-anisotropic interactions and quantum effects -in amorphous magnets has not been addressed. It is an open question -whether frustrated magnetic interactions on amorphous lattices can give -rise genuine quantum phases, i.e. to long-range entangled quantum spin -liquids (QSL)  [91–94].

-

In Chapter 4 I will introduce the Amorphous Kitaev model, a -generalisation of the Kitaev honeycomb model to random lattices with -fixed coordination number three. We will show that this model is a -soluble chiral amorphous quantum spin liquid. The model retains its -exact solubility but, as with the Yao-Kivelson model  [66], -the presence of plaquettes with an odd number of sides leads to a -spontaneous breaking of time reversal symmetry. We will confirm prior -observations that the form of the ground state can be written in terms -of the number of sides of elementary plaquettes of the model  [64,95]. We unearth a rich phase diagram -displaying Abelian as well as a non-Abelian chiral spin liquid phases. -Furthermore, I show that the system undergoes a finite-temperature phase -transition to a conducting thermal metal state and discuss possible -experimental realisations.

-

The next chapter, Chapter 2, will introduce some necessary background -to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and -localisation. Then Chapter 3 introduces and studies the Long Range -Falikov-Kimball Model in one dimension while Chapter 4 focusses on the -Amorphous Kitaev Model.

-

Next Chapter: 2 -Background

+

Spin-orbit coupling is a relativistic effect, that very roughly corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei look like magnetic fields to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy. This anisotropy will be how we frustrate the Mott insulators  [48,49]. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral \(t\) so what we need to see strong frustration is a material with strong spin-orbit coupling \(\lambda\) relative to its bandwidth \(t\).

+

In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour  [44,50–53].

+

At this point we can sketch out a phase diagram like that of fig. 3. When both electron-electron interactions \(U\) and spin-orbit couplings \(\lambda\) are small relative to the bandwidth \(t\) we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See  [54] for a much more expansive version of this diagram.

+

The Kitaev Honeycomb model  [55] was the first concrete spin model with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases  [56] and a finite temperature phase transition to a thermal metal state  [57]. It been proposed that its non-Abelian excitations could be used to support robust topological quantum computing  [58–60].

+

As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated \(z=3\) graph which can be 3-edge-coloured. Indeed many generalisations of the model to   [61–65]. Notably, the Yao-Kivelson model  [66] introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry, likely because edge-colouring and finding the ground state become much harder without it. Finding the ground state flux sector and understanding the QSL properties can still be challenging  [67,68]. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev Model. This might would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.

+

Amorphous materials do no have long-range lattice regularities but covalent compounds can induce short-range regularities in the lattice structure such as fixed coordination number \(z\). The best examples being amorphous Silicon and Germanium with \(z=4\) which are used to make thin-film solar cells  [69,70]. Recently is has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterized by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants  [71–77]. However, research on amorphous electronic systems has been mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity  [78–82] in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs  [83].

+

Amorphous magnetic systems has been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems  [84–87] and with numerical methods  [88,89]. Research on classical Heisenberg and Ising models has been shown to account for observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour  [90]. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed. It is an open question whether frustrated magnetic interactions on amorphous lattices can give rise genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL)  [91–94].

+

In chapter 4 I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model  [66], the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model  [64,95]. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.

+

The next chapter, Chapter 2, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then chapter 3 introduces and studies the Long Range Falikov-Kimball Model in one dimension while chapter 4 focusses on the Amorphous Kitaev Model.

