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)

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)

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] =
 6.36182015857375e-310

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 => [6.36182e-310;;]

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 => [6.36182e-310 6.36182e-310; 6.36182e-310 6.36182e-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.