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- 2. Useful Resources#
-Other useful resources for learning about tensor networks include (but are certainly not limited to):
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- Symmetries in Quantum Many-Body Physics
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- 11. Symmetries in Quantum Many-Body Physics#
-The goal of this section is to give a very gentle introduction to the concept of symmetries -in quantum many-body physics, and the notion of symmetric tensors. The general mathematical -framework of symmetries in physics (or at least the framework we will restrict to) is that -of group - and representation theory. Our goal is not to take this framework as a given and -illustrate it, but rather to first discuss a couple of important applications of symmetries -in the context of some concrete models and gradually build up to the more general framework. -We will finish our discussion with an outlook to generalizations of the framework presented -here. It goes without saying that we will only scratch the surface of this vast topic. The -interested reader is referred to the immense literature on this topic, or to a more -specialized course.
-
-
-11.1. Examples and Applications#
-
-
-11.1.1. Symmetry Breaking, Order Parameters and Phases#
-Recall the one-dimensional transverse field Ising model defined above. Its degrees of -freedom are qubits ordered on a one-dimensional lattice, and its Hamiltonian reads
-
-\[H = -\sum_{i} \sigma^z_i\sigma^z_{i+1} -h_x\sum_i\sigma^x_i.\]
-Let us simply consider periodic boundary conditions. Besides the obvious translation -symmetry, which we will discuss below, this model is also invariant under flipping all spins -simultaneously in the Z-direction, i.e. in the Pauli Z basis: -\(\ket{\uparrow}\leftrightarrow\ket{\downarrow}\). That this operation constitutes a symmetry -is clear from the Hamiltonian as the energy of the first term only depends on the -neighbouring spins being (anti-)aligned, which is clearly spin flip-invariant. The second -spin is trivially invariant as this models an external magnetic field which is orthogonal to -the Z-direction.
-This spin flip is “implemented”, or more correctly “represented”, by the unitary operator -\(P=\bigotimes_i \sigma^x_i\). Notice that \(P^2=1\) in accordance with our intuition that -flipping all the spins twice is equivalent with leaving all spins untouched. The fact that -this operator represents a symmetry of the model then translates to \([H,P]=0\), or -equivalently \(P^\dagger HP=H\). Notice that the identity operator is also trivially a -symmetry (of every model) and thus the set \(\{1,P\}\) is closed under taking the product.
-Even though the Hamiltonian has the symmetry regardless of the value of the parameter \(h_x\), -you might know from a previous course that the ground state or ground state subspace are not -necessarily invariant under the symmetry, a phenomenon known as spontaneous symmetry -breaking (SSB) or symmetry breaking for short. Let us investigate the ground state subspace -of the transverse field Ising model in the extremal case of vanishing and infinite magnetic -field.
--
-
\(h_x\rightarrow \infty\) In this case the model effectly reduces to a paramagnet. The -unique ground state is the product state \(\ket{\Psi_+}=\ket{+}^{\otimes N}\) where -\(\ket{+}=\frac{1}{\sqrt{2}}(\ket{\uparrow}+\ket{\downarrow})\) is the unique eigenvalue 1 -eigenvector of \(\sigma^x\). Notice that this state is invariant under the symmetry operator -\(P\), \(P\ket{\Psi_+}=\ket{\Psi_+}\). In other words, the ground state in this case is -symmetric. For reasons mentioned below this state is also considered to be disordered.
-\(h_x=0\) In this case the energy is minimized by aligning all the spins and the model -behaves as a classical ferromagnet. Obviously, two distinct ground states are -\(\ket{\Psi_\uparrow}=\ket{\uparrow\uparrow...\uparrow}\) and -\(\ket{\Psi_\downarrow}=\ket{\downarrow\downarrow...\downarrow}\). Contrary to the previous -case they span a two-dimensional ground state subspace, and these states are not -symmetric. In fact, under the action of \(P\) they get mapped onto the other: -\(P\ket{\Psi_\uparrow}=\ket{\Psi_\downarrow}\) and vice versa. The ground state in this case -is thus symmetry broken, or ordered.