+

Next Chapter: 2 Background

Bibliography

-
-
[1]
A. -J. King and D. J. Sumpter, Murmurations, Current Biology -22, R112 (2012).
+
+
[1]
A. J. King and D. J. Sumpter, Murmurations, Current Biology 22, R112 (2012).
-
-
[2]
M. -Ballerini et al., Interaction Ruling Animal -Collective Behavior Depends on Topological Rather Than Metric Distance: -Evidence from a Field Study, Proceedings of the National -Academy of Sciences 105, 1232 (2008).
+
+
[2]
M. Ballerini et al., Interaction Ruling Animal Collective Behavior Depends on Topological Rather Than Metric Distance: Evidence from a Field Study, Proceedings of the National Academy of Sciences 105, 1232 (2008).
-
-
[3]
P. -W. Anderson, More Is -Different, Science 177, 393 (1972).
+
+
[3]
P. W. Anderson, More Is Different, Science 177, 393 (1972).
-
-
[4]
S. -Kivelson and S. A. Kivelson, Defining Emergence -in Physics, Npj Quant Mater 1, 1 (2016).
+
+
[4]
S. Kivelson and S. A. Kivelson, Defining Emergence in Physics, Npj Quant Mater 1, 1 (2016).
-
-
[5]
W. -M. Saslow, A History of -Thermodynamics: The Missing Manual, Entropy (Basel) -22, 77 (2020).
+
+
[5]
W. M. Saslow, A History of Thermodynamics: The Missing Manual, Entropy (Basel) 22, 77 (2020).
-
-
[6]
D. -Flamm, History and -Outlook of Statistical Physics, arXiv:physics/9803005.
+
+
[6]
D. Flamm, History and Outlook of Statistical Physics, arXiv:physics/9803005.
-
-
[7]
N. -W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, -Rinehart and Winston, 1976).
+
+
[7]
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).
-
-
[8]
E. -Ising, Beitrag zur -Theorie des Ferromagnetismus, Z. Physik 31, -253 (1925).
+
+
[8]
E. Ising, Beitrag zur Theorie des Ferromagnetismus, Z. Physik 31, 253 (1925).
-
[9]
L. -D. Landau and E. M. Lifshitz, Fluid Mechanics: Landau and Lifshitz: -Course of Theoretical Physics, Volume 6, Vol. 6 (Elsevier, -2013).
+
[9]
L. D. Landau and E. M. Lifshitz, Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics, Volume 6, Vol. 6 (Elsevier, 2013).
-
-
[10]
G. -Jaeger, The -Ehrenfest Classification of Phase Transitions: Introduction and -Evolution, Arch Hist Exact Sc. 53, 51 -(1998).
+
+
[10]
G. Jaeger, The Ehrenfest Classification of Phase Transitions: Introduction and Evolution, Arch Hist Exact Sc. 53, 51 (1998).
-
-
[11]
F. -Bloch, Über die -Quantenmechanik der Elektronen in Kristallgittern, Z. Physik -52, 555 (1929).
+
+
[11]
F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Physik 52, 555 (1929).
-
-
[12]
S. -Blundell, Magnetism in Condensed Matter (OUP Oxford, -2001).
+
+
[12]
S. Blundell, Magnetism in Condensed Matter (OUP Oxford, 2001).
-
-
[13]
+
+
[13]
X.-G. Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford University Press, Oxford, 2007).
-
-
[14]
E. -Morosan, D. Natelson, A. H. Nevidomskyy, and Q. Si, Strongly Correlated -Materials, Adv. Mater. 24, 4896 (2012).
+
+
[14]
E. Morosan, D. Natelson, A. H. Nevidomskyy, and Q. Si, Strongly Correlated Materials, Adv. Mater. 24, 4896 (2012).
-
-
[15]
J. -Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic Theory of -Superconductivity, Phys. Rev. 106, 162 -(1957).
+
+
[15]
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic Theory of Superconductivity, Phys. Rev. 106, 162 (1957).
-
-
[16]
D. -E. Feldman and B. I. Halperin, Fractional Charge and -Fractional Statistics in the Quantum Hall Effects, Rep. Prog. -Phys. 84, 076501 (2021).
+
+
[16]
D. E. Feldman and B. I. Halperin, Fractional Charge and Fractional Statistics in the Quantum Hall Effects, Rep. Prog. Phys. 84, 076501 (2021).
-
-
[17]
N. -F. Mott, The -Basis of the Electron Theory of Metals, with Special Reference to the -Transition Metals, Proc. Phys. Soc. A 62, 416 -(1949).
+
+
[17]
N. F. Mott, The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals, Proc. Phys. Soc. A 62, 416 (1949).
-
-
[18]
