Introduction
Topological phases and transport phenomena
One of the most successful paradigms for describing phases of matter is the Landau-Ginzburg theory of second-order phase transitions. Bloch's statement about the absence of net electric currents in equilibrium systems in the case of the energy currents.
Generalizations of the Berry curvature and the Thouless pump
Each of the operators 𝐽𝑝𝑞𝑁 is local in the sense above (it commutes with operators whose supports are sufficiently far from both 𝑝 and 𝑞). If the Chern class of the Berry curvature Ω(2) is nontrivial, one cannot write Ω(2) = 𝑑𝜔(1) for a globally defined 1-form𝜔(1).
Hydrodynamics of thermoelectric coefficeints
Generalities
Note that in a time-reversal-invariant situation, the tensors 𝜎 and 𝜅 are required to be symmetric, but 𝜈 and 𝜂 can have both a symmetric and an anti-symmetric part. On the other hand, both symmetric and anti-symmetric parts of𝜈𝑘𝑚 and𝜂𝑘𝑚 contribute to dissipation.
Relative and absolute transport coefficients
It is also invariant to changes in the functions 𝑓𝜇(𝑝) if they have the same asymptotic behavior as 𝜃(𝑥𝜇(𝑝)). Furthermore, when both 𝑝and𝑞 are near 𝑥=3𝐿, the 1-forms ˜𝑇𝑝𝑞(1) are exponentially close to the 1-forms 𝑇𝑝𝑞(1) for the Hamiltonian 𝐻(𝜇).
Lattice systems
Introduction
While most of it is simple, some operators, such as current and magnetization, become functions of different lattice locations. However, we will avoid using it where possible to make the presentation more accessible.
Electric currents on a lattice
While the above definition of electric current seems natural, it is not entirely unique. In the lattice case, it is not clear that the only ambiguity in the current definition is (3.5).
Energy currents and energy magnetization on a lattice
It is not difficult to check that 𝐽𝑁(𝐴, 𝐵) does not change if one replaces 𝐽𝑝𝑞𝑁 with 𝐽0𝑁𝑝𝑞 defined in (3.5). This is because𝐽𝑁(𝐴, 𝐵) is physical: it is equal to minus the rate of change of electric charge in the region.
Magnetization
While the magnetization itself suffers from ambiguity and depends nonlocally on the boundary conditions, the change in magnetization with respect to the parameters of the Hamiltonian is local. Using the Kubo matching property (see Appendix A), we can easily verify the identity (3.19).
Equilibrium conditions and driving forces
The above result means that the integral of the 3-form Ω(3)(𝑓) will be the same regardless of whether 𝑎in 𝑓(𝑝) =𝜃(𝑝−𝑎) satisfies 𝑎 0 or 𝑎 0. As explained in the main text of the thesis , it suffices to show that the 1-form 𝜇𝐸(𝛿 𝑓 ∪𝛿𝑔) is exact.
Energy Bloch theorem
Introduction
However, the Bloch-Bohm argument does not immediately apply to the flow of energy, since it is essentially based on the quantization of the particle number, which has no analogue in the case of energy. In the case of the𝑈(1)symmetry, this implies that a lattice system with an on-site𝑈(1) symmetry or a system of non-relativistic particles cannot flow to a CFT with a non-zero𝑅𝑘−𝑘𝐿.
Assumptions
Nevertheless, it is widely believed that a system of particles with short-range interactions or a lattice system with short-range interactions cannot flow to a CFT with a non-zero𝑐𝑅 −𝑐𝐿. An immediate consequence is that lattice systems or systems of particles with finite-range interactions cannot flow to a 1+1d CFT with a non-zero𝑐𝑅 −𝑐𝐿.
Energy currents in lattice systems
Now, any deformation of the Hamiltonian can be decomposed into a sum of two deformations: one that vanishes for 𝑝 0 and the other that vanishes for 𝑝 0. The linearity of the response to the infinitesimal deformation implies that the change in the actual expectation value is the sum of the variations corresponding with two deformations.
Energy currents in particle systems
It is important for what follows that a quasi-local energy flow can be constructed for an arbitrary symmetric finite-range potential𝑉(𝑥, 𝑦). It is sufficient to show that the energy flow is unchanged under the two separately.
By setting 𝑓(𝑥) = −𝜇, we seem to find that the equilibrium current vanishes at any 𝜇, contradicting (4.2). The deformation (4.28) can be imagined as connecting the theory with the external electric potential 𝜑= 𝑓(𝑥).
Applications
A cohomology class of a form is a topological invariant of the family (it cannot change under deformations).
Thermal Hall Effect
Introduction
Annihilation is achieved only if the Hall thermal conductivity of the vacuum is normalized to zero. This allows giving a completely general formula for the derivatives of the thermal Hall conductivity with respect to arbitrary parameters of the Hamiltonian.
Electric Hall conductance
However, if the system is inhomogeneous, the electrical Hall conductance of the two regions may be different, and the net electrical Hall current will be given by . The function 𝑔 describes the profile of the electric potential and is thus a smoothed step function with a non-zero width.
