Lattice is a way of discretizing a quantum field theory for numerical simulations.

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What is the form of the kinetic energy operator on a one-dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the Hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
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Preparatory knowledge to study Quantum Gravity

I am an old (41-Years-old) physics graduate. I pretty much quit to "frequent physics" 10 years ago and my ability to manage physics is a bit rusty. I was fascinated with Quantum Gravity (QG) theory ...
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Lattice QFT: Non-homogenous lattice spacing

I am interested why we fix the lattice spacing, $a$ to be homogenous in all dimensions. After a Wick rotation, $a=i \epsilon$ where $\epsilon=t_{i+1} - t_{i}$ and with euclidean time given by $\tau = ...
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How to understand the entanglement in a lattice fermion system?

Topological insulator is a fermion system with only short-ranged entanglement, what does the entanglement mean here? For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is $V_s=V_1\...
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55 views

Partition sum for $SO(N)$ one-dimensional lattice model

I'm looking for derivation of explicit form of partition function for $SO(N)$ one-dimensional lattice model. The initial expression is $$ Z = \int \limits_{-\infty}^{\infty}d\sigma_{1}...d\sigma_{N}\...
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Sum rule and dense limit

Is the finite size of lattice constant important for derivation of sum rules, for example, the sum rule for diagonal conductivity $\sigma_{xx}$: $$ \int \limits_{-\infty}^{\infty} d\omega\sigma_{xx}(\...
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Is the U(1) gauge theory in 2+1D dual to a U(1) or an integer XY model?

The compact U(1) lattice gauge theory is described by the action $$S_0=-\frac{1}{g^2}\sum_\square \cos\left(\sum_{l\in\partial \square}A_l\right),$$ where the gauge connection $A_l\in$U(1) is defined ...
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Triangular and Kagome lattice anti ferromagnet at zero temperature

The triangular lattice with anti ferromagnetically coupled nearest neighbour ising spins has a power law ordered zero temperature state at the three sublattice wavevector. Kagome lattice, with the ...
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Is it possible to do a path integral between two boundaries analytically on a quantum lattice?

I have been trying to perform some path integral between two boundaries for a massless scalar field $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
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21 views

Clockwise and Anti-Clockwise next-nearest hopping on honeycomb lattice?

I have the difficulty of understanding, How we can distinguish that which next-nearest hopping on honeycomb lattice is clockwise or anticlockwise?
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Fermi-level of lattice model

I have a chain of $N$ lattice points with periodic boundary conditions. Every lattice point only has one orbital for an electron to occupy and spin is not included for simplicity. The lattice points ...
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54 views

Which groups can be lattice gauge groups?

Let me state first off that in this question I am most interested in lattice gauge theories, and not necessarily with Fermion couplings. But if Fermions and continuum gauge theories can also be ...
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43 views

Field theory on a lattice, what is this?

I only have the training of undergraduate quantum mechanics and solid state physics. For me, field theory is defined on a continuous space. So, what does lattice field theory mean? Is it similar to ...
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24 views

How to determine the basis of any crystal

In the following example, I have a rutile TiO2. Here the basis of O with respect to Ti is (0.3*x, 0.3*y, 0*z), where a,b,c are the lattice vectors. So in order to build a crystal, I need the basis and ...
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Edge states of Kitaev chain [closed]

I am reading paper about Kitaev chain of electrons, which can exhibit famous Majorana fermions at ends of wire. The Hamiltonian (his Eq. (6)) reads $H = \frac{i}{2} \sum_j - \mu c_{2j-1}c_{2j} +(w+|\...
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122 views

Reciprocal lattice and bloch waves

I have a simple question (but I precise that I am very a beginner in solid states physics). I know that the reciprocal vectors have this form : $ \overrightarrow{G}=n_1\overrightarrow{b_1}+n_2\...
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40 views

Continuum of lattice QCD is free?

I am having a hard time getting to grips with the statement that $$g_{0}(a) \to 0 \text{ as } a \to 0$$ where $g_{0}$ is the bare coupling in lattice QCD and $a$ the lattice spacing. How come this ...
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What does “sites” mean in the lattice language?

I acknowledge that this question is quite trivial. But in the lattice jargon, what does a $N$-sites lattice mean? it's a lattice $N\times N$ or it's a lattice with $N$ vertices? another option ...
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Reference request: 2D conformal field theory and functions on the triangular lattice

I don't have much of a physics background and was wondering if anyone knows what is meant by "conformally invariant" functions defined on the plaquettes of the honeycomb lattice (ie functions defined ...
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43 views

Snyder's Lorentz invariant discrete space-time

Can theories which say space-time is fundamentally discrete be compatible with Lorentz invariance? And if the answer is yes, in what sense is space-time no longer continuous? I'm sure this has been ...
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284 views

Entanglement entropy for $U(1)$ lattice gauge theory

Can someone please let me know if there is some reference for the calculation of entanglement entropy of $U(1)$ lattice gauge theory? I have seen a few references where Z2 lattice gauge theory has ...
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Obtaining the correlator of harmonic oscillator

The two point function for harmonic oscillator can be written as $$\langle \langle x(t_1)x(t_2) \rangle \rangle =\frac{\int Dx(t) x(t_1)x(t_2) e^{-S(x)}}{\int Dx(t) e^{-S(x)}} \tag{21} \, .$$ In ...
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Kitaev chain for an odd number of electrons

Kitaev shows in "Unpaired Majorana fermions in quantum wires" how unpaired Majorana fermions arise at both ends of a chain of $N$ electrons, where $N$ is even. After a quick literature search, I ...
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What is a bulk phase transition?