M. -P. A. Fisher, Mott -Insulators, Spin Liquids and Quantum Disordered -Superconductivity, in Aspects Topologiques de La Physique -En Basse Dimension. Topological Aspects of Low Dimensional Systems, -edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69 -(Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp. -575–641.
+
+
[18]
M. P. A. Fisher, Mott Insulators, Spin Liquids and Quantum Disordered Superconductivity, in Aspects Topologiques de La Physique En Basse Dimension. Topological Aspects of Low Dimensional Systems, edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David, Vol. 69 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp. 575–641.
-
-
[19]
J. -H. de Boer and E. J. W. Verwey, Semi-Conductors with -Partially and with Completely Filled 3d-Lattice Bands, Proc. -Phys. Soc. 49, 59 (1937).
+
+
[19]
J. H. de Boer and E. J. W. Verwey, Semi-Conductors with Partially and with Completely Filled 3d-Lattice Bands, Proc. Phys. Soc. 49, 59 (1937).
-
-
[20]
N. -F. Mott and R. Peierls, Discussion of the -Paper by de Boer and Verwey, Proc. Phys. Soc. -49, 72 (1937).
+
+
[20]
N. F. Mott and R. Peierls, Discussion of the Paper by de Boer and Verwey, Proc. Phys. Soc. 49, 72 (1937).
-
-
[21]
J. -G. Bednorz and K. A. Müller, Possible highTc -Superconductivity in the Ba−La−Cu−O System, Z. Physik B - -Condensed Matter 64, 189 (1986).
+
+
[21]
J. G. Bednorz and K. A. Müller, Possible highTc Superconductivity in the Ba−La−Cu−O System, Z. Physik B - Condensed Matter 64, 189 (1986).
-
-
[22]
P. -A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott Insulator: -Physics of High-Temperature Superconductivity, Rev. Mod. Phys. -78, 17 (2006).
+
+
[22]
P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott Insulator: Physics of High-Temperature Superconductivity, Rev. Mod. Phys. 78, 17 (2006).
-
-
[23]
M. -C. Gutzwiller, Effect of Correlation -on the Ferromagnetism of Transition Metals, Phys. Rev. Lett. -10, 159 (1963).
+
+
[23]
M. C. Gutzwiller, Effect of Correlation on the Ferromagnetism of Transition Metals, Phys. Rev. Lett. 10, 159 (1963).
-
-
[24]
J. -Kanamori, Electron -Correlation and Ferromagnetism of Transition Metals, Progress -of Theoretical Physics 30, 275 (1963).
+
+
[24]
J. Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Progress of Theoretical Physics 30, 275 (1963).
-
-
[25]
Hubbard, J., Electron Correlations in -Narrow Energy Bands, Proceedings of the Royal Society of -London. Series A. Mathematical and Physical Sciences -276, 238 (1963).
+
+
[25]
Hubbard, J., Electron Correlations in Narrow Energy Bands, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238 (1963).
-
-
[26]
J. -Hubbard and B. H. Flowers, Electron Correlations in -Narrow Energy Bands III. An Improved Solution, Proceedings of -the Royal Society of London. Series A. Mathematical and Physical -Sciences 281, 401 (1964).
+
+
[26]
J. Hubbard and B. H. Flowers, Electron Correlations in Narrow Energy Bands III. An Improved Solution, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 281, 401 (1964).
-
-
[27]
N. -Mott, Metal-Insulator -Transitions (CRC Press, London, 1990).
+
+
[27]
N. Mott, Metal-Insulator Transitions (CRC Press, London, 1990).
-
-
[28]
K. -T. Law and P. A. Lee, 1t-TaS2 as a Quantum Spin -Liquid, Proceedings of the National Academy of Sciences -114, 6996 (2017).
+
+
[28]
K. T. Law and P. A. Lee, 1t-TaS2 as a Quantum Spin Liquid, Proceedings of the National Academy of Sciences 114, 6996 (2017).
-
-
[29]
A. -Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y. Dagan, and A. -Kanigel, Gapless Excitations in -the Ground State of -$1t\text{\ensuremath{-}}{\Mathrm{TaS}}_{2}$, Phys. Rev. B -96, 195131 (2017).
+
+
[29]
A. Ribak, I. Silber, C. Baines, K. Chashka, Z. Salman, Y. Dagan, and A. Kanigel, Gapless Excitations in the Ground State of $1t\text{\ensuremath{-}}{\Mathrm{TaS}}_{2}$, Phys. Rev. B 96, 195131 (2017).
-
-
[30]