-
Since the ground state degeneracy is necessarily an integer, it is clear that it can not -change smoothly from two to one when the magnetic field is slowly turned on from -\(h_x = 0 \rightarrow \infty\). Therefore the Ising model for small \(h_x\) and large \(h_x\) are -said to belong to different phases, and for some finite value of \(h_x\) a phase transition -where the ground state degeneracy changes abruptly is expected to take place. As it turns -out, this change happens for \(h_x = 1\), at which point the Ising model becomes critical.
-Inspired by the credo of symmetry we can introduce a local operator which probes the phase -and can witness the phase transition. In the case of the Ising model this order parameter is -the local magnetisation on every site: \(O=\sum_i\sigma^z_i\). It is clear that this order -parameter anticommutes with the symmetry, \(P^\dagger OP=-O\), from which it follows that in -the symmetric phase the expectation value of the order parameter vanishes, -\(\braket{\Psi_+|O|\Psi_+}=0\), while in the ferromagnetic phase -\(\braket{\Psi_\uparrow|O|\Psi_\uparrow}>0\) and -\(\braket{\Psi_\downarrow|O|\Psi_\downarrow}<0\). Notice however that for the latter we could -also have chosen the ground state \(\ket{\Psi_\uparrow}+\ket{\Psi_\downarrow}\) in which case -the expectation value of \(O\) becomes 0. So it seems that the expectation value of the order -parameter is ill-defined is this phase. This can be remedied by first adding a small -symmetry breaking term \(\lambda\sum_i\sigma^z_i\) to the Hamiltonian which, depending on the -sign of \(\lambda\), selects one of the ground states \(\ket{\Psi_{\uparrow/\downarrow}}\) after -which the limit \(\lambda\rightarrow 0\) is taken.
-The synopsis of this example is thus the following. Symmetries in quantum many-body physics -(but also in single-particle quantum mechanics) are represented by unitary operators which -are closed under multiplication. Depending on the parameters in the Hamiltonian, part of -these symmetries can be broken by the ground state subspace, and this pattern of symmetry -breaking is a hallmark feature of different phases of the model. Different phases can be -probed by a local order parameter which does not commute with the symmetries. This paradigm -of classifying phases based on symmetry principles was first put forward by Landau -[Landau, 1937], and since then bears his name.
-
-
-11.1.2. Noether and Conserved Quantitites#
-You might remember Noether’s theorem from a course on field theory. It states that every -continuous symmetry of a system (in field theory most often defined via its Lagrangian) -gives rise to a conserved current. In the context of quantum physics Noether’s theorem -becomes almost trivial and states that the expectation value of every operator that commutes -with the Hamiltonian has a conserved expectation value:
-
-\[[H, O] = 0 \implies \frac{d}{dt}\braket{\Psi(t)|O|\Psi(t)} = 0.\]
-The proof is almost trivial and is left as a simple exercise.
--
-
The simplest example of this principle is obviously the Hamiltonain that trivially -commutes with itself. The consequence is that the expectation value of the total energy is -conserved.
-Another example is that of translation symmetry. Translation symmetry is implemented by -the operator \(T\) that acts on local operators \(O_i\) via \(T^\dagger O_iT=O_{i+1}\). Since -for a system with \(N\) sites we obviously have the identity \(T^N=1\), and \(T\) is unitary, -the eigenvalues of \(T\) are phases \(\exp(2\pi ip/N)\) where the quantum number -\(p=0,1,...,N-1\) is the momentum. By virtue of Noether, translation invariance is -understood to give rise to conservation of momentum, and thus momentum acts as a good -quantum number for the eigenstates of translationally invariant models.