I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of ...
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Dirac string and Nielsen–Ninomiya theorem

Nielsen–Ninomiya theorem states that in a lattice system one can not have just one chiral fermion. Fermions necessarily come in pairs of opposite chirality. I am wondering if one can "explain" this ...
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Bose-Hubbard in momentum space

The Bose-Hubbard-Hamiltonian reads: $ H=-t\sum_{<i,j>} c_i^\dagger c_j+\frac{1}{2}U\sum c_i^\dagger c_i^\dagger c_i c_i -\mu\sum c_i^\dagger c_i $ I can use a FT to get from space to momentum ...
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phonons with next-nearest harmonic interactions

Given an account of the propagation of atomic vibrations along a monatomic chain of atoms, mass $m$, in which the spacing between atoms is $a$ and in which atoms, distance $na$ from each other are ...
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Dualities in 2+1D lattice gauge theories

A nice way to understand $\mathbb{Z}_2$ gauge theories is via duality transformations. For example, it is illustrated in http://arxiv.org/abs/1202.3120 that a $\mathbb{Z}_2$ gauge theory (with Ising ...
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How to calculate thd average of a Wilson loop in LQCD?

I'm am trying to do the excercise in page 35 from Lepage's LQCD notes: http://arxiv.org/abs/hep-lat/0506036. I want to compute the thermal average of the Wilson loop of size $a\times a$, where $a$ is ...
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156 views

SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their ...
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Why in the first order phase transition, the chiral susceptibilities is volume dependence?(Lattice)

This question is about the chiral phase transition at zero baryon chemical potential. Because of the chiral susceptibilities is not volume dependence, so the chiral phase transition is second order ...
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Lattice QCD and the 5th dimension

I was digging into the Nielsen-Ninomiya Theorem and doubler fermions, as well as solutions to these problems using Domain Wall Fermions and overlap lattice fermions, both of which make effective use ...
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Discrete translational invariance of lattice systems and conserved quantities [duplicate]

Imagine a crystal lattice with discrete translational symmetry. Is there any way to obtain local periodic conserved quantities by taking a derivative (deliberately left abstract)? The discretised ...
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60 views

String operator in the string-net model

The string operator is a way to study the quasiparticle excitations in the string-net model http://arxiv.org/abs/cond-mat/0404617. It is claimed in the above reference (Eq.(19), p.9) that for string ...
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36 views

Lattice propagator computation

I am reading this lecture on random surface theory by Thordur Jonsson. I feel like I should describe the problem a little for those who don't want to read the lecture. We are ultimately interested in ...
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Do lattice gauge theories with discrete gauge groups have sensible continuum limits?

In lattice gauge theories the only gauge invariant observables are constructed from Wilson loops and local field strength observables are reconstructed as zero size limits of Wilson loops. Furthermore ...
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When do gauge theories have protected gapless excitations?

Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by ...
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211 views

What is continuum limit (low energy limit) in condense matter physics?

In condensed matter theory, I can sometimes encounter such a term as continuum limit, also known as low energy limit. I have a question about this term, let me illustrate my question through an ...
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What is the fundamental reason of the fermion doubling?

Recall that the fermion doubling is the problem in taking the $a \to 0$ limit of a naively discretized fermionic theory (defined on a lattice with lattice spacing $a$). After such a limit one finds ...
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Holographic dual of pure-classical systems

There are classical systems (eg. see Sections VII and VIII of Kogut's review) that shares many of the properties of a pure-gauge SU(N) quantum theory including factorization and mass-gap, but with ...
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Modelling discrete spacetime

Supposed space and time were to be discrete, then how would i go about modelling this inside a computer simulation? In a simple 2D world, taking a square for example with side length $A$, then if ...
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Number of classical oscillation modes of a Lattice and number of quantum phonons

In solving the Classical model for lattice dynamics [Rossler pag 38] we find that the lattice admits $$d\cdot N\cdot r = \#modes$$ where $d=$dimension of the problem $N=$ number of atoms $r=$ ...
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How to obtain the asymptotic behavior of Green's function?

This question arose from Eq.(9.135) and Eq.(9.136) in Fradkin's Field theories of condensed matter physics (2nd Ed.). The author mapped quantum-dimer models to an action of monopole gas in $(2+1)$ ...
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Is Elitzur's theorem valid only in lattice field theory?

Elitzur's theorem, stating that spontaneous breakdown of a gauge symmetry is impossible, was originally proved for a lattice gauge theory. Is it valid in continuum field theory? Any ref?
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Why do some rectangular lattices turn into cubic lattices upon heating?

Lattices that are rhombic or rectangular have been known to expand in the two shorter axes directions when heated until a cubic structure is reached before expanding linearly in all directions. Why is ...
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When do optical phonon branches arise?

As far as I understand from what I'm studying, only when you have more than one kind of atom in the lattice structure, will the phonon have an optical branch, or else it should not have an optical ...
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Quantization of energy of phonons

when taking into account of energy of photons, the relationship $E=nh\nu$ stands because it is said that the Energy is proportional to the frequency of the electromagnetic wave, however when the ...
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Help with the analytical solution for the dispersion relation of an elastic wave with wave vector k

I was studying Phonons and Lattice vibrations and came across the equation. I want the mathematical solution for this. $$M\frac{d^2u_n}{dt^2}=C[u_{n+1}+u_{n-1}-2u_n]$$ from here it is argued that ...
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Interacting fermions on a lattice

My rough understanding about lattice simulations of bosonic quantum field theories is that the partition function can be approximated by explicitly summing over a large number of field configurations, ...
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Introduction material for lattice field simulation

I am looking for introductory material of lattice field theory simulation. It is better start from simple example (e.g. \phi^4) and include some source code. Is there any on-line or book which is ...