J. -C. Slater, Magnetic -Effects and the Hartree-Fock Equation, Phys. Rev. -82, 538 (1951).
+
+
[30]
J. C. Slater, Magnetic Effects and the Hartree-Fock Equation, Phys. Rev. 82, 538 (1951).
-
-
[31]
M. -Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum Phase Transition from a -Superfluid to a Mott Insulator in a Gas of Ultracold Atoms, -Nature 415, 39 (2002).
+
+
[31]
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum Phase Transition from a Superfluid to a Mott Insulator in a Gas of Ultracold Atoms, Nature 415, 39 (2002).
-
-
[32]
A. -Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, -F. Grusdt, E. Demler, D. Greif, and M. Greiner, A Cold-Atom Fermi–Hubbard -Antiferromagnet, Nature 545, 462 (2017).
+
+
[32]
A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A Cold-Atom Fermi–Hubbard Antiferromagnet, Nature 545, 462 (2017).
-
-
[33]
D. -Belitz and T. R. Kirkpatrick, The Anderson-Mott -Transition, Rev. Mod. Phys. 66, 261 -(1994).
+
+
[33]
D. Belitz and T. R. Kirkpatrick, The Anderson-Mott Transition, Rev. Mod. Phys. 66, 261 (1994).
-
-
[34]
D. -M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal–Insulator -Transition in a Weakly Interacting Many-Electron System with Localized -Single-Particle States, Annals of Physics 321, -1126 (2006).
+
+
[34]
D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal–Insulator Transition in a Weakly Interacting Many-Electron System with Localized Single-Particle States, Annals of Physics 321, 1126 (2006).
-
-
[35]
U. -Brandt and C. Mielsch, Thermodynamics and Correlation -Functions of the Falicov-Kimball Model in Large Dimensions, Z. -Physik B - Condensed Matter 75, 365 (1989).
+
+
[35]
U. Brandt and C. Mielsch, Thermodynamics and Correlation Functions of the Falicov-Kimball Model in Large Dimensions, Z. Physik B - Condensed Matter 75, 365 (1989).
-
-
[36]
C. -Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, Ground States of the -Spinless Falicov-Kimball Model, Phys. Rev. B -41, 2198 (1990).
+
+
[36]
C. Gruber, J. Iwanski, J. Jedrzejewski, and P. Lemberger, Ground States of the Spinless Falicov-Kimball Model, Phys. Rev. B 41, 2198 (1990).
-
-
[37]
T. -Kennedy and E. H. Lieb, An Itinerant -Electron Model with Crystalline or Magnetic Long Range Order, -Physica A: Statistical Mechanics and Its Applications -138, 320 (1986).
+
+
[37]
T. Kennedy and E. H. Lieb, An Itinerant Electron Model with Crystalline or Magnetic Long Range Order, Physica A: Statistical Mechanics and Its Applications 138, 320 (1986).
-
-
[38]
A. -E. Antipov, E. Gull, and S. Kirchner, Critical Exponents -of Strongly Correlated Fermion Systems from Diagrammatic Multiscale -Methods, Phys. Rev. Lett. 112, 226401 -(2014).
+
+
[38]
A. E. Antipov, E. Gull, and S. Kirchner, Critical Exponents of Strongly Correlated Fermion Systems from Diagrammatic Multiscale Methods, Phys. Rev. Lett. 112, 226401 (2014).
-
-
[39]
T. -Ribic, G. Rohringer, and K. Held, Nonlocal Correlations -and Spectral Properties of the Falicov-Kimball Model, Phys. -Rev. B 93, 195105 (2016).
+
+
[39]
T. Ribic, G. Rohringer, and K. Held, Nonlocal Correlations and Spectral Properties of the Falicov-Kimball Model, Phys. Rev. B 93, 195105 (2016).
-
-
[40]
J. -K. Freericks and V. Zlatić, Exact Dynamical -Mean-Field Theory of the Falicov-Kimball Model, Rev. Mod. Phys. -75, 1333 (2003).
+
+
[40]
J. K. Freericks and V. Zlatić, Exact Dynamical Mean-Field Theory of the Falicov-Kimball Model, Rev. Mod. Phys. 75, 1333 (2003).
-
-
[41]
A. -J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, Nonequilibrium -Dynamical Cluster Approximation Study of the Falicov-Kimball -Model, Phys. Rev. B 94, 245114 (2016).
+
+
[41]
A. J. Herrmann, N. Tsuji, M. Eckstein, and P. Werner, Nonequilibrium Dynamical Cluster Approximation Study of the Falicov-Kimball Model, Phys. Rev. B 94, 245114 (2016).
-
-
[42]
