commit 36f583c6f407515c537159a82464b9507a292382
parent 41a344a32de519d61b69bd2f493bc0d843e5b6bb
Author: miksa <milutin@popovic.xyz>
Date: Tue, 3 May 2022 09:02:03 +0200
started workout
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diff --git a/qmbs_sem/build/quantum_phases.pdf b/qmbs_sem/build/quantum_phases.pdf
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diff --git a/qmbs_sem/quantum_phases.tex b/qmbs_sem/quantum_phases.tex
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+\include{preamble.tex}
+
+\begin{document}
+\maketitle
+\tableofcontents
+\section{Introduction}
+MPS or PEPS (in 2D) describe ground states of gapped local Hamiltonian of
+quantum many body systems. We will use this fact to generalize Landau's
+theory (to do: what is Landau's theory brief explanation, How do we describe
+these with MPS in particular). Here we consider only ground states that can
+be represented by MPS exactly. Two systems are in the same phase, if and only
+if the can be connected by a smooth path of local Hamiltonian's on the
+manifold of the parameters $\lambda$, where the local Hamiltonian's $h_i =
+h_i(\lambda)$ are all dependent on the parameters $\lambda$, intuitively this
+would look the figure below are all dependent on the parameters $\lambda$,
+intuitively this would look the figure below.
+
+\begin{figure}[H]
+ \centering
+\begin{tikzpicture}[]
+ \node[thick] at (3.8, 0.2) {$\lambda$};
+
+ \draw[-, thick] (0,0)--(4,0);
+ \draw[-, thick] (4,0)--(4,3);
+ \draw[-, thick] (4,3)--(0,3);
+ \draw[-, thick] (0,3)--(0,0);
+ \draw[-, thick] (1,3)--(3,0);
+
+
+ \filldraw[black] (1, 0.5) circle (0.1cm);
+ \filldraw[black] (0.5, 2.1) circle (0.1cm);
+
+% \filldraw[black] (1.5, 1.3) circle (0.1cm);
+% \filldraw[black] (2.4, 1.8) circle (0.1cm);
+% \draw[thick, dotted] (1.5, 1.3) to[out=20, in=-80] (2.4, 1.8);
+
+ \filldraw[black] (3.4, 2.4) circle (0.1cm);
+ \filldraw[black] (3.1, 1.25) circle (0.1cm);
+
+ \draw[thick, dotted] (3.1, 1.25) to[out=20, in=-80] (3.4, 2.4);
+ \draw[thick, dotted] (1, 0.5) to[out=20, in=-80] (0.5, 2.1);
+
+\end{tikzpicture}
+\caption{Two systems in the same phase are connected by a ''smooth path of local
+Hamiltonians``}
+\end{figure}
+Where along such paths the physical properties of the sate smoothly change.
+The Hamiltonian needs to be ''gapped``, meaning that there is a clear
+separation of the ground state and the first excited state. The loss of a
+gap in the Hamiltonian leads mostly to discontinuous ``behavior'' of the ground
+state and affection of global properties of the system. If we introduce
+symmetries along the path of such Hamiltonian we can derive a refined
+classification of phases. Additionally if such symmetries exist we can
+generalize gapped quantum phases to systems with symmetry breaching!
+%\printbibliography
+\section{Matrix Product States (MPS)}
+In the following we only consider translation-invariant systems on a finite
+chain of length $N$, with periodic boundary condition
+\begin{mydef}
+ Consider a spin chain $(\mathbb{C}^{d})^{\otimes N}$. A
+ translation-invariant MPS $\ket{\mu[\mathcal{P}]}$ of bond dimension $D$
+ on $(\mathbb{C}^{d})^{\otimes N}$ is constructed by placing maximally
+ entangled pairs $\ket{\omega_D}$, as
+ \begin{align}
+ \ket{\omega_D} := \sum_{n=1}^{D} \ket{i, i}
+ \end{align}
+ between adjacent sites and applying a linear map $\mathcal{P}:
+ \mathbb{C}^{D} \otimes \mathbb{C}^{D} \rightarrow \mathbb{C}^{d}$. In
+ graphical notation it would represent the figure below
+ \begin{figure}[H]
+ \centering
+ \begin{tikzpicture}[]
+ \filldraw[black] (0, 0) circle (0.1cm);
+ \filldraw[black] (0.5, 0) circle (0.1cm);
+