-
Let us consider another non-trivial example to illustrate the implications of this theorem. -Recall the spins \(s\) XXZ Heisenberg model whose Hamiltonian reads
-
-\[H = -J\sum_i S^x_iS^x_{i+1}+S^y_iS^y_{i+1}+\Delta S^z_iS^z_{i+1}.\]
-The spin operators are \(2s + 1\)-dimensional and satisfy the \(\mathfrak{su}(2)\) commutation -relations
-
-\[[\sigma^a_i,\sigma^b_j]=i\delta_{i,j}\sum_c \varepsilon_{abc}S^c_i \]
-Let us define the total spin
-
-\[S^a = \sum_i S^a_i.\]
-From a direct computation it follows that in the case where \(\Delta=1\), and the model thus -reduces to the Heisenberg XXX model, \(H\) commutes with all \(S^a\), \([H,S^a]=0\), \(a=x,y,z\). -However, when \(\Delta\neq 1\) only the Z component \(S^z\) commutes with \(H\), \([H, S^z]=0\). -Notice the difference with the Ising model where the same symmetry was present for all -values of \(h_x\).
-This means that in the \(\Delta=1\) case the Hamiltonian is symmetric under the full \(SU(2)\) -(half integer s) or \(SO(3)\) (integer s) symmetry (see below), whereas when \(\Delta\neq 1\) -only an \(SO(2)\simeq U(1)\) symmetry generated by \(S^z\) is retained. If \(H\) commutes with -\(S^z\) it follows that it automatically also commutes with \(\exp(i\theta S^z)\), -\(\theta\in[0,2\pi)\). This operator has an interpretation as a rotation around the Z-axis -with an angle \(\theta\).
-According to Noether the Heisenberg model thus has conserved quantities associated with -these operators. Regardless of \(\Delta\) the Z component of the total spin is conserved, and -for \(\Delta=1\) all components of the total spin are conserved. In particular this means that -the eigenvalue \(M_z\) of \(S^z\) and \(S(S+1)\) of \(\vec{S}\cdot\vec{S}\) are good quantum numbers -to label the eigenstates of the Heisenberg Hamiltonian.
-
-
-11.2. Group and Representation Theory#
-Motivated by the examples from above, we will gently introduce some notions of group - and -representation theory that form the backbone of a general theory of symmetries.
-
-
-11.2.1. Group Theory#
-Roughly speaking a group \(G\) is a set of symmetry operators and a multiplication rule on how -to compose them. Let us motivate the definition one step a time.
-First of all notice that a model can have a finite or infinite (discrete or continuous) -number of symmetries. Clearly, the spin flip symmetry of the Ising model consists of only -one non-trivial symmetry operation, namely flipping all spins. The operator carrying out -this transformation is \(P=\bigotimes_i\sigma^x_i\). The XXZ model however has a continuous -symmetry, namely rotations around the Z-axis, that is implemented via \(\exp(i\theta S^z)\), -\(\theta\in[0,2\pi)\), where we should really think about every value of \(\theta\) as labeling -a different symmetry operation.
-These symmetries can be composed or multiplied to form a new symmetry operation. Take for -example flipping all the spins. Flipping all spins twice results in not flipping any spins -at all, which is trivially also a symmetry of the Hamiltonian. Next, consider also the -\(U(1)\) symmetry of the XXZ model. First rotating over \(\theta_2\) and then over \(\theta_1\) -gives a new rotation over \(\theta_1+\theta_2\): \(\exp(i\theta_1 S^z)\exp(i\theta_2 -S^z)=\exp(i(\theta_1+\theta_2) S^z)\). This leads to the first part of the definition of what -a group is.
--
-
A group \(G\) is a set \(G=\{g_1,g_2,...\}\) endowed with a multiplication \(G\times -G\rightarrow G\). There exists an identity \(1\in G\) for the multiplication such that -\(1g=g1=g, \forall g\in G\).
-
Note that this multiplication is not necessarily abelian. A simple example is the full -\(SU(2)\) symmetry of the XXX model defined above. However, the composition of symmetries is -still associative:
--
-
For all group elements \(g,h,k\) we have that \(g(hk)=(gh)k\).