P. -W. Anderson, Resonating Valence -Bonds: A New Kind of Insulator?, Materials Research Bulletin -8, 153 (1973).
+
+
[42]
P. W. Anderson, Resonating Valence Bonds: A New Kind of Insulator?, Materials Research Bulletin 8, 153 (1973).
-
-
[43]
C. -Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. -Senthil, Quantum -Spin Liquids, Science 367, eaay0668 -(2020).
+
+
[43]
C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum Spin Liquids, Science 367, eaay0668 (2020).
-
-
[44]
S. -Trebst and C. Hickey, Kitaev -Materials, Physics Reports 950, 1 -(2022).
+
+
[44]
S. Trebst and C. Hickey, Kitaev Materials, Physics Reports 950, 1 (2022).
-
-
[45]
L. -Balents, M. P. A. Fisher, and C. Nayak, Nodal Liquid Theory of -the Pseudo-Gap Phase of High-Tc Superconductors, Int. J. Mod. -Phys. B 12, 1033 (1998).
+
+
[45]
L. Balents, M. P. A. Fisher, and C. Nayak, Nodal Liquid Theory of the Pseudo-Gap Phase of High-Tc Superconductors, Int. J. Mod. Phys. B 12, 1033 (1998).
-
-
[46]
L. -Balents, M. P. A. Fisher, and C. Nayak, Dual Order Parameter for -the Nodal Liquid, Phys. Rev. B 60, 1654 -(1999).
+
+
[46]
L. Balents, M. P. A. Fisher, and C. Nayak, Dual Order Parameter for the Nodal Liquid, Phys. Rev. B 60, 1654 (1999).
-
-
[47]
H.-H. Lin, L. Balents, and M. P. A. Fisher, -Exact SO(8) -Symmetry in the Weakly-Interacting Two-Leg Ladder, Phys. Rev. B -58, 1794 (1998).
+
+
[47]
H.-H. Lin, L. Balents, and M. P. A. Fisher, Exact SO(8) Symmetry in the Weakly-Interacting Two-Leg Ladder, Phys. Rev. B 58, 1794 (1998).
-
-
[48]
G. -Jackeli and G. Khaliullin, Mott Insulators in -the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum -Compass and Kitaev Models, Phys. Rev. Lett. -102, 017205 (2009).
+
+
[48]
G. Jackeli and G. Khaliullin, Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models, Phys. Rev. Lett. 102, 017205 (2009).
-
-
[49]
G. -Khaliullin, Orbital -Order and Fluctuations in Mott Insulators, Progress of -Theoretical Physics Supplement 160, 155 (2005).
+
+
[49]
G. Khaliullin, Orbital Order and Fluctuations in Mott Insulators, Progress of Theoretical Physics Supplement 160, 155 (2005).
-
[50]
G. -Jackeli and G. Khaliullin, Mott Insulators in -the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum -Compass and Kitaev Models, Physical Review Letters -102, 017205 (2009).
+
[50]
G. Jackeli and G. Khaliullin, Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models, Physical Review Letters 102, 017205 (2009).
-
-
[51]
M. -Hermanns, I. Kimchi, and J. Knolle, Physics -of the Kitaev Model: Fractionalization, Dynamic Correlations, and -Material Connections, Annual Review of Condensed Matter Physics -9, 17 (2018).
+
+
[51]
M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections, Annual Review of Condensed Matter Physics 9, 17 (2018).
-
[52]
S. -M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. -Gegenwart, and R. Valentí, Models and Materials for Generalized -Kitaev Magnetism, Journal of Physics: Condensed Matter -29, 493002 (2017).
+
[52]
S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and Materials for Generalized Kitaev Magnetism, Journal of Physics: Condensed Matter 29, 493002 (2017).
-
[53]
H. -Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, -Concept and Realization of Kitaev Quantum Spin Liquids, Nature -Reviews Physics 1, 264 (2019).
+
[53]
H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and Realization of Kitaev Quantum Spin Liquids, Nature Reviews Physics 1, 264 (2019).
-
-
[54]
W. -Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated -Quantum Phenomena in the Strong Spin-Orbit Regime, Annual -Review of Condensed Matter Physics 5, 57 (2014).
+
+
[54]
W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime, Annual Review of Condensed Matter Physics 5, 57 (2014).
-
-
[55]
A. -Kitaev, Anyons -in an Exactly Solved Model and Beyond, Annals of Physics -321, 2 (2006).
+