+ \filldraw[black] (1.5, 0) circle (0.1cm);
+ \filldraw[black] (2, 0) circle (0.1cm);
+
+ \filldraw[black] (3, 0) circle (0.1cm);
+ \filldraw[black] (3.5, 0) circle (0.1cm);
+
+ \draw[-, thick] (-0.5, 0) -- (0, 0);
+ \draw[-, thick] (0.5, 0) -- (1.5, 0);
+ \draw[-, thick] (2, 0) -- (3, 0);
+ \draw[-, thick] (3.5, 0) -- (4, 0);
+
+ \draw[very thick] (0.25,0) ellipse (0.4cm and 0.3cm);
+ \draw[very thick] (1.75,0) ellipse (0.4cm and 0.3cm);
+ \draw[very thick] (3.25,0) ellipse (0.4cm and 0.3cm);
+
+ \draw[->, very thick] (0.25, -0.5) -- (0.25, -1.5) node[midway, right] {$\mathcal{P}$};
+ \draw[->, very thick] (1.75, -0.5) -- (1.75, -1.5) node[midway, right] {$\mathcal{P}$};
+ \draw[->, very thick] (3.25, -0.5) -- (3.25, -1.5) node[midway, right] {$\mathcal{P}$};
+
+ \filldraw[black] (0.25, -1.8) circle (0.15cm);
+ \filldraw[black] (1.75, -1.8) circle (0.15cm);
+ \filldraw[black] (3.25, -1.8) circle (0.15cm);
+
+ \end{tikzpicture}
+ \end{figure}
+ \begin{align}
+ \ket{\mu[\mathcal{P}]} := \mathcal{P}^{\otimes N}\ket{\omega_D}^{\otimes N}
+ \end{align}
+\end{mydef}
+We note that the MPS as defined above is robust under blocking sites, we are
+essentially blocking $k$-sites into one ''super``-site of dimension $d^k$,
+which gives a new MPS with the same bond dimension in the lines of the
+projector (which is not a projection but a simple linear map)
+\begin{align}
+ \mathcal{P}' = \mathcal{P}^{\otimes k} \ket{\omega_D}^{\otimes (k-1)}.
+\end{align}
+By this blocking and using of the gauge degrees of freedom (including the
+variability of $D$) any MPS which is well defined in the Thermodynamic limit
+($\beta \rightarrow 0$) can be brought into a so called \textbf{Standard
+form}, where the linear map $\mathcal{P}$ is supported on a block-diagonal
+space, i.e diagonalisation of the
+\begin{align}
+ \text{ker}(\mathcal{P})^{\perp} =
+ \mathcal{H}_1 \oplus \cdots \oplus \mathcal{H}_{\mathcal{A}},
+\end{align}
+where
+\begin{align}
+ \mathcal{H}_\alpha = \text{span}\left\{ \ket{i, j}: \zeta_{\alpha- 1}<
+ i,j \le \zeta_\alpha \right\},
+\end{align}
+for $0 = \zeta_0 < \cdots < \zeta_\mathcal{A} = D$, and gives the
+partitioning $1, \ldots, D$ for $D_i = \zeta_i - \zeta_{i-1}$. The case of
+$\mathcal{A}=1$ we have an injective map $\mathcal{P}$. The $\mathcal{A} >1$
+is the non-injective case.
+
+All in all we assume that $\mathcal{P}$ is \textbf{surjective}, which is
+backed by the restriction of the state space $\mathbb{C}^{d}$ to the image of
+$\mathcal{P}$ (by definition).
+\subsection{Parent Hamiltonian}
+Given an MPS in the standard form, we can construct local,
+translation-invariant \textbf{parent Hamiltonians}, which have the given MPS
+as the \textbf{ground state}
+\begin{align}\label{eq: hamiltonian}
+ H = \sum_{i=1}^{N} h(i,i+1).
+\end{align}
+The local terms $h(i, i+1) \ge 0$ act on one MPS object $(i, i+1)$ mapped by
+$\mathcal{P}$. The kernels of these local terms support the reduced density
+operator of the corresponding MPS, that is the kernel can be written as
+\begin{align}
+ \text{ker}(h(i,i+1)) = (\mathcal{P} \otimes
+ \mathcal{P})(\mathbb{C}^{D}\otimes \ket{\omega} \otimes \mathbb{C}^{D}).
+\end{align}
+Note that by the definition we have first that $H \ge 0$, and that
+$H\ket{\mu[\mathcal{P}]} = 0$, because the system $\ket{\mu[\mathcal{P}]}$ is
+the ground state of $H$.
+\newline
+
+To summarize, given a matrix product state (MPS) there exists a unique gapped
+local parent Hamiltonian, where the given MPS is in the groundstate
+(Perez-Garcia et al. 2007). Also, backed up by the fact that the groundstate
+of any one dimensional, gapped Hamiltonian can be well approximated by an MPS
+(proven by Hastings 2007).