-
A property we also would like to formalize is the fact that every symmetry transformation -can be undone. Take for example a \(U(1)\) rotation \(\exp(i\theta S^z)\), if we compose it with -the opposite rotation \(\exp(i(2\pi-\theta) S^z)\) we get the identity. Hence:
--
-
Every group element \(g\) has a unique inverse \(g^{-1}\): \(gg^{-1}=g^{-1}g=1\).
-
Together 1. 2. and 3. constitute the definition of a group. Before mentioning some examples -let us also introduce the concept of a subgroup. As the name suggests, a subgroup is a -subset of a group which itself constitutes a group. Note for example that a rotation over -\(\pi\), \(\exp(i\pi S^z)\), together with the identity, generates a subgroup of \(\{\exp(2\pi -i\theta S^z|\theta\in[0,2\pi)\}\) with two elements.
-The concept of subgroups lies at the heart of symmetry breaking. Recall that in the -ferromagnetic phase, the Ising model breaks the spin flip symmetry. In Landau’s paradigm we -say that the pattern of symmetry breaking is \(\mathbb{Z}_2\rightarrow \{1\}\) (see below for -an explanation of the notation). In other words, the full symmetry group (\(\mathbb{Z}_2\)) is -broken in the ferromagnetic phase to a subgroup (the trivial group). More generally, a -theory with a \(G\) symmetry can undergo a pattern of symmetry breaking \(G\rightarrow H\) where -\(H\) is a subgroup of \(G\). The meaning of this symbolic expression is that the ground states -keep an H symmetry, and the ground state degeneracy is \(|G|/|H|\).
-
-
-11.2.1.1. Examples#
--
-
The trivial group is a group with only one element that is than automatically also the -identity, and a trivial multiplication law. Above, it was denoted by \(\{1\}\).
-\(\mathbb{Z}_N\) is the additive group of integers modulo \(N\). The group elements are the -integers \(\{0,1,...,N-1\}\) and the group multiplication is addition modulo \(N\). Hence it -is clearly a finite group. In particular, the spin flip symmetry from above corresponds to -the group \(\mathbb{Z}_2\). Notice that for all \(N\) \(\mathbb{Z}_N\) is abelian.
-Another abelian group is \(U(1)\). This group is defined as -\(U(1)=\left\{z\in\mathbb{C}:|z|^2 = 1\right\}\), with group multiplication the -multiplication of complex numbers. Note we encountered this group in the XXZ model as -being the rotations around the Z axis: \(\{\exp(2\pi i\theta S^z|\theta\in[0,2\pi)\}\).
-\(SU(2)\) is the group of unimodular unitary \(2\times 2\) matrices:
--\[SU(2) := \left\{U \in \mathbb{C}^{2\times 2} | - \det U = 1, UU^\dagger = U^\dagger U = \mathbb{I}\right\}.\]-The group multiplication is given by group multiplication. Similarly, one defines -\(SU(N),N\geq 2\). Note that none of these groups are abelian.
-
-The 3D rotation group or special orthogonal group \(SO(3)\) is the group of real \(3\times 3\) -orthogonal matrices with unit determinant:
--\[SO(3) := \left\{M\in\mathbb{R}^{3\times 3}|MM^T=M^TM=\mathbb{I},\det M=1\right\}.\]-Similarly, one defines \(SO(N),N\geq 2\). Note that only \(SO(2)\) is abelian.
-
-
-
-11.2.2. Representation Theory#
-In the above examples, we were dealing with the question which symmetry transformations -leave the Hamiltonian (and in the absence of symmetry breaking also the ground states) -invariant. These symmetry representations were implemented (represented) by invertible -linear operators, non-singular matrices, that form a closed set under multiplication. This -multiplication structure is what we identified as a group. What we could now do, is to take -a group as given, and wonder which linear transformations we can come up with that multiply -according to these multiplication rules. This is exactly the underlying idea of -representation theory. Representation theory deals with the question how groups can linearly -act on vector spaces.
-This immediately raises a plethora of questions such as if we can classify all -representations (up to some kind of equivalence), if there exists ‘minimal’ representations -and how we can construct new representations of known ones. A minimal answer to these -questions is the goal of this section.