+
[55]
A. Kitaev, Anyons in an Exactly Solved Model and Beyond, Annals of Physics 321, 2 (2006).
-
-
[56]
J. -Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Dynamics of -Fractionalization in Quantum Spin Liquids, Phys. Rev. B -92, 115127 (2015).
+
+
[56]
J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Dynamics of Fractionalization in Quantum Spin Liquids, Phys. Rev. B 92, 115127 (2015).
-
-
[57]
C. -N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, Thermally Induced -Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of -the Kitaev Model with Parity Projection, Phys. Rev. B -99, 045142 (2019).
+
+
[57]
C. N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, Thermally Induced Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of the Kitaev Model with Parity Projection, Phys. Rev. B 99, 045142 (2019).
-
-
[58]
A. -Yu. Kitaev, Fault-Tolerant -Quantum Computation by Anyons, Annals of Physics -303, 2 (2003).
+
+
[58]
A. Yu. Kitaev, Fault-Tolerant Quantum Computation by Anyons, Annals of Physics 303, 2 (2003).
-
-
[59]
M. -Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological Quantum -Computation, Bull. Amer. Math. Soc. 40, 31 -(2003).
+
+
[59]
M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological Quantum Computation, Bull. Amer. Math. Soc. 40, 31 (2003).
-
-
[60]
C. -Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian Anyons and -Topological Quantum Computation, Rev. Mod. Phys. -80, 1083 (2008).
+
+
[60]
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80, 1083 (2008).
-
[61]
G. -Baskaran, S. Mandal, and R. Shankar, Exact Results for -Spin Dynamics and Fractionalization in the Kitaev Model, Phys. -Rev. Lett. 98, 247201 (2007).
+
[61]
G. Baskaran, S. Mandal, and R. Shankar, Exact Results for Spin Dynamics and Fractionalization in the Kitaev Model, Phys. Rev. Lett. 98, 247201 (2007).
-
[62]
G. -Baskaran, D. Sen, and R. Shankar, Spin- S Kitaev Model: -Classical Ground States, Order from Disorder, and Exact Correlation -Functions, Phys. Rev. B 78, 115116 -(2008).
+
[62]
G. Baskaran, D. Sen, and R. Shankar, Spin- S Kitaev Model: Classical Ground States, Order from Disorder, and Exact Correlation Functions, Phys. Rev. B 78, 115116 (2008).
-
[63]
Z. -Nussinov and G. Ortiz, Bond Algebras and -Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems, -Physical Review B 79, 214440 (2009).
+
[63]
Z. Nussinov and G. Ortiz, Bond Algebras and Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems, Physical Review B 79, 214440 (2009).
-
[64]
K. -O’Brien, M. Hermanns, and S. Trebst, Classification of -Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models, -Phys. Rev. B 93, 085101 (2016).
+
[64]
K. O’Brien, M. Hermanns, and S. Trebst, Classification of Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models, Phys. Rev. B 93, 085101 (2016).
-
[65]
M. -Hermanns, K. O’Brien, and S. Trebst, Weyl Spin Liquids, -Physical Review Letters 114, 157202 (2015).
+
[65]
M. Hermanns, K. O’Brien, and S. Trebst, Weyl Spin Liquids, Physical Review Letters 114, 157202 (2015).
-
-
[66]
H. -Yao and S. A. Kivelson, An Exact Chiral -Spin Liquid with Non-Abelian Anyons, Phys. Rev. Lett. -99, 247203 (2007).
+
+
[66]
H. Yao and S. A. Kivelson, An Exact Chiral Spin Liquid with Non-Abelian Anyons, Phys. Rev. Lett. 99, 247203 (2007).
-
-
[67]
T. -Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. -Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev -Spin Liquid, Physical Review Research 1, 032011(R) -(2019).
+
+
[67]
T. Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev Spin Liquid, Physical Review Research 1, 032011(R) (2019).
-
[68]
V. -Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R. -Moessner, and R. Thomale, Non-Abelian Chiral -Spin Liquid on a Simple Non-Archimedean Lattice, Phys. Rev. B -101, 041114 (2020).
+
[68]
V. Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R. Moessner, and R. Thomale, Non-Abelian Chiral Spin Liquid on a Simple Non-Archimedean Lattice, Phys. Rev. B 101, 041114 (2020).
-
[69]
D. -Weaire and M. F. Thorpe, Electronic Properties of -an Amorphous Solid. I. A Simple Tight-Binding Theory, Phys. -Rev. B 4, 2508 (1971).
+
[69]
D. Weaire and M. F. Thorpe, Electronic Properties of an Amorphous Solid. I. A Simple Tight-Binding Theory, Phys. Rev. B 4, 2508 (1971).
-
-
[70]
G. -Betteridge, A Possible Model of Amorphous Silicon and -Germanium, Journal of Physics C: Solid State Physics -6, L427 (1973).
+
+
[70]
G. Betteridge, A Possible Model of Amorphous Silicon and Germanium, Journal of Physics C: Solid State Physics 6, L427 (1973).
-
-
[71]
N. -P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, -Amorphous -topological insulators constructed from random point sets, -Nature Phys 14, 380 (2018).
+
+
[71]
N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Amorphous topological insulators constructed from random point sets, Nature Phys 14, 380 (2018).
-
-
[72]
A. -Agarwala, Topological Insulators in Amorphous Systems, in -Excursions in Ill-Condensed Quantum Matter (Springer, 2019), -pp. 61–79.
+
+
[72]
A. Agarwala, Topological Insulators in Amorphous Systems, in Excursions in Ill-Condensed Quantum Matter (Springer, 2019), pp. 61–79.
-
-
[73]
Q. -Marsal, D. Varjas, and A. G. Grushin, Topological Weaire-Thorpe -Models of Amorphous Matter, Proc. Natl. Acad. Sci. U.S.A. -117, 30260 (2020).
+
+
[73]
Q. Marsal, D. Varjas, and A. G. Grushin, Topological Weaire-Thorpe Models of Amorphous Matter, Proc. Natl. Acad. Sci. U.S.A. 117, 30260 (2020).
-
[74]
M. -Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. -Fazzio, Toward Realistic Amorphous Topological Insulators, Nano -Letters 19, 8941 (2019).
+
[74]
M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Toward Realistic Amorphous Topological Insulators, Nano Letters 19, 8941 (2019).
-
-
[75]
A. -Agarwala, V. Juričić, and B. Roy, Higher-Order Topological -Insulators in Amorphous Solids, Physical Review Research -2, 012067 (2020).
+
+
[75]
A. Agarwala, V. Juričić, and B. Roy, Higher-Order Topological Insulators in Amorphous Solids, Physical Review Research 2, 012067 (2020).
-
-
[76]
H. -Spring, A. Akhmerov, and D. Varjas, Amorphous Topological Phases -Protected by Continuous Rotation Symmetry, SciPost Physics -11, 022 (2021).
+
+
[76]
H. Spring, A. Akhmerov, and D. Varjas, Amorphous Topological Phases Protected by Continuous Rotation Symmetry, SciPost Physics 11, 022 (2021).
-
-
[77]
P. -Corbae et al., Evidence for Topological Surface States in Amorphous -Bi _ {2} Se _ {3}, arXiv Preprint arXiv:1910.13412 (2019).
+
+
[77]
P. Corbae et al., Evidence for Topological Surface States in Amorphous Bi _ {2} Se _ {3}, arXiv Preprint arXiv:1910.13412 (2019).
-
-
[78]
W. -Buckel and R. Hilsch, Einfluß Der Kondensation Bei Tiefen -Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für -Verschiedene Metalle, Zeitschrift Für Physik 138, -109 (1954).
+
+
[78]
W. Buckel and R. Hilsch, Einfluß Der Kondensation Bei Tiefen Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für Verschiedene Metalle, Zeitschrift Für Physik 138, 109 (1954).
-
-
[79]
W. -McMillan and J. Mochel, Electron Tunneling Experiments on Amorphous -Ge 1- x Au x, Physical Review Letters 46, 556 -(1981).
+
+
[79]
W. McMillan and J. Mochel, Electron Tunneling Experiments on Amorphous Ge 1- x Au x, Physical Review Letters 46, 556 (1981).
-
-
[80]
L. -V. Meisel and P. J. Cote, Eliashberg Function in -Amorphous Metals, Phys. Rev. B 23, 5834 -(1981).
+
+
[80]
L. V. Meisel and P. J. Cote, Eliashberg Function in Amorphous Metals, Phys. Rev. B 23, 5834 (1981).
-
-
[81]
G. -Bergmann, Amorphous Metals and Their Superconductivity, Physics -Reports 27, 159 (1976).
+
+
[81]
G. Bergmann, Amorphous Metals and Their Superconductivity, Physics Reports 27, 159 (1976).