+\subsection{Definition of quantum phases}
+We arrive at the definition of quantum phases, where we initially pose a
+question whether two systems are in the same phase. Two systems are in the
+same phase if they can be connected by a continuous path of gapped local
+Hamiltonians
+\subsubsection{Phases without symmetries}
+Let $H_1, H_2$ be a family of translation-invariant gapped local Hamiltonians
+on a ring (i.e. periodic boundary conditions). We say that $H_1$ and $H_2$
+are in the same phase, if and only if exists an finite $k$, when blocking $k$
+sites both $H_1$ and $H_2$ are two local and can be written as
+\begin{align}
+ H_p = \sum_{i=1}^{N} h_p(i,i+1) \qquad p=0,1.
+\end{align}
+Additionally to this there exists a translation-invariant path of local
+gapped Hamiltonians
+\begin{align}
+ H_\gamma = \sum_{i=1}^{N} h_\gamma(i,i+1) \qquad \gamma \in [0, 1],
+\end{align}
+where $h_\gamma$ is acting locally with the following properties
+\begin{itemize}
+ \item $h_0 = h_{\gamma=0}$ ; $h_1 = h_{\gamma=1}$
+ \item $\|h\|_{op} \le 1$
+ \item $h_\gamma$ is continuous w.r.t. $\gamma \in [0, 1]$
+ \item $H_\gamma$ has a spectral gap above the ground state manifold,
+ bounded below by some constant $\Delta >0$ independent of $N$ and
+ $\gamma$.
+\end{itemize}
+We can say that $H_0$ and $H_1$ are in the same phase if they are connected
+by a local, bound-strength, continuous and gapped path, which applies to both
+Hamiltonians with unique and degenerate ground states.
+\subsubsection{Phases with symmetries}
+Let $H_p$, with $p \in \left\{ 0, 1 \right\} $ be a Hamiltonian acting on the
+space $\mathcal{H}^{\otimes N}_p$ where $\mathcal{H}_p=\mathbb{C}^{d_p}$ and
+$U_g^p$ be a linear unitary representation of some group $G \ni g$ of
+$\mathcal{H}_p$. Now, $U_g$ is a symmetry of a family of local gapped
+Hamiltonians $H_p$, if
+\begin{align}
+ [H_p, (U_g^p)^{\otimes N}] = 0 \qquad \forall g\in G,
+\end{align}
+where $U_g^p$ is a strictly one dimensional representation of the group $G$
+as
+\begin{align}
+ U_g^p \leftrightarrow e^{i\phi_g^p}U_g^p.
+\end{align}
+We can say that $H_1$ and $H_2$ are in the same phase under symmetry $G$, if
+there exists a phase gauge of $U_g^0$ and $U_g^1$ and a representation
+\begin{align}
+ U = U_g^0 \oplus U_g^1 \oplus U_g^{\alpha} \qquad \alpha \in (0, 1)
+\end{align}
+on the Hilbertspace $\mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus
+\mathcal{H}_\alpha$ with the properties of the previous section, such that
+\begin{align}
+ [H_\gamma, U_g^{\otimes N}] = 0
+\end{align}
+and $H_p$ is supported on $\mathcal{H}_p$ for $p=0, 1$ respectively.
+\subsubsection{Robust definition of Phases}
+It is usually required for a phase to be \textbf{robust}, meaning that the
+phase is an open set in the space of allowed Hamiltonians. For all
+Hamiltonians \ref{eq: hamiltonian} there exists and $\varepsilon >0$, such
+that
+\begin{align}
+ H = \sum_{i=1}^{N} \left( h(i,i+1) + \varepsilon k(i, i+1) \right)
+\end{align}
+is in the same phase for any bound-strength $k(i,i+1)$ which obeys the
+symmetries of the system.
+\subsubsection{Restriction to parent Hamiltonians}
+Indeed we want a classification of phases of gapped local Hamiltonians with an
+exact MPS ground state. We are in luck because for every MPS we can find such
+a Hamiltonian, the parent Hamiltonian which is sufficient enough to classify
+the phases.
+
+For two gapped Hamiltonians $H, H'$ with some ground state subspace, the
+interpolating path
+\begin{align}
+ \gamma H + (1-\gamma)H'
+\end{align}
+has all the desired properties and it is gapped. Indeed all parent
+Hamiltonians for a given MPS are interchangeable!
+
+\begin{figure}[H]
+ \centering
+\begin{tikzpicture}[]
+ \node[thick] at (0, 0) {$\ket{\psi_0}$};
+ \node[thick] at (2, 0) {$\ket{\psi_1}$};
+ \node[thick] at (0.25, 1) {$\ket{\hat{\psi}_0}$};
+ \node[thick] at (1.75, 1) {$\ket{\hat{\psi}_1}$};
+
+ \node[thick] at (4, 0) {$H_0$};
+ \node[thick] at (6, 0) {$H_1$};
+ \node[thick] at (4.25, 1) {$\hat{H_0}$};
+ \node[thick] at (5.75, 1) {$\hat{H_1}$};
+
+ \draw[thick, dotted] (0.3, 0) to[out=20, in=160] (1.7, 0);
+ \draw[thick, dotted] (4.3, 0) to[out=20, in=160] (5.7, 0);
+
+ \draw[line width=0.1cm, opacity=0.5] (0, 0.2) to[out=90, in=90,
+ distance=1.7cm] (2, 0.2);
+ \draw[line width=0.1cm, opacity=0.5] (4, 0.2) to[out=90, in=90,
+ distance=1.7cm] (6, 0.2);
+
+ \draw[-stealth, line width=0.1cm] (2.5, 0.5) -- (3.5, 0.5);
+
+
+
+\end{tikzpicture}
+\caption{Interchangeability of MPS and parent Hamiltonians}
+\end{figure}
+\subsection{The isometric Form}
+\subsubsection{Reduction to a standard form}
+Given two MPS $\ket{\mu [\mathcal{P}_p]}$ with $p=0,1$ together with their
+nearest neighbor parent Hamiltonians $H_p$. Our goal is to see weather $H_1$
+and $H_2$ are in the same phase. This is achieved by interpolating
+$\mathcal{P}_0$ and $\mathcal{P}_1$ along $\mathcal{P}_\gamma$, such that the
+result is the path $H_\gamma$ in the space of parent Hamiltonians satisfying
+all the requirements (continuity and gap).
+\subsubsection{The isometric form}
+The isometric form of a MPS captures the essential entanglement, long
+range properties of the sate and forms a fixed point of a renormalization
+procedure. Given an MPS state $\ket{\mu[\mathcal{P}]}$ we decompose
+$\mathcal{P}$ by the \textbf{Polar-decomposition} of
+$\mathcal{P}|_{(\text{ker}\mathcal{P})^\perp}$ as
+\begin{align}
+ \mathcal{P} = QW,
+\end{align}
+where $WW^\dagger= \mathbbm{1}$ and $Q > 0$. And w.l.o.g. we assume $0<Q\le
+\mathbbm{1}$ which can be achieved by rescaling of $\mathcal{P}$. The
+isometry form of $\ket{\mu[\mathcal{P}]}$ is $\ket{\mu[W]}$, where the MPS
+described by $W$ is the isometric part of the tensor $\mathcal{P}$. To see
+that $\ket{\mu[\mathcal{P}]}$ and $\ket{\mu[W]}$ are in the same phase, we
+essentially define an interpolating path in terns of $Q_\gamma$
+\begin{align}
+ \mathcal{P}_\gamma = Q_\gamma W \quad \text{where} \quad
+ Q_\gamma = \gamma Q + (1-\gamma)\mathbbm{1},
+\end{align}
+for $\gamma \in [0, 1]$. No consider the parent Hamiltonian of
+$\ket{\mu[\mathcal{P}_0]}$
+\begin{align}
+ H_0 = \sum_{i=1}^{N}h_0 (i, i+1)
+\end{align}
+where $h_0$ is a projector and we define $\Lambda_\gamma =
+(Q^{-1}_\gamma)^{\otimes 2}$ for a $\gamma$-deformed Hamiltonian
+\begin{align}
+ H_\gamma = \sum_{i=1}^{N} h_\gamma(i, i+1) \quad \text{where} \quad
+ h_\gamma =
+ \Lambda_\gamma h_0 \Lambda_\gamma \ge 0.
+\end{align}
+Now we have that $\ket{\mu[\mathcal{P}_0]} = 0$ is equivalent to
+$\ket{\mu[\mathcal{P}_\gamma]} = 0$, i.e. $H_\gamma$ is a parent Hamiltonian
+of $\ket{\mu[\mathcal{P}_\gamma]}$. All we need to show now is that
+$H_\gamma$ is uniformly gapped, that there exists a constant $\Delta >0$
+which $H_\gamma$ by bellow independent of $\gamma$ and $N$. By this we would
+have that the whole set of $\ket{\mu[\mathcal{P}_\gamma]}$ for $\gamma \in
+[0, 1]$ are indeed in the same phase.
+
+Additional observation is that the lower bound of the gapped parent
+Hamiltonians is bound by correlation length $\xi$ of the gap $H_\gamma$,
+restricted to $\xi$ sites and since both depend smoothly, positive definite
+on $\gamma$ and $\xi \rightarrow 0$ as $\gamma \rightarrow 0$ we have a
+uniform lower bound on the gap.
+\end{document}
+