-
-
-11.2.2.1. Definition#
-For the sake of these notes, a representation of a group \(G\) is thus a set of matrices -indexed by the group elements, \(\{X_g|g\in G\}\) that multiply according to the -multiplication rule of \(G\):
-
-\[X_gX_h = X_{gh}\]
-Note that the identity is always mapped to the identity matrix!
-We call the dimension of the matrices \(X_g\) the dimension of the representation.
-
-
-11.2.2.1.1. Examples#
--
-
Every group can be trivially represented by mapping every group element to the ‘matrix’ -(1). Obviously, this representation is one-dimensional and is called the trivial -representation.
-Probably the simplest non-trivial representation, is the representation of \(\mathbb{Z}_2\) -that maps the non-trivial element to -1. Concretely, \(X_0=1, X_1=-1\), and indeed -\(X_1X_1=(-1)^2=X_0\). This representation is called the sign representation.
-Let us construct a two-dimensional representation of \(\mathbb{Z}_2\). Since the Pauli -matrix \(\sigma^x\) (as any other Pauli matrix) squares to the identity, \(\sigma^x\) together -with the two-dimensional identity matrix constitutes a two-dimensional representation of -\(\mathbb{Z}_2\). In the notation from above, \(X_0=\mathbb{I}_2\), \(X_1=\sigma^x\). This -representation is called the regular representation of \(\mathbb{Z}_2\).
-
-
-11.2.2.2. Complex Conjugate Representation, Tensor Product and Direct Sum Representation#
-Given a representation \(\{X_g|g\in G\}\), the complex conjugate representation \(\bar X\) is -defined as \(\bar X=\{\bar X_g|g\in G\}\) which satisfies the defining property of -representations via \(\bar X_g\bar X_h= \overline{X_gX_h}=\bar X_{gh}\).
-Given two representations of \(G\), \(X\equiv\{X_g|g\in G\}\) and \(Y\equiv\{Y_g|g\in G\}\), there -are two obvious ways to construct a new representation.
-The first one is the tensor product representation defined via the Kronecker product of -matrices:
-
-\[\{X_g\otimes Y_g|g\in G\}.\]
-You should check that these still satisfy the defining property of a representation. The -dimension of the tensor product is the product of the dimensions of the two representations -\(X\) and \(Y\).
-The other one is the direct sum:
-
-\[\{X_g\oplus Y_g|g\in G\}.\]
-Its dimension is that the sum of the dimensions of \(X\) and \(Y\).
-
-
-11.2.2.3. Irreducible Representations#
-It is clear that physical observables should not depend on any choice of basis. Therefore -two representations are (unitarily) equivalent when there is a unitary basis transformation -\(U\) such that \(X_g' =UX_gU^\dagger\). Note that \(U\) is independent of \(g\).
-
-
-11.2.2.3.1. Example#
-Consider again the two-dimensional regular representation of \(\mathbb{Z}_2\) from above. The -basis transformation
-
-\[\begin{split}H=\frac{1}{\sqrt 2}
-\begin{pmatrix}
- 1 & 1\\
- 1 & -1
-\end{pmatrix}\end{split}\]
-shows that this representation is equivalent to one where the non-trivial element of -\(\mathbb{Z}_2\) is represented by \(H\sigma^x H^\dagger=\sigma^z\). This illustrates that the -regular representation is equivalent to the direct sum of the trivial representation and the -sign representation!
-The crux of this example is the following. Some representations can, by an appropriate -choice of basis, be brought in a form where all \(X_g\) are simultaneously block-diagonal:
-
-\[\begin{split}X_g'=UX_gU^\dagger=
-\begin{pmatrix}
- \fbox{$X^1_g$} & 0 &\cdots\\
- 0& \fbox{$X^2_g$} & \cdots\\
- \vdots & \vdots & \ddots
-\end{pmatrix}.\end{split}\]
-These blocks correspond to invariant subspaces of the representation, i.e. subspaces that -transform amongst themselves under the action of the group.
-An irreducible representation, irrep for short, can then be defined as a representation that -can not be brought in a (non-trivial) block-diagonal form by any change of basis.
-It can be shown that every finite group has a finite number of irreps. The sum of the -dimensions squared is equal to the number of elements in the group: \(\sum_\alpha -d_\alpha^2=|G|\) where the sum is over all irreps labeled by \(\alpha\) and \(d_\alpha\) denote -their respective dimensions.
-One of the key questions of representation theory is what the irreps of a given group are -and how the tensor product of irreps (which is in general not an irrep!) decomposes in a -direct sum of irreps. The latter are sometimes known as the fusion rules. The basis -transformation that reduce a given representation in a direct sum of irreps is sometimes -called the Clebsch-Gordan coefficients, and are for some groups known explicitly. Before -discussing the example of \(SU(2)\), let us first state the most important result in -representation theory which is due to Schur.
-[Schur’s lemma] If a matrix \(Y\) commutes with all representation matrices of an irreducible -representation of a group G, \(X_gY=YX_g\) \(\forall g\in G\), then \(Y\) is proportional to the -identity matrix.
-
-
-11.2.2.4. Example#
-The answer to the questions posed above is very well understood for the case of \(SU(2)\). You -probably know the answer from a previous course on quantum mechanics.
-The irreps of \(SU(2)\) can be labeled by its spin, let us call it \(s\), that takes values -\(s=0,1/2,1,3/2,...\). The dimension of the spin \(s\) representation is equal to \(2s+1\), so -there is exactly one irrep of every dimension. The spin \(s=0\) irrep corresponds to the -trivial representation.
-The fusion rules can be summarized as
-
-\[s_1\otimes s_2 \simeq \bigoplus_{s=|s_1-s_2|}^{s_1+s_2}s.\]
-For example: \(\frac{1}{2}\otimes\frac{1}{2}\simeq 0\oplus 1\). The Clebsch-Gordan -coefficients for \(SU(2)\) have been computed analytically, and for low-dimensional irreps -have been tabulated for example -here.
-
-
-11.3. Symmetric Tensors#
-In physics we are often dealing with tensors that transform according to the tensor product -representation of a given group \(G\). A symmetric tensor can then be understood as a tensor -that transforms trivially under the action of \(G\), or more concretely under the tensor -product representation \(X\otimes\bar Y\otimes\bar Z\):
-
-
-
-
-This has strong implications for the structure of the tensor \(T\). Notice that we didn’t -assume the representations \(X,Y\) and \(Z\) to be irreducible. As we argued above, an -appropriate change of basis can bring the representations \(X,Y\) and \(Z\) in block-diagonal -form where every block corresponds to an irrep of the group and every block can appear -multiple times, which we call the multiplicity of an irrep in the representation. Schur’s -lemma then implies that in this basis, the tensor becomes block-diagonal. In an appropriate -matricization of \(T\) we can thus write \(T=\bigoplus_c B_c\otimes\mathbb{I}_c\) where the -direct sum over \(c\) represents the decomposition of \(X\otimes\bar Y\otimes\bar Z\) in irreps -\(c\) that can appear multiple times. In other words, the generic symmetric tensor \(T\) can be -stored much more efficiently by only keeping track of the different blocks \(B_c\).
-TensorKit is particularly well suited for dealing with symmetric tensors. What TensorKit -does is exactly what was described in the previous paragraph, it keeps track of the block -structure of the symmetric tensor, hereby drastically reducing the amount of memory it takes -to store these objects, and is able to efficiently manipulate them by exploiting its -structure to the maximum.