-
-
[82]
S. -Manna, S. K. Das, and B. Roy, Noncrystalline Topological -Superconductors, arXiv:2207.02203.
+
+
[82]
S. Manna, S. K. Das, and B. Roy, Noncrystalline Topological Superconductors, arXiv:2207.02203.
-
-
[83]
S. -Kim, A. Agarwala, and D. Chowdhury, Fractionalization and Topology -in Amorphous Electronic Solids, arXiv Preprint arXiv:2205.11523 -(2022).
+
+
[83]
S. Kim, A. Agarwala, and D. Chowdhury, Fractionalization and Topology in Amorphous Electronic Solids, arXiv Preprint arXiv:2205.11523 (2022).
-
-
[84]
A. -Aharony, Critical Behavior of Amorphous Magnets, Physical -Review B 12, 1038 (1975).
+
+
[84]
A. Aharony, Critical Behavior of Amorphous Magnets, Physical Review B 12, 1038 (1975).
-
[85]
G. -A. Petrakovskii, Amorphous -Magnetic Materials, Soviet Physics Uspekhi 24, -511 (1981).
+
[85]
G. A. Petrakovskii, Amorphous Magnetic Materials, Soviet Physics Uspekhi 24, 511 (1981).
-
-
[86]
T. -Kaneyoshi, Introduction to Amorphous Magnets (World Scientific -Publishing Company, 1992).
+
+
[86]
T. Kaneyoshi, Introduction to Amorphous Magnets (World Scientific Publishing Company, 1992).
-
[87]
T. -Kaneyoshi, editor, Amorphous Magnetism (CRC Press, Boca Raton, -2018).
+
[87]
T. Kaneyoshi, editor, Amorphous Magnetism (CRC Press, Boca Raton, 2018).
-
[88]
M. -Fähnle, Monte Carlo Study of Phase Transitions in Bond-and -Site-Disordered Ising and Classical Heisenberg Ferromagnets, -Journal of Magnetism and Magnetic Materials 45, 279 -(1984).
+
[88]
M. Fähnle, Monte Carlo Study of Phase Transitions in Bond-and Site-Disordered Ising and Classical Heisenberg Ferromagnets, Journal of Magnetism and Magnetic Materials 45, 279 (1984).
-
[89]
J. -Plascak, L. E. Zamora, and G. P. Alcazar, Ising Model for Disordered -Ferromagnetic Fe- Al Alloys, Physical Review B 61, -3188 (2000).
+
[89]
J. Plascak, L. E. Zamora, and G. P. Alcazar, Ising Model for Disordered Ferromagnetic Fe- Al Alloys, Physical Review B 61, 3188 (2000).
-
-
[90]
J. -Coey, Amorphous Magnetic Order, Journal of Applied Physics -49, 1646 (1978).
+
+
[90]
J. Coey, Amorphous Magnetic Order, Journal of Applied Physics 49, 1646 (1978).
-
[91]
P. -W. Anderson, Resonating Valence Bonds: A New Kind of -Insulator?, Mater. Res. Bull. 8, 153 (1973).
+
[91]
P. W. Anderson, Resonating Valence Bonds: A New Kind of Insulator?, Mater. Res. Bull. 8, 153 (1973).
-
[92]
J. -Knolle and R. Moessner, A Field -Guide to Spin Liquids, Annual Review of Condensed Matter -Physics 10, 451 (2019).
+
[92]
J. Knolle and R. Moessner, A Field Guide to Spin Liquids, Annual Review of Condensed Matter Physics 10, 451 (2019).
-
[93]
L. -Savary and L. Balents, Quantum Spin Liquids: A Review, Reports -on Progress in Physics 80, 016502 (2017).
+
[93]
L. Savary and L. Balents, Quantum Spin Liquids: A Review, Reports on Progress in Physics 80, 016502 (2017).
-
[94]
C. -Lacroix, P. Mendels, and F. Mila, editors, Introduction to -Frustrated Magnetism, Vol. 164 (Springer-Verlag, Berlin Heidelberg, -2011).
+
[94]
C. Lacroix, P. Mendels, and F. Mila, editors, Introduction to Frustrated Magnetism, Vol. 164 (Springer-Verlag, Berlin Heidelberg, 2011).
-
-
[95]
T. -Eschmann, P. A. Mishchenko, K. O’Brien, T. A. Bojesen, Y. Kato, M. -Hermanns, Y. Motome, and S. Trebst, Thermodynamic -Classification of Three-Dimensional Kitaev Spin Liquids, Phys. -Rev. B 102, 075125 (2020).
+
+
[95]
T. Eschmann, P. A. Mishchenko, K. O’Brien, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Motome, and S. Trebst, Thermodynamic Classification of Three-Dimensional Kitaev Spin Liquids, Phys. Rev. B 102, 075125 (2020).
diff --git a/_thesis/2_Background/2.1_FK_Model.html b/_thesis/2_Background/2.1_FK_Model.html index cf3dc3b..f98595a 100644 --- a/_thesis/2_Background/2.1_FK_Model.html +++ b/_thesis/2_Background/2.1_FK_Model.html @@ -27,14 +27,12 @@ image: