Calculus of continuous matrix product states

I Introduction

Many revolutions and breakthroughs in quantum physics, and quantum many body physics in particular, were stimulated by guessing a suitable variational ansatz that captures the relevant correlations for the systems under consideration. Feynman’s ansatz for the roton in superfluid HeliumFeynman (1954); Feynman and Cohen (1956), the Bardeen-Cooper-Schrieffer wave function for superconductivityBardeen, Cooper, and Schrieffer (1957) and the Laughlin wave function for the fractional quantum Hall effectLaughlin (1983) are only a few prominent examples. For gapped one-dimensional quantum spin systems, the set of matrix product statesAffleck et al. (1987, 1988); Fannes, Nachtergaele, and Werner (1992); Verstraete, Murg, and Cirac (2008); Cirac and Verstraete (2009) is a very general ansatz that can describe a range of different phenomena and different physical phases, including normal symmetric and symmetry broken phases as well as the more exotic symmetry-protected topologically ordered phases such as the Haldane phaseHaldane (1983a, b); Pollmann et al. (2010). Indeed, with the benefit of hindsight, we now understand White’s powerful density matrix renormalization group algorithmWhite (1992, 1993) as a variational optimization over the set of matrix product statesÖstlund and Rommer (1995); Rommer and Östlund (1997).

Until recently, few equally general ansatzes that surpass mean field theory were available for extended quantum systems in the continuum, i.e. quantum fields. Numerical approaches require a finite number of degrees of freedom in order to fit the problem in the memory of a computer. For compact systems such as nuclei, atoms and molecules, an expansion in terms of a finite-dimensional basis is possible, but for extended systems this eventually results in a discretization to an effective lattice system. A new variational ansatz field theories in $d=1$ spatial dimensions was developed by Verstraete and Cirac in 2010 Verstraete and Cirac (2010). This ansatz is formulated in the continuum and does not require an underlying lattice approximation. It can be considered to be the continuum limit of a special subclass of matrix product states (MPS) and is therefore called the continuous matrix product state (cMPS) class.

The aim of the current paper is to discuss in greater detail the properties of cMPS. Section II reviews the different definitions and representations of these states in the current literature. We then derive a set of regularity conditions that become relevant in the case of systems with multiple particle species in Section III. Section V discusses how to (efficiently) evaluate expectation values with respect to these states. Section VI is devoted to the gauge invariance and the existence of canonical forms in the continuous matrix product state representation for generic systems without translation invariance. We also discuss uniform continuous matrix product states in the thermodynamic limit and illustrate how continuous matrix product states possess a natural ultraviolet cutoff in Section VII. Finally, Section VIII provides an intuitive construction of tangent vectors to the variational set and discusses their representation properties as well, both for finite systems and in the thermodynamic limit. These tangent states are relevant when studying time evolution or elementary excitations along the lines of analogous MPS algorithms Haegeman et al. (2011, 2012); Pirvu, Haegeman, and Verstraete (2012); Milsted et al. (2012). We do not strive for absolute mathematical rigor, but merely attempt to explain in full detail the prerequisites for using cMPS in numerical algorithms. For example, due to the intrinsic difficulty of the various infinite-dimensional function spaces involved, we do not include a rigorous proof that the set of continuous matrix product states constitutes a smooth (complex) manifold and that the construction of a tangent space is justified.

II Various definitions of the variational class

Consider a quantum system defined on a one-dimensional continuum $\mathcal{R}=[-L/2,+L/2]$ with length $\lvert\mathcal{R}\rvert=L$ that accommodates $q$ bosonic and/or fermionic particle species, which are labeled by the greek index $\alpha=1,\ldots,q$. Throughout this paper, we restrict to non-relativistic systems. A state of the quantum system containing $N_{\alpha}$ particles of type $\alpha$ is then described by a square integrable function on $\prod_{\alpha=1}^{q}\mathcal{R}^{(N_{\alpha})}_{\eta_{\alpha}}$, where $\eta_{\alpha}=+1$ ($-1$) if particle species $\alpha$ is bosonic (fermionic) and $\mathcal{R}^{(N_{\alpha})}_{+}$ ($\mathcal{R}^{(N_{\alpha})}_{-}$) corresponds to the symmetric (antisymmetric) subspace of $\mathcal{R}^{N}$, the Cartesian product of $N$ copies of $\mathcal{R}$. The space of the square integrable functions on this domain is a Hilbert space that is denoted as

Following the principles of second quantization, we now define the Fock space

where $\eta_{\alpha,\beta}=-1$ if both $\alpha$ and $\beta$ represent fermionic particles and $\eta_{\alpha,\beta}=1$ when at least one of the two particles species $\alpha$ or $\beta$ is bosonic. Clearly $\eta_{\alpha,\alpha}=\eta_{\alpha}$. We always write sums over the species index $\alpha$ explicitly and do not use Einstein’s summation convention with respect to this index.

II.2 Original definition

A cMPS is defined to be the state Verstraete and Cirac (2010)

A special case is obtained for $\lambda=0$, since this requires us to redefine $Q(x)$ as $Q^{\prime}(x)=Q(x)-\infty\openone_{D}$. Hence, the null state is not contained within $\mathcal{V}_{\text{cMPS}(D)}$ but only in its closure. Correspondingly, the variational set $\mathcal{V}_{\text{cMPS}(D^{\prime})}$ with $D^{\prime}<D$ is not a subset of $\mathcal{V}_{\text{cMPS}(D)}$. For example, if the boundary matrices are fixed to $B^{\prime}=\openone_{D^{\prime}}$ and $B=\openone_{D}$ (periodic boundary conditions), then a representation of the cMPS $\ket{\Psi^{\prime}[Q^{\prime},\{R_{\alpha}^{\prime}\}]}$ with bond dimension $D^{\prime}$ as a cMPS $\ket{\Psi[Q,\{R_{\alpha}\}]}$ with bond dimension $D>D^{\prime}$ requires $Q=Q^{\prime}\oplus(-\infty\times\openone_{D-D^{\prime}})$ and $R_{\alpha}=R_{\alpha}^{\prime}\oplus(0\times\openone_{D-D^{\prime}})$, hence $\mathcal{V}_{\text{cMPS}(D^{\prime})}$ is only included in the closure of $\mathcal{V}_{\text{cMPS}(D)}$. Note that this differs from the case of MPS on the lattice, where $\mathcal{V}_{\text{MPS}(D^{\prime})}\subset\mathcal{V}_{\text{MPS}(D)}$ for $D\geq D^{\prime}$.

II.3 Fock space embedding

We can then expand the round brackets and reorder the sum in terms of the actual number of created particles by grouping subsequent occurrences of the $Q$ term, so as to obtain

only when $x_{1}\leq x_{2}\leq\cdots\leq x_{N}$. It can be extended to any other order of the arguments by reordering the annihilation operators in Eq. (7) according to the given commutation or anticommutation relations in Eq. (3). The non-relativistic kinetic energy requires that these functions are sufficiently regular, which together with the extension to arbitrary order of the arguments imposes certain non-trivial constraints on the matrix functions $Q$ and $R_{\alpha}$ that are to be discussed in Section III.

II.4 The continuum limit of matrix product states

The cMPS $\ket{\Psi[Q,\{R_{\alpha}\}]}$ was originally constructed in Ref. Verstraete and Cirac, 2010 as the continuum limit of a certain subset of MPS, where the subset was selected in such a way as to obtain a valid continuum limit. We explore this construction in greater detail and elaborate on some of the non-trivial implications regarding ultraviolet cutoffs and correlation lengths (infrared cutoffs).

We approximate the continuum $\mathcal{R}=[-L/2,L/2]$ by a lattice $\mathcal{L}$ with lattice spacing $a$ and $N=L/a$ sites, where we send $a\to 0$. On every site of the lattice we can create and annihilate particles of type $\alpha$ by acting with the creation and annihilation operators $\hat{c}_{\alpha}^{\dagger}(n)$ and $\hat{c}_{\alpha}(n)$. We can relate them to the field operators by

and its hermitian conjugate. The local basis on site $n$ thus consists of the states $\ket{0}_{n}$ (no particles), $\ket{\alpha}_{n}=c_{\alpha}^{\dagger}(n)\ket{0}_{n}$, $\ket{\alpha,\beta}_{n}=c_{\alpha}^{\dagger}(n)c_{\beta}^{\dagger}(n)\ket{0}_{n}$, … On this lattice, we can define an MPS $\ket{\Psi[A]}$ with matrices $A^{s}(n)$ where $s$ can take values , $\alpha$, $(\alpha,\beta)$, … If the local basis is infinite-dimensional, this MPS definition is only formal, i.e. it cannot be used for practical computations. In the limit $a\to 0$, the number of sites $L/a$ in the lattice $\mathcal{L}$ goes to infinity.

together with $\ket{\Omega}=\ket{\bm{0}}=\otimes_{n\in\mathcal{L}}\ket{0}_{n}$, $\forall n=-L/2a,-L/2a+1,\ldots,+L/2a-1$. This equivalence can be obtained from a Taylor expansion of the $\exp$-operator, although this is only completely rigorous when the entries of $Q$ and $R_{\alpha}$ are finite and the operators $\hat{\psi}^{\dagger}(x)$ are bounded (i.e. not for bosons). Most results for cMPS in the remainder of this chapter can be derived from this correspondence with MPS, but we attempt to derive these results directly in the continuum as much as possible.

II.5 Alternative construction through continuous measurement

Rather than trying to construct a cMPS as the continuum limit of a MPS, we could also try to directly define the continuum limit of the processes that define MPS. Unfortunately, the process of sequential Schmidt decompositions has no straightforward generalization to the continuum and neither has the definition of valence bond solids. One can however define a continuum version of the sequential generation process that creates MPSSchön et al. (2005), based on the paradigm of continuous measurement Caves and Milburn (1987). The resulting process for creating cMPS is described in Ref. Osborne, Eisert, and Verstraete, 2010, and is here summarised for the sake of completeness.

We recall that $K(x)$ is a Hermitian matrix. Generic cMPS can be brought into this form by using the gauge invariance of the cMPS representation, as discussed in Section VI.

From a physical perspective, this construction is important as it clearly sketches the holographic properties of the cMPS. The physical state of a one-dimensional system is described by a zero-dimensional boundary theory. The spatial coordinate of the physical system acts as a time coordinate in the boundary theory. The physical state is created because the boundary theory interacts with the physical system, where the position of the interaction shifts linearly in time. This interaction results in the boundary theory not being at equilibrium. Instead, the boundary theory is subject to dissipative dynamics, as will become clear in the following section. This holographic property is of course strongly related with the intrinsic area law for entanglement that is present in cMPS.

II.6 Path integral representation

Recently, it has also been illustrated that we can break up the path ordered exponential in the definition of $\ket{\Psi[Q,\{R_{\alpha}\}]}$ and insert resolutions of the identity in order to obtain a path integral description of the same stateBrockt et al. . The easiest way to insert an identity is by first introducing a second quantized version of the ancilla by making the substitution

Following the standard recipe, we can then obtain the path integral description of $\ket{\Psi[Q,\{R_{\alpha}\}]}$ as

where $\phi(x)$ is a $D$-dimensional vector function with components $\phi_{j}(x)$, $j=1,\ldots,D$. This path integral representation can serve as a useful starting point for generalizations of the cMPS, e.g. by replacing the second quantized auxiliary system by a true field theory, so that this becomes the cMPS analogon of the construction in Ref. Cirac and Sierra, 2010; Nielsen, Sierra, and Cirac, 2011. If this field theory is a conformal field theory, it is then very close in spirit to some model states for Quantum Hall SystemsMoore and Read (1991); Dubail, Read, and Rezayi .

III Regularity conditions

In Eq. (7) we have defined the $N$-particle wave functions $\phi_{\alpha_{1},\ldots,\alpha_{N}}(x_{1},\ldots,x_{N})$. For $x_{1}\leq\cdots\leq x_{N}$ these are completely specified by Eq. (8). However, for general choices of the matrix functions $Q$ and $R_{\alpha}$, the extension of Eq. (8) to all orders of its arguments does not automatically satisfy the required properties that a physical $N$-particle wave function should satisfy. For example, the $N$-particle wave functions should be differentiable in each of its arguments if the state has to produce a finite non-relativistic kinetic energy.

However, there is no need to work with the Fock space expansion of Eq. (6). We can check the regularity of the $N$-particle wave functions by immediately evaluating the kinetic energy in second quantization. For further reference, we first define

In order to compute any expectation value, which is the topic of the next section, we need to be able to act with the field annihilation operators $\hat{\psi}_{\alpha}(x)$ on the state $\ket{\Psi[Q,\{R_{\alpha}\}]}$. If we are able to drag $\hat{\psi}_{\alpha}(x)$ through the path-ordered exponential, it then acts on $\ket{\Omega}$, which is annihilated by any field operator. We can now use Eq. (106) as derived in Appendix A, where $\hat{B}=\hat{\psi}_{\alpha}(x)$, $\hat{A}_{1}(z)$ contains both $Q(z)\otimes\hat{\openone}$ and any term $R_{\beta}(z)\otimes\hat{\psi}^{\dagger}_{\beta}(z)$ for which $\eta_{\alpha,\beta}=1$, and $\hat{A}_{2}(z)$ contains the terms $R_{\beta}(z)\otimes\hat{\psi}^{\dagger}_{\beta}(z)$ for which $\eta_{\alpha,\beta}=-1$. We then obtain

Hence, acting with an annihilation operator of type $\alpha$ at position $x$ not only lowers a matrix $R_{\alpha}(x)$, but also transforms the path ordered exponential $\hat{U}(-L/2,x)$ into $\hat{U}_{\alpha}(-L/2,x)$, because we had to take the particle statistics into account for bringing $\hat{\psi}_{\alpha}(x)$ to the position where it could lower $R_{\alpha}(x)$.

The non-relativistic kinetic energy operator $\hat{T}$ is given by

where the kinetic energy density $\hat{t}(x)$ at position $x$ is given by

Differentiability of the wave function is sufficient for a finite kinetic energy, which is by far the most important physical requirement of the wave function. We can also impose higher regularity constraints on the $N$-particle wave functions. Since these do in general not arise from physical considerations, we postpone this discussion to Appendix B. While the resulting conditions are interesting from an algebraic point of view, they are in general hard to satisfy with finite-dimensional matrices. For practical applications, satisfying the original condition in Eq. (25), as imposed by the finiteness of the kinetic energy, should be sufficient.

We conclude this subsection by investigating what else can be learned from the physical considerations concerning particle statistics. The regularity conditions [Eq. (25)] already require that the matrices $R_{\alpha}$ behave as the corresponding operators $\hat{\psi}_{\alpha}$ in terms of commutation and anticommutation relations. In a physical system, we should not have fermionic condensates, i.e. $\braket{\Psi}{\hat{\psi}_{\alpha}(x)}{\Psi}=0$ if particle species $\alpha$ is fermionic. This is a consequence of the invariance of an physical Hamiltonian $\hat{H}$ under the action of the parity operator $\hat{P}$, which flips the sign of any fermionic operator ($\hat{P}\hat{\psi}_{\alpha}(x)\hat{P}=\eta_{\alpha,\alpha}\hat{\psi}_{\alpha}(x)$) and is thus idempotent ($\hat{P}=\hat{P}^{-1}=\hat{P}^{\dagger}$). We can construct $\hat{P}$ as

with $D^{(+)}+D^{(-)}=D$. The required transformation behavior of $Q$ and $R_{\alpha}$ then imposes the following decomposition

In the cMPS $\ket{\Psi[Q,\{R_{\alpha}\}]}$, all contributions with either an even or an odd number of fermions in Eq. (6) drop out, depending on the boundary matrices $B$. If only states with an even number of fermions are allowed, $B$ should have a decomposition as

is required to select only states with an odd number of fermions.

IV Boundary conditions

We have already mentioned in Section II that the type of boundary conditions —open or periodic— is encoded in the rank of the boundary matrix $B$. For a system with periodic boundary conditions, $B$ has full rank and is typically chosen to be the identity ($B=\openone_{D}$). Since periodic boundary conditions identify the points $x=-L/2$ and $x=+L/2$, it is natural to assume that the matrix functions $Q$ and $R_{\alpha}$ are also single-valued, i.e. $Q(-L/2)=Q(+L/2)$ and $R_{\alpha}(-L/2)=R_{\alpha}(+L/2)$ for all $\alpha=1,\ldots,q$.

A more detailed discussion of these conditions is presented in Ref. Verstraete , where a partial differential equation for the evolution of $Q$ and $R_{\alpha}$ under real or imaginary time dynamics is derived. In order to solve this partial differential equation, it needs to be complemented by the proper boundary conditions as given above. Throughout the remainder of this manuscript, we assume that we are working with cMPS where the matrix functions $Q$ and $R_{\alpha}$ satisfy the required conditions.

V Computation of expectation values

This section is concerned with the computation of expectation values of normally ordered operators. We have already illustrated how to act with annihilation operators and derivatives thereof in the Section III. With a MPS, the computation of expectation values boils down to a contraction of the physical indices in the network. In the continuum, however, the intuitive notion of physical indices is a bit lost. We therefore start by computing the overlap of two cMPS $\ket{\Psi[Q,\{R_{\alpha}\}]}$, $\ket{\Psi[Q^{\prime},\{R_{\alpha}^{\prime}\}]}$, which are given as an expansion in Fock space [Eq. (6)]. It is clear that the basis states $\hat{\psi}^{\dagger}_{\alpha_{1}}(x_{1})\cdots\hat{\psi}^{\dagger}_{\alpha_{N}}(x_{N})\ket{\Omega}$ are automatically orthogonal for different $N$, and further that

due to the ordering of the arguments $x_{1}\leq\cdots\leq x_{N}$ and $y_{1}\leq\cdots\leq y_{N}$. We thus obtain

Using trivial direct product identities such as $\operatorname{tr}[A]\operatorname{tr}[B]=\operatorname{tr}[A\otimes B]$, $(AB)\otimes(CD)=(A\otimes B)(C\otimes D)$ and $\exp(A)\otimes\exp(B)=\exp(A\otimes\openone_{D}+\openone_{D}\otimes B)$ for $D\times D$ matrices $A$, $B$, $C$ and $D$, the previous expression can be rewritten as

Reverting the expansion of the path ordered exponential that lead to Eq. (6), results in

The expectation value of any normally ordered operator $\hat{O}=:O[\{\hat{\psi}^{\dagger}_{\alpha}\},\{\hat{\psi}_{\beta}\}]:$ can now be computed by first acting with all annihilation operators $\hat{\psi}_{\alpha}(x)$ on the ket $\ket{\Psi[Q,\{R_{\beta}\}]}$ as we did in the Section III, and similarly acting with the creation operators on the bra. The result of this is the insertion of some operators acting on the virtual system at the corresponding positions, with operators $\hat{U}(x,y)$, $\hat{U}_{\alpha}(x,y)$ or $\hat{U}_{\alpha,\beta}(x,y)$ connecting them. The expectation value is obtained by “contracting the physical indices”, which results in the inserted virtual operators in the ket combining with those in the bra at the same positionIf there is no insertion at the same position, we can always insert a unit operator $\openone_{D}$, whereas the contraction of the part in between the local insertions result in a path ordered exponential of the transfer matrix. However, to incorporate the particle statistics, we also need to define generalized transfer operators as

All quantities in this expression, if we could store and manipulate variables with a fully continuous $x$-dependence, are $D^{2}\times D^{2}$ matrices. Since such matrices need to be multiplied, this is an operation with computational complexity of $\operatorname{\mathscr{O}}(D^{6})$, or $\operatorname{\mathscr{O}}(D^{5})$ if we exploit the tensor-product structure.

From the correspondence with completely positive maps, it can be shown that the solution $l(x)$ and $r(x)$ of Eq. (42) starting from positive definite initial conditions $l(-L/2)$ and $r(+L/2)$ are positive for any $x\in\mathcal{R}$ (see Theorem 3 in Ref. Lindblad, 1976). The norm is thus guaranteed to be positive. Note that, for the special parameterization of $Q(x)$ in the continuous measurement interpretation [Eq. (12)], we can write the determining differential equation for $r(x)$ as

with $\delta\ /\delta J_{\alpha}$ the functional derivative with respect to $J_{\alpha}$, and

which for a system with open boundary conditions results in

Let us now illustrate this approach by defining a generic Hamiltonian for a single-boson system with open boundary conditionsWhile we mentioned in Section IV that we always assume the matrix functions $Q$ and $R_{\alpha}$ to satisfy the proper boundary conditions, we do not have to use the condition in Eq. (33) at any point in deriving the expectation value of the Hamiltonian $\hat{H}$ in Eq. (48).

describing particles with mass $m$ that interact with an external potential $v(x)$ and with each other through two-particle interaction $w(x,y)$.

Using Eq. (45) we find (henceforth omitting the arguments $Q$ and $R$ in the state $\ket{\Psi}$)

Defining $R^{(l)}_{x}(x)=R(x)^{\dagger}l(x)R(x)$ for every $x\in[-L/2,+L/2]$ and solving

for every $y\in[x,L/2]$, we can write the expectation value of the potential and interaction energy as

To evaluate the expectation value of the kinetic energy, we compute

where we used the same trick. Note that derivatives with respect to the Heaviside functions (which would produce a diverging contribution $\delta(x-y)$) nicely cancel for both derivatives with respect to $y$ and to $x$. As noted in the Section III, the regularity condition Eq. (25) is automatically fulfilled for the case of a single boson. We thus obtain

VI Gauge invariance

While we prefer the continuum derivation, these transformation formulas can also be obtained by using the correspondence with MPS [Eq. (10)] and the well-known gauge transformations for MPS Haegeman et al.

Clearly, this differential equation only determines $g(x)$ up to a unitary prefactor. Put differently, for any solution $g(x)$ of this equation, $g^{\prime}(x)=u(x)g(x)$ with $u(x)$ a unitary matrix is an equally valid solution. We can use the remaining gauge freedom $u(x)\in\mathsf{U}(D)$ to diagonalize $r(x)$ at every point $x$, hence obtaining the left-canonical form.

Alternatively, we can also impose the right orthonormalization condition, which boils down to

Clearly, for any solution $g(x)$, we obtain a family of solutions $g^{\prime}(x)=g(x)u(x)$ with $u(x)\in\mathsf{U}(D)$. This unitary freedom can be fixed by diagonalizing $l(x)$, resulting in the right-canonical form. As for the left-canonical form, one has to pay careful attention to the boundary conditions that need to be satisfied by $g$. For a system with open boundary conditions, the easiest solution is again to include one of the boundary vectors in the set of the variational parameters and also transform it under the action of the gauge transform.

Note that we can also define a gauge transformation $g(x)$ for the cMPS $\ket{\Psi[Q,\{R_{\alpha}\}]}\in\mathcal{M}_{\text{cMPS}}$ so that

This formulation is close in spirit to the bosonic mean field ansatz

VII Translation invariance and the thermodynamic limit

In expectation values of local operators, this overall factor always appears, but the rest of the expression will not depend on $B$. Hence, the $B$-dependence is cancelled by considering normalized expectation values, or by henceforth choosing $B$ such that $\braket{\Psi(Q,\{R_{\alpha}\})}{\Psi(Q,\{R_{\alpha}\})}=(l|B\otimes\overline{B}|r)=1$.

We can now evaluate correlation functions as

We define the Fourier transformed correlation function

then it is easy to see why we have used the suggestive notation $n_{\alpha,\beta}$ for the Fourier transform of $C_{\alpha,\beta}$. We obtain

These results can be used to show that both the second and third term in the expansion vanish when they are plugged into Eq. (68). The first non-vanishing term is thus of order $p^{-4}$. Because $n_{\alpha,\beta}(p)$ is a dimensionless quantity, this asymptotic behavior allows us to introduce a momentum cutoff $\Lambda$ as

we have now defined a UV cutoff scale $\Lambda$ based on the large momentum behavior of the momentum occupation number $n_{\alpha,\beta}(p)$.

Let us also illustrate how to compute the expectation value of a translation invariant Hamiltonian. The generic Hamiltonian in Eq. (48) becomes translation invariant for $v(x)=v$ and $w(x,y)=w(y-x)$ with $w(x)=w(-x)$. Since the uniform cMPS is extensive, expectation values are proportional to the volume and it makes more sense to compute the expectation values of the kinetic, potential and interaction energy densities $\hat{t}$, $\hat{v}$ and $\hat{w}$. We obtain

VIII Tangent vectors of continuous matrix product states

For MPS, a new algorithm for time evolution and variational optimization (via imaginary time evolution) was recently constructed using the time-dependent variational principleHaegeman et al. (2011). An essential ingredient of this algorithm is the study of (infinitesimally) small variations of MPS, i.e. the set of MPS tangent vectors. Indeed, it was rigorously proven that the set of MPS can be given the structure of a variational manifold with a well-defined tangent spaceHaegeman et al. by eliminating some singular points or regions. While we do expect the same theorems to hold for cMPS, the infinite dimensionality of the parameter space and Hilbert space might require a different proof strategy, especially in the absence of translation invariance. As noted several times before, this would be beyond the scope of this paper.

where the maps $\mathscr{M}_{\Phi}^{[Q]}$ and $\mathscr{N}_{\alpha,\Phi}^{[R_{\alpha}]}$ ($\forall\alpha=1,\ldots,N$) are given by

We now restrict to the case of open boundary conditions and discard the explicit reference to the base point $\ket{\Psi[Q,\{R_{\alpha}\}]}$ in the notation of tangent vectors. To take full advantage of the gauge freedom, we noted in Section VI that is better to include one of the boundary vectors in the set of variational parameters. We thus generalize our definition of tangent vectors by also including variations with respect to e.g. the right boundary vector $\bm{v}_{\text{R}}$. We write

The overlap between two tangent vectors is given by

which together with the boundary condition $h(-L/2)=0$ results in the solution

Alternatively, we can also impose a right gauge fixing condition

VIII.2 Uniform case

We can also compute the overlap between two of these tangent vectors and obtain

If we again resort to the decomposition of Eq. (VII), we can further evaluate this to

The momentum eigenstates $\ket{\Phi_{p}(V,\{W_{\alpha}\})}$ cannot be normalized to unity in the thermodynamic limit, but rather satisfy a $\delta$-normalization. For $p=p^{\prime}=0$, there is an additional divergence which is stronger than the $\delta$-normalization. It can be related to the overlap between the $\ket{\Phi_{p}(V,\{W_{\alpha}\})}$ and the original cMPS $\ket{\Psi(Q,\{R_{\alpha}\})}$, which is given by

IX Conclusion and outlook

This manuscript provides a detailed description of a variational class of wave functions for one-dimensional quantum field theories, that goes by the name of “continuous matrix product states”. We reviewed different alternative constructions that produce the same class of states and have their own merits, e.g. in offering clear hints on how to generalize this class to different settings such as open quantum systems or higher-dimensional theories.

We illustrated how to formulate the cMPS ansatz for the most general class of theories including an arbitrary number of bosonic and fermionic particles, and were naturally led to a set of constraints that the variational parameters needed to satisfy in order to produce a finite kinetic energy density. We also discussed other physical constraints such as fermion parity. We then proceeded by explaining in detail how to compute expectation values, in particular for the case of systems with open boundary conditions. We provided some additional details for the case of systems with translation invariance, where we can use the expectation value of a correlation function to define an ultraviolet cutoff within the cMPS state.

We also discussed the important topic of gauge invariance in the cMPS representation. Finally we introduced the concept of cMPS tangent vectors, and discussed how the gauge invariance allows to represent them in such a way that the metric of the cMPS manifold simplifies tremendously.

While we have not introduced any practical algorithms or recipes for finding cMPS approximations of ground states or for describing other physical phenomena, we have introduced all necessary definitions and concepts in order to comfortably work with cMPS. This set of definitions can now be used in follow-up papers that will focus on new algorithms. As such, the current paper provides a stepping stone that will hopefully spur more research in the context of variational methods for quantum field theories in one dimension and beyond.

References

Appendix A A useful formula

Consider an operator $\hat{U}(x,y)$ defined as

where $\hat{A}$ is not necessarily antihermitian. This operator satisfies

For the derivatives of the inverse operator $\hat{U}(x,y)^{-1}$ we can use the general result

Now define the following operator quantity depending on an arbitrary operator $\hat{B}$

By taking the derivative with respect to $y$, we obtain

We then multiply this equality with $\hat{U}(x,y)$ to the right and make use of the obvious identity $\hat{U}(x,y)=\hat{U}(x,z)\hat{U}(z,y)$ for any $x<z<y$ in the integral of the right hand side in order to obtain our final result

We can further generalize this result. Suppose we have two operators $\hat{U}_{\pm}(x,y)$ defined as

for arbitrary $\hat{A}_{1,2}(z)$. If we consider the quantity

using a similar derivation. Continuing along the same line results in

Appendix B Higher order regularity conditions

We can also impose regularity of the mixed derivatives of the $N$-particle wave function, by first evaluating $\hat{\psi}_{\alpha}(x)\hat{\psi}_{\beta}(y)\ket{\Psi[Q,\{R_{\gamma}\}]}$

where a new set of operators $\hat{U}_{\alpha,\beta}(x,y)$ ($\alpha,\beta=1,\ldots,q$) was introduced as

Note that the regularity condition in Eq. (25) is sufficient for the annihilation of two particles $\hat{\psi}_{\alpha}(x)\hat{\psi}_{\beta}(y)\ket{\Psi[Q,\{R_{\gamma}\}]}$ to be continuous at $x=y$. By first differentiating with respect to $x$, we obtain

where we have assumed the regularity condition in Eq. (25) to hold. This allows one to eliminate the fixed insertion of particles at position $x$ as well as the terms obtained from differentiating the Heaviside functions (i.e. the terms proportional to $\delta(x-y)$). Such terms would indeed arise if $\hat{\psi}_{\alpha}(x)\hat{\psi}_{\beta}(y)\ket{\Psi[Q,\{R_{\gamma}\}]}$ were not continuous at $x=y$. If we now also differentiate with respect to $y$, we obtain a divergent contribution

If we differentiated with respect to $y$ first, and then to $x$, the divergent contribution is

The higher order regularity conditions derived in this appendix put very strong constraints on $Q$ and $R_{\alpha}$ that might be hard to satisfy with finite-dimensional matrices. As mentioned in the main text, satisfying the original condition in Eq. (25), as imposed by the finiteness of the kinetic energy, should be sufficient for most practical applications.