Thermal Hall Conductance
Its physical meaning is the energy magnetization differential in the region where both 𝑓 and 𝑔 change significantly. Therefore, the form 𝜅𝑥𝑦 can be extended to an open subset of the augmented parameter space given by 𝑇 > 0.
Concluding remarks
For 𝐷 = 1, one can hope that the change in the expectation of the charge on the half-line 𝑝 > 𝑎 is limited. In the main part of the thesis, we showed how to define a closed formΩ(𝐷+2) on the parameter spaceM.
Thermoelectric Hall Effects
Introduction
Recently, much attention has been devoted to non-dissipative time-reversal odd transport coefficients, such as electrical Hall conductivity and Hall viscosity. These are microscopic formulas for transport coefficients in terms of correlators of flows of conserved quantities.
Microscopic formulas for thermoelectric coefficients
To use this symmetry, we need to argue that 𝑓 can be replaced by a smeared step function in the 𝑥 direction. The determination of the relative transport coefficient in this case would correspond to the integration over a path in the parameter space from an irrelevant insulator to the material of interest.
Discussion
Assume that the parameters of the Hamiltonian vary slowly in the 𝑦 direction while maintaining a large gap. The integral of this form- on a cycle𝐷 in parameter space is invariant under deformations of the cycle.
Nernst and Ettingshausen effects in gapped quantum materials . 56
Thermoelectric coefficients of gapped materials and the Third Law . 58
One standard formulation of the Third Law ("the Nernst unattainability principle states that it is impossible to lower the entropy of a body to its. According to the Nernst heat theorem, the entropy𝑆 of the cylinder must approach a 𝐵-independent constant at𝑇 =0.
Flux insertion argument for vanishing of the Nernst coefficient
With this notation the change in expected energy value is given by Δ𝐸 =Tr[𝐻𝑈 𝜌0𝑈†] −Tr(𝐻 𝜌0). Since the flux insertion process does not introduce any mass excitation, we know that 𝜌𝑓 must be of the form
Concluding remarks
To these data one can associate a differential 2-formΩ on the parameter space: the curvature of the Berry connection [8]. Thus integrals of 𝑄(𝐷) over 𝐷 cycles in the parameter space are a natural generalization of the Thouless charge pump to dimension𝐷.
Higher-dimensional generalization of the Berry curvature
Introduction
Moreover, the degeneracy points detected by Berry's curvature are stable to Hamiltonian deformations. This form is a higher-dimensional generalization of the Berry curvature and can be used to detect phase transition locations in parameter space.
Effective action considerations
But in the case of translational invariant tight-binding free fermion Hamiltonians in 1d we show that the cohomology class of Ω(𝐷+2) is determined by the curvature of the Berry-Bloch connection. More generally, any globally defined Berry connection can be made zero by a suitable deformation of the gap-preserving Hamiltonian.
Higher Berry curvature for gapped 1d systems
The closure of Ω(3)(𝑓) implies that its cohomology class is a topological invariant of the family of systems with gaps. Since the formula for the 3-form is local, for 𝑎 0 the integral of the 3-form coincides with.
Higher Berry curvature for gapped systems in any dimension
This property implies that a family of systems parameterized by a closed 3-manifold Σ3 with a non-zero value of the integral∫. For 𝑛 = 3, it can be interpreted as a variation of a local susceptibility with respect to a variation of the Hamiltonian elsewhere.
Discussion
In the case 𝑓(𝑝) =𝜃(𝑝−𝑎) the 1-form𝑄(1)(𝑓) has the meaning of the regulated differential of the charge on the half-line 𝑝 > 𝑎. Such a map is topologically the same as a loop in the parameter space with a base point corresponding to the value of the parameters at𝑥.
Higher-dimensional generalizations of the Thouless charge pump 87
Effective field theory
9.4) Such topological terms in the 𝑈(1) current have been discovered by Goldstone and Wilczek in their work on soliton charges [22]. In the bosonic case, this follows from the fact that for 𝐷 ≤ 6, every 𝐷 cycle is an advance of the basic homology class of a closed oriented 𝐷 manifold [54].
Thouless charge pump for 1d lattice systems
Let𝐽𝑁(𝑎) be the operator corresponding to the current through a point𝑎 ∈R 𝐽𝑁(𝑎) =Õ. The total charge passing through the point𝑎for one period𝑇 is given by Δ𝑄=. 9.14) where we dropped the terms which are small in the adiabatic expansion. The remaining terms can be rewritten using (9.11) as Δ𝑄 =𝑖. 9.15) Here
A static formula for the Thouless charge pump
This agrees with the intuition that the time reversal reverses the sign of the Tholess charge pump. The Tholess charge pump for any loop is the estimate of that cohomology class on the homology class of the loop.
Descendants of the Thouless charge pump
In Appendix F.1, we argue that the integral of𝑄(1)(𝑓) over any loop in the parameter space is an integer. Note that the chain𝑇(0) takes values in 0-forms on the parameter space, while the chain𝑇(1) takes values in 1-forms on the parameter space.