-As a simple exercise, let us construct a rank 3 \(SU(2)\) symmetric tensor as above. For -example the spin \(1/2\) and spin \(1\) representation can be called via respectively
-
-
-
-
-
-
-using TensorKit
-
-s = SU₂Space(1/2 => 1)
-l = SU₂Space(1 => 1)
-
-
--Show code cell output - -
-
-
-
-
-Rep[SU₂](1=>1)
-Here, => 1 essentially means that we consider only one copy (direct summand) of these
-representations. If we would want to consider the direct sum \(\frac{1}{2}\oplus\frac{1}{2}\)
-we would write
-
-
-
-
-
-ss = SU₂Space(1/2 => 2)
-
-
--Show code cell output - -
-
-
-
-
-Rep[SU₂](1/2=>2)
-A symmetric tensor can now be constructed as
-
-
-
-
-
-
-A = TensorMap(l ← s ⊗ s)
-
-
-
-
-TensorMap(Rep[SU₂](1=>1) ← (Rep[SU₂](1/2=>1) ⊗ Rep[SU₂](1/2=>1))):
-* Data for fusiontree FusionTree{Irrep[SU₂]}((1,), 1, (false,), ()) ← FusionTree{Irrep[SU₂]}((1/2, 1/2), 1, (false, false), ()):
-[:, :, 1] =
- 0.0
-This tensor then has, by construction, the symmetry property that it transforms trivially
-under \(1\otimes\bar{\frac{1}{2}}\otimes\bar{\frac{1}{2}}\). The blocks can then be inspected
-by calling blocks on the tensor, and we can also check that the dimensions of the domain
-and codomain are as expected:
-
-
-
-
-
-@assert dim(domain(A)) == 4
-@assert dim(codomain(A)) == 3
-blocks(A)
-
-
-
-
-TensorKit.SortedVectorDict{SU2Irrep, Matrix{Float64}} with 1 entry:
- 1 => [0.0;;]
-We see that this tensor has one block that we can fill up with some data of our liking. Let -us consider another example
-
-
-
-
-
-
-B = TensorMap(s ← s ⊗ s)
-blocks(B)
-
-
-
-
-TensorKit.SortedVectorDict{SU2Irrep, Matrix{Float64}}()
-This tensor does not have any blocks! This is compatible with the fact that two spin 1/2’s -cannot fuse to a third spin 1/2. Finally let us consider a tensor with with more blocks:
-
-
-
-
-
-
-C = TensorMap(ss ← ss)
-blocks(C)
-
-
-
-
-TensorKit.SortedVectorDict{SU2Irrep, Matrix{Float64}} with 1 entry:
- 1/2 => [2.3342e-313 6.15379e-313; 3.39519e-313 6.92768e-310]
-This tensor has four non-trivial entries.
-
-
-11.4. Outlook and generalizations#
-Let us conclude with an outlook and some generalizations.
--
-
Besides the “global” symmetries we considered here, you might also be familiar with gauge -symmetries from another course. Gauge theories are ubiquitous in physics and describe a -plethora of interesting physical phenomena. Gauge symmetries should however not be thought -of as actual symmetries transforming physically different states into each other, but -rather describe a redundancy in the description of the system. Nevertheless, group theory -also lies at the heart of these theories.
-In this brief overview we mostly neglected spatial symmetries. Spatial symmetries can be -understood as transformations that translate, rotate or reflect the lattice. These kind of -symmetries thus don’t act “on site” anymore. The full classification of spatial symmetry -groups is notoriously rich and beautiful, especially in higher dimensions, and exploiting -them in algorithms can result in tremendous speedup and stability. We already encountered -the example of translation symmetry. One of the benefits of exploiting this symmetry in -tensor networks is e.g. that if the ground state of an infinite one-dimensional model does -not break translation invariance, this ground state can be well modelled by a uniform -matrix product state, a matrix product state consisting of one tensor repeated -indefinitely.
-Inspired by the discovery of topological phases of matter and their anyonic excitations, -there has been a growing fascination with the exploration of non-invertible, or -categorical symmetries. These symmetries are beyond the scope of these notes. These -categorical symmetries are not described by groups but by more general and intricate -algebraic structures called fusion categories, of which (finite) groups and their -representations are specific examples. For an example of how spin chains with categorical -symmetries can be constructed, see for example [Feiguin et al., 2007]. TensorKit -allows for an efficient construction and storage of tensors which are symmetric with -respect to these more general kind of symmetries.
-