Physical interpretation of the 2d Thouless charge pump
This arises from the insensitivity of shape correlators. on the Hamiltonian far from the support 𝐴in𝐵[60]. We now use the same approach to understand the physical meaning of the Tholess invariant charge pump for 2d systems.
Higher Thouless charge pump for systems of free fermions in 2d
Then the Bloch wavefunctions of the valence bands form a vector bundle over the product of the Brillouin zone𝑆1×𝑆1 and the parameter space Σ. The integral of the degree 4 component of the Chern character of this connection over the product of the Brillouin zone and the parameter space.
Discussion
A special case of this is the electrical or energy flow from region 𝐵 to region 𝐴 which is denoted 𝐽𝑁(𝐴, 𝐵) or 𝐽𝐸(𝐴, 𝐵) in the body of the thesis. To answer this question we need to know the homology of the complex of controlled chains in degree 1.
Kubo canonical pairing
Dynamic response
The change in state of the system can be found from the Liouville equation as follows. 0 𝑑𝑡hh𝐴; Δ𝐻¤(−𝑡)ii, (B.9), where we used the Kubo pairing notation (see Appendix B.1) and Δ𝐴 is an explicit variation of the operator.
Exponential decay of certain correlators in a gapped phase
Using the explicit form of the perturbation (B.5) and the properties of the Kubo link, we can rewrite this formula as. Splitting 𝐶 (which is assumed to be a sum of local operators) into two parts𝐶 =𝐶𝐴+𝐶𝐵where the support of𝐶𝐴 is far away from 𝐵and the support of𝐶𝐵 is far away from 𝐴, we get.
On the path-independence of the relative thermal Hall conductance . 126
To show that the integral (B.28) defining 𝐼(M,M0) converges and is independent of the exact choice of endpoints, consider a variation of the Hamiltonian supported in a quadrant of R2. Following the same logic as before, we can move 𝑓 , 𝑔inΨ(𝑓 , 𝑔) away from the support of the variation introducing an error that is exponentially small in temperature.
Free fermion systems
The state of the system at a temperature 𝑇 = 1/𝛽 is defined via Wick's theorem and the Gibbs distribution. The value of energy magnetization 𝜇𝐸 on a 2-cochain𝛿 𝑓 ∪𝛿𝑔 can be found as 𝜇𝐸(𝛿 𝑓 ∪𝛿𝑔) = 1. where 𝑑ℎ is the variation of the 1-particle Hamiltonian.
Onsager reciprocity revisited
Invariance under Hamiltonian density redefinition
Now let's look at the effect of redefinition of Hamiltonian density on energy flow. We find that the net energy flow transforms as follows under a redefinition of the Hamiltonian density:
Thermoelectric coefficients for free fermions
2hh𝐽𝑁(𝛿𝑔);𝐴(𝛿 𝑓)ii (C.36) We see that the variation of this term cancels the variations of the Kubo parts. Note that since in the limit𝑇 →0 the Fermi-Dirac distribution𝔣(𝑧) becomes a step function, and since 𝑐1(0) = 𝑐1(1) = 0, both 𝜈𝐴(𝑇) and 𝜂𝐴(𝑇) 𝑇 =0 regardless of the choice of the Hamiltonianℎ.
Středa formulas
Proving Eqs. (7.49), (7.50)
Likewise, for each 𝑗, 𝑗0 there exists an operator𝑂𝑡𝑗0𝑗 supported near the top edge, such that𝑂𝑗0𝑗|𝑖, 𝑗i=|𝑖, 𝑗0ifor all𝑖. 𝑂˜𝑏𝑖0𝑖 =𝑈†𝑂𝑖𝑏0𝑖𝑈, 𝑂˜𝑡𝑗0𝑗 =𝑈†𝑂𝑏𝑗0140) form (7.44), we know that 𝑂˜𝑖𝑏0𝑖 and ˜𝑂𝑡𝑗0𝑗 are supported near the bottom and top edges, respectively (this follows from Lieb-Robinson bounds [26]).
Quantization of higher Berry curvatures
Therefore, if the cohomology class Ω(𝐷+2) is not trivial, it is impossible to find a family with gaps of boundary conditions for 𝐻[s] that is defined in the entirety of 𝑆3 and varies smoothly with s. For 𝐷 =0, the analogous statement is that the cohomology class of the Berry curvature is an obstacle to finding a family of ground states on the entire parameter space that is integrally dependent on the parameters.
Higher Berry curvature for 1d insulators of class A
In particular, we can consider the degree 4 component of the Chern character of the Berry-Bloch connection and its integral over 𝑆1×Σ:. We assume that, in general, for class A insulators in dimensions of the form Ω(𝐷+2) is in the same cohomology class as the integral of degree 2𝐷+2 of the Chern character component of the Bloch-Berry connection over the Brillouin zone.
Quantization of the Thouless charge pump and its descendants
There is no topological order for 1d systems and thus all gap systems of the family are in the same SRE phase. Such a substitution introduces an error of order 𝐿−∞, because the sums are sensitive only to the Hamiltonian of the systems in the region where𝐻+𝑝[𝜆].
